TOPOLOGY, GEOMETRY AND PHYSICS: BACKGROUND FOR THE WITTEN CONJECTURE PART I Gregory L. Naber Department of Mathematics, California State University Chico, CA 95929-0525, U.S.A. Abstract The profound, beautiful and, at times, rather mysterious symbiosis between mathematics and physics has always been a source of wonder, but, in the past twenty years, the intensity of the mutual interaction between these two has become nothing short of startling. Our objective here is to provide an introduction, in terms as elementary as possible, to one small aspect of this relationship. Toward this end we shall tell a story. Although we make no attempt to relate it chronologically, the story can be said to begin with the efforts of Yang and Mills [49] to construct a nonabelian generalization of classical electromagnetic theory, and to culminate in a remarkable conjecture of Ed Witten [48] concerning the Donaldson invariants of a smooth 4-manifold.

1

Instantons and 4-Manifolds

The central characters in our story are all “classical gauge theories” and we will eventually introduce them in some generality (Section 3), but we would like to begin by getting to know a few of them personally. For this we first recall the construction of the quaternionic Hopf bundle π

Sp ( 1 ) ,→ S 7 −→ IH IP 1 .

(1.1)

Here Sp ( 1 ) is the Lie group of unit quaternions (those g ∈ IH satisfying | g | = 1 ). As a manifold it is diffeomorphic to S 3 , but it is also isomorphic to ¯T ) the Lie group SU ( 2 ) of 2×2 complex matrices U that are unitary ( U −1 = U and satisfy det ¶ U = 1. Indeed, every such U can be written in the form µ U=

α

β

− β¯

α ¯

| , where α, β ∈ C satisfy | α |2 + | β |2 = 1 and the map

1

Ã

α − β¯

β α ¯

!

¡ ¢ −→ α + β j = α1 + α2 i + β 1 + β 2 i j = α1 + α2 i + β 1 j + β 3 k

(1.2)

is an isomorphism. We will allow ourselves the luxury of adopting whichever view of this Lie group is most convenient in any given situation. We will usually think of S 7 as { p = ( q 1 , q 2 ) ∈ IH 2 : | q 1 |2 + | q 2 |2 = 1 }. Then we can define a smooth right action σ : S 7 × Sp ( 1 ) → S 7 of Sp ( 1 ) on S 7 by ¡ ¢ ¡ ¢ σ ( p , g ) = p · g = q1 , q2 · g = q1 g , q2 g . This action is clearly free and the orbits p·Sp ( 1 ) are submanifolds of S 7 diffeomorphic to S 3 . The orbit space S 7 / Sp ( 1 ) is, by definition, the quaternionic projective line IH IP 1 . We will denote by π : S 7 → IH IP 1 the projection map which carries p = ( q 1 , q 2 ) onto its orbit π ( p ) = π ( q 1 , q 2 ) = [ q 1 , q 2 ]. Obviously, π ( p · g ) = π ( p ) for all p ∈ S 7 and g ∈ Sp ( 1 ). IH IP 1 is given the quotient topology determined by π and the differentiable structure determined by the atlas consisting of the two charts ( Uk , ϕk ), k = 1, 2, defined as follows: © £ ¤ ª Uk = x = q 1 , q 2 ∈ IH IP 1 : q k 6= 0 , k = 1, 2 ϕk : Uk −→ IH ∼ = IR 4 , k = 1, 2 ¡ £ 1 2¤¢ ¡ ¢−1 ϕ1 ( x ) = ϕ1 q ,q = q2 q1 ϕ2 ( x ) = ϕ2

¡ £

q1 , q2

¤¢

(1.3)

¡ ¢−1 = q1 q2 .

−1 Clearly, ϕ1 and ϕ2 map onto IH , ϕ−1 1 ( q ) = [ 1, q ], ϕ2 ( q ) = [ q, 1 ] and, on

ϕ1 ( U1 ∩ U2 ) = ϕ2 ( U1 ∩ U2 ) = IH − { 0 } , −1 −1 . ϕ1 ◦ ϕ−1 2 ( q ) = ϕ2 ◦ ϕ1 ( q ) = q

(1.4)

It follows from this that we have, indeed, defined a differentiable structure. One can further verify that, relative to this structure, π : S 7 → IH IP 1 is smooth and each of the maps Ψk : π −1 ( Uk ) −→ Uk × Sp ( 1 ) , k = 1, 2 ¡

1

Ψk ( p ) = Ψk q , q

2

¢

=

¡

k

k

π(p), q / |q |

¢

(1.5)

is a diffeomorphism that is equivariant with respect to the given right action of Sp ( 1 ) on S 7 and the natural right action of Sp ( 1 ) on Uk × Sp ( 1 ) (this means that, writing Ψk ( p ) = ( π ( p ), ψk ( p ) ), where ψk ( p ) = q k / | q k | , we have Ψk ( p · g ) = ( π ( p ), ψk ( p )g ) = ( π ( p ), ψk ( p ) ) · g ). To summa2

rize, what we have just shown is that S 7 is a smooth principal Sp ( 1 ) ( i.e., SU ( 2 ) )-bundle over IH IP 1 with local trivializations given by (1.5). We note | in passing that one can replace the quaternions IH by the complex numbers C everywhere above and thereby construct the complex Hopf bundle | IP 1 . U ( 1 ) ,→ S 3 −→ C

(1.6)

There are, of course, also the obvious higher dimensional generalizations Sp ( 1 ) | IP n−1 for n > 2. ,→ S 4n−1 → IH IP n−1 and U ( 1 ) ,→ S 2n−1 → C Remark: Let ϕS and ϕN be the stereographic projection maps from the north and south poles of S 4 , respectively, and let ϕ1 and ϕ2 be as in (1.3). If −1 ϕ¯1 denotes the map ϕ¯1( x ) = ϕ1( x ), then both ϕ−1 ¯1 are S ◦ ϕ2 and ϕN ◦ ϕ

diffeomorphisms of IH IP 1 minus a point to S 4 minus a point. On the intersection of their domains they agree and so determine a global diffeomorphism of IH IP 1 onto S 4 . Composing with π : S 7 → IH IP 1 gives a principal bundle Sp ( 1 ) ,→ S 7 −→ S 4 which is also often referred to as the quaternionic Hopf bundle. Some caution is advised, however, since reversing the roles of ϕ1 and ϕ2 above gives another identification of IH IP 1 with S 4 , but the corresponding Sp( 1 )-bundle over S 4 is not equivalent to the one we just described. This is most readily shown by computing their Chern numbers which turn out to be 1 in the first case and −1 in the second (we will briefly review these calculations shortly). Now we focus our attention on a fixed point p = ( q 1 , q 2 ) ∈ S 7 ⊆ IH 2 . The orbit of our Sp( 1 )-action containing p ( i.e., the fiber of π : S 7 → IH IP 1 above π( p ) ) is a copy of S 3 . The subset of the tangent space Tp ( S 7 ) to S 7 at p consisting of vectors tangent to this fiber is called the vertical space at p and denoted Vertp ( S 7 ). It is a 3-dimensional linear subspace of Tp ( S 7 ) which, in turn, can be identified with a linear subspace of T ( IH 2 ) ∼ = IR 8 . = T ( IR 8 ) ∼ p

p

Thus, relative to the usual Euclidean inner product on IR 8 , Vertp ( S 7 ) has an orthogonal complement and we shall call the intersection of this orthogonal complement with Tp ( S 7 ) the horizontal space at p and denote it Horp ( S 7 ). Thus, at each p ∈ S 7 we have a natural decomposition ¡ ¢ ¡ ¢ ¡ ¢ Tp S 7 ∼ = Vertp S 7 ⊕ Horp S 7 . If one fixes a g ∈ Sp ( 1 ) and explicitly computes the derivative of the diffeomorphism σg : S 7 → S 7 , σg ( p ) = σ ( p, g ) = p · g, it is easy to see that ¡

σg

¢ ∗p

¡

¡ ¢ ¡ ¢¢ = Horp·g S 7 Horp S 7 3

and so the distribution { Horp ( S 7 ) : p ∈ S 7 } of 4-dimensional subspaces satisfies the conditions required of a connection on the Hopf bundle. This is π called the natural connection on Sp( 1 ) ,→ S 7 −→ IH IP 1 . Now, any connection arises from a connection 1-form, i.e., a Lie algebra-valued 1-form on the principal bundle space whose kernel at each point is the horizontal space at that point. Identifying the Lie algebra of Sp ( 1 ) with the pure imaginary quaternions Im IH and defining an Im IH -valued 1-form on IH 2 by ¡ ¢ ˜ = Im q¯1 d q 1 + q¯2 d q 2 , ω it is not difficult to see that the connection 1-form ω for the natural connection ˜ to S 7 , i.e., on the Hopf bundle is the restriction of ω ˜, ω = ι∗ ω

(1.7)

where ι : S 7 ,→ IH 2 is the inclusion map (the proof is on page 295 of [34] ). In the physics lliterature it is more common to describe connections (there called gauge potentials ) locally on the base manifold by pulling back the connection 1-form by sections corresponding to some trivializing cover of the bundle. For the trivializations ( Uk , Ψk ), k = 1, 2, of the Hopf bundle given by (1.5) each Uk covers all but one point of IH IP 1 and it follows that the connection ω is uniquely determined by either one of the corresponding pullbacks. For example, it is easy to verify that the inverse of Ψ1 : π −1 ( U1 ) → U1 × Sp ( 1 ) is given by Ψ1−1 ( [ q 1 , q 2 ] , g ) = ( | q 1 | g , q 2 ( q 1 / | q 1 | )−1 g ) ∈ S 7 ⊆ IH 2 and so the associated section s1 : U1 → π −1 ( U1 ) is ¡ £ 1 2¤¢ ¡ £ 1 2¤ ¢ s1 ( x ) = s1 q ,q = Ψ−1 q ,q ,1 1 ³ ¯ ¯ ¯ ¯ ¢−1 ´ ¡ = ¯ q1 ¯ , q2 q1 / ¯ q1 ¯ . Since U1 is also the domain of the standard chart ( U1 , ϕ1 ) on IH IP 1 we can write the pullback s∗1 ω in terms of these coordinates on IH IP 1 . More precisely, we pull s∗1 ω back to IH ∼ = IR 4 by ϕ−1 1 . These calculations are carried out in detail on pages 256-258 of [34] and yield µ ¶ ¢∗ ¡ q¯ A1 = s1 ◦ ϕ−1 ω = Im dq (1.8) 1 1 + |q|2 which we now simply regard as an Im IH -valued 1-form on IH . Oddly enough, this 1-form on IR 4 first appeared in the physics literature [4] where it was initially referred to as a pseudoparticle. We will have more to say about this shortly. Thus far we know only one connection on the Hopf bundle (the “natural” one) and we would now like to produce some more. Recall that an automor4

π

phism of Sp( 1 ) ,→ S 7 −→ IH IP 1 is a diffeomorphism f : S 7 → S 7 of S 7 onto itself that respects the group action ( f ( p · g ) = f ( p ) · g ) and that each such automorphism induces a diffeomorphism fIH IP1 : IH IP 1 → IH IP 1 of IH IP 1 onto itself by π ◦ f = fIH IP1 ◦ π. If fIH IP1 happens to be the identity on IH IP 1 , then f is called a ( global ) gauge transformation. Now, if f is any automorphism and ω is any connection 1-form, then the pullback f ∗ ω is also a connection 1-form. If f is a gauge transformation, then the connections ω and f ∗ ω are said to be gauge equivalent. Remark: The motivation here is as follows. In physics, a (local) section π s : U → π −1 ( U ) of a principal bundle G ,→ P −→ X is called a choice of gauge on U and is regarded as a selection, at each x ∈ U , of a frame (coordinate system) in some internal space. The gauge principle asserts that the laws of physics should be invariant under an arbitrary change of gauge and, more particularly, that quantities with the same set of gauge representations are to be regarded as physically equivalent. For example, if ω is a connection and f is a gauge transformation, then, for any section s, f ◦ s is also a section and s∗ ( f ∗ ω ) = ( f ◦s )∗ ω. Thus, ω and f ∗ ω have the same set of gauge potentials and so are taken to be “equivalent”. For future reference we note that a local gauge transformation on U can be identified with a map g : U → G which gives rise to a new section/gauge sg : U → π −1 ( U ) defined by sg ( x ) = s ( x )·g ( x ) and that if ω is any connection, A = s∗ ω and Ag = ( sg )∗ ω, then Ag = g −1 A g + g −1 d g . Although not entirely obvious, one can show (pages 297-303 of [34] ) that, by judiciously choosing automorphisms of the Hopf bundle by which to pull back the natural connection, one can produce a new connection ω λ,n for each ( λ, n ) ∈ ( 0, ∞ ) × IH that is uniquely determined by the gauge potential µ ¶ q¯ − n ¯ Aλ,n = Im dq (1.9) λ2 + | q − n | 2 on IH ( thus, A1 = A1,0 ). For reasons that we will discuss shortly, Aλ,n is called the BPST-instanton with center n and spread λ. Although all of these differ from the natural connection by an automorphism we will see that distinct pairs ( λ, n ) give rise to connections that are not gauge equivalent. Any connection ω has a curvature Ω that can be calculated from the Cartan Structure Equation Ω = d ω + 21 [ ω, ω ] and is uniquely determined by a family of pullbacks F = s∗ Ω , called gauge field strengths, by sections corresponding to some trivializing cover. For the connection ω λ,n on the Hopf bundle the curvature Ωλ,n is uniquely determined by the single gauge field strength F λ,n = d Aλ,n + 12 [ Aλ,n , Aλ,n ]. A rather tedious, but routine 5

calculation (pages 284-289 of [34] ) gives F λ,n = =

λ2 ( λ2 + | q − n | 2 )

2

³ ¡

2λ2 ( λ2 + | q − n | 2 ) ¡

d q¯ ∧ d q

2

¢ d x1 ∧ d x2 − d x3 ∧ d x4 i +

(1.10)

¢ ¡ ¢ ´ d x1 ∧ d x3 + d x2 ∧ d x4 j + d x1 ∧ d x4 − d x2 ∧ d x3 k

where x1 , x2 , x3 and x4 are standard coordinates on IR 4 . The Im IH -valued 2-forms F λ,n on IH ( ∼ = IR 4 ) have a number of crucial properties. If ∗ denotes the Hodge star operator on IR 4 arising from the usual orientation and inner product and if we extend this operator to Im IH -valued forms componentwise, then each F λ,n is anti-self-dual (ASD) in the sense that ∗ F λ,n = − F λ,n (1.11) (page 333 of [34] ). The Hodge star also gives a (pointwise) inner product on the spaces Ωp ( IR 4 ), 0 ≤ p ≤ 4, of real-valued p-forms on IR 4 (if µ and ν are in Ωp ( IR 4 ), then ∗ν is in Ω4−p ( IR 4 ) so µ ∧ ∗ ν is in Ω4 ( IR 4 ) and so is a multiple of the metric volume form vol on IR 4 and one defines h µ , ν i by µ ∧ ∗ ν = h µ , ν i vol ). Combined with the Killing form on the Lie algebra this will give a pointwise inner product on any space of Lie algebra-valued forms on IR 4 . Remark: The conventions we adopt are as follows: The Lie algebra Im IH is isomorphic to the Lie algebra su ( 2 ) of 2 × 2 complex matrices A that are skew-Hermitian ( A¯T = − A ) and tracefree ( tr Aµ = 0 ). ¶ We take as µ a basis for ¶ su ( 2 ) the matrices Tj = − µ and σ3 =

1

0

0

−1

1 2

i σj , where σ1 =



0

1

1

0

, σ2 =

0

−i

i

0

are the Pauli spin matrices. Thus, every element of

su ( 2 ) can be written in the form 1 A = A T1 + A T2 + A T3 = − 2 1

2

Ã

3

A3 i

A2 + A1 i

− A2 + A1 i

− A3 i

! . (1.12)

As the inner product associated with the Killing form of su( 2 ) we take hA, Bi = −2 tr ( AB ) so that { T1 , T2 , T3 } is an orthonormal basis. The structure constants for this basis are given by [ Ti , Tj ] = εijk Tk ( εijk is the Levi-Civita symbol with ε123 = 1 ). The structure constants for the basis { 12 i , 21 j , 12 k } for Im IH are the same ( [ x, y ] = xy −yx = 2 Im ( xy ) on Im IH ) so an isomor6

phism of Im IH onto su ( 2 ) is a1i+a2j+a3k → ( 2a1 )T1 +( 2a2 )T2 +( 2a3 )T3 . In particular, our Lie algebra inner product is four times the usual IR 3 inner product in Im IH . Now, if F = F 1 i + F 2 j + F 3 k is an Im IH -valued form on IR 4 , each component has a squared norm k F i k 2 = h F i , F i i given by the Hodge dual as above. Our Lie algebra squared norm k F k 2 for F is then taken to be four times the sum of these. 3 X ° i °2 °F ° . kF k = 4 2

(1.13)

i=1

Writing F as an su ( 2 ) matrix of complex-valued forms in the manner indicated above, ∗F is computed entrywise. Defining the wedge product F ∧ ∗F to be the matrix product with entries multiplied by the ordinary wedge, a simple calculation shows that − tr ( F ∧ ∗F ) =

1 2 k F k vol . 2

(1.14)

To compute k F λ,n k 2 for the BPST gauge field strength given by (1.10) one observes that, for example, ¡ 1 ¢ ¡ ¢ dx ∧ dx2 − dx3 ∧ dx4 ∧∗ dx1 ∧ dx2 − dx3 ∧ dx4 = ¡ ¢ ¡ ¢ − dx1 ∧ dx2 − dx3 ∧ dx4 ∧ dx1 ∧ dx2 − dx3 ∧ dx4 = 2dx1 ∧ dx2 ∧ dx3 ∧ dx4 = 2 vol and similarly for the rest so à à !! 4 ° °2 96 λ4 8 λ ° F λ,n ° = 4 3 = 4 4 . (1.15) ( λ2 + | q − n | 2 ) ( λ2 + | q − n | 2 ) Notice that k F λ,n k 2 has a maximum value of 96/λ4 at q = n and that, for a fixed n, its variation with λ (illustrated in the Figure) is such that the “total field strength” Z Z ° ° 1 48 λ4 ° F λ,n ° 2 vol = vol 2 2 4 2 IR 4 IR 4 ( λ + | q − n | ) (1.16) = 8π

2

remains constant at 8 π 2 . Thus, the gauge potential Aλ,n on IR 4 has field strength that is “centered” at n in IR 4 with a “spread” that is determined by λ (hence the terminology introduced earlier), and a total field strength that is 7

independent of λ and n (the reason for this is, as we will see shortly, deeper than it might seem). Let us now temporarily suppress from our minds where the potentials Aλ,n came from (i.e., the Hopf bundle) and regard them simply as Lie algebra-valued 1-forms on IR 4 . Any such Lie algebra-valued 1-form A on IR 4 can be thought of as a gauge potential for a connection on the trivial Sp ( 1 ) - ( or SU ( 2 ) - ) bundle over IR 4 and so has a gauge field strength F = dA + 21 [ A , A ] on IR 4 . We define the Yang-Mills action YM ( A ) of A by Z Z 1 2 ∗ k F k vol . (1.17) YM ( A ) = − tr ( F ∧ F ) = 4 4 2 IR IR This integral might well be infinite, of course, but if it is not we will say that A has finite action and think of YM ( A ) as the total field strength of the gauge potential A. In an attempt to describe the isotopic spin of a nucleon, Yang and Mills [49] devised a non-abelian generalization of classical electromagnetic theory in which the electromagnetic potential was replaced by an su ( 2 ) -valued 1-form A on IR 4 (actually, on Minkowski spacetime, but we will restrict our attention to the Euclidean version). The field strength for their potential was our F and the “action” (i.e., Lagrangian) of the theory was YM ( A ) . The field equations were the Euler-Lagrange equations for YM ( A ) under variations of A and it is not difficult to show that these are d A ∗F = 0 ,

(1.18)

where d A ∗ F = d ∗ F + [ A , ∗ F ] is the covariant exterior derivative of ∗ F 8

associated with A. Equations (1.17) are the Yang-Mills equations on IR 4 . Quite independently of YM, any field strength F satisfies a purely geometrical condition called the Bianchi identity dA F = 0,

(1.19)

(page 268 of [34] ). Now notice that if the field strength F of A happens to be ASD ( ∗ F = − F ), then (1.19) implies that (1.18) is automatically satisfied. Thus, a gauge potential A on IR 4 with ASD field strength F is a solution to the Yang-Mills equations (it is shown on page 325 of [34] that these actually give absolute minima for the Yang-Mills action). This is the context in which the BPST instantons Aλ,n were first discovered. Belavin, Polyakov, Schwarz and Tyupkin [4] sought finite action solutions to the Yang-Mills equations (1.18) on IR 4 and found them via the simpler anti-self-dual equations ∗

F = −F .

(1.20)

A finite action gauge potential A on IR 4 with ASD field strength is called an instanton on IR 4 . We have described a family of instantons Aλ,n parametrized by ( λ , n ) ∈ ( 0 , ∞ ) × IR 4 . Now, k F k 2 is invariant under gauge transformation. This is essentially because it is defined in terms of the trace (1.14), which is invariant under conjugation, and a local gauge transformation g : U −→ G ( see the Remark preceding (1.9) ) conjugates field strengths, i.e., F g = g −1 F g. Thus, we conclude from (1.15) that Aλ,n and Aλ0 ,n0 can be gauge equivalent if and only if ( λ0 , n0 ) = ( λ , n ), i.e., distinct BPST instantons are gauge inequivalent. Nevertheless, they all have the same total field strength YM ( Aλ,n ) and we must now investigate this “coincidence” more closely. Lately we have been thinking of the Aλ,n simply as Lie algebra-valued 1-forms on IR 4 and forgetting where they came from. They are, of course, much more. They are pullbacks to IR 4 of the connections ω λ,n on Sp ( 1 ) ,→ S 7 → IH IP 1 . Let us now identify IH IP 1 with S 4 in the manner described at the beginning of the Remark after (1.6). Each ω λ,n is then regarded as a connection 1-form on Sp ( 1 ) ,→ S 7 −→ S 4 . (1.21) Their pullbacks to S 4 by the induced sections of (1.21), when written in coordinates on IR 4 obtained by stereographic projection from the north pole of S 4 , are the gauge potentials Aλ,n . Summarizing, we find that the Aλ,n (connection 1-forms on the trivial Sp ( 1 )-bundle over IR 4 ) “come from” connection 1-forms ω λ,n on the nontrivial Hopf bundle (1.21) over S 4 . Turning matters about, one might say that the connections on the trivial bundle over IR 4 “extend to S 4 ” in the sense that S 4 = IR 4 ∪ { ∞ } is the 1-point compactification of IR 4 and, due to their asymptotic behavior as | x | → ∞ in IR 4 , the connections extend to the point at infinity. Note, however, that the extension process involves not 9

only the connection, but the bundle on which it is defined as well. Now, a remarkable theorem of Karen Uhlenbeck [43] asserts that this interpretation is not as fanciful as it might sound. Indeed, a very special case of this result states that if A is any finite action Im IH -valued gauge potential on IR 4 with ASD field strength F , then there exists a unique (up to equivalence) Sp ( 1 )-bundle Sp ( 1 ) ,→ P → S 4 over S 4 and a connection ω on P whose pullback by some section s of P is A when written in stereographic coordinates. Furthermore, the bundle to which A “extends” is uniquely determined by the Yang-Mills action YM ( A ) of A as we shall now explain. π An Sp ( 1 )-bundle Sp ( 1 ) ,→ P −→ X over a compact, oriented, smooth 4manifold X is uniquely determined by a certain characteristic cohomology class, called the second Chern class c2 ( P ) and constructed as follows. Choose a connection ω on P . The curvature Ω of ω is a Lie algebra-valued 2-form on P . One can show (Section 6.3 of [35] ) that 8 π1 2 tr ( Ω ∧ Ω ) is a real-valued 4-form on P which descends to (i.e., is the pullback by π of) a closed 4-form on X whose cohomology class c2 ( P ) ∈ H 4 ( X; IR ) does not depend on the choice of ω. Two Sp ( 1 )-bundles over X are known to be equivalent if and only if they have the same second Chern class and, indeed, if and only if they have the same second Chern number, defined by Z c2 ( P ) [ X ] = c2 ( P ) (1.22) X

(which is always an integer). Now take X to be S 4 . Stereographic projection from the north pole of S 4 is an orientation preserving diffeomorphism from S 4 minus a point onto IR 4 so c2 ( P ) [ S 4 ] can be computed by integrating pullbacks over IR 4 , i.e., Z £ 4¤ 1 c2 ( P ) S = tr ( F ∧ F ) , (1.23) 8 π 2 IR 4 where F is the corresponding field strength. Now let us consider an instanton A on IR 4 with field strength F . Since F is ASD, ∗F = −F and so − tr ( F ∧ ∗F ) = tr ( F ∧ F ). Uhlenbeck’s Theorem guarantees that A extends to a connection on some principal Sp ( 1 )-bundle Sp ( 1 ) ,→ P → S 4 over S 4 and a comparison of (1.17) and (1.23) shows that the second Chern number of this bundle is given by £ ¤ c2 ( P ) S 4 =

1 YM ( A ) . 8 π2

(1.24)

Thus, the Yang-Mills action of an instanton A on IR 4 is directly encoded in the topology of the bundle over S 4 to which A extends. Notice, however, that the value of YM ( A ) is entirely determined by the asymptotic behavior of the field strength F on IR 4 so it is this physical characteristic of the gauge field 10

that is represented by the Chern number. Physicists call −c2 ( P ) [ S 4 ] the instanton number, or topological charge, of A. The “reason” that all of the BPST instantons Aλ,n have the same Yang-Mills action is now clear: they all extend to (i.e., come from) the same Sp ( 1 )-bundle over S 4 , i.e., the Hopf bundle (1.21), which (1.16) now shows to have Chern number 1 (as promised in the Remark following (1.6) ). Notice also that the topological charge of an instanton, being an integer, cannot be altered by a continuous variation of the field and so is “conserved”, but for purely topological reasons unlike the more common Noether conserved quantities. Such topological conservation laws play a crucial role in understanding modern gauge field theories. | IP 1 one defines Remark: For the complex Hopf bundle U ( 1 ) ,→ S 3 → C the natural connection geometrically exactly as in the quaternionic case. The | corresponding connection 1-form ω is the restriction to S 3 of the Im C | 2. valued 1-form i Im ( z¯1 dz 1 + z¯2 dz 2 ) on C Choosing a section and coordinates analogous to those in the quaternionic case gives the gauge poten∼ | | IP 1 with S 2 in the two tial i Im ( 1+ |z¯z | 2 dz ) on C = IR 2 . Identifying C

ways indicated for IH IP 1 and S 4 gives two inequivalent U ( 1 )-bundles over S 2 (those with “first Chern number” ± 1 ). On each of these one obtains an induced connection, uniquely determined by a gauge potential which, when written in spherical (rather than stereographic) coordinates, takes the form − 12 ni ( 1 − cos φ )dθ, where n = ± 1. More generally, one has, for each n ∈ ZZ , a U ( 1 )-bundle U ( 1 ) ,→ Pn → S 2 over S 2 with first Chern number n and on it a connection uniquely determined by the gauge potential An = −

1 ni ( 1 − cos φ ) dθ . 2

This potential (and the corresponding connection) represent the field of a Dirac monopole of magnetic charge −n. Just as for the instanton number, magnetic charge is “topological” in that it is encoded in the topology of the bundle on which the connection lives and is conserved for topological reasons. We will have more to say about Dirac monopoles in Section 7. There is another perspective on the topological nature of instantons which we now briefly describe. Observe that YM ( A ) < ∞ implies that k F k 2 must approach zero sufficiently fast as | x | → ∞ in IR 4 . This, together with F = dA + 12 [ A , A ] would seem to require a similar decay for the components of A and their first derivatives. However, due to the gauge freedom available in the choice of A ( k F k 2 is gauge invariant), this is not the case. All that is necessary is that there exist some local gauge transformation g : U → Sp ( 1 ), defined for sufficiently large | x | , such that the potential Ag in this new gauge has components that decay sufficiently fast ( g need only be defined for large 11

| x | because the integral over any compact set in IR 4 is necessarily finite). If 3 such a g exists and SR is a 3-sphere in IR 4 of sufficiently large radius R that it is contained in the domain of g , then 3 3 g | SR : SR −→ Sp ( 1 )

can be regarded as a map from the 3-sphere to itself and so determines an element of the homotopy group π3 ( S 3 ). But π3 ( S 3 ) ∼ = ZZ and an isomorphism is provided by the Brouwer degree deg. Thus, the asymptotic behavior of F 3 ] ∈ π3 ( S 3 ) and this gives an integer determines g, which determines [ g | SR 3 deg ( g | SR ) (the restrictions of g to two such spheres are clearly homotopic and so have the same degree). Thus, the various possible asymptotic behaviors for finite action field strengths on IR 4 fall into “homotopy classes”, each labeled by an integer. If F is ASD so that A extends to a principal Sp ( 1 )-bundle over S 4 , then these integers also characterize the bundles. Remark: Briefly, the reason for this is as follows. S 4 = IR 4 ∪ { ∞ } consists of two copies of the closed 4-dimensional disc (upper and lower hemispheres) glued together along the equator which is a copy of S 3 and which we can take to 3 be SR . The restriction of any bundle over S 4 to either of these discs is trivial because the disc is contractible. This provides a trivializing cover of S 4 consisting of just two trivializations and hence essentially one transition function g. This one transition function determines the bundle up to equivalence and its restriction to the equator is a map from S 3 to Sp ( 1 ) ∼ = S 3 . Furthermore, any 3 3 map from S to S can be regarded as such a restriction and homotopic maps correspond to equivalent bundles. In particular, this is true of the restricted 3 3 gauge transformation g | SR so the integer deg ( g | SR ) uniquely determines 4 an Sp ( 1 )-bundle over S . Now let us consider somewhat more generally the Hopf bundle Sp ( 1 ) ,→ S → S 4 over S 4 with Chern number 1. Any connection ω on this bundle is uniquely determined by the gauge potential A on IR 4 obtained by pulling back by the natural section on S 4 minus the north pole and then again by the inverse of stereographic projection from the north pole. This stereographic projection is an orientation preserving conformal diffeomorphism and so preserves the Hodge dual. We will say that the connection ω is anti-self-dual (ASD) if the gauge potential A is ASD (we will see shortly how to extend this notion to bundles over more general 4-manifolds). The set of ASD connections is invariant under global gauge transformations of the bundle so we may consider the set M of gauge equivalence classes of ASD connections on Sp ( 1 ) ,→ S 7 → S 4 . This then is the same as the set of gauge equivalence classes of ASD potentials A on IR 4 with YM ( A ) = 8 π 2 (regarded as connection forms on the trivial bundle). Each BPST instanton Aλ,n determines a point [ Aλ,n ] in M and we have 7

12

already observed that distinct pairs ( λ, n ) give distinct points in M. A remarkable, and very deep result of Atiyah, Hitchin and Singer [2] asserts that, in fact, every element of M is represented by some Aλ,n and so the map ( λ , n ) ∈ ( 0 , ∞ ) × IR 4 −→

£

¤ Aλ,n ∈ M

is a bijection. This picture of M as the half-space ( 0 , ∞ ) × IR 4 in IR 5 , as simple and pleasing as it is, is not the most informative. An alternative arises from the fact that there is an orientation preserving conformal diffeomorphism of ( 0 , ∞ ) × IR 4 onto the open 5-dimensional ball B 5 in IR 5 . Indeed, one can (pages 337-341 of [34] ) introduce “spherical coordinates” on M that yield a picture of M as B 5 with [ A1,0 ] at its center. Proceeding radially outward from [ A1,0 ] toward a point on ∂B 5 = S 4 one encounters potentials all of which have the same center n, but which become more and more concentrated, i.e., for which the spread λ → 0. A particularly pleasing aspect of this picture is that the base manifold S 4 of the bundle emerges as the boundary of the moduli ¯ 5 = B 5 ∪ S 4 ) and its space M in a compactification of M ( M ∼ = B 5 ,→ B points can be identified intuitively with “delta function” potentials. One sees quite clearly in this example how the topologies of the underlying 4-manifold S 4 and the moduli space M of ASD connections on the bundle Sp ( 1 ) ,→ S 7 → S 4 are inextricably linked. We will conclude by briefly describing an amazing generalization of this scenario due to Simon Donaldson [10]. We let X denote a compact, oriented, simply connected, smooth 4-manifold. H2 ( X ; ZZ ) will denote its second homology group (with integer coefficients) and QX : H2 ( X ; ZZ ) × H2 ( X ; ZZ ) −→ ZZ its intersection form. Remark: H2 ( X ; ZZ ) is a finitely generated, free Abelian group and each of its elements can be identified with a certain equivalence class represented by a smoothly embedded, oriented, closed surface (2-manifold with boundary) P in X. Very roughly, the definition of QX goes as follows: For α1, α2 ∈ P P H2 ( X ; ZZ ) one can select surfaces intersect 1, 2 representing them that P transversely ( i.e., at each intersection point the tangent spaces to 1 and P span the tangent space to X ). An intersection point p is assigned the 2 P value 1 if an oriented basis for Tp ( 1 ) together with an oriented basis for P Tp ( 2 ) gives an oriented basis for Tp ( X ); otherwise it is assigned the value -1. Then QX ( α1 , α2 ) is the (necessarily finite) sum of these values over all the intersection points. QX is a symmetric, bilinear form and is, moreover, unimodular, i.e., if α1 , . . . , αt is a basis for H2 ( X ; ZZ ) over ZZ , then the

13

matrix ( QX ( αi , αj ) ) has determinant ±1. Here are a few examples: X S

H2 ( X ; ZZ )

QX

0



ZZ

(1)

4

| IP 2 C | IP C

2

2

S × S

ZZ 2

Ã

ZZ ⊕ ZZ

( − 1) 0

1

1

0

!

| IP 2 is the orbit space of S 5 = { ( z 1 , z 2 , z 3 ) ∈ C | 3 : | z1 | 2 + | z2 | 2 + Here C | z 3 | 2 = 1 } by the U ( 1 )-action ( z 1 , z 2 , z 3 ) · g = ( z 1 g, z 2 g, z 3 g ). It is

2

| IP naturally a complex 2-manifold and so has a canonical orientation. C is the same manifold with the opposite orientation. A less pedestrian example is the Kummer surface which we will denote K3 and which can be defined | IP 3 (same definition as C | IP 2 , but begin as the complex algebraic surface in C | 4 ) whose homogeneous coordinates z 1 , z 2 , z 3 , z 4 satisfy ( z 1 )4 + with S 7 ⊆ C ( z 2 )4 +( z 3 )4 +( z 4 )4 = 0. The rank of H2 ( K3 ; ZZ ) is 22 and the intersection form is à ! à ! à ! 0 1 0 1 0 1 ⊕ ⊕ ⊕ ( − E8 ) ⊕ ( − E8 ) , 1 0 1 0 1 0

where E8 is given by  2 −1   −1 2    0 −1   0  0   0  0   0 0    0 0  0

0

0

0

0

0

0

−1

0

0

0

0

2

−1

0

0

0

−1

2

−1

0

0

0

−1

2

−1

0

0

0

−1

2

−1

0

0

0

−1

2

0

0

−1

0

0

0



 0    0    0  .  −1   0    0   2

The intersection form can also be defined for topological 4-manifolds, but we will not enter into this here. It has been known for some time that the intersection form is a basic invariant for compact 4-manifolds. In 1949, Whitehead proved that two compact, simply connected 4-manifolds X1 and X2 have the same homotopy type if and only if 14

their intersection forms are equivalent (i.e., there exist bases for H2 ( X1 ; ZZ ) and H2 ( X2 ; ZZ ) relative to which QX1 and QX2 have the same matrix). In 1982, Freedman [16] showed that every unimodular, symmetric, integer bilinear form is the intersection form of at least one (and at most two) compact, simply connected topological 4-manifold(s). In particular, this is true of the vast, impenetrable maze of positive definite forms (when the rank is 40 there are at least 1051 equivalence classes of definite forms). Donaldson has shown that the differential topologist need not venture into this maze because only one positive definite, unimodular, symmetric, integer bilinear form can arise as the intersection form of a compact, simply connected smooth 4-manifold. Donaldson’s 1983 Theorem: Let X be a compact, oriented, simply connected, smooth 4-manifold with positive definite intersection form QX . Then QX is standard, i.e., there is a basis for H2 ( X ; ZZ ) over ZZ relative to which the matrix of QX is the identity matrix. Donaldson’s Theorem is remarkable, but still more remarkable is its proof, which is a byproduct of the analysis of an instanton moduli space for X. We (very, very) briefly sketch the idea. Consider the bundle over X analogous to the Hopf bundle over S 4 , i.e., the Sp( 1 )-bundle Sp ( 1 ) ,→ P −→ X over X with Chern number 1. Next, choose a Riemannian metric g on X. Both the bundle and the metric are to be regarded as auxiliary structures to facilitate the study of X. From g and the given orientation for X one obtains a Hodge star operator and thereby a notion of ASD connection on P . In somewhat more detail, the definition is as follows. Let ω be a connection on Sp( 1 ) ,→ P → X and Ω and its curvature. Then Ω is a globally defined Lie algebra-valued 2-form on P . It is horizontal in the sense that it vanishes when either of its arguments is vertical (tangent to a fiber of the principal bundle). The corresponding local gauge field strengths F = s∗ Ω on X are related by the adjoint representation of Sp( 1 ) on its Lie algebra and so these patch together to give a globally defined 2-form F ω on X with values in the adjoint bundle ad ( P ) of P (the vector bundle associated to P by the adjoint representation). The 2-form F ω is very often also called the curvature of ω. It’s advantage is that it is defined on the 4-manifold X so its Hodge dual 2-form ∗ F ω is also a 2-form and it makes sense to say that the connection ω is antiself-dual (ASD) if ∗F ω = −F ω . Remark: For the record we point out that anti-self-dual connections can exist only on bundles with positive Chern number, whereas self-dual connections ( ∗F ω = F ω ) can exist only if the Chern number is negative. The discussion to follow can be carried at equally well with self-dual connections on the bundle with Chern number -1. 15

Now, in general, there is no reason to believe that such ASD connections exist, but a deep result of Taubes [41] asserts that, for manifolds satisfying the hypotheses we have assumed of X, the bundle Sp( 1 ) ,→ P → X actually admits ASD connections. Thus, we may introduce the moduli space M( g ) of ASD connections on P . This moduli space does depend on the choice of g and, for a randomly chosen Riemannian metric, little can be said about its structure. One can show, however, that, for some choice of g (indeed, for a “generic” choice of g ), the moduli space M has all of the following properties. 1. If m denotes half the number of homology classes α ∈ H2 ( X ; ZZ ) for which QX ( α , α ) = 1, then there exist points p1 , . . . , pm ∈ M such that M − { p1 , . . . , pm } is a smooth, orientable 5-manifold. 2. Each pi , i = 1, . . . , m, has a neighborhood in M that is homeomorphic 2 | IP 2 with p | to a cone over C i at the vertex (the cone over C IP is the 2 | IP quotient of C × [ 0, 1 ] obtained by identifying all points of the form ( p, 1 ) ). 3. There is a compact set K ⊆ M such that M − K is a submanifold of M − { p1 , . . . , pm } diffeomorphic to X × ( 0, 1 ).

Now we build a new space M0 from M by cutting off the top half of each cone and the bottom half of the cylinder X × ( 0, 1 ). M0 is compact (because K is compact). It is also a manifold with boundary whose boundary consists of the | IP 2 . disjoint union of a copy of X and m copies of C 16

Now, in general, if X1 and X2 are two n-manifolds and if there exists an ( n + 1 )-manifold M with boundary whose boundary is a disjoint union of X1 and X2 , then M is called a cobordism between X1 and X2 . X1 and X2 are then said to be cobordant. Thus, M0 is a cobordism between X and a | IP 2 ’s. As it happens, the signature of the intersection form disjoint union of C of a 4-manifold is a cobordism invariant. This fact,Ftogether with the positive | IP 2 and a bit of integer definiteness of QX , the known intersection form of C linear algebra suffice to produce a basis for H2 ( X ; ZZ ) relative to which the matrix for QX is the identity (page 347 of [34] ). Donaldson’s 1983 Theorem was the first gauge-theoretic assault on a problem in the topology of smooth 4-manifolds. Subsequent developments in what came to be known as Donaldson Theory yielded spectacular results, but at a cost in labor that seemed to grow exponentially with each new success (we will describe some of the most basic elements of Donaldson theory in Sections 4 and 8). A breakthrough occurred in 1993 when Kronheimer and Mrowka isolated the (apparently large) class of 4-manifolds of “simple type” and proved that, for these at least, Donaldson theory had some realistic chance of becoming effectively computable. At precisely this moment, however, fate (or rather Ed Witten) intervened and the subject of smooth 4-manifold topology took an entirely new turn. This is the story we would like to tell.

2

SU ( 2 ) Yang-Mills-Higgs Theory on IR 3

The notion of a “classical gauge theory”, which we have promised to define carefully in Section 3, is not adequately motivated by the examples of the preceding section (which would be classified by physicists as “pure” Yang-Mills theories). In general, gauge fields are coupled to (i.e., interact with) what we shall call “matter fields”. For example, electromagnetic fields ( U ( 1 )-gauge fields) are 17

coupled to charged particles and, in the original proposal of Yang and Mills [49], SU ( 2 )-gauge fields interact with nucleons and, at least in the absence of electromagnetic fields, govern the evolution of their isotopic spin (proton/neutron) state. In this section we will consider a concrete example which may seem a bit more abstruse, but which has proven to be very important and which has the added advantage of being derivable from either purely mathematical considerations (“dimensional reduction”) or in the manner more familiar to physicists (“field content plus action”). We will describe both. Let us briefly return to the ASD equations on IR 4 (it is traditional, and will be convenient in this section, to think in matrix terms so that we now identify Sp ( 1 ) with SU ( 2 ) and Im IH with sp ( 1 ) in the manner described in (1.2) and the Remark following (1.11) ). Thus, we have a 1-form Aˆ = Aˆ1 d x1 + Aˆ2 d x2 + Aˆ3 d x3 + Aˆ4 d x4 = Aˆα d xα on IR 4 with each Aˆα a smooth map on IR 4 taking values in su ( 2 ). The gauge potential Aˆ gives rise ˆ = dAˆ + 1 [ Aˆ , Aˆ ] which, in coordinates, is given to a gauge field strength F 2 by ˆ = 1 Fˆ d xα ∧ d xβ F 2 αβ h i´ 1³ = d xα ∧ d xβ , ∂α Aˆβ − ∂β Aˆα + Aˆα , Aˆβ 2

(2.1)

ˆ is given by ∗F ˆ = 1 ∗Fˆ d xα∧ where ∂α means ∂ / ∂ xα . The Hodge dual of F αβ 2 P4 1 β ∗ˆ ˆ d x , where Fα β = 2 γ,δ=1 εα β γ δ Fγ δ and εα β γ δ is the totally antisymmetric Levi-Civita symbol with ε1234 = 1. Thus, the ASD equations (1.20) can be written 4 1 X Fˆα β = − εα β γ δ Fˆγ δ , 2

α, β = 1, 2, 3, 4 .

(2.2)

γ,δ=1

There are many duplications in this list of equations (e.g., α = 3, β = 4 and α = 1, β = 2 both reduce to Fˆ12 = − Fˆ34 ). Indeed, all of the equations in (2.2) are easily seen to be contained in Fˆij = −

3 X

εijk Fˆk4 ,

i, j = 1, 2, 3 ,

(2.3)

k=1

where εi j k is totally anti-symmetric and ε123 = 1 (e.g., Fˆ12 = − Fˆ34 is equivP3 alent to Fˆ12 = − ε123 Fˆ34 = − k=1 ε12 k Fˆk4 ). Now, the finite action solutions to (2.3) are just what we have called instantons on IR 4 . We wish now to abandon the finite action condition and seek solutions Aˆ to (2.3) that are static, i.e., independent of x4 . With this as18

sumption, (2.1) gives Fˆk4 = ∂k Aˆ4 + [ Aˆk , Aˆ4 ] so (2.3) becomes Fˆij = −

3 X

³ εijk

i ´ h ∂k Aˆ4 + Aˆk , Aˆ4 ,

i, j = 1, 2, 3 .

(2.4)

k=1

Let us now “reduce to IR 3 ” as follows: Fix (arbitrarily) some value x40 of x4 and consider the submanifold IR 3 × { x40 } of IR 4 (henceforth written simply IR 3 ). For i = 1, 2, 3 we let A = Aˆ | IR 3 and then define A = A d x1 + i

1

i

A2 d x2 + A3 d x3 = Ai d xi on IR 3 . The gauge potential A on IR 3 has a corresponding field strength F = 12 Fij d xi ∧ d xj , with components Fij = Fˆ | IR 3 . Note that Aˆ does not enter into either the potential or the field 4

ij

strength on IR 3 . However, if we define ψ : IR 3 → su ( 2 ) by ¯ ¯ ψ = Aˆ4 ¯ IR 3 , then the static ASD equations (2.4), when restricted to IR 3 , become Fˆij = −

3 X

εijk ( ∂k ψ + [ Ak , ψ ] ) ,

i, j = 1, 2, 3 ,

(2.5)

k=1

or, in even more detail, 3 X £ ¤ ∂i Aj − ∂j Ai + Ai , Aj = − εijk ( ∂k ψ + [ Ak , ψ ] ) .

(2.6)

k=1

These we regard as field equations for an SU ( 2 )-gauge potential A coupled to a matter field ψ whose wavefunction takes values in su ( 2 ). Somewhat more precisely, A corresponds to a connection on the trivial SU ( 2 )-bundle over IR 3 and ψ is a section of the (likewise trivial) adjoint bundle. Thus, ( ∂k ψ + [ Ak , ψ ] ) d xk is the corresponding covariant exterior derivative d A ψ of ψ and the sum on the right-hand side of (2.6) gives the components of the IR 3 -Hodge dual of d A ψ. Consequently, (2.6) can be written F = − ∗d A ψ .

(2.7)

In whatever form they are written these are called the Bogomolny monopole ˆ =F ˆ on IR 4 we equations (beginning instead with the self-dual equations ∗F would have arrived at F = ∗d A ψ and these go by the same name). Before explaining the origin of the “monopole” terminology we will describe another path leading to the same set of equations. We begin with the underlying base manifold IR 3 with its usual Riemannian metric and orientation and with standard coordinates x1 , x2 and x3 . Consider the trivial SU ( 2 )-bundle SU ( 2 ) ,→ IR 3 × SU ( 2 ) ,→ IR 3 19

over IR 3 (the right action of SU ( 2 ) on IR 3 × SU ( 2 ) being given by p · g = ( x, h ) · g = ( x, hg ) ). Thus, we have a natural global section s : IR 3 −→ IR 3 × SU ( 2 ) s(x) = (x, e) µ where e =

1

0

0

1

¶ is the identity element of SU ( 2 ).

Any other global

section is then of the form sg : IR 3 −→ IR 3 × SU ( 2 ) sg ( x ) = s ( x ) · g ( x ) = ( x , g ( x ) ) for some smooth map

g : IR 3 −→ SU ( 2 )

which we identify with a global gauge transformation on the bundle. A connection on IR 3 × SU ( 2 ) is an su( 2 )-valued 1-form ω on IR 3 × SU ( 2 ). Since the bundle is trivial, ω is uniquely determined by the gauge potential A = s∗ ω . Moreover, any su( 2 )-valued 1-form on IR 3 is the pullback by s of some connection on IR 3 × SU ( 2 ) so we may restrict attention entirely to globally defined gauge potentials A on IR 3 . A gauge transformation g : IR 3 → SU ( 2 ) gives a new gauge representation ∗

Ag = ( sg ) ω = g −1 Ag + g −1 d g , where d g is the entrywise exterior derivative of g : IR 3 → SU ( 2 ). The curvature Ω = dω + 21 [ ω , ω ] is likewise determined by the field strength F = s∗ Ω = dA + 12 [ A , A ] and a gauge transformation gives ∗

F g = ( sg ) Ω = g −1 F g . We wish to construct a field theory in which one of the fields is an SU ( 2 )gauge potential A on IR 3 as described above and the other, to which A will be coupled, is a so called “Higgs field”. Now, in general, a “matter field”, when quantized, represents a particle. The matter field itself is represented by a wavefunction which takes values in some vector space and which transforms under a gauge transformation by some representation of the structure group (in our case SU ( 2 ) ) on that vector space (we will expand upon these points in Section 3). More precisely, the matter field is a section of the vector bundle associated to the given principal bundle by some representation of the structure group on a vector space. When the vector space is the Lie algebra of the structure group and the representation is the adjoint representation (so that 20

the vector bundle is the adjoint bundle) the matter field is called a Higgs field. The adjoint bundle of the trivial bundle SU ( 2 ) ,→ IR 3 × SU ( 2 ) → IR 3 is likewise trivial and any section of it can be identified with a smooth map Ψ : IR 3 × SU ( 2 ) → su( 2 ) that is equivariant, i.e., satisfies Ψ ( p · g ) = g −1 · Ψ ( p ) Ψ ( ( x , h ) · g ) = g −1 · Ψ ( x , h ) Ψ ( x , hg ) = g −1 Ψ ( x , h ) g . Since the bundle is trivial, Ψ is uniquely determined by ψ = s∗ Ψ = Ψ ◦ s ψ(x) = Ψ(x, e) (because Ψ ( x, g ) = Ψ ( ( x, e ) · g ) = g −1 ψ ( x )g ) and we will focus our attention on ψ. A gauge transformation g : IR 3 → SU ( 2 ) gives another gauge representation ∗ ψ g = ( sg ) Ψ = g −1 ψ g of the Higgs field. We now write down an action (analogous to the Yang-Mills action (1.17) ) the Euler-Lagrange equations of which will govern the interaction of A and ψ. The integrand will contain three terms. The first is the Yang-Mills term − tr ( F ∧ ∗ F ) just as in (1.17). The second is called the interaction term ³ ´ − tr d A ψ ∧ ∗ d A ψ , where d A ψ = dψ+[ A , ψ ] is the covariant exterior derivative of ψ (physicists would say that this term reflects the principle of minimal coupling). Finally, there is a term intended to describe the internal self-interaction energy of the matter field ψ. The precise form of this term must be postulated by choosing some non-negative, smooth, invariant, real-valued function V on su ( 2 ) and composing with ψ. We take V to be the familiar “Mexican hat” potential, i.e., V : su ( 2 ) −→ IR ´2 λ³ 2 V (A) = kAk − 1 , 8 where λ ≥ 0 is a constant and k A k 2 = h A , A i = −2 tr ( A2 ), as in the Remark following (1.11). We will write V ◦ ψ as V ( ψ ) = λ8 ( k ψ k 2 − 1 )2 and think of it as a 0-form on IR 3 so that its Hodge dual is λ 8

λ∗ 8 (

k ψ k 2 − 1 )2 =

( k ψ k 2 − 1 ) vol. With this we can write down the so-called Yang-Mills-

21

Higgs action on IR 3 : Z ³ ³ ´ YMH ( A , ψ ) = − tr ( F ∧ ∗ F ) − tr d A ψ ∧ ∗ d A ψ IR 3

´2 ¶ λ ∗³ 2 + kψk −1 (2.8) 8 Z µ °2 ° ´2 ¶ λ³ 1 ° ° 2 2 kF k + °dAψ° + kψk −1 vol . = 2 IR 3 4 This action is gauge invariant, i.e., A → Ag , F → F g and ψ → ψ g leaves the integral unchanged (we have already seen that F g = g −1 Ag and ψ g = g −1 ψg so k F g k 2 = k F k 2 and k ψ g k 2 = k ψ k 2 and a short calculation gives ³ ´ g d A ψ g = g −1 d A ψ g as well). We will generally be interested only in finite action field configurations ( A , ψ ), i.e., those for which (2.8) is finite. Now, the requirement that YMH ( A , ψ ) < ∞ implies that, as k x k → ∞ in IR 3 , k A k → 0, k d A ψ k → 0 and, at least if λ 6= 0, k ψ k −→ 1 .

(2.9)

Indeed, it is shown in [22] that each of these limits is achieved uniformly on IR 3 . Now, when λ = 0 there is no reason to suppose that finite action implies k ψ k → 1 as k x k → ∞. However, it is also shown in [22] that, for stationary configurations (i.e., those satisfying the Euler-Lagrange equations for YMH ( A , ψ ) ) one loses nothing by restricting attention to those that satisfy (2.9), even when λ = 0. More precisely, we have the following: For any finite action critical point of YMH ( A , ψ ) with λ = 0 there exists a constant c ≥ 0 such that k ψ k → c uniformly as k x k → ∞. If c = 0, then ( A , ψ ) is trivial. If c 6= 0, then one can rescale to obtain a new configuration ( A0 , ψ 0 ) given by ( A0 ( x ) , ψ 0 ( x ) ) = ( c−1 A ( c−1 x ) , c−1 ψ ( c−1 x ) ). Then ( A0 , ψ 0 ) is also a finite action critical point for YMH ( A , ψ ) with λ = 0 and it satisfies k ψ 0 k → 1 uniformly as k x k → ∞. We intend to focus our attention on a certain limiting case of the YangMills-Higgs action (that in which λ → 0 in (2.8) ), but we retain a “virtual” self-interaction in the form of the boundary condition (2.9) ). First, however, we describe an important general feature of the full action (2.8). Notice that it has some obvious absolute minima. Indeed, YMH ( A , ψ ) , which is non-negative, is zero when A = 0 and ψ = ψ0 is a constant in su ( 2 ) with k ψ0 k = 1. Such an absolute minimum is regarded as a ground state of the system. The corresponding quantum state of lowest energy is called a vacuum state and physicists perform perturbation calculations about such vacuum states. The 22

point here is that these ground states are not unique. A specific choice of ψ0 is said to break the symmetry from SU ( 2 ) to U ( 1 ). The rationale behind the terminology is as follows: A gauge transformation g : IR 3 → SU ( 2 ) acts on ψ by ψ → ψ g = g −1 ψg. If the ground state is to be gauge invariant, then we must have g −1 ψ0 g = ψ0 and this occurs only if g is in the isotropy subgroup (stabilizer) of ψ0 in SU ( 2 ) under the adjoint action. We claim that this isotropy subgroup is a copy of U ( 1 ). Briefly, the argument is as follows. Let H = { g ∈ SU ( 2 ) : g −1 ψ0 g = ψ0 } be the isotropy subgroup. Obviously, −g ∈ H if and only if g ∈ H. Now, identifying SU ( 2 ) / ± e with SO ( 3 ) (page 374 of [34] ) and su ( 2 ) with IR 3 , the adjoint action is just the natural action of SO ( 3 ) on IR 3 , i.e., rotation (see the Appendix of [34] ). This natural action of SO ( 3 ) is transitive on S 2 and the isotropy subgroup of ψ0 ∈ S 2 is H 0 = H / ± e. Since SO ( 3 ) is compact, SO ( 3 ) / H 0 is homeomorphic to S 2 (Theorem 1.6.6 of [34] ). Thus, ( SU ( 2 ) / ± e ) / ( H / ± e ) ∼ = S 2 so 2 ∼ SU ( 2 ) / H = S . Consequently, H is 1-dimensional. Being closed in SU ( 2 ), H is also compact. Now, a compact, 1-dimensional smooth manifold is a disjoint union of circles (Section 5.11 of [33] ) so, being a subgroup of SU ( 2 ), H must be a single copy of the circle U ( 1 ). What we have just witnessed is an instance of the phenomenon of spontaneous symmetry breaking in which a field theory with an exact symmetry group G gives rise to ground states that are invariant only under some proper subgroup H of G. With this digression behind us we return to the limiting case of the YangMills-Higgs action described above. Thus, we consider the action Z ³ ³ ´´ A(A, ψ ) = − tr ( F ∧ ∗ F ) − tr d A ψ ∧ ∗ d A ψ IR 3

=

1 2

Z

µ

IR 3

and take as our configuration space ( C =

(2.10)

° °2 ¶ ° ° 2 kF k + °dAψ° vol

( A , ψ ) : A ( A , ψ ) < ∞ , lim

sup

R→∞ k x k ≥R

¯ ¯ ¯ ¯ ¯ 1−kψk ¯ = 0

) . (2.11)

The Euler-Lagrange equations for the action A ( A , ψ ) are the Yang-MillsHiggs equations  £ ¤  ∗ d A ∗F = d A ψ , ψ (2.12)  ∗ A∗ A d d ψ = 0 and we seek solutions to these in C. Any configuration ( A , ψ ) satisfying

23

(2.12) also satisfies an analogue of the Bianchi identity (1.19) which we write as   dAF = 0 . (2.13)  A A d d ψ = [F , ψ] Now, just as in the case of the Yang-Mills action, one can find a simpler set of first order equations whose solutions give absolute minima for the action A ( A , ψ ) in (2.10) and so, in particular, satisfy the Yang-Mills-Higgs equations (2.12). To see how these arise we reason as follows. On IR 3 , both F and ∗d A ψ are 2-forms and the Hodge dual is an isometry so k ∗d A ψ k 2 = k d A ψ k 2 . Now observe that ° °2 ° °2 ° ° ° ° 2 2 k F k + ° d A ψ ° = k F k + ° ∗d A ψ ° ° °2 D E ° ° = ° F + ∗ d A ψ ° − 2 F , ∗d A ψ

(2.14)

and similarly, ° °2 ° °2 D E ° ° ° ° 2 k F k + ° d A ψ ° = ° F − ∗ d A ψ ° + 2 F , ∗d A ψ . It follows that A ( A , ψ ) will achieve its absolute minimum value (i.e., 0) when F = ± ∗d A ψ and these we recognize as the Bogomolny monopole equations introduced by quite different means earlier. The appellation “monopole” derives from a certain exact solution to F = ∗ A − d ψ discovered by ´t Hooft, Polyakov, Prasad and Sommerfeld. In spherical coordinates on IR 3 this solution is given by A = A1 T 1 + A2 T 2 + A3 T 3 , ψ = ψ 1T 1 + ψ 2T 2 + ψ 3T 3 (see the Remark following (1.11) ), where A1 = − A2 =

ρ ( sin θ dφ + cos θ sin φ dθ ) sinh ρ

ρ ( cos θ dφ − sin θ sin φ dθ ) sinh ρ

A3 = − ( 1 − cos φ ) dθ ψ1 = ψ2 = 0 ψ 3 = coth ρ −

1 ρ

(the derivation of this solution is carried out in considerable detail on pages 141150 of [35] ). Notice that, despite appearances to the contrary, this configuration ( A , ψ ) is a globally defined, smooth object on all of IR 3 (the component functions are actually real analytic everywhere, even at ρ = 0 ). Furthermore, 24

when viewed from a distance (i.e., as ρ → ∞ ) the Higgs field approaches the constant value T3 (since coth ρ − ρ1 → 1 ) and the first two components of A approach zero (since ρ / sinh ρ → 0 ). On the other hand, A3 does not depend on ρ so it remains fixed at −( 1 − cos φ )dθ. Thus, for large ρ, A is effectively −( 1 − cos φ )dθ T 3 . Under the isomorphism of su ( 2 ) onto Im IH described in the Remark following (1.11), this becomes − 21 k ( 1 − cos φ )dθ. Since the span | of k in Im IH is just a copy of Im C we recognize here just the potential for a Dirac monople (see the Remark following (1.24) ). Thus, the ´t Hooft-PolyakavPrasad-Sommerfield monopole is a smooth field configuration of SU ( 2 ) YangMills-Higgs theory which “looks like” a Dirac monopole from afar. The most interesting thing about the appearance of the Dirac monopole here is that it was entirely voluntary. In classical electromagnetic theory magnetic monopoles are, but certainly need not be, inserted by hand, whereas in SU ( 2 ) YangMills-Higgs theory, they appear of their own accord (we return to this point at the end of the section). We remark that the same potential A paired with the Higgs field −ψ gives a solution to the “other” Bogomolny monopole equation F = ∗ d A ψ. Thus motivated we will refer to any ( A , ψ ) ∈ C satisfying (2.7) as a monopole and will now associate with it a “monopole number”. Notice that if F = − ∗d A ψ, then (2.14) becomes k F k 2 + k d A ψ k 2 = − 2h F , ∗d A ψ i so, for monopoles, Z D E A(A, ψ ) = − F , ∗d A ψ vol Z

IR 3

=

³ ´ 2 tr F ∧ ∗∗ d A ψ

IR 3

Z =

³ ´ 2 tr F ∧ d A ψ

IR 3

Z =

³ ´ Tr F ∧ d A ψ ,

IR 3

where we now use Tr = 2 tr. Computing this integral for the ´ t Hooft-PolyakovPrasad-Sommerfeld monopole gives a value of 4π. We normalize the action and define the monopole number of any ( A , ψ ) ∈ C satisfying (2.7) by Z ³ ´ 1 N (A, ψ) = Tr F ∧ d A ψ . (2.15) 4 π IR 3 Like the instanton number introduced in Section 1, this monopole number is, in fact, an integer and one can see this in at least two different ways. Perhaps the easiest to describe is as follows (consult [35] for details on the rest that we have to say about N ( A , ψ ) ): Since k ψ k → 1 as k x k → ∞ in IR 3 there exists an R0 > 0 such that k x k > R0 implies k ψ ( x ) k > 12 . For 25

k x k > R0 we can therefore define −1 ψˆ ( x ) = k ψ ( x ) k ψ(x)

and, for any R > R0 ,

¯ ¯ 2 ψˆR = ψˆ ¯ SR ,

2 = { x ∈ IR 3 : k x k = R }. Now, ψˆR can be regarded as a map from where SR S 2 to S 2 and so determines an element [ ψˆ ] of the homotopy group π ( S 2 ). 2

R

Moreover, since ψˆ is smooth on k x k > R0 , its restrictions to any two such spheres are clearly homotopic so [ ψˆR ] is independent of R > R0 and we will denote it simply [ ψˆ∞ ] (physicists would refer to ψˆ∞ as the restriction of ψˆ to the “sphere at infinity”). Now, π ( S 2 ) ∼ = ZZ and an isomorphism is provided 2

by the Brouwer degree. One can show that the monopole number N ( A , ψ ) is equal to the degree of any ψˆR , R > R0 , written ³ ´ N ( A , ψ ) = deg ψˆ∞ , (2.16) and so, in particular, is an integer. Monopoles fall into distinct topological types according to the homotopy type of the (normalized) Higgs field on large spheres (these topological types are actually the connected components of the space of solutions to (2.7) in C which are sometimes referred to as topological sectors in physics). This is, of course, entirely analogous to our earlier description of the instanton number in terms of the homotopy type of a gauge transformation on large (3-) spheres. We will conclude by briefly sketching a description of the monopole number as the Chern number of a U ( 1 )-bundle over S 2 obtained by breaking the SU ( 2 ) symmetry to U ( 1 ) through the selection of some ground state ψ0 . Fix some R > R0 as above. ψ is the pullback s∗ Ψ by the standard section s of SU ( 2 ) ,→ IR 3 × SU ( 2 ) → IR 3 of an equivariant map Ψ : IR 3 × SU ( 2 ) → 2 su ( 2 ). The restriction of this trivial SU ( 2 )-bundle over IR 3 to SR is the 2 trivial SU ( 2 )-bundle over SR : π

2 2 . SU ( 2 ) ,→ SR × SU ( 2 ) −→ SR

(2.17)

2 ˆ = k Ψ k −1 Ψ . Both are equivariant Now let ΨR = Ψ | SR × SU ( 2 ) and Ψ R R R 2 ˆ takes values in S 2 and Ψ R su ( 2 ) = { A ∈ su ( 2 ) : k A k = 1 }. Furthermore, ˆ by the standard section of the trivial bundle (2.17). ψˆR is the pullback of Ψ R Thus, ψˆ is the standard gauge representation of a Higgs field on the bundle R

(2.17). 2 Now break the symmetry, i.e., select some ground state ψ0 ∈ Ssu ( 2 ) . The isotropy subgroup of ψ0 (with respect to the adjoint action of SU ( 2 ) on su ( 2 ) ) is, as we have seen, a copy of U ( 1 ) and we will denote it simply 26

ˆ −1 ( ψ ) is a submanifold of S 2 × SU ( 2 ) U ( 1 ). Now, one can show that Ψ 0 R R which is invariant under the action of U ( 1 ) and that, moreover, the restriction 2 of π to this submanifold gives a principal U ( 1 )-bundle over SR : ˆ −1 ( ψ ) π|Ψ 0 R

ˆ −1 ( ψ ) U ( 1 ) ,→ Ψ 0 R

- S2 . R

(2.18)

The U ( 1 )-bundle (2.18) is called a reduction of the structure group of (2.17) to U ( 1 ). Now, U ( 1 )-bundles over S 2 are classified by their first Chern number (the integral over S 2 of the first Chern class) which is always an integer. The result of interest to us here is that by choosing an appropriate connection on (2.18) and writing down the formula for the first Chern number using this connection one arrives at the expression (2.15) for the monopole number of ( A , ψ ). We will conclude our discussion of SU ( 2 )-monopoles by very briefly discussing an issue which must surely be troubling the reader. In classical electromagnetic theory magnetic monopoles must be inserted by hand. One of Maxwell’s equations explicitly forbids the existence of “magnetic charges” and, in order to understand the consequences of their possible existence, Dirac [9] was forced to abandon (or, rather, modify) this equation and postulate the existence of a magnetic analogue of the electric charge. Certain SU ( 2 )-monopoles “look like” Dirac monopoles from a distance. One might wonder as to the “source” of their magnetic charge. A naive hint concerning the source of the magnetic charge of SU ( 2 )-monopoles can be found in our earlier view of them as static, ASD potentials on IR 4 . Recall that any solution to the ASD equations on IR 4 also satisfies the full Yang-Mills equations d A ∗F = 0 and the Bianchi identity d A F = 0 and that these are regarded as a nonabelian generalization of Maxwell’s equations. The static version of Maxwell’s equations that contains both electric and magnetic charge densities ( ρe and ρm , respectively) is, in appropriate units and on IR 3 , d ∗F = 0, d F = ∗ ρm and ∇2 ψ = ρe , where F is the magnetic field 2-form and ψ is the electric potential. Noting that d A F = 0 is equivalent to dF = −[A, F ] one can view the commutator term as playing the role of a magnetic charge density. The role of the Higgs field ψ (or, more to the point, the boundary condition k ψ k → 1 as k x k → ∞ ) is to break the symmetry at large distances from SU ( 2 ) down to U ( 1 ), thus turning the SU ( 2 ) theory into a U ( 1 ), i.e., electromagnetic, theory.

3

Classical Gauge Theories

Abstracting the salient features of the examples in the preceding sections, we now propose to enumerate a sequence of basic mathematical ingredients which together will serve as our working definition of a classical gauge theory. 27

(1) A smooth, oriented, (semi-) Riemannian manifold X. Generally, this will be space ( IR 3 ), a spacetime (e.g., Minkowski spacetime IR 1,3 ), a Euclidean (“Wick rotated”) version of a spacetime (e.g., IR 4 ), a compactification of one of these (e.g., S 4 = IR 4 ∪ { ∞ } ), an open submanifold of one of these (e.g., IR 3 − { ( 0, 0, 0 ) } ), or some homotopy equivalent (e.g., S 2 ' IR 3 − { ( 0, 0, 0 ) } ). The particles and fields which it is the ultimate goal of gauge theory to describe “live” in X. (2) A finite dimensional vector space V equipped with an inner product h , i (positive definite if V is real and Hermitian if V is complex). The particles have wavefunctions that take values in V. The choice of V is dictated by the internal structure of the particle (charge, spin, isospin, etc.) | and so V is called the internal space. Typical examples are C (spin zero 4 8 | | charged particles), C (Dirac electrons), C (nucleons), or the Lie algebra G of some Lie group G (Higgs fields). From the inner product h , i one computes squared norms of V-valued functions, forms, etc. and from these formulates action principles that govern the dynamics (see (8) below). (3) A matrix Lie group G and a representation ρ : G → GL( V ) of G on V that is orthogonal with respect to the inner product h , i i.e., h ρ(g ) (v ) , ρ(g ) (w) i = hv , wi for all g ∈ G and v, w ∈ V. G will generally be one of the classical groups (e.g., U ( 1 ), SU ( 2 ) , SO ( 4 ), | ), etc.) or a product of these. In general, G describes a symmetry of SL ( 2, C the physical system under consideration, while ρ describes the particular type of invariance that a particle’s wavefunction exhibits under this symmetry. More specifically, the Lie group G plays the following dual roles: (a) The inner product h , i on V determines a class of orthonormal bases, or frames, in V and these are related by the elements of G, i.e., if P is the collection of all such frames, then there is a (right) action of G on P which sends any frame p ∈ P to a new frame p · g ∈ P . By fixing (arbitrarily) some frame at the outset one can therefore identify the elements of G with the frames. (b) G also acts on V (on the left) via the representation ρ ( v → ρ ( g ) ( v ) = g · v ) and so acts on the wavefunction at each point. If ψ ( p ) is a value of the wavefunction described relative to the frame p ∈ P , then its description relative to the frame p · g is ψ ( p · g ) = g −1 · ψ ( p ) . 28

(3.1)

The right action of G on P transforms frames in the internal space and the left action of G on V describes the corresponding transformation law for the wavefunction. π

(4) A smooth principal G-bundle G ,→ P −→ X over X. Typical examples are trivial bundles (e.g., SU ( 2 ) ,→ IR 4 × SU ( 2 ) → IR 4 ) and Hopf bundles (e.g., U ( 1 ) ,→ S 3 → S 2 and SU ( 2 ) ,→ S 7 → S 4 ). At each x ∈ X the fiber π −1 ( x ) is a copy of G, thought of as the set of all frames in the internal space at x ∈ X. A local section s : U → π −1 ( U ) ⊆ P ( U open in X and π ◦ s = idU ) is a smooth selection of an internal frame at each point of U relative to which wavefunctions can be described on U . Such a local section is also called a local gauge. π

(5) A connection ω on G ,→ P −→ X with curvature Ω. As motivation for (5) we recall that, in classical electrodynamics, an electromagnetic field is generally modeled by a 2-form F defined on space or spacetime (i.e., on X ). The corresponding potential is a 1-form A with F = dA. F is globally defined on X, but, in general, potentials are only locally defined so that a complete description of F will require a number of potentials with domains that cover X. In nonabelian gauge theories even the field strengths are, in general, only locally defined on X. However, by virtue of the manner in which these locally defined forms on X are related on the intersections of their domains (the local gauge transformation laws) one can piece them together into globally defined forms on the bundle space P of some principal bundle (characterized by transition functions that are simply read off from the transformation laws). These are the connection ω and its curvature Ω. On the other hand, given ω and Ω one retrieves the physical potentials and fields by choosing a local gauge/section s : U → P and pulling back to X : A = s∗ ω is the local gauge potential and F = s∗ Ω is the local gauge field strength (both in gauge s ). Another local gauge s0 : U 0 → P with U ∩ U 0 6= ∅ will be related to s by s0 ( x ) = s ( x ) · g ( x ), where g : U ∩ U 0 → G and · is the right action in the principal bundle. One generally writes sg rather that s0 to explicitly display the so-called transition function g. The corresponding potential and field strength are written Ag = ( sg )∗ ω and F g = ( sg )∗ Ω and are given by

and

Ag = g −1 A g + g −1 d g

(3.2)

F g = g −1 F g

(3.3)

on U ∩ U 0 . The change of gauge s → sg = s · g is a local gauge transformation and can be identified with the map g : U ∩ U 0 → G. The gauge principle, or principle of local gauge invariance, is a cornerstone of modern 29

theoretical physics and asserts that such a gauge transformation alters only the appearance and not the physics of a situation, e.g., that Ag and F g represent the same potential and field strength as A and F , only written in different internal coordinates. Remark: Before recording the next item in our list of ingredients for a classical gauge theory we recall several facts from geometry (see Section 5.7 of [34] for π more details). Given a principal G-bundle G ,→ P −→ X and a left action π

G of G on some manifold F one can construct a fiber bundle P ×G F −→ X associated to the principal bundle by the left action whose typical fiber is F . In particular, if F is a vector space V and the left action of G on V arises from a representation ρ : G → GL ( V ) of G on V one obtains an associated vector bundle, usually written P ×ρ V. A typical example is the adjoint bundle ad P = P ×ad G, where V is the Lie algebra G of the structure group G and ρ = ad is the adjoint representation of G on G ( ad ( g ) ( A ) = g A g −1 ). We will need to use the fact (page 356 of [34] ) that there are two equivalent ways of viewing a section of an associated bundle P ×G F , i.e., either as a map ψ from X to P ×G F for which πG ◦ ψ is the identity, or as a map ψ from P to F that is equivariant ( ψ ( p · g ) = g −1 · ψ ( ρ ) ). The latter view and (3.1) should motivate

(6) A global section ψ of the vector bundle P ×ρ V associated to G ,→ π P −→ X by the representation ρ : G → GL ( V ) (or, equivalently, an equivariant map ψ : P → V ). Particles coupled to (i.e., experiencing the effects of) the gauge field determined by ω have locally defined wavefunctions taking values in V that are obtained by solving field equations (see (8) below) that involve the local potentials A. A change of gauge changes the wavefunction by the representation ρ (see (3.1) ) so these local wavefunctions piece together into a globally defined object called a matter field that can be described in either of the two equivalent ways referred to in (6). Remark: It is entirely possible that more than one matter field is coupled to the gauge field, but we will phrase our basic scheme for classical gauge theories assuming that there is just one and leave it to the reader to add on more terms if necessary. (7) A smooth, non-negative, real-valued function V : V → IR on V that is invariant under the action of G on V ( V ( g · v ) = V ( v ) ). V is regarded as a potential function with V ◦ ψ = V ( ψ ) describing the selfinteraction energy of the matter field ψ. Typically, this will depend only on k v k 2 = h v , v i , e.g., λ8 ( k v k 2 − 1 )2 , or 12 m k v k 2 , where λ and m are non-negative constants.

30

(8) An action (energy) functional A ( ω , ψ ), the stationary points of which are the physically significant field configurations ( ω , ψ ). The EulerLangrange equations for A ( ω , ψ ) are the field equations (or, equations of motion) for the classical gauge theory. When X is Riemannian (as it is in cases of topological interest) one can generally expect an action of the form Z ³ ´ 2 2 A(ω , ψ ) = c k F ω k + c k d ! ψ k + c2 V ( ψ ) vol , (3.4) X

where c is some normalizing constant, c1 and c2 are “coupling constants”, F ω is the globally defined 2-form on X with values in the adjoint bundle ad ( P ) which locally reduces to the gauge field strengths F = s∗ Ω, d ! ψ is the covariant exterior derivative of the matter field ψ and the norms arise from the metric on X, the inner product on V and some ad-invariant inner product on G. We have already seen several examples of classical gauge theories that are of particular interest to us because of the topological nature of certain solutions to their field equations. Later (Sections 5 and 7) we will see other, rather more complicated examples whose impact on topology and geometry has been much more profound. Of course, most examples of interest in physics are not topological in nature at all, but we will nevertheless pause briefly to describe one of the simplest of these (more details and still more examples are to be found in Chapter 2 of [35] ). The situation we intend to model (at the classical level) is the interaction of an electromagnetic field with a charged, spin zero particle (e.g., a π − -meson). Remark: Certain technical complications, which we do not wish to become involved in, arise for more familiar charged particles like the electron and proton. The reason is that these have spin 12 and so, according to Dirac, have | ), wavefunctions that transform under a certain representation of SL ( 2 , C whereas the electromagnetic field to which it is coupled is a U ( 1 )-gauge theory. To fit this interaction into the general framework we have described would | )-bundle together into a require “splicing” a U ( 1 )-bundle and an SL ( 2 , C | single U ( 1 ) × SL ( 2 , C )-bundle on which both objects may be thought to live. This can be done and the process is carried out in more detail in Section 2.4 of [35]. The arena within which electrodynamics is done is Minkowski spacetime IR 1,3 . As a differentiable manifold IR 1,3 is just IR 4 , but, rather than the usual Riemannian metric on IR 4 we introduce the semi-Riemannian Minkowski metric given, relative to standard coordinates x0 , x1 , x2 , x3 by ηα β d xα ⊗ d xβ

31

where

    ηα β =

  

1, −1, 0,

α = β = 0 α = β = 1, 2, 3 . α 6= β

One thinks of the elements of IR 1,3 as events whose standard coordinates are the time ( x0 ) and spatial ( x1 , x2 , x3 ) coordinates by which the event is identified by some fixed, but arbitrary inertial observer. The entire history of a (point) object can then be identified with a continuous sequence of events (i.e., a curve) in IR 1,3 called its worldline. Remark:

We will denote by η the 4 × 4 matrix ( ηα β ) and, even though

−1

η is actually equal to η, we will write η −1 = ( η α β ) to facilitate use of the Einstein summation convention. Now we let X denote some open submanifold of IR 1,3 (the charges creating our electromagnetic field live in IR 1,3 and we intend to carve out their worldlines and consider only the source free Maxwell equations on the resulting open submanifold of IR 1,3 ). Traditionally, an electromagnetic field on X is modeled by a globally defined, real-valued 2-form F on X that satisfies the source free Maxwell equations dF = 0

and d ∗F = 0 ,

where ∗ is the Hodge star on IR 1,3 determined by the usual orientation of IR 1,3 as IR 4 and the Minkowski metric (specifically, if F = 12 Fα β d xα ∧ d xβ , then ∗F =

1∗ 2 Fα β

d xα ∧ d xβ , where ∗Fα β =

1 2

εαβγδ F γ δ and F γ δ =

η µ γ η ν δ Fµ ν ). An electromagnetic potential for F is a 1-form A (generally only locally defined) that satisfies dA = F on its domain. In the gaugetheoretic formulation we propose now these will both acquire a (trivial) Liealgebra factor of −i (i.e., we will deal instead with F = −i F and A = −i A ). Now we build the classical gauge theory model by introducing items (1)-(8). X, as we have said, will be an open submanifold of IR 1,3 , with the induced orientation and semi-Riemannian metric. Since the particle we have in mind is charged and has spin zero, physics dictates that its wavefunction should have one complex component so we take V to be the (2-dimensional, real) vector | space C with the usual positive definite inner product h , i, which can be written 1 h z 1 , z2 i = ( z1 z¯2 + z¯1 z2 ) . (3.5) 2 The matrix Lie group G of (3) is taken to be U ( 1 ). Now, every irreducible

32

| representation of U ( 1 ) on C is of the form | ) ρn : U ( 1 ) −→ GL ( C

(3.6) ρn ( g ) ( z ) = g · z = g n z for some integer n and all of these are easily seen to be orthogonal with respect to h , i . Since electric charge is quantized we can measure it in multiples of the electron’s charge, i.e., by an integer. We identify the n in (3.6) with the charge of the spin zero particle we have under consideration. π Now let U ( 1 ) ,→ P −→ X be a principal U ( 1 )-bundle over X and ω a connection on the bundle with curvature Ω = d ω (since U ( 1 ) is abelian, all brackets are zero). For any section s : U → P we can write the corresponding | -valued) in gauge potential A and field strength F (which are u( 1 ) = Im C terms of real-valued forms A and F , respectively, as follows: A = s∗ ω = Aα d xα = −i Aα d xα = −i A . F = s∗ Ω =

1 1 Fα β dxα ∧ dxβ = − i Fα β dxα ∧ dxβ = −i F . 2 2

(3.7)

(3.8)

¡ ¢ Fα β = ∂α Aβ − ∂β Aα = −i ∂α Aβ − ∂β Aα . If s0 U 0 → P is another section with U ∩ U 0 6= ∅ and if, on U ∩ U 0 , s0 = s · g, where g : U ∩ U 0 → U ( 1 ) is the local gauge transformation, then the corresponding potential and field strength are given by Ag = g −1 A g + g −1 dg = A + g −1 dg and

F g = g −1 F g = F

on U ∩ U 0 (again we use the fact that U ( 1 ) is abelian). Notice that F g = F is the reason that field strengths in abelian gauge theories are globally defined on the base manifold X. | A matter field (item (6) ) can be identified with a map ψ : P → C that is equivariant, i.e., satisfies ψ ( p · g ) = g −1 · ψ ( p ) = g −n ψ ( p )

(3.9)

for all p ∈ P and all g ∈ U ( 1 ), or, equivalently, with the corresponding section | of the vector bundle P ×ρn C (we will use the same symbol ψ for both). As a potential function (item (7) ) we take | V : C −→ IR

(3.10) 1 1 1 2 V ( z ) = m h z , z i = m z z¯ = m | z | , 2 2 2 33

where m > 0 is a constant (ultimately identified with the mass of the particle). Since ρn is orthogonal with respect to h , i , V is invariant under the ac| , as required. Finally, we must specify an action (energy) tion of U ( 1 ) on C functional (item (8) ). Remark: Since the metric on X is now semi-Riemannian, inner products of forms need no longer be positive definite and we will refrain from writing norms as we did in (3.4). Since we have thus far dealt only with su( 2 ) (i.e., Im IH )valued forms we briefly recall that if α and β are two p-forms with values in some vector space with an inner product, then one defines the (pointwise) inner product of α and β as follows: Select a basis { Ta } and write α = αa Ta and β = β b Tb , where αa and β b are real-valued p-forms. These realvalued forms have (pointwise) inner products h αa , β b i defined by αa ∧ ∗β b = h αa , β b i vol and we define h α , β i by D E D E h α , β i = αa Ta , β b Tb = αa , β b h Ta , Tb i (we rely upon the reader to decide which inner product is intended by looking at what is inside). The result is independent of the choice of { Ta } . Applying | -valued 2-form F of (3.8) with the standard inner product this to the Im C | (3.5) on Im C reveals that F ∧ ∗ F = − h F , F i vol =

1 1 Fα β F α β vol = − Fα β F α β vol . 2 2

(3.11)

| -valued p-form µ one finds, again using the standard inner Similarly, for any C | product (3.5) on C and writing µ = µ1 + µ2 i , that

¯ = µ1 ∧ ∗ µ1 + µ2 ∧ ∗ µ2 = h µ , µ i vol . µ ∧∗µ

(3.12)

Finally, we remark that the switch to Minkowski spacetime necessitates a sign change in the Yang-Mills term F ∧ ∗ F of the action in order to ensure that the energy of the field (which is related to its spatial integrals) is positive. Now, as was the case for SU ( 2 ) Yang-Mills-Higgs theory, our action will contain a Yang-Mills term, an interaction term and a potential term. Only the interaction term remains to be discussed and it, once again, is determined by “minimal coupling”. In somewhat more detail, let us (temporarily) think of | -valued map on P . Then the covariant the matter field ψ as an equivariant C ! exterior derivative d ψ is just dψ acting on ω-horizontal parts of tangent | , ψ is determined by vectors. As a section of the vector bundle P ×ρn C the pullbacks of the equivariant map and the corresponding covariant exterior derivative is determined by the pullbacks of d ! ψ. These are given locally on X and in standard coordinates by ( ∂α + n Aα ) ψ dxα = ( ∂α − i n Aα ) ψ dxα , 34

(3.13)

where A = Aα dxα = − i Aα dxα is the corresponding gauge potential (we are thinking of the matter field as a section now and so write ψ rather than s∗ ψ = ψ ◦ s ). Now, (3.12) gives d ! ψ ∧ ∗ d ! ψ = h d ! ψ , d ! ψ i vol which, when written out locally in coordinates with (3.13) yields ¢ ¡ h d ! ψ , d ! ψ i = ( ∂ ψ + n A ψ ) ∂ α ψ¯ − n Aα ψ¯ α

α

¡ ¢ = ( ∂α ψ − i n Aα ψ ) ∂ α ψ¯ + i n Aα ψ¯

(3.14)

where Aα = η α β Aβ , Aα = η α β Aβ and ∂ α = η α β ∂β . With this we can write down a proposed action for our system consisting of a scalar field of mass m and charge n coupled to an electromagnetic field determined by the local gauge potential A = Aα dxα = − i Aα dxα as Z ³ ´ 1 2 A(ω , ψ ) = F ∧∗F + d! ψ ∧∗d! ψ + m∗ | ψ | 2 X Z µ 1 = − Fα β F α β (3.15) 4 X ¶ ¡ α ¢ 1 1 2 α ¯ ¯ + ( ∂α ψ − i n Aα ψ ) ∂ ψ + i n A ψ + m | ψ | vol. 2 2 The corresponding Euler-Lagrange equations are ( ∂α − i n A α ) ( ∂ α − i n A α ) ψ + m 2 ψ = 0

(3.16)

d ∗F = 0 ,

(3.17)

where ∗ denotes the Minkowski spacetime Hodge star. Since − i F is the pullback of a curvature form it satisfies the Bianchi identity and this gives dF = 0 .

(3.18)

The last two equations are just the sourcefree Maxwell equations, while (3.16) is the Klein-Gordon equation coupling our scalar field to the electromagnetic field.

4

The Zero-Dimensional Donaldson Invariant

We have seen in Section 1 that pure Yang-Mills theory, which arose from attempts by physicists to understand elementary particles, has deep consequences in differential topology (Donaldson’s 1983 Theorem). Coupling a gauge field to matter fields, as in SU ( 2 ) Yang-Mills-Higgs theory, also yields some rather 35

tantalizing connections with topology, as we saw in Section 2. This is, however, just the beginning of our story. From 1983 to 1994 the study of smooth 4-manifolds was dominated by the ideas of Simon Donaldson who showed how to extend the techniques behind his theorem on intersection forms to construct remarkably sensitive differential topological invariants for such manifolds (we describe the simplest of these in this section). In 1988, Witten [46], prompted by Atiyah, produced a classical gauge theory in the sense of Section 3 which, upon quantization, was found to contain certain observables whose expectation values were precisely these Donaldson invariants (the simplest of these invariants is the partition function of the quantum field theory and we will “derive” it in Section 5). This construction of Witten’s was a remarkable achievement and provided the most direct sort of link between topology and physics. However, the most extraordinary aspect of all of this did not emerge until the Fall of 1994 when his then recent work with Seiberg on supersymmetric gauge theories led Witten [48] to conjecture that all of the topological information contained in the Donaldson invariants could be extracted also from the vastly simpler set of invariants now known as Seiberg-Witten invariants (at least for a certain large class of 4-manifolds). This part of the story will be related in Sections 7 and 8. We begin our journey down this road by outlining the construction of the socalled zero-dimensional Donaldson invariant. Throughout this section B will denote a compact, simply connected, oriented, smooth 4-manifold (when the need arises somewhat later we will recall the definition of b+ 2 ( B ) and impose π additional assumptions regarding it). Every SU ( 2 )-bundle SU ( 2 ) ,→ P −→ B over B has a second Chern number c2 ( P ) [ B ] ∈ ZZ which can be written as Z 1 c2 ( P ) [ B ] = tr ( F ω ∧ F ω ) , (4.1) 8 π2 B where ω is any connection on the bundle and F ω is its curvature (thought of as a 2-form on B with values in the adjoint bundle ad ( P ) ). Such bundles are characterized up to equivalence by this integer and we shall denote by π

k SU ( 2 ) ,→ Pk −→ B

the bundle with c2 ( Pk ) [ B ] = k. Shortly we will explain why we are interested only in those bundles with k > 0. C( Pk ) will denote the set of all connection 1-forms on Pk and G( Pk ) is the gauge group of all (global) gauge transformations of Pk (diffeomorphisms f of Pk onto itself satisfying πk ◦f = πk and f ( p · g ) = f ( p ) · g for all p ∈ Pk and g ∈ SU ( 2 ) ). G( Pk ) acts on C( Pk ) on the right by pullback ( ω −→ ω · f = f ∗ ω ). Two connections ω , ω 0 ∈ C( Pk ) are said to be gauge equivalent if there is an f ∈ G( Pk ) such that ω 0 = f ∗ ω and we will denote by [ ω ] the gauge equivalence class of ω. The set of all such gauge equivalence classes is called the moduli space of connections on Pk and written B ( Pk ) = C ( Pk ) / G ( Pk ) = { [ ω ] : ω ∈ C ( Pk ) } . 36

It is this moduli space that we wish to study. Unfortunately, it has no reasonable mathematical structure in the smooth context in which we have just introduced it so one must replace the smooth objects just defined with appropriate Sobolev completions. This will require that some of the definitions be recast in other, but equivalent forms. Remark: Let us briefly recall a convenient means of defining Sobolev completions for a space of sections of a vector bundle. Begin with a compact Lie group π G and a principal G-bundle G ,→ P −→ X over some compact, oriented manifold X. Let V be a finite-dimensional real vector space with a positivedefinite inner product and ρ : G → GL( V ) an orthogonal representation of G on V. Let E = P ×ρ V be the associated vector bundle (any vector bundle over X can be represented in this way). Let Ωi ( X, E ) be the space of i-forms on X with values in E. In particular, Ω0 ( X, E ) is the space of sections of E. Choosing a Riemannian metric g on X one obtains natural inner products on each Ωi ( X, E ). Choosing a connection ω on P induces covariant exterior differentiation operators d!

d!

d!

Ω0 ( X , E ) −→ Ω1 ( X , E ) −→ Ω2 ( X , E ) −→ · · · .

(4.2)

Now suppose ξ ∈ Ω0 ( X , E ). For each m = 0, 1, 2, . . . one defines the Sobolev m-norm k ξ k m of ξ by m Z ° ³ °2 ´ X m ° ° 2 k ξ km = ° d ! ◦ · · · ◦ d ! ( ξ ) ° vol , j=0

X

where vol is the metric volume form of g. This is, indeed, a norm on Ω0 ( X , E ) and different choices of the Riemannian metric g , the connection ω and the inner product on V give rise to equivalent norms. The completion of Ω0 ( X , E ) relative to this norm is actually a Hilbert space L2m ( E ). Sobolev embedding theorems guarantee that, by choosing m sufficiently large, one can achieve any desired degree of smoothness for the elements of L2m ( E ). More precisely, if l is a non-negative integer and m > 21 ( dim X ) + l, then L2m ( E ) embeds in the space C l ( X , E ) of l-times continuously differentiable sections of E. Also note that each Ωi ( X , E ) is itself a space of sections of some vector bundle and so has Sobolev completions. Before returning to the main development we remark for future reference that, unlike the ordinary exterior derivative, the sequence (4.2) of covariant exterior derivatives is generally not a complex. Indeed, when E = ad P the composition of the first two d ! ◦ d ! : Ω0 ( X , ad P ) −→ Ω2 ( X , ad P ) is given by

d! ◦ d! (· ) = [F! ,

·]

.

(4.3)

Now we refashion our earlier definitions in such a way that we can define their Sobloev completions in the manner described in the Remark. Denote by 37

Ωi ( Pk , su( 2 ) ) the vector of space of i-forms on Pk with values in the Lie algebra su( 2 ). Then Ωiad ( Pk , su( 2 ) ) will denote the subspace consisting of all ϕ ∈ Ωi ( Pk , su( 2 ) ) that are tensorial of type ad, i.e., satisfy the following two conditions: 1. ϕ is horizontal in the sense that it vanishes whenever one of its arguments is vertical (tangent to a fiber in Pk ). 2. For each g ∈ SU ( 2 ) σg∗ ϕ = g −1 · ϕ = g −1 ϕ g, where σg : Pk → Pk is the diffeomorphism σg ( p ) = p · g. Finally, let Ωi ( B , ad Pk ) denote the space of i-forms on B with values in the adjoint bundle ad Pk . One easily shows that Ωiad ( Pk , su ( 2 ) ) and Ωi ( B , ad Pk ) are isomorphic (pull back elements of Ωiad ( Pk , su( 2 ) ) by sections of Pk and show, using (1) and (2), that these piece together to give elements of Ωi ( B , ad Pk ) ). For example, the curvature Ω of a connection ω is an su( 2 )-valued 2-form that is tensorial of type ad and the corresponding element of Ω2 ( B , ad Pk ) is what we have been denoting F ω . Our interest in this vector space is accounted for by the following proposition. Proposition 4.1 C( Pk ) is an affine space modeled on the vector space Ω1ad ( Pk , su( 2 ) ) ∼ = Ω1 ( B , ad Pk ), i.e., if ω 0 is any element of C( Pk ), then ª © (4.4) C ( Pk ) = ω 0 + ϕ : ϕ ∈ Ω1ad ( Pk , su ( 2 ) ) . The proof is simple since one need only show that if ω and ω 0 are in C( Pk ), then ω − ω 0 is tensorial of type ad. Now, each Ωi ( B , ad Pk ) is a space of sections of a vector bundle and so has Sobolev completions Ωim ( B , ad Pk ) for m = 0, 1, 2, 3, . . . For sufficiently large m its elements are all continuous sections so the isomorphism Ωi ( B , ad Pk ) ∼ = Ωiad ( Pk , su( 2 ) ) serves to define the Sobolev completions Ωiad, m ( Pk , su( 2 ) ). Thus, we can define a Sobolev space of connections on Pk for each such m by © ª Cm ( Pk ) = ω 0 + ϕ : ϕ ∈ Ω1ad, m ( Pk , su ( 2 ) ) , where ω 0 is any fixed, smooth connection on Pk . For our purposes it will suffice to take m = 3. C3 ( Pk ) =

©

ω 0 + ϕ : ϕ ∈ Ω1ad, 3 ( Pk , su ( 2 ) )

ª

.

(4.5)

To define Sobolev completions of the gauge group G ( Pk ) we consider the nonlinear adjoint bundle Ad Pk . This is the fiber bundle associated to SU ( 2 ) ,→ Pk → B by the adjoint (conjugation) action of SU ( 2 ) on itself. In particular, its typical fiber is the group SU ( 2 ), although it is not a principal bundle. Let Ω0 ( B , Ad Pk ) be the set of smooth sections of Ad Pk . It is a

38

group under pointwise multiplication in the fibers and is easily seen to be isomorphic to the group Ω0Ad ( Pk , SU ( 2 ) ) of smooth maps ψ : Pk → SU ( 2 ) that are equivariant, i.e., satisfy σg∗ ψ = g −1 · ψ, or, equivalently, ψ( p · g ) = g −1 ψ( p )g (here the group operation is pointwise multiplication in SU ( 2 ) ). We care about these groups for the following reason. Proposition 4.2 G ( Pk ) ∼ = Ω0Ad ( Pk , SU ( 2 ) ) ∼ = Ω0 ( B , Ad Pk ) . Once again the proof is simple. A gauge transformation f : Pk → Pk preserves the fibers of Pk and satisfies f ( p · g ) = f ( p ) · g so, for each p ∈ Pk there is a unique ψ( p ) ∈ SU ( 2 ) for which f ( p ) = p · ψ( p ) and this defines the appropriate ψ ∈ Ω0Ad ( Pk , SU ( 2 ) ). Now, unfortunately, Ω0 ( B , Ad Pk ) consists of sections of a fiber bundle with fiber SU ( 2 ) and not a vector bundle so it is not immediately clear how to define its Sobolev completions. However, if we regard | ) of 2 × 2-complex matrices, SU ( 2 ) as a subset of the vector space M2×2 ( C | ), where ρ is then Ad Pk embeds in the vector bundle E = Pk ×ρ M2×2 ( C | the representation of SU ( 2 ) on M2×2 ( C ) corresponding to conjugation. But the Sobolev spaces L2m ( E ) are defined (and C 1 for sufficiently large m ) so we can take © ª Gm ( Pk ) = s ∈ L2m ( E ) : s ( B ) ⊆ Ad Pk for such m. It will suffice for our purposes to take m = 4 and we will abuse the notation somewhat and write G4 ( Pk ) = Ω04 ( B , Ad Pk )

(4.6)

(the Sobolev index for G must be one greater than that for C to ensure a smooth action of G on C ). One can show ( [15] or [30] ) that G4 ( Pk ) is a Hilbert Lie group with Lie algebra (tangent space at the identity 11 ) that can be identified with T11 ( G4 ( Pk ) ) = Ω04 ( B , ad Pk ) (4.7) (this is at least believable since the sections in Ω04 ( B , ad Pk ) can be exponentiated pointwise to give elements of Ω04 ( B , Ad Pk ) ). Now, the action of G ( Pk ) on C ( Pk ) extends to an action of G4 ( Pk ) on C3 ( Pk ) (same formulas since the elements of G4 ( Pk ) are C 1 and those of C3 ( Pk ) are continuous). It is shown in [15] and [30] that this action is actually smooth and that, if ω ∈ C3 ( Pk ) is fixed, the map of G4 ( Pk ) to C3 ( Pk ) given by f −→ ω · f has a derivative at 11 that can be identified with d ! : Ω04 ( B , ad Pk ) −→ Ω13 ( B , ad Pk ) .

39

(4.8)

Remark: Differential operators extend to bounded operators on Sobolev completions and this is the meaning of d ! here and henceforth. In particular, the tangent space at ω to the orbit ω · G4 ( Pk ) of ω under G4 ( Pk ) is given by ¡ ¢ Tω ( ω · G4 ( Pk ) ) = Im ( d ! ) = d ! Ω04 ( B , ad Pk ) . (4.9) Now, the moduli space B3 ( Pk ) = C3 ( Pk ) / G4 ( Pk ) of (Sobolev index 3) connections on Pk is the set of gauge equivalence classes of the elements of C3 ( Pk ) modulo the action of G4 ( Pk ). Since C3 ( Pk ) is an affine space (by its very definition (4.5) ) it has a natural topology and we provide B3 ( Pk ) with the quotient topology, which one can show is Hausdorff ( [15] or [30] ). Our next objective is to study the local structure of B3 ( Pk ). Ideally, we would like a local manifold structure at each [ ω ], but we will find that this is possible only for what are called “irreducible” connections ω. The definition is as follows. For any ω ∈ C3 ( Pk ) the stabilizer (or isotropy subgroup) of ω is the subgroup stab ( ω ) of G4 ( Pk ) that leaves ω fixed, i.e., stab ( ω ) = { f ∈ G4 ( Pk ) : ω · f = ω } . Any such stabilizer contains the subgroup ZZ 2 of G4 ( Pk ) generated by ± 11 and if this is all it contains, i.e., if stab ( ω ) = ZZ 2 , then ω is said to be irreducible; otherwise ω is reducible. The following characterization of reducibility is proved in [15] and [30] (indeed, these will be our references for everything further we have to say about the moduli spaces). Theorem 4.3 The following are equivalent for any ω ∈ C3 ( Pk ). (a) ω is reducible, i.e., stab ( ω )/ ZZ 2 is nontrivial. (b) stab ( ω )/ ZZ 2 ∼ = U ( 1 ). (c) d ! : Ω04 ( B , ad Pk ) −→ Ω13 ( B , ad Pk ) has nontrivial kernel. We will denote by Cˆ3 ( Pk ) the subset of C3 ( Pk ) of irreducible connections (it is, in fact, an open subset) and by Bˆ3 ( Pk ) = Cˆ3 ( Pk ) / G4 ( Pk ) the moduli space of irreducible (Sobolev index 3) connections on Pk . The latter is an open subspace of B3 ( Pk ). Now we turn to the local structure of these moduli spaces. First consider

40

an ω ∈ Cˆ3 ( Pk ) so that [ ω ] ∈ Bˆ3 ( Pk ). We will produce a “slice” of the Gˆ4 ( Pk )-action on C3 ( Pk ) near ω, i.e., a submanifold O of C3 ( Pk ) such that T! ( C3 ( Pk ) ) = T! ( ω · G4 ( Pk ) ) ⊕ T! ( O )

(4.10)

and such that the restriction to O of the projection into the moduli space is injective near ω. Then the local structure of the moduli space near [ ω ] is the same as that of O near ω. To produce this O we will first produce an “orthogonal decomposition” of T! ( C3 ( Pk ) ) into T! ( ω · G4 ( Pk ) ) plus “something” and then use the affine structure (4.5) of C3 ( Pk ) to define a submanifold having this “something” as its tangent space at ω. Choosing a Riemannian metric g on B and an ad-invariant inner product h , i on su( 2 ) gives rise to natural inner products on all of the vector spaces Ωi ( B , ad Pk ) so that the operator d ! : Ω0 ( B , ad Pk ) −→ Ω1 ( B , ad Pk ) has a formal adjoint δ ! : d !- 1 Ω0 ( B , ad Pk ) ¾ Ω ( B , ad Pk ) δ! (in fact, δ ! = − ∗d ! ∗ , where ∗ is the Hodge dual corresponding to g and the given orientation of B ). It turns out that δ ! ◦ d ! : Ω0 ( B , ad Pk ) −→ Ω0 ( B , ad Pk ) is a (formally self-adjoint) elliptic operator. We use the same symbols for the extensions of these operators to the Sobolev completions Ω04 ( B , ad Pk ) and Ω13 ( B , ad Pk ). Elliptic theory (the generalized Hodge Decomposition Theorem) implies that (a) ker ( δ ! ◦ d ! ) = ker ( d ! ) is finite-dimensional, (b) Im ( d ! ) = ker ( δ ! )⊥ , (c) d ! has closed range, and (d) there is an orthogonal decomposition Ω13 ( B , ad Pk ) = Im ( d ! ) ⊕ ker ( δ ! ) , i.e.,

T! ( C3 ( Pk ) ) = T! ( ω · G4 ( Pk ) ) ⊕ ker ( δ ! )

(4.11)

(by (4.5) and (4.9) ). Now, for any ε > 0, the submanifold O!,ε = { ω + A : A ∈ ker ( δ ! ) , k A k 3 < ε } clearly satisfies

¡ ¢ T! O!,ε = ker ( δ ! ) 41

(4.12)

so

¡ ¢ T! ( C3 ( Pk ) ) = T! ( ω · G4 ( Pk ) ) ⊕ Tω O!,ε .

(4.13)

We claim that, for sufficiently small ε > 0, O!,ε projects injectively into the moduli space. To prove this last claim one first observes that, since ω ∈ Cˆ3 ( Pk ) and Cˆ3 ( Pk ) is open in C3 ( Pk ) we can take ε > 0 small enough to ensure that O!,ε ⊆ Cˆ3 ( Pk ). Now consider the map Ψ : O!,ε × G4 ( Pk ) −→ Cˆ3 ( Pk ) Ψ ( ω0 , f ) = ω0 · f . The derivative of Ψ at ( ω , 11 ) is computed to be ( d Ψ )( !,11 ) : ker ( δ ! ) ⊕ Ω04 ( B , ad Pk ) −→ ker ( δ ! ) ⊕ Im ( d ! ) ³ ´ ( d Ψ )( !,11 ) = id ker ( δ ω ) , d! . This is certainly surjective and, because ω is assumed irreducible, Theorem 4.3 (c) implies that it is also injective. By the Open Mapping Theorem (a bounded, surjective, linear operator between Banach spaces is an open mapping), ( d Ψ )( !,11 ) is an isomorphism. Thus, the Inverse Function Theorem for Banach manifolds (see [29] ) implies that, near ( ω , 11 ), Ψ is a local diffeomorphism. More precisely, for some (perhaps smaller) ε > 0 there is an open neighborhood U! of ω in Cˆ3 ( Pk ) and an open set U11,ε = { f ∈ G4 ( Pk ) : k 11 − f k 4 < ε } in G4 ( Pk ) such that the restriction Ψ : O!,ε × U11,ε −→ U! is a diffeomorphism. In particular, no two things in O!,ε are gauge equivalent by any gauge transformation that is within ε of 11. A “bootstrapping ” argument then shows that, for a (possibly) still smaller ε > 0 , no two things in O!,ε are gauge equivalent by any gauge transformation. For such an ε > 0 , O = O!,ε projects injectively into Bˆ3 ( Pk ) and so is our slice and provides a local manifold structure for Bˆ ( P ) near [ ω ]. In particular, Bˆ ( P ) has the 3

3

k

k

structure of a smooth Hilbert manifold. If ω ∈ C3 ( Pk ) is reducible the analysis is similar except that to get an injective projection into the moduli space one must first factor out the action of the stabilizer of ω . More precisely, defining O!,ε as in (4.12) and g ( ω ) = stab ( ω )/ ZZ ∼ stab 2 = U ( 1 ) one finds that, for sufficiently small ε > 0 , the projection g ( ω ) −→ B ( P ) O!,ε / stab 3 k is a homeomorphism onto an open neighborhood of [ ω ] in B3 ( Pk ) which, 42

g ( ω ). There are in fact, is a diffeomorphism outside the fixed point set of stab generally singularities, where there is no local smooth structure, at the images of these fixed points (e.g., these account for the cones in our picture of the moduli space used in the proof of Donaldson’s 1983 Theorem). Now, the objects of real interest in Donaldson theory are certain subspaces of B3 ( Pk ) and Bˆ3 ( Pk ) which we now introduce. Begin by selecting some Riemannian metric g on B. Together with the orientation of B this gives a Hodge star operation ∗ on smooth forms defined on B. Since B is 4dimensional, ∗ : Ω2 ( B , ad Pk ) −→ Ω2 ( B , ad Pk ) and, since the elements of Ω22 ( B , ad Pk ) are continuous, this extends to ∗ : Ω22 ( B , ad Pk ) −→ Ω22 ( B , ad Pk ) . The curvature map F : C ( Pk ) −→ Ω2 ( B , ad Pk ), ω → F ! , also extends to a smooth map F : C3 ( Pk ) −→ Ω22 ( B , ad Pk ) ω −→ F ! so we may say that an ω ∈ C3 ( Pk ) is g-anti-self-dual ( g-ASD) if ∗

F! = −F! .

Remark:

The Chern number k of our bundle can be written as Z 1 k = c 2 ( Pk ) [ B ] = tr ( F ! ∧ F ! ) 8π 2 B Z ³ ° − °2 ° + °2 ´ 1 °F! ° − °F! ° = vol 2 8π B

1 ∗ where F ± ! = 2 ( F ! ± F ! ) are the self-dual and anti-self-dual parts of F ! . Consequently, when k < 0 we must have F + ! 6= 0 and anti-self-dual connections cannot exist. When k = 0 it is possible for anti-self-dual connections ω to exist, but they must be flat ( F ! = 0 ) because F + ! = 0 and k = 0 gives + − F− = 0 and F = F + F . We will see shortly that these are not partic! ! ! ! ularly interesting and this will account for our restriction to bundles with k > 0 .

Now we define Asd3 ( Pk , g ) =

©

ω ∈ C3 ( Pk ) : ω is g - ASD

ª

and d (P , g) = Asd 3 k

n

ω ∈ Cˆ3 ( Pk ) : ω is g - ASD

43

o

and the corresponding moduli spaces M ( Pk , g ) = Asd3 ( Pk , g ) / G4 ( Pk ) d (P , g) / G (P ) ˆ ( Pk , g ) = Asd M 3 k 4 k of g-ASD and irreducible g-ASD connections. Donaldson theory is built on the analysis of these moduli spaces. Remark: Note that the Sobolev indices have been dropped on M ( Pk , g ) ˆ ( P , g ). The reason is that, for any ω ∈ Asd ( P , g ) , elliptic regand M 3 k k ularity implies that there is an f ∈ G4 ( Pk ) such that ω · f is a smooth connection (see Section 5 of [15] ). Thus, these moduli spaces do not depend on the choice of (sufficiently large) Sobolev index. For a given B, g and k, Asd3 ( Pk , g ) (and therefore M ( Pk , g ) ) might | IP 2 well be empty. This is the case, for example, when B is either S 2 ×S 2 or C | with their standard orientations and metrics (Fubini-Study in the case of C IP 2 ) and k = 1. Changing the orientation of the manifold can have a dramatic ef2 | IP fect, e.g., the k = 1 bundle over C (also with the Fubini-Study metric) has a moduli space of ASD connections that one can describe as explicitly as we did for S 4 in Section 1 (for more details on this and many more examples, see [12] ). A general result of considerable interest was proved by Taubes [41]. Through an ingenious “grafting” procedure using the k = 1 instantons on S 4 described in Section 1 he was able to prove that the k = 1 bundle over any B with b+ 2 ( B ) = 0 admits g-ASD connections for any Riemannian metric g (the definition of b+ 2 ( B ) follows). Remark: Since it will play a recurrent role from this point on we recall the definition of b+ 2 ( B ) for a compact, simply connected, oriented, smooth 4manifold B. In Section 1 we introduced the intersection form QB : H2 ( B ; ZZ ) × H2 ( B ; ZZ ) −→ ZZ . It is an integer-valued, symmetric, bilinear form on the finitely-generated, free abelian group H2 ( B ; ZZ ). If b2 ( B ) is the rank of H2 ( B ; ZZ ), then one can write − b2 ( B ) = b+ 2 ( B ) + b2 ( B ) , − where b+ Z) 2 ( B ) ( b2 ( B ) ) is the maximal dimension of a subspace of H2 ( B ; Z on which QB is positive (negative) definite. One can show that b+ ( B ) 2 − ( b2 ( B ) ) is also the dimension of the space of self-dual (anti-self-dual) harmonic 2-forms on B (for any choice of a Riemannian metric on B ) and this accounts for the role it plays in the study of ASD connections.

ˆ ( P , g ) for k > 0 Before proceeding with the study of M ( Pk , g ) and M k 44

we explain our earlier comment that the k = 0 case is “not particularly interesting” ( k < 0 is definitely not interesting since the moduli spaces are empty). The k = 0 bundle SU ( 2 ) ,→ P0 → B is trivial and, as we observed earlier, any ASD connection on it is necessarily flat. Conversely, any flat connection is certainly ASD ( F ! = 0 implies F + ! = 0 ). Since flat connections exist on any trivial bundle (page 92 of Vol I of [24] ), the moduli space M ( P0 , g ) is nonempty (for any g ). Since B is simply connected, any two flat connections on B are gauge equivalent (Proposition 2.2.3 of Vol I of [24] ) so M ( P0 , g ) is, in fact, just a single point. Now we return to the general study of the moduli spaces M ( Pk , g ) and ˆ ( P , g ). For this we consider the smooth map M k pr+ ◦ F : C3 ( Pk ) −→ Ω2+,2 ( B , ad Pk ) , where F is the curvature map ( F ( ω ) = F ! ) and pr+ projects onto the self-dual part. Thus, ¡ ¢ pr+ ◦ F ( ω ) = F + ! and Asd3 ( Pk , g ) =

¡

pr+ ◦ F

¢−1

(0) .

(4.14)

At any ω ∈ C3 ( Pk , g ) the derivative of this map can be identified with 2 ! 1 d! + = pr+ ◦ d : Ω3 ( B , ad Pk ) −→ Ω+,2 ( B , ad Pk ) ,

(see page 54 of [15] ). Now, we have already observed that, in general, d! ◦ d! is not zero, but it follows from (4.3) that, when ω is ASD, ! = £F+ , ·¤ = [0, ·] = 0 d! + ◦ d ! so

¡ ¢ ¡ ¢ Im ( d! ) = d! Ω04 ( B , ad Pk ) ⊆ ker d! + .

(4.15)

Thus, we have associated with every ω ∈ Asd3 ( Pk , g ) a complex E ( ω ) d! d+! - 2 0 −→ Ω04 ( B , ad Pk ) ¾ Ω13 ( B , ad Pk ) ¾ Ω+,2 ( B , ad Pk ) −→ 0 , δ! δ+! where we have included also the adjoints δ ! and δ+! of d ! and d+! , respectively. This complex is, in fact, elliptic and the entire analysis of the local structure of the moduli space near [ ω ] rests on an analysis of the structure of E ( ω ) (the so-called fundamental elliptic complex associated with ω ∈ Asd3 ( Pk , g ) ). We begin by simply enumerating some consequences of the generalized Hodge Decomposition Theorem for elliptic complexes.

45

(1) The Laplacians ∆0! = δ ! ◦ d !

∆1! = d ! ◦ δ ! + δ+! ◦ d+! ∆2! = d+! ◦ δ+!

are all self-adjoint, elliptic operators. (2) The spaces ker ( ∆k! ), k = 0, 1, 2, of harmonic forms are finite-dimensional and consist of smooth forms (smoothness follows from “elliptic regularity”). (3) Each of the cohomology groups H 0 ( ω ) = ker ( d ! ) ¡ ¢ H 1 ( ω ) = ker d+! / Im ( d ! )

¡ ¢ H 2 ( ω ) = Ω2+,2 ( B , ad Pk ) / Im d+!

associated with E ( ω ) contains a unique harmonic representative. In particular, Hk ( ω ) ∼ = ker ( ∆k! ) , k = 0, 1, 2 so all of these cohomology groups are finite-dimensional and we may define the index of the complex E ( ω ) by ¡ ¢ ¡ ¢ ¡ ¢ Ind ( E ( ω ) ) = dim H 0 ( ω ) − dim H 1 ( ω ) + dim H 2 ( ω ) ¡ ¢ ¡ ¢ ¡ ¢ = dim ker ( ∆0! ) − dim ker ( ∆1! ) + dim ker ( ∆2! ) . (4) There are orthogonal decompositions Ω04 ( B , ad Pk ) ∼ = Im ( δ ! ) ⊕ ker ( d ! ) Ω13 ( B , ad Pk ) ∼ = Im ( d ! ) ⊕ ker ( δ ! ) ¡ ¢ ¡ ¢ Ω2+,2 ( B , ad Pk ) ∼ = Im d+! ⊕ ker δ+! . Now we put all of this information to use in the following way. Fix ω ∈ Asd3 ( Pk , g ) and restrict the map pr+ ◦ F to one of the sets O!,ε (see (4.12) ) ¯ pr+ ◦ F ¯ O!,ε : O!,ε −→ Ω2+,2 ( B , ad Pk ) . Then

¡

pr+ ◦ F

¢−1

( 0 ) = Asd3 ( Pk , g ) ∩ O!,ε

and the derivative at ω is ¯ ! ! 2 ¯ d! + ker ( δ ) : ker ( δ ) −→ Ω+,2 ( B , ad Pk ) . 46

(4.16)

(4.17)

Before proceeding we recall a few facts from analysis. Remark: Recall that a bounded linear map T : H1 → H2 between two Hilbert spaces is said to be Fredholm if either of the following two equivalent conditions is satisfied: (a) dim ( ker T ) < ∞ , dim ( ker T ∗ ) < ∞ and Im T is closed. (b) H1 ∼ = ker T ⊕ Im T ∗ and H2 ∼ = ker T ∗ ⊕ Im T . We will soon appeal to the following infinite-dimensional version of the Implicit Function Theorem: Let X and Y be Hilbert manifolds, F : X → Y a smooth map and x0 ∈ X a point at which the derivative Dfx0 : Tx0 ( X ) → TF (x ) ( Y ) is a surjective Fredholm map. Then there exists an open neighbor0

hood of x0 in F −1 ( F ( x0 ) ) that is a smooth (finite-dimensional) manifold of dimension ³ ³ ´´ dim ker Dfx0 .

! We will show that the map d! + | ker( δ ) is always Fredholm and determine conditions under which it is surjective, thus setting up an application of the Implicit Function Theorem to obtain a smooth manifold structure for Asd3 ( Pk , g ) ∩ O!,ε near ω . If, in addition, the projection of Asd3 ( Pk , g ) ∩ O!,ε into the moduli space is injective (i.e., if ω is irreducible) this will give a finite-dimensional smooth manifold structure for the moduli space near [ ω ]. ! ! To see that d! + | ker( δ ) is Fredholm we reason as follows. First, ker( d+ | ! ! ! 1 ker( δ ) ) = ker( d+ ) / Im ( d ) = H ( ω ) by (4) and (3) so this is finite! ∗ = δ ! | Im ( d! | dimensional by (3). Next observe that ( d! + + | ker ( δ ) ) + ! ! ∗ ! | Im ( d! ) ) ! ! ! ker( δ ) ) = δ+ | Im ( d+ ) so ker( ( d+ | ker( δ ) ) ) = ker( δ+ + ! ) = H 2 ( ω ) is finite-dimensional. which is finite-dimensional because ker( δ+ ! ⊥ by (4) so this is closed. ! ! Finally, Im ( d! + | ker( δ ) ) = Im ( d+ ) = ( ker( δ+ ) ) ! We have verified the requirements of (a) in the Remark above so d! + | ker( δ ) is Fredholm. ! ! Now we determine when d! + | ker( δ ) is surjective. Since d+ acts on 1 ! ! ! Ω3 ( B , ad Pk ) ∼ = Im ( d ) ⊕ ker ( δ ) and since d+ vanishes identically on ! ! ! Im ( d ) by (4.15), d! + | ker( δ ) is surjective if and only if d+ is surjective, ! i.e., if and only if Im ( d+ ) = Ω2+,2 ( B , ad Pk ). But, by (3), this is the case if and only if H 2 ( ω ) is trivial. Noting that H 0 ( ω ) is trivial if and only if ker( d! ) is trivial and, by Theorem 4.3(c), this is the case if and only if ω is irreducible we arrive at the following interpretations of the cohomology groups of E ( ω ). H 0 ( ω ) = 0 ⇐⇒ ω is irreducible

47

(4.18)

H 1 ( ω ) = ker

¡

¯ ! ¢ ¯ d! + ker ( δ )

= kernel of the derivative of ¯ pr+ ◦ F ¯ O!, ε at ω

¯ ! ¯ H 2 ( ω ) = 0 ⇐⇒ d! + ker ( δ ) is surjective .

(4.19)

(4.20)

Recalling that when ω is irreducible the projection into the moduli space is injective near ω we can summarize all of this in the following theorem. Theorem 4.4 At ω ∈ Asd3 ( Pk , g ) the map ¯ pr+ ◦ F ¯ O!, ε : O!, ε −→ Ω2+,2 ( B , ad Pk ) has derivative ¯ ! ! 2 ¯ d! + ker ( δ ) : ker ( δ ) −→ Ω+,2 ( B , ad Pk ) that is Fredholm. The derivative is surjective if and only if H 2 ( ω ) = 0 and, in this case, ¯ ¡ ¢−1 pr+ ◦ F ¯ O!, ε ( 0 ) = Asd3 ( Pk , g ) ∩ O!, ε is a smooth manifold of dimension ³ ´ ¯ ¡ ! ¢ = dim ¡ H 1 ( ω ) ¢ ¯ dim ker d! + ker ( δ ) near ω. If, in addition, H 0 ( ω ) = 0 , then the projection into the moduli space M ( Pk , g ) gives a chart of dimension dim ( H 1 ( ω ) ) near [ ω ]. Notice that if H 0 (ω) = 0 and H 2 (ω) = 0 are both trivial, then Ind(E(ω) ) = − dim ( H 1 ( ω ) ) so, if we could calculate the index of the elliptic complex E ( ω ), we would have (minus) the dimension of the moduli space near [ ω ]. The Atiyah-Singer Index Theorem gives the index of E ( ω ) in terms of topological data on B and SU ( 2 ) ,→ Pk → B. In our present context the result is ¢ ¡ Ind ( E ( ω ) ) = − 8k + 3 1 + b+ (4.21) 2 (B) (see pages 267-271 of [12] ). Note, in particular, that the result is independent of ω so we obtain the following consequence of Theorem 4.4. Corollary 4.5 If H 0 ( ω ) = 0 and H 2 ( ω ) = 0 for every ω ∈ Asd3 ( Pk , g ), ˆ ( P , g ) is a smooth manifold of dimension 8k − 3( 1 + then M ( Pk , g ) = M k + b2 ( B ) ). 48

For a given choice of the Riemannian metric g it may or may not be the case that H 0 ( ω ) = 0 and H 2 ( ω ) = 0 for every ω ∈ Asd3 ( Pk , g ). We will see shortly, however, that for “almost all” choices of g, H 2 ( ω ) will be trivial for all ω ∈ Asd3 ( Pk , g ) and, with one additional restriction on the topology of B, the same is true H 0 ( ω ). First, however, we note that if ω ∈ Asd3 ( Pk , g ) and H 2 ( ω ) = 0, then the slice Asd3 ( Pk , g ) ∩ O!,ε still has a local manifold structure near ω, but if H 0 ( ω ) 6= 0 one can only obtain a one-to-one projection into the moduli space by first factoring out the action of the stabilizer of ω. The consequence is that, near [ ω ], M ( Pk , g ) is not smooth but has a neighborhood homeomorphic to this quotient, which turns | IP 4k−2 with [ ω ] at the vertex. Reducible connections, out to be a cone over C when they exist, give rise to cone-like “singularities” in the moduli space Next we will require a brief discussion of various “generic metrics” theorems which assert that, under certain circumstances, “almost all” choices for the Riemannian metric g give rise to “nice” moduli spaces of ASD connections. Begin by considering the space R of all Riemannian metrics on B. This is a space of sections of a fiber bundle over B and can be given the structure of a (pathwise connected) Hilbert manifold. With this structure one can show that 1. There is a dense Gδ -set in R such that, for every g in this set, any g-ASD connection ω on Pk , k > 0, satisfies H 2 ( ω ) = 0. 2. If b+ 2 ( B ) > 0, then there is a dense Gδ -set in R such that, for every g in this set, any g-ASD connection ω on Pk , k > 0, satisfies ( H 2 ( ω ) = 0 and ) H 0 ( ω ) = 0. ˆ ( P , g ) is (either empty or) a In short, for “generic” g , M ( Pk , g ) = M k + smooth manifold of dimension 8k − 3( 1 + b2 ( B ) ). We will not attempt to sketch proofs of these last two results, but, very ˆ ( P , g ) is smooth roughly, here is how one might go about showing that M k for a generic choice of g. Consider the so-called parametrized moduli space n o ˆ ( Pk , R ) = M ( [ ω ] , g ) ∈ Bˆ3 ( Pk ) × R : ω is g - ASD . This is an infinite dimensional smooth submanifold of Bˆ3 ( Pk ) × R. One shows that the projection map ˆ ( Pk , R ) −→ R M is smooth with Fredholm derivative at each point. The Sard-Smale Theorem (infinite dimensional version of Sard’s Theorem) implies that the set of regular values of the projection is a dense Gδ -set in R . But then, for any g in this set, the inverse image ³ ´ ˆ ( Pk , R ) = M ˆ ( Pk , g ) Bˆ3 ( Pk ) × { g } ∩ M ˆ P , R ). is a smooth submanifold of M( k 49

Remark: The restriction on b+ 2 ( B ) arises because the subset of R consisting of those g for which reducible g-ASD connections on Pk exist is a countable + union of smooth submanifolds of codimension b+ 2 ( B ). If b2 ( B ) = 0, then reducibles are generically unavoidable. As we saw in Section 1 this is a good, not a bad thing as it leads to Donaldson’s theorem on definite intersection forms. Crudely put, the idea behind defining the Donaldson polynomial invariants ˆ ( P , g ) as cycles over which to integrate is to regard the moduli spaces M k certain carefully selected differential forms. In order to carry out such a program these moduli spaces must be orientable and, if the result is to be a differential topological invariant, the integrals must be independent of the choice of (generic) g. We now record two results that guarantee this. Theorem 4.6 Suppose g is a Riemannian metric on B for which H 2 ( ω ) = 0 for every g-ASD connection ω on Pk , k > 0. Then the moduli space ˆ ( P , g ) is orientable. An orientation for M ˆ ( P , g ) can be uniquely specM k k ified by choosing an orientation for B and an orientation for the vector space 2 H+ ( B; IR ) of self-dual 2-forms on B. The proof amounts to constructing an explicit model for the determinant line ˆ ( P , g ) from a family of differential operbundle (top exterior power) of M k ators on B and exhibiting a nonzero section (see Sections 5.4.1 and 7.1.6 of [12] ). To state the final result of this section we consider two metric g 0 and g 1 in the dense Gδ -set in R on which H 2 ( ω ) = 0 for all g-ASD connections. Since R is pathwise connected we can join them with a path { g t : 0 ≤ t ≤ 1 } in R . Define the parametrized moduli space n o ˆ ( Pk , { g t } ) = ˆ ( Pk , g t ) . M ( [ ω ] , t ) ∈ Bˆ3 ( Pk ) × [ 0 , 1 ] : [ ω ] ∈ M Theorem 4.7 If g 0 and g 1 are in the dense Gδ -set in R on which H 2 ( ω ) = 0 for all g-ASD connections ω, then, for a generic path { g t : 0 ≤ t ≤ 1 } ˆ P , { g t } ) is a smooth, orientable submanifold of Bˆ ( P ) × joining them, M( 3 k k 2 [ 0, 1 ] with boundary. A choice of orientation µ for H+ ( B; IR ) determines an ˆ P , {g t}). Moreover, the oriented boundary of M( ˆ P , {g t}) orientation for M( k k ˆ is the disjoint union of M( Pk , g 1 ) with the orientation induced by µ and ˆ P , g ) with the orientation opposite to that induced by µ. M( k

0

ˆ P , g ) within a single homology In short, a generic variation of g varies M( k ˆ ˆ class. Even if M( Pk , g 0 ) and M( Pk , g 1 ) are smooth manifolds one canˆ P , g t ) are not, in general, arrange that the intermediate moduli spaces M( k all smooth. There may be finitely many values of t for which one encounters 50

reducible connections. Even this can be avoided, however, if one is willing to assume of B that b+ 2 ( B ) > 1. In this case a generic path { g t } from g 0 to ˆ P , g t ) is a smooth, orientable manifold. g 1 has the property that each M( k With the apparatus we have now assembled one is almost (but not quite) in a position to begin building the Donaldson invariants. The remaining obstacle ˆ P , g ) are generally (and it is a serious one) is that the moduli spaces M( k not compact so one cannot integrate over them (more precisely, they do not determine a fundamental homology class with which to pair cohomology classes). ˆ P , g ) with what is known as its One overcomes this obstacle by replacing M( k “Uhlenbeck compactification”. We intend to not overcome, but circumvent the obstacle by considering only a special case (an outline of the general situation is available in Section 8 ). Notice that, by an appropriate arrangement of b+ 2 ( B ) and the Chern number k, it is entirely possible for the dimension 8k − 3( 1 + b+ 2 ( B ) ) of the ˆ P , g ) to come out zero. In this case, M( ˆ P , g ) is generic moduli space M( k k 2 a 0-dimensional, oriented manifold (given an orientation of H+ ( B; IR ) ), i.e., it is a set of isolated points [ ω ] each equipped with a sign which we will write ( −1 )[ ! ] . As it happens, the moduli space is necessarily compact in this case so we can add these signs to obtain an integer X [!] γ0 ( B ) = ( −1) . (4.22) ˆ P ,g) [ ! ]∈ M( k

One can show (from the homology result Theorem 4.7 and the remarks following it) that if b+ 2 ( B ) > 1, then this integer does not depend on the choice of (generic) metric g and is, in fact, an orientation preserving diffeomorphism 2 invariant of B (assuming the orientation of H+ ( B; IR ) is fixed). Under all of these circumstances the integer γ0 ( B ) is called the 0-dimensional Donaldson invariant of B. Our concern here is not with calculating γ0 ( B ) nor with using it to obtain topological information (for a nontrivial calculation and application see Section 9.1 of [12] ). Rather we would like to show that, by adoping a slightly different perspective, γ0 ( B ) is in many ways analogous to an Euler characteristic. This, in turn, will eventually lead us to formulas for γ0 ( B ) that evolve into the partition function for the topological quantum field theory of Witten referred to at the beginning of this section. Before describing our new perspective on γ0 ( B ) we will breifly review some standard material on the Euler number of a vector bundle. We consider an oriented, real vector bundle πE : E → X of even rank (fiber dimension) 2k over a compact, oriented manifold X of dimension 2k, e.g., the tangent bundle of a compact, oriented 2k-manifold. The typical fiber of E will generally be denoted V and will usually come equipped with a positive definite inner product (e.g., from a fiber metric on E ). The exterior algebra of V will be denoted V L ∞ Vp V = p=0 V and should be thought of as the graded algebra of polynominals with real coefficients in the odd (anti-commuting) variables ψ 1 , . . . , ψ 2k , 51

where { ψ 1 , . . . , ψ 2k } is some fixed oriented, orthonormal basis for V . The volume form for V corresponding to { ψ 1 , . . . , ψ 2k } is vol = ψ 1 · · · ψ 2k , where V we omit the customary wedge ∧ and write the product in V by juxtaposition. There are a number of ways to approach the definition of the Euler number of the vector bundle E, several of which will be important to us. We proceed as follows. The Euler number χ ( E ) of the vector bundle πE : E → X is defined by Z χ(E ) = e(E ) (4.23) X 2

where e ( E ) ∈ H ( X; IR ) is the “Euler class” of E, for which we offer two (equivalent) definitions. The Euler class of E can be defined by the Chern-Weil procedure in a manner entirely analogous to the familiar definitions of Chern and Pontryagin classes. We briefly review the ideas behind this procedure. Let G ,→ πP P −→ X be a principal G-bundle with connection ω and curvature Ω. Let { ξ1 , . . . , ξn } be a basis for the Lie algebra G of G and write ω = ω a ξa and Ω = Ωa ξa , where ω a ∈ Ω1 ( P ) and Ωa ∈ Ω2 ( P ), a = 1, . . . , n , are real| [ G ]G be the algebra of complex-valued polynomials P on valued forms. Let C G that are ad G-invariant ( P ( g −1 ξg ) = P ( ξ ) for all g ∈ G and ξ ∈ G ). | [ G ]G concretely as follows: Let { x1 , . . . , xn } be the basis One can realize C ∗ for G dual to { ξ1 , . . . , ξn } and think of the xi as linear functions on G. The symmetric algebra S( G ∗ ) can be identified with the polynomial algebra | | [ x1 , . . . , xn ] from which we IR [ x1 , . . . , xn ] and tensoring with C gives C G | select the ad-invariant elements to get C [ G ] . Next we denote by Ω∗ ( P )BAS the graded algebra of real-valued forms on P that are basic, i.e., G-invariant ( σg∗ ϕ = ϕ for every g ∈ G, where σg ( p ) = p · g ), and horizontal ( ιW ϕ = 0 for all vertical vector fields W , where ιW denotes interior multiplication by W ). These are precisely the forms on P which descend to X, i.e., for which ¯ ∈ Ω∗ ( X ) such that πP∗ ϕ ¯ = ϕ. Now there is a map there is a ϕ | [G ] CW! : C

G

−→ Ω∗ ( P )BAS ,

called the Chern-Weil homomorphism, defined by “evaluating the polyno| [ G ]G has degree k and mial on the curvature of ω.” More precisely, if P ∈ C if we denote by P also the corresponding k-multilinear map on G (obtained by polarization), then ³ ´ CW! ( P ) = P ( Ω ) = P Ωa1 ξa1 , . . . , Ωak ξak ³ =P

´ ξa1 , . . . , ξak

Ωa1 ∧ · · · ∧ Ωak .

¯ Ω ) on X which Being basic, P( Ω ) is the pullback by πP of a form P( ¯ Ω)] ∈ can be shown to be closed and whose (de Rham) cohomology class [ P( 2k H ( X; IR ) does not depend on the choice of ω. Making specific choices for 52

P gives rise to various characteristic classes of the bundle. For example, if G = U ( 1 ), then 2iπ [ tr ( Ω ) ] is the first Chern class and, if G = SU ( 2 ), then − 8 1π2 [ tr ( Ω ∧ Ω ) ] is the second Chern class (see Chapter XII, Vol. II, of [24] for more details). Now, to define the Euler class of our vector bundle πE : E → X by this procedure we will require a principal bundle and an invariant polynomial. For the former we select a fiber metric on πE : E → X and consider the corresponding oriented, orthonormal frame bundle SO ( 2k ) ,→ FSO ( E )

πSO - X.

(4.24)

For the SO( 2k )-invariant polynomial we select the Pfaffian P f : so ( 2k ) −→ IR defined as follows: To each skew-symmetric matrix Q = ( qij ) ∈ so( 2k ) we associate an element X 1 qij ψ i ψ j = ψ T Q ψ 2 i
µ

¶k

1 T ψ Qψ 2

= P f ( Q ) vol .

(4.25)

One can show that P f is ad( SO( 2k ) )-invariant (in fact, it is a square root of the determinant, i.e., ( P f ( Q ) )2 = det Q for each Q in so( 2k ) ). Now choose a connection ω on the frame bundle (4.24), denote its curvature by Ω and define the Euler class e( E ) of πE : E → X by e( E ) = ( −2π )−k [ P f ( Ω ) ], which we prefer to write as ¤ −k £ e ( E ) = ( 2π ) Pf ( − Ω ) . (4.26) Locally, e( E ) is given by −k

e ( E ) = ( 2π )

[ P f ( − s∗ Ω ) ] ,

(4.27)

where s is any section of the frame bundle (4.24). With this our definition of the Euler number (4.23) is complete. Remark: The famous Gauss-Bonnet-Chern Theorem asserts that when E is the tangent bundle T X of X, then the Euler number χ( T X ) is, in fact, the Euler characteristic (alternating sum of Betti numbers) of X and so is a 53

topological invariant of X. Example: We consider the 2-sphere S 2 with its usual orientation and Riemannian metric and its tangent bundle π : T S 2 → S 2 . The corresponding oriπ

SO ented, orthonormal frame bundle is SO( 2 ) ,→ FSO ( T S 2 ) −→ S 2 . If θ and φ are the usual spherical coordinates on S 2 , then { e1 , e2 } = { ∂∂φ , sin1 φ ∂∂θ } is

an oriented, orthonormal frame field on S 2 , i.e., a section s of FSO ( S 2 ). The dual oriented, orthonormal field of 1-forms is { e1 , e2 } = { dφ , sin φ dθ } so the metric volume form is e1∧e2 = sin φ dφ∧dθ. One computes de1 = 0 = 0( e1∧e2 ) and de2 = cos φ dφ ∧ dθ = cot φ( e1 ∧ e2 ) so 0 · e1 + ( cot φ )e2 = cot φ dθ. Thus, the Levi-Civita connection ω on the frame bundle is determined by à ! 0 cos φ dθ ∗ s ω = − cos φ dθ 0 so its curvature Ω = dω ( SO( 2 ) is abelian) is à ! 0 − sin φ dφ ∧ dθ ∗ . s Ω = sin φ dφ ∧ dθ 0 A representative of the Euler class is therefore ( 2π )

−1

P f ( − s∗ Ω ) =

1 sin φ dφ ∧ dθ . 2π

Notice that the Euler number ¡ ¢ 1 χ T S2 = 2π

Z sin φ dφ ∧ dθ = 2 S2

which is, of course, the Euler characteristic of S 2 . There is an alternative description of the Euler class that will be important ∗ to us soon. Denote by HCV ( E; IR ) the compact-vertical cohomology of E (generated by the differential forms on E whose restriction to each fiber of πE : E → X has compact support). One can show that there is a unique 2k element U ( E ) ∈ HCV ( E; IR ) whose integral over each fiber is 1. This is called the Thom class of πE : E → X and it has the property that, if s : X → E is any section of the vector bundle, e.g., the 0-section, then e ( E ) = s∗ U ( E ) .

(4.28)

It is not clear from either of these definitions, but the Euler number χ( E ) is actually an integer. An alternative description of χ( E ) in which its integrality is manifest is contained in the so-called Poincar´ e-Hopf Theorem. To state 54

this we recall that, for any section s : X → E, the image s( X ) is a submanifold of E diffeomorphic to X. This is, in particular, true of the zero section s0 : X → E and one often identifies X with s0 ( X ). Then s( X ) ∩ s0 ( X ) is the set of zeros of s (if E = T X, s is a vector field on X and these are its singularities). We will say that s is generic if s( X ) intersects s0( X ) transversely (meaning that, for any s( x ) ∈ s( X ) ∩ s0( X ), Ts( x ) ( E ) = Ts( x ) ( s( X ) ) ⊕ Ts( x ) ( s0( X ) ) ). According to the Thom Transversality Theorem, generic sections are dense in the space of all sections. For such a section s, s( X ) ∩ s0( X ) is necessarily a finite set of isolated points and we attach a sign to each such point p as follows: sign ( p ) = 1 if an oriented basis for Tp ( s( X ) ) together with an oriented basis for Tp ( s0( X ) ) is an oriented basis for Tp ( E ); otherwise, sign ( p ) = −1. The intersection number of s( X ) and s0( X ) is the sum of these signs over all points in s( X ) ∩ s0( X ). The Poincar´e-Hopf Theorem asserts that the Euler number χ( E ) is the intersection number of any generic section. With this digression behind us we may return to the new perspective on the Donaldson invariant γ0( B ) promised earlier. For this and all subsequent discussions we intend to employ a more economical notation, dropping all references to Sobolev indices, writing P for Pk G for G4 ( Pk ), etc. The gauge group G does not act freely on the space Aˆ of irreducible connections since even irreducible connections have a ZZ 2 stabilizer. However, Gˆ = G / ZZ 2 does act freely on Aˆ so we have an infinite-dimensional principal bundle Gˆ ,→ Aˆ −→ Bˆ (4.29) ˆ over the Banach manifold Bˆ (note that the orbits in Aˆ of the G-action are the ˆ same as those of the G-action so the quotient is still B ). We build a vector bundle associated to this principal bundle as follows. Consider the (infinitedimensional) vector space Ω2+ ( B , ad P ) of self-dual 2-forms on B with values in the adjoint bundle. We claim that there is a smooth left action of Gˆ on Ω2+ ( B , ad P ). To see this we think of G as the group of sections of the nonlinear adjoint bundle Ad P under pointwise multiplication. Since the elements of Ω2+ ( B , ad P ) take values in the su( 2 ) fibers of ad P , G acts on these values by conjugation. Moreover, conjugation has the same effect at ± f ∈ G so this ˆ G-action on Ω2+ ( B , ad P ) descends to a G / ZZ 2 = G-action. Thus, we have an associated vector bundle Aˆ ×Gˆ Ω2+ ( B , ad P ) , the elements of which are equivalence classes [ ω , γ ] = [ ω · f , f −1 · γ ] with ˆ γ ∈ Ω2 ( B , ad P ) and f ∈ G. ˆ ω ∈ A, + Now recall that sections of associated vector bundles can be identified with equivariant maps from the principal bundle space into the vector space. In our case we have an obvious map from Aˆ into Ω2+ ( B , ad P ), i.e., the self-dual

55

curvature map: F + : Aˆ −→ Ω2+ ( B , ad P ) 1 F ( ω ) = Fω = ( Fω +∗Fω ) . 2 +

(4.30)

+

Since the action of Gˆ on Aˆ is by conjugation and curvatures transform by conjugation under a gauge transformation, F + is equivariant: −1 + −1 −1 F+ (ω · f ) = F+ · F+ · F+ (ω ) ω = f ω ·f = f F ω f = f

F + can therefore by identified with a section s+ : Bˆ −→ Aˆ ×Gˆ Ω2+ ( B , ad P ) of our vector bundle, given explicitly by s+ ( [ ω ] ) =

£

ω , F+ ω

¤

ˆ This section is Fredholm in the sense that its local reprefor every [ ω ] ∈ B. sentatives, thought of as sections of trivial bundles, i.e., Ω2+ ( B , ad P )-valued ˆ have derivatives at each point maps on open subsets of the Banach manifold B, ˆ of that are (linear) Fredholm maps. Notice now that the moduli space M + anti-self-dual connections ( F ω = 0 ) is precisely the zero set of the section s+. Identifying Bˆ with the image of the zero section s0 : Bˆ −→ Aˆ ×Gˆ Ω2+ ( B , ad P ) we conclude that

ˆ = s ( Bˆ ) ∩ s ( Bˆ ) . M + 0

(4.31)

ˆ is 0-dimensional so that each point of the intersection In the case in which M (4.31) acquires a sign and the Donaldson invariant γ0 ( B ) is the sum of these signs one sees quite clearly the sense in which γ0 ( B ) can be regarded (at least formally) as an infinite-dimensional analogue of the Poincar´e-Hopf version of an Euler number. Taking this analogy seriously would suggest the possibility of an integral representation of γ0 ( B ) modeled on our definition (4.23) of the Euler number. Notice, however, that such an “integral” would necessarily be over the infinitedimensional moduli space Bˆ and such integrals are notoriously difficult to define rigorously. But, as Hitchin [21] has phrased it, “This is such stuff as quantum field theory is made of.” Indeed, it was Edward Witten who first produced such an integral representation of γ0 ( B ), not directly, but as what is called the “partition function” of the quantum field theory introduced in [46]. We intend to produce Witten’s partition function, but not from the quantum field theory arguments of [46]. Rather we will follow Atiyah and Jeffrey [3] who showed that an integral formula for the Euler number of a (finite-dimensional) vector bundle proved by Mathai and Quillen[32], when formally applied to the vector bundle Aˆ ×Gˆ Ω2+ ( B , ad P ), yields precisely this partition function. 56

5

Mathai-Quillen Formalism and Witten’s Partition Function

We begin by having a closer look at the expression (4.27) for the Euler class of the oriented, real vector bundle πE : E → X. Recall that we denote by V the typical fiber of the bundle, which we assume has dimension 2k and a positive definite inner product. We fix, once and for all, an oriented, orthonormal basis { ψ 1 , . . . , ψ 2k } for V . We regard the elements of the exterior algebra V V as polynomials with real coefficients in the odd (anti-commuting) variables V 1 ψ , . . . , ψ 2k and provide V with its usual ZZ 2 -grading à ∞ ! à ∞ ! ∞ M M V2i+1 V Vi ∼ M V2i V = V = V ⊕ V i=0

i=0

=

³V

i=0

´ V

0



³V

´ V

1

.

V

V is thereforeVa supercommutative superalgebra. One can define the exponential map on V by the usual V power series, noting that the series eventually terminates for any element of V due to the anti-commutativity of the multiplication. V The Berezin (or fermionic) integral of an element f of V is the (real) coefficient of ψ 1 · · · ψ 2k = vol in the polynomial f and we will write this as Z f D ψ = fvol . For example, our definition (4.25) of the Pfaffian of Q ∈ so( 2k ) can be written Z T 1 e 2 ψ Qψ D ψ = Pf ( Q ) . (5.1) In particular, (4.27) now gives representatives of the Euler class as Berezin integrals of the form Z T ∗ 1 −k (2π) e 2 ψ ( −s Ω ) ψ D ψ . (5.2) We will also need to extend this notion of Berezin integration in the following way. Let A be any otherVsupercommutative superalgebra V and consider the (super) tensor product A⊗ V . Regard the elements of A⊗ V as polynomials in the odd variables ψ 1 , . . .V, ψ 2k with coefficients in A and define the Berezin integral of such an F ∈ A⊗ V to be the coefficient (in A) of ψ 1 · · · ψ 2k = vol . Z F D ψ = Fvol . As an example we introduce coordinates u1 , . . . , u2k on V corresponding to 57

the basis ψ 1 , . . . , ψ 2k . Thus, { u1 , . . . , u2k } is the basis for V ∗ dual to { ψ 1 , . . . , ψ 2k }. Let A = Ω∗ ( V ) be the algebra of differential forms on V (which, throughout this section, we take to be complex-valued). Thus, each duj is in Ω1 ( V ) and −i duj ψ j = i ψ j duj (sum over j = 1, . . . , 2k ) is in V Ω∗ ( V ) ⊗ V . Writing du for ( du1 · · · du2k )> we will show that Z Z i ψ j duj i ψ T du e Dψ = e D ψ = du1 · · · du2k (5.3) (by which we mean du1 ∧ · · · ∧ du2k ∈ Ω2k ( V ) ). Indeed, Z Z T i ψ j duj e i ψ du Dψ = e Dψ Z e i( ψ

=

1

du1 + ··· + ψ 2k du2k )



Z eiψ

=

¡

Z = Z =

¡ ¡

1

du1

··· eiψ

2k

du2k



the elements ψ j duj are even ´ ^ in Ω∗ ( V ) ⊗ V and so commute

1 + iψ 1 du1 iψ 1 du1

¢

¢

¡ ¢ · · · 1 + iψ 2k du2k Dψ

¡ ¢ · · · iψ 2k du2k Dψ

( only this product contributes

= (i)

2k

¢ to the coefficient of ψ 1 · · · ψ 2k Z 1 ( 2k ) ( 2k+1 ) du1 · · · du2k ψ 1 · · · ψ 2k Dψ ( −1) 2

= du1 · · · du2k . Notice that if we write k u k 2 = u1 2 + · · · + u2k2 ∈ Ω0 ( V ) and identify V this with k u k 2 ⊗ 1 ∈ Ω∗ ( V ) ⊗ V , then Z 2 T 2 1 1 −k −k (2π) e − 2 k u k + iψ du Dψ = ( 2 π ) e − 2 k u k du1 · · · du2k , (5.4) which is a form on V that integrates to 1 over V . It does not have compact support on V , but one can think of it as a “Gaussian representative” of the Thom class of the vector bundle over a point whose fiber is V (the compact vertical cohomology of a vector bundle is isomorphic to the cohomology of forms that are “rapidly decreasing” in the fibers and the usual discussion of the Thom class extends easily to this context). Shortly we will introduce the so-called 58

“universal Thom form” of Mathai and Quillen [32] which adds one more term to the exponent in (5.4) to produce what is called an “equivariant differential form.” For this though we require a brief digression. Equivariant cohomology arose from attempts to understand the topology of the orbit space M / G of a topological space on which some topological group G acts. We will be concerned only with the case in which M is a smooth manifold and G is a compact, connected matrix Lie group acting smoothly on M on the left. For this action we will write σ : G × M −→ M σ ( g , m ) = g · m = σg ( m ) = σm ( g ) . | [ G ] the algebra of complex-valued We denote by G the Lie algebra of G, C ∗ polynomials on G and Ω ( M ) the algebra of complex-valued differential forms | [ G ] ⊗ Ω∗ ( M ), every element of on M . We consider the tensor product C | [ G ] and which is a sum of terms of the form α = P ⊗ ϕ, where P ∈ C ∗ ∗ ϕ ∈ Ω ( M ). These are best thought of as Ω ( M )-valued polynomials on G (e.g., α( ξ ) = ( P ⊗ ϕ ) ( ξ ) = P( ξ )ϕ for each ξ ∈ G ). Rather than the usual | [ G ]⊗Ω∗ ( M ) we will, for reasons that will become tensor product grading on C | [ G ]. More precisely, if α = P ⊗ ϕ we clear shortly, “double the degrees” in C define deg α = deg ( P ⊗ ϕ ) = 2 deg P + deg ϕ , (5.5)

where deg P is the algebraic degree of the polynomial P and deg ϕ is the cohomological degree of the form ϕ . Thus, M | [ G ] ⊗ Ω∗ ( M ) = | i [ G ] ⊗ Ωj ( M ) . C C 2i+j=k

The action of G on M together with the adjoint action of G on G give a | [ G ] ⊗ Ω∗ ( M ), i.e., if α = P ⊗ ϕ and g ∈ G, then natural action of G on C | g · α is the element of C [ G ] ⊗ Ω∗ ( M ) whose value at any ξ ∈ G is ¡ ¢ ∗ ϕ. ( g · α ) ( ξ ) = ( g · ( P ⊗ ϕ ) ) ( ξ ) = P g −1 ξ g σg−1 | [ G ] ⊗ Ω∗ ( M ) is said to be G-invariant if g · α = α for An element α of C ∗ every g ∈ G. This is easily seen to be equivalent to α( gξg −1 ) = σg−1 α( ξ ) for every g ∈ G and every ξ ∈ G. The algebra of all G-invariant elements of | [ G ] ⊗ Ω∗ ( M ) is denoted C | [ G ] ⊗ Ω∗ ( M ) ] Ω∗G ( M ) = [ C

G

and its elements are called G-equivariant differential forms on M . Our | [ G ] ⊗ Ω∗ ( M ) gives grading of C Ω∗G ( M ) =

∞ M k=0

ΩkG ( M ) =

M 2i+j=k

59

£

| i [ G ] ⊗ Ωj ( M ) C

¤G

and we will take ΩkG ( M ) to be trivial for k < 0. If α ∈ Ω∗G ( M ), then, for each ξ ∈ G, α( ξ ) is an element of Ω∗ ( M ) and so can be written α ( ξ ) = α ( ξ )[ 0 ] + α ( ξ )[ 1 ] + · · · + α ( ξ )[ n ] , where α( ξ )[ k ] ∈ Ωk ( M ) and n = dim M . Similarly, we will write α[ k ] | [ G ]G (those P ∈ C | [G ] for the part of α in ΩkG ( M ). Notice that both C −1 ∗ G satisfying P( g ξ g ) = P( ξ ) for all g ∈ G and ξ ∈ G ) and Ω ( M ) (those ∗ ϕ ∈ Ω∗ ( M ) satisfying σg−1 ϕ = ϕ for all g ∈ G ) can be identified with

subalgebras of Ω∗G ( M ) via P → P ⊗ 1 and ϕ → 1 ⊗ ϕ, respectively. Next we define the G-equivariant exterior derivative dG on Ω∗G ( M ) | [ G ] ⊗ Ω∗ ( M ) and any ξ ∈ G we define as follows: For any α ∈ C ( dG α ) ( ξ ) = d ( α ( ξ ) ) − ι ξ # ( α ( ξ ) ) ,

(5.6)

where ξ # is the vector field on M defined, at each m ∈ M , by ξ# ( m ) =

d ( exp ( − tξ ) · m ) | t=0 dt

(5.7)

and ιξ# denotes interior multiplication by ξ # (the minus sign in (5.7) is due to the fact that G acts on M on the left and ensures that the map ξ → ξ # is a homomorphism of Lie algebras). Alternatively, if { ξ1 , . . . , ξn } is a basis for G and if we write ιa for ι # , then ξa

dG = 1 ⊗ d − xa ⊗ ιa ,

(5.8)

where { x1 , . . . , xn } is a basis for G ∗ dual to { ξ1 , . . . , ξn } and we regard | [ G ]. It is enough to verify (5.8) for elements of each xa as an element of C the form α = P ⊗ ϕ and this is straightforward. Now, for any element of | [ G ] ⊗ Ω∗ ( M ) of the form α = P ⊗ ϕ we have deg α = 2 deg P + deg ϕ and C so deg ( ( 1 ⊗ d ) ( α ) ) = 2 deg P + ( deg ϕ + 1 ) = deg α + 1 and deg ( ( xa ⊗ ιa ) ( α ) ) = 2 ( deg P + 1 ) + ( deg ϕ − 1 ) = deg α + 1 imply that deg( dG α ) = deg α+1 (this is the reason for the peculiar grading on ∗ | [ G ]⊗Ω∗ ( M ) ). One also verifies that d C G preserves the subalgebra ΩG ( M ) ∗ | of invariant elements and satisfies, for any α ∈ C [ G ]⊗Ω ( M ) and any ξ ∈ G, ( ( dG ◦ dG ) ( α ) ) ( ξ ) = − Lξ# ( α ( ξ ) ) ,

(5.9)

where Lξ# is the Lie derivative with respect to the vector field ξ # . Since an | [ G ] ⊗ Ω∗ ( M ) satisfies L invariant element α of C ( α( ξ ) ) = 0 for every ξ# ξ ∈ G we obtain from (5.9) that 60

dG ◦ dG = 0 on Ω∗G ( M ) . Thus

(5.10)

( Ω∗G ( M ) , dG )

is a cochain complex. The cohomology of this complex is called the Cartan ∗ ( M ). In model of the G-equivariant cohomology of M and is denoted HG ∗ somewhat more detail, an element α of ΩG ( M ) is said to be G-equivariantly closed if dG α = 0 and G-equivariantly exact if α = dG β for some β ∈ Ω∗G ( M ). Writing dkG for the restriction of dG to ΩkG ( M ) we have Ωk−1 (M ) G

dk−1 G - ΩkG ( M )

dkG

- Ωk+1 G (M )

= 0 so that with dkG ◦ dk−1 G ∗ HG (M ) =

∞ M k=0

k HG (M ) =

∞ M

¡ ¢ ¡ ¢ ker dkG / Im dk−1 . G

k=0

Notice that if M is a single point (connected 0-dimensional manifold), then | [ G ]G . Each every element of Ω∗G ( M ) is of the form P ⊗ 1 for some P ∈ C of these is G-equivariantly closed, but none is G-equivariantly exact so G ∗ | [G ] ( pt ) ∼ . HG = C

(5.11)

Notice also that if G is trivial, then so is the Lie algebra G so there are only constant polynomials on G. Everything is G-invariant so one can identify Ω∗G ( M ) with Ω∗ ( M ). Furthermore, ιξ# = ι0 = 0 so dG agrees with d and we conclude that ∗ ∗ G = { 1 } =⇒ HG (M ) ∼ = Hde Rham ( M ) .

(5.12)

Example: To gain some familiarity with these definitions we will compute just one equivariant cohomology group from scratch. We consider the standard action of G = S 1 on M = S 3 that gives rise to the complex Hopf bundle. Specifically, we consider o n ¡ ¯ ¯2 ¯ ¯2 ¢ | 2 : ¯ z1 ¯ S3 = z1 , z2 ∈ C + ¯ z2 ¯ = 1 and define a left action of S 1 = { ei θ : θ ∈ IR } on S 3 by ¡ ¢ ¡ ¢ ei θ · z 1 , z 2 = ei θ z 1 , ei θ z 2 . The action is clearly free and the orbit space S 3 / S 1 is, by definition, the | IP 1 , which is diffeomorphic to S 2 . Since S 1 is 1complex projective line C dimensional, its Lie algebra has a single generator. Choose one such and denote | [ G ] can it ξ1 . We denote by x1 the corresponding dual basis vector so that C 61

| [ x1 ] of polynomials with complex coefficients be identified with the algebra C 1 in the single “variable” x . Since S 1 is abelian, all of these polynomials are S 1 -invariant so

£

£ 1¤ ¡ ¢ ¤S 1 £ 1¤ ¡ ¢S 1 | | C x ⊗ Ω∗ S 3 x ⊗ Ω∗ S 3 = C .

(5.13)

| We will leave it to the reader to show that HS0 1 ( S 3 ) ∼ and HS1 1 ( S 3 ) is =C 2 3 trivial so that we may turn our attention to HS 1 ( S ). Thus, we consider

¡ ¢ Ω1S 1 S 3

d1S 1 ¡ ¢ - Ω2S 1 S 3

d2S 1 ¡ ¢ - Ω3S 1 S 3

and compute ker( d2S 1 ) / Im( d1S 1 ). Now notice that (5.13) together with the grading we have defined on Ω∗S 1 ( S 3 ) imply that every element of Ω1S 1 ( S 3 ) can be written in the form 1 ⊗ η, (5.14) 1

where η ∈ Ω1 ( S 3 )S , and every element of Ω2S 1 ( S 3 ) can be written in the form 1 ⊗ ω + x1 ⊗ f , (5.15) 1

where ω ∈ Ω2 ( S 3 )S and f is a complex-valued function on S 3 that is con˜ ∈ Ω2S 1 ( S 3 ) be S 1 -equivariantly closed. stant on each S 1 -orbit. Now, let ω 1 ˜ = 1 ⊗ ω + x ⊗ f as in (5.15). Then Write ω ¢¡ ¢ ¡ ˜ = 1 ⊗ d − x1 ⊗ ι1 1 ⊗ ω + x1 ⊗ f 0 = dS 1 ω = 1 ⊗ dω + x1 ⊗ ( df − ι1 ω ) implies that dω = 0 and df = ι1 ω .

(5.16) 1

| We show first that there exists an a ∈ C and an η ∈ Ω1 ( S 3 )S such that ¡ ¢ ¡ ¢ 1 ⊗ ω + x1 ⊗ f − a x1 ⊗ 1 = d1S 1 ( 1 ⊗ η ) , (5.17)

i.e., 1 ⊗ ω + x1 ⊗ ( f − a ) = 1 ⊗ dη − ι1 η .

(5.18)

Now, in order for (5.18) to be satisfied we must have dη = ω and ι1 η = a − f so we will simply solve these equations. Since dω = 0 by (5.16) and since ∗ 3 Hde Rham ( S ) = 0, ω must be exact in the de Rham sense, i.e., ω is d of something in Ω1 ( S 3 ). To see that we can choose this element of Ω1 ( S 3 ) to be S 1 -invariant we require a general lemma. Lemma 5.1 If ω ∈ Ωk+1 ( M )G is (de Rham) exact, then there exists an η ∈ Ωk ( M )G with dη = ω. 62

The proof of the lemma proceeds in the following way. One shows that any α ∈ Ω∗ ( M ) can be “G-invariantized” in the sense that there is a cochain map I : Ω∗ ( M ) → Ω∗ ( M )G which reduces to the identity on Ω∗ ( M )G ⊆ Ω∗ ( M ) (“cochain map” means d ◦ I = I ◦ d ). This map is constructed by “averaging over the group G”. In somewhat more detail, one chooses an invariant measure dG on G and, for α ∈ Ωk ( M ), p ∈ M and v1 , . . . , vk ∈ Tp ( M ), defines Z ¡ ∗ ¢ σg α p ( v1 , . . . , vk ) dG ( I ( α ) ) p ( v1 , . . . , v k ) = G

Z =

G

αg·p

³ ¡

σg

¢ ∗p

´ ¡ ¢ ( v1 ) , . . . , σg ∗p ( vk ) dG .

Now, if ω ∈ Ωk+1 ( M )G is exact there is an α ∈ Ωk ( M ) with dα = ω. But I( α ) ∈ Ωk ( M )G and ω = I( ω ) = I( dα ) = d( I( α ) ) as required. 1 Returning to the proof of (5.18), we can now select an η ∈ Ω1 ( S 3 )S with ω = dη and so the first equation is satisfied. Furthermore, since η is S 1 invariant, 0 = L1 η = ( d ◦ ι1 + ι1 ◦ d ) η = d ( ι1 η ) + ι1 ω = d ( ι1 η + f ) . Since S 3 is connected this implies that ι1 η + f is some constant function a, i.e., ι1 η = a − f so, for this a, the second condition is satisfied as well. This completes the proof of (5.18) and therefore of (5.17). To understand the conclusion to be drawn from (5.17) we observe that x1 ⊗1 is S 1 -equivariantly closed and so determines an S 1 -equivariant cohomology class ˜ = 1⊗ω+ in HS2 1 ( S 3 ). Thus, (5.17) implies that the cohomology class of ω 1 1 ˜ was x ⊗ f is a multiple (by a) of the cohomology class of x ⊗ 1. Since ω an arbitrary S 1 -equivariantly closed element of Ω2S 1 ( S 3 ) we conclude that HS2 1 ( S 3 ) is generated by the class of x1 ⊗ 1. We conclude by showing that x1 ⊗ 1 is not dS 1 -exact so that this class is nontrivial and therefore ¡ ¢ | . HS2 1 S 3 ∼ (5.19) = C ˜ in Ω1S 1 ( S 3 ) To prove this we assume to the contrary that there is an element η 1

˜ = x1 ⊗ 1. η ˜ can be written as η ˜ = 1 ⊗ η, where η ∈ Ω1 ( S 3 )S . with d1S 1 η Thus, d1S 1 ( 1 ⊗ η ) = x1 ⊗ 1, i.e., 1 ⊗ dη − x1 ⊗ ι1 η = x1 ⊗ 1 , so we must have dη = 0 and ι1 η = −1 .

(5.20)

1 3 But dη = 0 and Hde Rham ( S ) = 0 imply that η is de Rham exact and 1 Lemma 5.1 implies that there is an f ∈ Ω0 ( S 3 )S with η = df . Thus,

ι1 η = ι1 ( df ) = L1 f − d ( ι1 f ) = 0 − d ( 0 ) = 0 63

and so the second condition in (5.20) could not be satisfied. Thus, x1 ⊗1 cannot be dS 1 -exact and the proof of (5.19) is complete. We should point out that, for each of the examples we have described for HS∗ 1 ( S 3 ), the S 1 -equivariant cohomology group of S 3 agrees with the corresponding ordinary de Rham cohomology group (with complex coefficients) of the orbit space S 3 / S 1 ∼ = S 2 . That this is no accident is the content of a beautiful theorem of Henri Cartan (see [20] for a proof of a much more general result). Theorem 5.2 (Cartan) Let M be a smooth manifold and G a compact, connected Lie group. Suppose there is a smooth, free action of G on M on the ∗ left. Then the G-equivariant cohomology algebra HG ( M ) is isomorphic to the ∗ de Rham cohomology Hde Rham ( M/G ) with complex coefficients of the orbit manifold M/G. Finally, we must introduce a notion of integration for equivariant forms and cohomology classes. For this we now assume that M is compact and oriented and that the G-action on M preserves the orientation (each diffeomorphism σg : M → M is orientation preserving). For each α ∈ Ω∗G( M ) we define an R | [ G ]G by setting, for each ξ ∈ G, element M α ∈ C µZ ¶ Z Z α (ξ) = α ( ξ ) = definition α ( ξ )[ n ] , M

M

M

R

where n = dim M . Note that M α really is G-invariant since µZ ¶ Z ¡ ¢ ¡ ¢ −1 α gξg = α g ξ g −1 [ n ] M

Z

M

= Z

M

= M

σg−1∗

³

´ α ( ξ )[ n ]

α ( ξ )[ n ]

µZ



=

α

(ξ) .

M

Notice also that if α is dG-exact (say, α = dG β ), then, for each ξ ∈ G, R | [ G ]G . The α( ξ )[ n ] = d( β( ξ )[ n−1 ] ) so, by Stokes’ Theorem, M α = 0 ∈ C conclusion is that the integration map Z G | [G ] : Ω∗G ( M ) −→ C M

descends to cohomology: Z M

∗ | [G ] : HG ( M ) −→ C

64

G

.

Now we return to the general development. We wish to write out a specific representative of an equivariant cohomology class called the “universal Thom form” (for vector bundles with typical fiber V ). For this we take M = V (our real vector space of dimension 2k with a positive definite inner product) and G = SO( V ) with its defining action on V . Thus, we seek an element of £ ¤SO( V ) | [ so ( V ) ] ⊗ Ω∗ ( V ) Ω∗SO( V ) ( V ) = C V and we will obtain it as the Berezin integral of an element of A⊗ V , where A = | [ so( V ) ]⊗Ω∗ ( V ). Recalling the notation introduced earlier, { ψ 1 , . . . , ψ 2k } C is a fixed orthonormal basis for V , { u1 , . . . , u2k } is its dual basis of coordinate functions on V , { ξ1 , . . . , ξn } is a basis for the Lie algebra G = so( V ) and { x1 , . . . , xn } is its dual basis, regarded as linear functions on so( V ), | [ so( V ) ]. i.e., as elements of C Now define, for each ξ ∈ so( V ), a linear transformation Mξ : V −→ V by d Mξ ( ψ ) = ( exp ( t ξ ) · ψ dt

¯ ¯ ) ¯¯

t=0

for each ψ ∈ V . Write Ma for Mξa and notice that, for each ξ = xa ( ξ )ξa ∈ so( V ), Mξ = xa ( ξ ) Ma . Furthermore, if ( Mξ ) denotes the matrix of Mξ relative to { ψ 1 , . . . , ψ 2k }, then ( Mξ ) ∈ so( 2k ) and −

2k ¡ ¢ 1 X l a 1 ψ x ( ξ ) Ma ψ l = ψ > Mξ ψ 2 2 l=1

so (5.1) gives

Z 1

e− 2

P l

ψ l xa( ξ )Ma ψ l

¡ ¢ Dψ = P f Mξ .

(5.21)

P P l a 1 a l l l Now notice that − 12 l ψ x Ma ψ = l ( − 2 x ) ⊗ 1 ⊗ ( ψ ( Ma ψ ) ) can V ∗ | [ so( V ) ] ⊗ Ω ( V ) ⊗ be regarded as an element of C V and therefore so can 1

e− 2

P l

ψ l xa Ma ψ l

.

We now intend to include this factor in the integrand on the left-hand side of (5.4) to obtain what is called the Mathai-Quillen universal Thom form ν for V , defined by Z P l a l − 1 k u k 2 + i ψ j duj − 21 −k l ψ x Ma ψ Dψ ν = ( 2π ) e 2 (5.22) Ã ! Z X 1 −k − 12 k u k 2 = ( 2π ) e exp i ψ j duj − ψ l xa Ma ψ l Dψ . 2 l

65

For example, if one carries out the Berezin integration in (5.22) for V = IR 2 (usual orientation and inner product) and SO( V ) = SO( 2 ) the result is −1

ν = ( 2π )

1

2

e− 2 ( u1

+ u22 )

−1

du1 du2 + ( 2π )

1

2

x 1 e − 2 ( u1

+ u22 )

.

(5.23)

Note that each term in (5.23) is SO( 2 )-invariant, has degree 2 = dim V and | 0 [ so( 2 ) ] ⊗ Ω2 ( IR 2 ) ) integrates to 1 over IR 2 . In general, one the first ( in C can verify the following properties of the form ν given by (5.22). | [ so( V ) ] ⊗ Ω∗ ( V ) of degree 2k = 1. ν is an SO( V )-invariant element of C dim V .

2. ν is SO( V )-equivariantly closed ( dSO( V ) ν = 0 ) and so determines an equivariant cohomology class 2k [ ν ] ∈ HSO( V

)

(V ) .

| 0 [ so( V ) ] ⊗ Ω2k ( V )-part of) ν over V is 1. 3. The integral of (the C

ν is called a universal Thom form because, as we now show, one can produce from it a (Gaussian) representative of the Thom class for any vector bundle whose typical fiber is V . We recall that any such vector bundle can be regarded as the vector bundle associated to some principal G-bundle by a representation of G on V . Thus, let us suppose G is a Lie group and ρ : G −→ SO ( V ) is a representation of G on V . Then ρ induces a Lie algebra homomorphism ρ∗ : G −→ so ( V ) (just the derivative of ρ at the identity in G ). It then follows easily from the fact that ν is an SO( V )-equivariantly closed form on V that ! Ã Z 1 X l a −k − 21 k u k 2 l j νG = ( 2π ) e exp i ψ duj − ψ ( x ◦ ρ∗ ) Ma ψ Dψ 2 l (5.24) is a G-equivariantly closed form on V . π

P Now suppose that G ,→ P −→ X is a principal G-bundle over a compact, smooth manifold of dimension 2k = dim V . Then the representation ρ

πρ

determines an associated vector bundle P ×ρ V −→ X over X with typical fiber V . We show now that there is a generalization of the Chern-Weil map ∼ ∗ | [ G ]G → Ω∗ ( P ) CW! : C BAS = Ω ( X ) (Section 4) which associates with every G-equivariantly closed form on V an ordinary form on the vector bundle space P ×ρ V and that, when applied to νG , one obtains a (Gaussian) 66

representative of the Thom class of P ×ρ V . Begin by considering the commutative diagram P × V

prP

q ? P ×ρ V

- P πP ? - X

πρ

where prP is the projection onto the first factor and q is the map ¡ ¢ q ( p , ψ ) = q p · g , g −1 · ψ = [ p , ψ ] which projects P × V onto the orbit space P ×ρ V of the G-action ( p, ψ ) · g = ( p · g , g −1 · ψ ) = ( p · g , ρ( g −1 ) ( ψ ) ). Since the action of G on P is free, so is this action of G on P × V so we may regard q

G ,→ P × V −→ P ×ρ V as a principal G-bundle. In particular, we have an isomorphism ¡ ¢ Ω∗ ( P × V ) BAS ∼ = Ω∗ P ×ρ V between the algebras of ordinary forms on P ×ρ V and the forms on P × V that are basic with respect to the action of G on P × V . Thus, to specify a form on P ×ρ V (e.g., a Thom form) it is enough to specify a basic form on P ×V. Now choose a connection ω on G ,→ P → X. Then ω 0 = prP∗ ω is a connection on G ,→ P × V → P ×ρ V . Identifying T(p,ψ) ( P × V ) with Tp ( P ) ⊕ Tψ ( V ), the ω 0-horizontal spaces are clearly given by Hor(p,ψ) ( ω 0 ) ∼ = Horp ( ω ) ⊕ Tψ ( V ). The decomposition ¡ ¢ T(p,ψ) ( P × V ) ∼ = Horp ( ω ) ⊕ Tψ ( V ) ⊕ Vert(p,ψ) ( P × V ) determines a projection hω 0 of forms on P × V to ω 0-horizontal forms on P × V (evaluate on ω 0-horizontal parts of tangent vectors). Now, for any α = P ⊗ ϕ ∈ Ω∗G ( V ) one can evaluate the polynomial part P on the curvature Ω of the connection ω as in the ordinary Chern-Weil map to obtain P( Ω ) ⊗ ϕ ∈ Ω∗ ( P ) ⊗ Ω∗ ( V ). This one can identify with a form P( Ω ) ∧ ϕ on P × V which, because P ⊗ ϕ is G-invariant, is in Ω∗ ( P × V )G . It is generally not horizontal, however, so we compose with the horizontal projection hω 0 to define the generalized Chern-Weil homomorphism, also denoted ¡ ¢ CW! : Ω∗G ( V ) −→ Ω∗ ( P × V )BAS ∼ = Ω∗ P ×ρ V , 67

by CW! ( α ) = CW! ( P ⊗ ϕ ) = hω 0 ( P ( Ω ) ∧ ϕ )

(5.25)

for elements of the form P ⊗ ϕ and then by linearity on all of Ω∗G ( V ). One can show that CW! is actually a cochain map d ◦ CW! = CW! ◦ dG

(5.26)

and so carries G-equivariantly closed forms on V to ordinary closed forms on P × V which then descend to closed forms on P ×ρ V . Applying this procedure to the G-equivariantly closed form νG of (5.24) gives a closed, basic form CW! ( νG ) on P × V which one can write formally as the horizontal projection of à ! Z 1 X l a −k − 12 k u k 2 j l ( 2π ) e exp i ψ duj − ψ ( x ( ρ∗ Ω ) ) Ma ψ Dψ 2 l (5.27) which it is customary to write more compactly as µ ¶ Z 1 T −k − 12 k u k 2 T ( 2π ) e exp i ψ du + ψ ( ρ∗ Ω ) ψ Dψ . (5.28) 2 In this last expression ( ρ∗ Ω ) is to be interpreted as the skew-symmetric matrix image of the G-curvature under (the derivative of) the representation when so( V ) is identified with so( 2k ). We generally work directly with (5.28), but one obtains a (Gaussian) representative of the Thom class (which pulls back to an Euler form) by replacing Ω by − Ω , or, what amounts to the same thing, the transpose of Ω ; cf., (4.26). As an example of what the result of such a calculation might look like we return to the universal Thom form (5.23) for V = IR 2 and SO( V ) = SO( 2 ). For the vector bundle with fiber IR 2 we take the tangent bundle T S 2 of the 2-sphere. This we describe as an associated bundle in the following way. The usual orientation and Riemannian metric on S 2 give an oriented, orthonormal frame bundle ¡ ¢ πSO 2 SO ( 2 ) ,→ FSO S 2 −→ S . If ρ : SO( 2 ) → SO( 2 ) is the identity representation, ρ = idSO( 2 ) , then FSO ( S 2 ) ×ρ IR 2 is the tangent bundle of S 2 . Moreover, ρ∗ : so( 2 ) → so( 2 ) is also the identity so x1 ◦ ρ∗ = x1 and ¢ ¡ 2 2 1 −1 νSO( 2 ) = ( 2π ) e− 2 ( u1 + u2 ) x1 + du1 du2 . Choosing a connection ω = ω 1 ξ1 with curvature Ω = Ω1 ξ1 on the frame bundle FSO ( S 2 ) we have x1 ( ρ∗ Ω ) = Ω1 so CW! ( νSO( 2 ) ) is the horizontal projection (determined by ω 0 = prF ( S 2 ) ∗ω ) of SO

µ = ( 2π )

−1

e−

1 2

( u12 + u22 )

68

¡

Ω1 + du1 du2

¢

.

The horizontal projection of this form on FSO ( S 2 )×IR 2 can either be described by evaluating µ on ω 0-horizontal parts or explicitly calculated from the easily verified formula hω 0 ( µ ) = µ − ( prF ( S 2 ) ∗ω 1 ) ∧ ι # µ . Performing this ξ1 SO latter calculation gives ³ ´ ³ 2 2 1 −1 CW! νSO( 2 ) = ( 2π ) e− 2 ( u1 + u2 ) Ω1 + du1 du2 (5.29) ´ ´ ³ ∗ 1 ∧ ( u1 du1 + u2 du2 ) . + prF ( S 2 ) ω SO

In general, we will denote by U the horizontal projection of the form (5.28) and will, at least temporarily, write this as µ ¶ Z 1 T −k − 12 k u k 2 T U = ( 2π ) e exp i ψ du + ψ ( ρ∗ Ω ) ψ Dψ (5.30) 2 ( evaluated on horizontal parts ) . Since our primary concern, however, is with Euler forms we will want to pull the form to which U descends on P ×ρ V back by a section of the vector bundle. Now, any section of the associated bundle P ×ρ V can be written as x

( s, S ◦ s ) ¡ ¢ - s(x), S (s(x))

q

-

£

s(x), S (s(x))

¤

,

where s is a section of G ,→ P → X and S : P → V is an equivariant map ˜ , then ( S( p · g ) = ρ( g −1 ) ( s( p ) ) ). Thus, if we temporarily write U = q ∗ U ³ ´ ¡ ¢ ∗ ˜ ∗ ˜ = s∗ ( 1 , S )∗ U . ( q ◦ ( s, S ◦ s ) ) U = ( (1, S )◦ s ) q∗ U ˜ back by a section of P × V we compute ( 1, S )∗ U , which Thus, to pull U ρ simply pulls the V -factors of U back by the equivariant map S, and then pull this form on P back by a section of the principal bundle. We will illustrate with an example (taken from [28] ). Begin with the form CW! ( νSO( 2 ) ) in (5.29). As a section of the principal bundle FSO ( S 2 ) we choose the oriented, orthonormal frame field corresponding to the spherical coordinate chart: µ ¶ 1 s ( φ , θ ) = φ , θ , ∂φ , ∂θ . sin φ S

We choose an equivariant map FSO ( S 2 ) −→ IR 2 by beginning with a vector field on S 2 (section of T S 2 ). This can be chosen arbitrarily and we will take V = γ sin θ∂φ + γ cos θ cot φ∂θ , where γ is an arbitrary real parameter (with γ = 1 this is the infinitesimal generator for rotations about the x-axis). Relative to the frame field introduced above the components of V are γ sin θ and γ cos θ cot φ so we define S on the image of s by S ◦ s ( φ , θ ) = ( γ sin θ , γ cos θ cos φ ) 69

and elsewhere by equivariance. Pull back the IR 2 -parts of CW! ( νSO( 2 ) ) by S ◦ s by substituting u1 = γ sin θ and u2 = γ cos θ cos φ. One finds that −1

( 2π )



e

1 2

( u12 + u22 )

= ( 2π )

−1

e



1 2

γ 2 ( sin2 θ + cos2 θ cos2 φ )

,

du1 du2 = du1 ∧ du2 = γ 2 cos2 θ sin φ dφ ∧ dθ , and ¡ ¢ u1 du1 + u2 du2 = γ 2 sin θ cos φ sin2 φ dθ − cos2 θ sin φ cos φ dφ . As in our earlier computation of the Euler number of T S 2 we substitute the (transposed) Levi-Civita connection s∗ ω and curvature s∗ Ω so that s∗ Ω1 = sin φdφ ∧ dθ and s∗ ( prF ( S 2 ) ∗ω 1 ) = − cos φdθ. The result of all of these SO

substitutions is the following representative of the Euler class of S 2 : −1

( 2π )

1

e− 2 γ

2

( sin2 θ + cos2 θ cos2 φ )

¡ ¢ sin φ 1 + γ 2 cos2 θ sin2 φ dφ ∧ dθ. (5.31)

In particular, one obtains the not altogether obvious integral formula Z 2π Z π ¡ ¢ 2 2 2 2 1 1 e− 2 γ ( sin θ + cos θ cos φ ) sin φ 1 + γ 2 cos2 θ sin2 φ dφ dθ = 2 2π 0 0 (note that, for γ = 0, this reduces to our earlier computation of the Euler characteristic of S 2 ). We recall now that our interest in the Mathai-Quillen formalism stems from the fact that Atiyah and Jeffrey [3] have shown how it can be adapted and formally applied to the infinite-dimensional vector bundle Aˆ ×Gˆ Ω2+ ( B, ad P ) to yield an integral representation of the 0-dimensional Donaldson invariant which coincides with the partition function of Witten’s topological quantum field theory [46]. We begin now with the appropriate adaptation of the formula (5.30), still working in the finite-dimensional context. In addition to the assumptions we have made thus far we will henceforth assume that P is oriented and that the action of G on P preserves orientation (i.e., each of the diffeomorphisms σg is orientation preserving). We now make a specific choice of connection on G ,→ P → X. Remark: Before proceeding we must recall that for any action of a compact Lie group G on a manifold M it is always possible to construct a Riemannian metric h , iG on M that is G-invariant, i.e., for which the diffeomorphisms σg: M → M are all isometries. Roughly, this is done by selecting some Riemannian metric h , i on M and, at each point p ∈ M , averaging over G relative to some invariant measure dG on G, i.e., defining, for all Vp , Wp ∈ Tp ( M ), Z D ­ ® ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢E Vp , W p G = σg ∗p Vp , σg ∗p Wp dG . G

70

We fix, once and for all, a G-invariant Riemannian metric, denoted simply h , i, on P . At each p ∈ P this Riemannian metric defines an orthogonal complement to the vertical subspace of Tp ( P ) (tangent space to the G-orbit at p ) and, since G acts by isometries, these orthogonals are invariant under the action of G and so they determine a connection ω on P . Henceforth, we will use this connection on P exclusively. In particular, the pullback connection ω 0 = prP∗ ω on P × V has Hor(p,ψ) ( ω 0 ) ∼ = Tp ( p · G )⊥ ⊕ Tψ ( V ) at each point. Now we proceed with some cosmetic surgery on (5.30). First recall the Cartan formula Ω = dω + 12 [ ω, ω ] and notice that the second term vanishes on ω-horizontal vectors by definition. Since the formula for U in (5.30) is to be evaluated on ω 0-horizontal parts and since Hor(p,ψ) ( ω 0 ) ∼ = Horp ( ω )⊕Tψ ( V ), the result will be the same whether or not 21 [ ω, ω ] is present. Thus, we may write µ ¶ Z 2 1 1 −k U = ( 2π ) e− 2 k u k exp i ψ T du + ψ T ( ρ∗ ( dω ) ) ψ Dψ (5.32) 2 ( evaluated on horizontal parts ) . For the next manipulation of U we will require a few preliminaries. Begin by defining, at each, p ∈ P , a linear map Cp : G −→ Vertp ( P ) ⊆ Tp ( P ) by Cp ( ξ ) = ξ # ( p ) =

¯ d ( p · exp ( tξ ) ) ¯ t=0 . dt

This is an isomorphism onto Vertp ( P ), but we wish to regard it as a map into Tp ( P ). Now choose some ad-invariant inner product ( , ) on G. Tp ( P ) has an inner product h , ip arising from the Riemannian metric on P . Thus, Cp has an adjoint Cp∗ : Tp ( P ) −→ G defined by

­

w , Cp ( η )

® p

=

¡

Cp∗ ( w ) , η

¢

for all w ∈ Tp ( P ) and η ∈ G. In particular, ­ where

Cp ( ξ ) , C p ( η )

® p

³ =

´ ¡ ¢ ¡ ¢ Cp∗ Cp ( ξ ) , η = Rp ( ξ ) , η ,

Rp = Cp∗ ◦ Cp : G −→ G .

It is easy to see that Rp is self-adjoint and has trivial kernel so we have an inverse Rp−1 : G −→ G . 71

Now, since Cp carries G isomorphically onto Vertp ( P ) there is also an inverse Cp−1 : Vertp ( P ) −→ G and we claim that this agrees with ω p on Vertp ( P ), i.e., Cp−1 ( w ) = ω p ( w ) , w ∈ Vertp ( P ) .

(5.33)

Indeed, if w ∈ Vertp ( P ), then w = η # ( p ) = Cp ( η ) for a unique η ∈ G. ω is a connection form so ω p ( η # ( p ) ) = η for every η ∈ G. Thus, ω p ( w ) = η = Cp−1 ( w ). Now we define a 1-form θ ∈ Ω1 ( P, G ∗ ) on P with values in the dual ∗ G of G as follows: For p ∈ P and w ∈ Tp ( P ), θ p ( w ) ∈ G ∗ is the map θ p ( w ) : G → IR given by ¡ ¢ ­ ® ¡ ¢ θ p ( w ) ( ξ ) = Cp ( ξ ) , w p = ξ , Cp∗ ( w ) . (5.34) Note that θ p vanishes on horizontal vectors at p because ω-horizontal means h , i-orthogonal to the G-orbit through p, i.e., to Vertp ( P ), and Cp ( ξ ) is in Vertp ( P ). Now, if we use the inner product ( , ) to identify G ∗ and G, the last equality in (5.34) shows that θ p ( w ) is identified with Cp∗ ( w ). Thus, regarded as a G-valued 1-form on P , θ is just C ∗ so, in particular, C ∗ ∈ Ω1 ( P, G ) vanishes on horizontal vectors. We claim that C∗ = R ◦ ω ,

(5.35)

i.e., Cp∗ ( w ) = Rp ( ω p ( w ) ) for every w ∈ Tp ( P ). Since both sides vanish on horizontal vectors one need only verify (5.35) when w ∈ Vertp ( P ). But then (5.33) gives ¡ ¢ ¡ ¢ ¡ ¢ ¡ −1 ¢ Rp ω p ( w ) = Rp Cp−1 ( w ) = Cp∗ ◦ Cp Cp ( w ) = Cp∗ ( w ) . Fixing a basis for G we can identify each Rp with an invertible matrix and ω with a matrix of real-valued 1-forms on P so (5.35) becomes a matrix equation C ∗ = R ω which we write as ω = R−1 C ∗ . From this we compute dω = dR−1 ∧ C ∗ + R−1 dC ∗ . Noting that the first term vanishes on horizontal vectors we find that in the expression (5.32) for U we may replace dω by R−1 dC ∗ to obtain µ ¶ Z ¢¢ 1 T¡ ¡ −1 −k − 12 k u k2 T ∗ U = ( 2π ) e exp i ψ du + ψ ρ∗ R dC ψ Dψ (5.36) 2 ( evaluated on horizontal parts ) . 72

The next objective is to remove the explicit appearance of the inverse in (5.36) by using the Fourier inversion formula. We begin with a brief review of the Fourier transform. Let W Vbe an oriented real vector space of dimension n n with volume element dw ∈ W ∗ and let w1 , . . . , wn be coordinates on W with dw = dw1 · · · dwn . Let y1 , . . . , yn be coordinates on W ∗ dual to Vn w1 , . . . , wn and dy = dy1 · · · dyn ∈ W the volume element for W ∗ . Let S( W ) and S( W ∗ ) be the Schwartz spaces of rapidly decreasing functions in w1 , . . . , wn and y1 , . . . , yn, respectively. Finally, let h , i denote the natural pairing between W and W ∗ . The Fourier transform of f ∈ S( W ) is fˆ ∈ S( W ∗ ) defined by Z −n/2 ˆ e− i h w,y i f ( w ) dw . f ( y ) = ( 2π ) W

The Fourier inversion formula then asserts that Z −n/2 f ( w ) = ( 2π ) e i h w,y i fˆ ( y ) dy . W∗

Combining these two formulas gives Z Z −n f ( w ) = ( 2π ) e i h w,y i e− i h z,y i f ( z ) dz dy . W∗

W



Assuming now that W and W are identified via some inner product we will write this simply as Z Z −n f ( w ) = ( 2π ) e i h w,y i e− i h z,y i f ( z ) dz dy (5.37) with the understanding that both integrations are over W and the exponents are inner products. The situation to which we would like to apply (5.37) is as follows. If R is a positive self-adjoint matrix, then one can use the formula to compute f ( R−1 w ). To get an integral that does not explicitly involve the inverse, however, we also make the change of variable y → Ry. Then h R−1 w, Ry i = h w, y i and d( Ry ) = det R dy so Z Z ¡ ¢ −n f R−1 w = ( 2π ) e i h w,y i e− i h z,Ry i f ( z ) det R dz dy . (5.38) Now we return to our last expression (5.36) for U . Letting φ = ( φ1 , . . . , φn ) denote a Lie algebra variable in G we consider the function on G defined by µ ¶ Z 2 1 1 −k exp i ψ T du + ψ T ( ρ∗ ( φ ) ) ψ Dψ . f ( φ ) = ( 2π ) e− 2 k u k 2 Each value of f is an element of Ω∗ ( V ) whose components (relative to du1 , . . . , 73

duk ) are polynomials in φ. Remark: These polynomials are not in the Schwartz space S( G ), but we nevertheless propose to apply the Fourier formula (5.38) componentwise to f ( φ ). This is rather sloppy, of course, but could be made more precise by inserting a rapidly decaying test function e− ²( φ,φ ) and taking the limit as ² → 0. Since our objective is a formula to be applied formally in an infinite-dimensional situation where complete rigor is (for the time being, at least) out of the question anyway, we will not be scrupulous about such details. Letting λ = ( λ1 , . . . , λn ) be another Lie algebra variable in G we apply (5.38) with w = dC ∗ (i.e., with w = dC ∗ ( χ1 , χ2 ) for each pair ( χ1 , χ2 ) of tangent vectors) to get Z Z ¡ −1 ¢ ∗ −n ∗ U = f R dC = ( 2π ) e i( dC , λ ) e− i( φ, Rλ ) f ( φ ) det R dφ dλ Z Z −n

= ( 2π )

−k

Z e i( dC

( 2π )



,λ)

1

e− i( φ, Rλ ) e− 2 k u k

2

³ exp

i ψ T du

¶ 1 T + ψ ( ρ∗( φ ) ) ψ Dψ det R dφ dλ 2 Z Z Z −n

U = ( 2π )

−k

( 2π )

e



1 2

k u k2

µ 1 exp i ψ T du + ψ T( ρ∗( φ ) ) ψ 2

(5.39)

+ i ( dC ∗ , λ ) − i ( φ, Rλ ) ) det R Dψ dλ dφ ( evaluated on horizontal parts ) . Notice that this expression contains one fermionic and two ordinary (“bosonic”) integrations. Next we would like to include the parenthetical remark “evaluated on horizontal parts” in (5.39) directly into the integral expression for U . For this we require the notion of a “normalized vertical volume form” on a principal bundle, which is essentially an analogue of a Thom form on a vector bundle. πQ

Thus, we consider a principal G-bundle G ,→ Q −→ M with M compact and orientable and dim G = n. We assume the bundle itself to be orientable in the sense that there exists an n-form Ψ on Q such that, if m ∈ M and −1 ι m : πQ ( m ) ,→ Q is the inclusion, then ι∗m Ψ is an orientation form for the ∼ G. It follows that Q is orientable and we assume it to submanifold π −1 ( m ) = Q

∗ be oriented by the so-called local product orientation πQ ω M ∧ Ψ, where ω M is an orientation form for M . One can assume also that the action of G on Q is orientation preserving (these matters are discussed in detail in Chapter VII, Vol I, of [24] ). We will henceforth make these assumptions of our underlying π

P principal bundle G ,→ P −→ X as well. A normalized vertical volume −1 form for the bundle is an n-form W on Q such that, if ιm : πQ ( m ) ,→ Q

74

is the inclusion of a fiber, then Z −1 πQ

(m)

ι∗m W = 1 .

(5.40)

It is not difficult to show that one can construct such a form W from a connection ω on Q as follows. Choose a positive definite ad-invariant inner product ( , ) on G, normalized so that the volume of G (arising from the corresponding bi-invariant Riemannian metric on G ) is 1. Let { ξ1 , . . . , ξn } be an orthonormal basis for G relative to ( , ) and consistent with the orientation G inherits as a fiber of Q. Write ω = ω a ξa , where ω a ∈ Ω1 ( Q ), a = 1, . . . , n. One then shows that W = ω1 ∧ · · · ∧ ωn (5.41) πQ

is a normalized vertical volume form for G ,→ Q −→ M . Such a form has a number of properties of interest to us. For any top rank form β on M one has Z Z ∗ β = πQ β ∧ W (5.42) M

Q

(essentially Fubini’s Theorem together with (5.40) ). Furthermore, the process of evaluating an element ϕ of Ω∗ ( Q ) on ω-horizontal parts (i.e., of computing the ω-horizontal projection h! ( ϕ ) of ϕ ) can be accomplished as follows. An explicit formula for h! ( ϕ ) reads h! ( ϕ ) = ϕ − ω 1 ∧ ι1 ϕ − ω 2 ∧ ι2 ϕ − · · · − ω n ∧ ιn ϕ +

X

( −1)

1≤a1<···1

∧ ω ar ∧ where ιa = ι

r(r+1)/2

ω a1 ∧ · · ·

³

(5.43)

´ ιa1 ◦ · · · ◦ ιar

(ϕ),

. Thus h! ( ϕ ) ∧ W = ϕ ∧ W (by (5.41) ) so one can arrive at h! ( ϕ ) by wedge multiplying ϕ by W and integrating over the fibers to obtain Z h! ( ϕ ) ι∗m W = h! ( ϕ ) . # ξa

−1 πQ (m)

Notice that what is really going on here is that ϕ ∧ W kills all of the terms in ϕ with vertical parts (because W has a full contingent of n vertical coordinate differentials) and the surviving terms are just those of h! ( ϕ ) with an extra factor of W which we integrate out (i.e., ignore). The conclusion is that “evaluating on ω-horizontal parts” can be accomplished by “multiplying by the vertical volume form (5.41) and integrating over the fibers”. We wish to apply this observation to the form U of (5.39), where the prinq cipal bundle is G ,→ P × V −→ P ×ρ V . However, we would like to include the 75

procedure as part of the integration formula so we begin by showing that W can be written as a Berezin integral. Denote by { η1 , . . . , ηn } an orthonormal basis for G relative to the normalized G-invariant inner product ( , ) on G introduced above (we will be more specific V about the choice of this basis shortly). Regard these V as odd generators of ( G ) and consider the following element of Ω∗ ( Q ) ⊗ ( G ): e

Pn a=1

!a ηa = e!1 η1 · · · e!n ηn = ( 1 + ω η ) · · · ( 1 + ω η ) . (5.44) n n 1 1

Performing a Berezin integration with respect to η gives Z P Z n e a=1 !a ηa Dη = ( 1 + ω 1 η1 ) · · · ( 1 + ω n ηn ) Dη Z =

ω 1 η1 · · · ω n ηn Dη Z

= ( −1)

n( n−1 ) / 2

ω 1 · · · ω n η1 · · · ηn Dη .

If we now choose { η1 , . . . , ηn } to be the same as { ξ1 , . . . , ξn } if n( n − 1 ) / 2 is even and an odd permutation of { ξ1 , . . . , ξn } if n( n − 1 ) / 2 is odd, this gives Z P n e a=1 !a ηa Dη = ω 1 · · · ω n = ω 1 ∧ · · · ∧ ω n = W (5.45) πQ

for the normalized vertical volume form of G ,→ Q −→ M . We would now like to express the Berezin integral representation (5.45) for W without explicit reference to the connection forms ω a . For this we assume, π

P as for G ,→ P −→ X earlier, that Q is a Riemannian manifold and that ω is the connection on Q whose horizontal spaces are the orthogonal complements to the G-orbits. Thus, we have available the maps C, C ∗ and R and all of the results we have proved about them (note that this is true of the bundle q G ,→ P × V −→ P ×ρ V when the metric, connection and orientation are taken to be the pullbacks by prP : P × V → P of those we have chosen for P ). | 1 [ G ] ⊗ Ω1 ( Q ) whose value at any η ∈ G is the Now consider the element of C ∗ 1-form ( C , η ) on Q defined by ® ­ (5.46) ( C ∗ , η ) ( χ ) = ( C ∗ χ , η ) = h χ , Cη i = χ , η #

for any vector field χ on Q. Now let { A1 , . . . , An , . . . , Ad } be local coordinates on Q (where n = dim G ). Then ( C∗ , η ) =

d µ X

µ C∗

j=1

76

∂ ∂ Aj



¶ ,η

dAj .

Write C ∗ (

∂ ∂ Aj

)=



(C , η) =

Pn i=1

d X n X

aij ηi so that j

aij ( ηi , η ) dA =

j=1 i=1

n X

 

i=1

Define 1-forms β i on Q by β i = ( C∗ , η ) =

n X

Pd j=1

n X

 aij dA  ( ηi , η ) . j

j=1

aij dAj . Then

β i ( ηi , η ) =

i=1

=

d X

n X

β i ( ηi , · ) ( η )

i=1

β iη i ( η ) ,

i=1

where { η 1 , . . . , η n } = { ( η1 , · ) , . . . , ( ηn , · ) } is the basis for G ∗ dual to Pn V ∗ Pn i ∗ { η1 , . . . , ηn }. Now identify i=1 β i η ∈ Ω ( Q ) ⊗ ( G ) with i=1 β i ηi ∈ V ∗ Ω ( Q ) ⊗ ( G ) and compute the Berezin integral Z P n n( n−1 ) / 2 β1 ∧ · · · ∧ βn = e i=1 i ηi Dη = ( − 1 ) (−1)

n(n−1)/2

¡

¢ ¡ ¢ a11dA1 + · · · + a1ddAd ∧ · · · ∧ an1dA1 + · · · + anddAd = X n( n−1 ) / 2 ( −1) det ( aH ) d AH , H

where H = { h1 , . . . , hn }, 1 ≤ h1 < · · · < hn ≤ d, and  a1h ··· a1h n 1  . .  .. aH =  .. anh

···

1

anh

dAH = dAh1 ∧ · · · ∧ dAhn    . 

n

We compute the determinants det( aH ) as follows. For each j = 1, . . . , n, Pn C ∗ ( ∂ ∂Aj ) = i=1 aij ηi and so µ aij =

µ C



∂ ∂ Aj



Ã

¶ , ηi

=

µ R

µ ω

∂ ∂ Aj

!

¶¶ , ηi

˜ 1 η1 +· · ·+ ω ˜ n ηn . By the way we have chosen by (5.35). Now let us write ω = ω ˜1 , . . . , ω ˜ n } is at worst a permutation of { ω 1 , . . . , ω n } and, { η1 , . . . , ηn }, { ω in any case, ˜1 ∧ · · · ∧ ω ˜ n = ( − 1 )n( n−1 ) / 2 ω 1 ∧ · · · ∧ ω n = ( − 1 )n( n−1 ) / 2 W . ω

77

Now we have

µ

ω and

µ R

µ ω

∂ ∂ Aj

so that

µ

˜ aij = ω

∂ ∂ Aj

1



µ ˜1 = ω

¶¶

∂ ∂ Aj

µ ˜ = ω

1

∂ ∂ Aj



µ ˜n η1 + · · · + ω

∂ ∂ Aj



∂ ∂ Aj

µ ˜ R ( η1 ) + · · · + ω



µ ˜ ( R ( η 1 ) , ηi ) + · · · + ω

n

∂ ∂ Aj

n

¶ ηn

∂ ∂ Aj

¶ R ( ηn )

¶ ( R ( ηn ) , ηi ) .

Consequently, ¶ µ ¶  µ ∂ ∂ 1 1  ω ˜ ˜ · · · ω (R ( η1) , η1) · · · (R ( ηn) , η1)  ∂ Ah 1 ∂ Ahn        .. .. .. .. .  aH =  . . . .     µ ¶ ¶ µ   ∂ ∂ (R ( η1) , ηn) · · · (R ( ηn) , ηn) ˜n ˜n ω · · · ω h h ∂A 1 ∂A n (5.47) Thus, ˜H ) det ( aH ) = det R det ( ω (5.48) 

˜ H is the matrix shown in (5.47). Since where ω ˜j = ω

d X i=1

µ ˜j ω

∂ ∂ Ai

¶ dAi ,

j = 1,...,n ,

we find that Z P X n n( n−1 ) / 2 e i=1 i ηi Dη = ( − 1 ) det ( aH ) dAH H

= ( −1)

n( n−1 ) / 2

X

˜ H ) dAH det R det ( ω

H

= ( −1)

n( n−1 ) / 2

= ( −1)

n( n−1 ) / 2

det R

X

˜ H ) dAH det ( ω

H

˜1 ∧ · · · ∧ ω ˜n det R ω

= ( det R ) W . Thus,

Z −1

W = ( det R )

78

e

Pn i=1

i ηi Dη .

(5.49)

We have already seen that, with our identification of G ∗ with G via ( , ), Pn Pn ( C ∗ , · ) is identified with i=1 i ηi , i.e., ( C ∗ , η ) = i=1 i η i ( η ) for each η ∈ G. Substituting this into the Berezin integral (5.49) it is best (notationally) to retain reference to the fermionic variable η and write Z ∗ −1 W = ( det R ) e( C , η ) Dη . (5.50) Now we return to the expression (5.39) for U and enforce the horizontal projection by multiplying by W in the form (5.50). In this way the det R cancels and we simply add the term ( C ∗ , η ) to the exponent to obtain µ ZZZZ 1 1 −n −k 2 U = ( 2π ) ( 2π ) exp − k u k + iψ T du + ψ T( ρ∗ ( φ ) ) ψ 2 2 (5.51) ´ ∗ ∗ + i ( dC , λ ) − i ( φ , Rλ ) + ( C , η ) Dη Dψ dλ dφ . Remarks: One should keep in mind that we are here applying the result q (5.50) to the principal bundle G ,→ P × V −→ P ×ρ V . Moreover, in (5.51) there is also an implicit integration over the fibers to remove the vertical volume form after it has served its purpose of killing the vertical parts. The form U in (5.51), after integrating out the vertical parts, is a basic form on P × V which, regarded as a form on P ×ρ V , represents the Thom class. Pulling U back by a section of P ×ρ V gives a representative of the Euler class which, when integrated over X, gives the Euler number. We have already observed that every section of P ×ρ V is of the form x → [ s( x ), S( s( x ) ) ], where s is a section of P and S : P → V is an equivariant map and that pulling back by such a section amounts to pulling back the V -factors of U by S and then pulling back the resulting form on P by s. Thus, our Euler form is the pullback by s of µ ZZZZ 1 1 −n −k 2 ( 2π ) ( 2π ) exp − k S k + iψ T dS + ψ T( ρ∗ ( φ ) ) ψ 2 2 (5.52) ´ ∗ ∗ + i ( dC , λ ) − i ( φ , Rλ ) + ( C , η ) Dη Dψ dλ dφ , where we have written simply S for S ∗ u = u ◦ S. Integrating this over X gives the Euler number. On the other hand, if we refrain from pulling back by s (and from integrating out the vertical volume form), (5.52) gives a form on P whose integral over P is also the Euler number of P ×ρ V . We have one last bit of cosmetic surgery to perform on U . There is a common notational device in (supersymmetric) physics whereby the integral of a top rank form on a manifold is written as two successive integrations, one fermionic and one bosonic. Recall that the integral of a (properly decaying) 79

function ϕ on an oriented, Riemannian manifold P is defined by multiplying the volume form dω of P by ϕ and integrating this over P Z ϕ dω . P

Now, if α is any (properly decaying) form on P written in terms of local coordinates xi on P ( α = α( xi , dxi ) ) and if one introduces odd variables χi (generators for some exterior algebra), then one can define an element α( xi , χi ) of this exterior algebra by formally making the substitutions dxi → χi . Then the fermionic integral Z ¡ ¢ α xi , χi D χ is precisely the function one integrates (next to dω as above) to get the integral of α over P : Z Z Z ¡ ¢ α = α xi , χi Dχ dω . P

P

Applying this convention to the expression for the Euler number obtained by integrating over P gives the final formula toward which all of this has been leading us. For this we will explicitly indicate all of the dependences on the three fermionic ( χ, η, ψ ) and three bosonic ( λ, φ, ω ) variables (in particular, all of the terms giving rise to forms on P ( iψ T dS, i( dC ∗ , λ ), and ( C ∗ , η ) ) are regarded as functions of the new fermionic variable χ ). We will also (finally!) suppress all but one of the integral signs. ½ Z 1 1 −n −k 2 ( 2π ) ( 2π ) exp − k S ( ω ) k + ψ T( ρ∗ ( φ ) ) ψ 2 2 + iψ T dSω ( χ ) + i ( dCω∗ ( χ , χ ) , λ ) o − i ( φ , Rω λ ) + ( Cω∗ χ , η ) Dχ Dη Dψ dλ dφ dω

(5.53)

This we will call the Atiyah-Jeffrey formula for the Euler number of P ×ρ V . Our objective now is to formally apply it to the infinite-dimensional vector bundle Aˆ ×Gˆ Ω2+ ( B, ad P ) of Donaldson theory with S = F + as the equivariant map. The result will be, formally at least, an expression for an “Euler number” for the bundle (which, however, depends on the choice of S ) and also, as it happens, the partition function for Witten’s topological quantum field theory (i.e., the 0-dimensional Donaldson invariant of B ). We must emphasize at the outset that what we intend to do here is not mathematics (and certainly not physics). Our objective is to find, within the context of the infinite-dimensional vector bundle Aˆ ×Gˆ Ω2+ ( B, ad P ) associated with the Donaldson invariant, formal field-theoretic analogues of the various bosonic and fermionic variables appearing in (5.53) and natural identifications of the terms in the exponent of (5.53) with functions of these variables. In the process the (perfectly well-defined) 80

bosonic and fermionic integrals in (5.53) will metamorphose into Feynman integrals over spaces of fields with all of their attendant mathematical difficulties. The purist will argue that this is meaningless manipulation of symbols and we can offer no credible defense against the charge. The only mitigating circumstance is that such formal manipulations have proved extraordinarily productive for both physics and mathematics and promise to be even more so in the future as the two subjects continue to re-establish lines of communication. We begin with a brief summary of the notation accumulated in Section 4. Throughout the remainder of this section B will denote a compact, simply connected, oriented, smooth 4-manifold with b+ 2 ( B ) > 1 and we will consider only the structure group G = SU ( 2 ) with Lie algebra su( 2 ). g will deπ note a generic Riemannian metric on B and SU ( 2 ) ,→ P −→ B a smooth ˆ principal SU ( 2 )-bundle over B. A is the space of irreducible connections on P , G is the group of gauge transformations and Gˆ = G / ZZ 2 is G modulo its center. Then Bˆ = Aˆ / G ∼ = Aˆ / Gˆ is the moduli space of irreducible d P, g ) is the space of g-ASD conconnections on P . Next, Asd( P, g ) = Asd( nections on P and M = Asd( P, g )/G ∼ = Asd( P, g )/Gˆ is the moduli space of gauge equivalence classes of (irreducible) g-ASD connections on P . Then ˆ Gˆ ,→ Aˆ → Bˆ is a principal G-bundle and Gˆ acts on Ω2+ ( B, ad P ) on the left ˆ A section so we have an associated vector bundle Aˆ ×Gˆ Ω2+ ( B, ad P ) → B. + 2 is determined by the equivariant map S = F : Aˆ → Ω+ ( B, ad P ) defined 1 ∗ by S( ω ) = F + ( ω ) = F + ω = 2 ( F ω + F ω ). M is identified with the zero set of this section, i.e., with the intersection of the images of the corresponding section and the zero-section. We assume for the remainder of this section that π the Chern number of the bundle SU ( 2 ) ,→ P −→ B has been fixed so that the dimension of M is zero. Then the Donaldson invariant γ0 ( B ) given by (4.22) can be viewed as the intersection number of the section corresponding to S, i.e., as an “Euler number” for the bundle Aˆ ×Gˆ Ω2+ ( B, ad P ). The formal

extension of (5.53) to this new infinite-dimensional context “should” provide an integral (Gauss-Bonnet-Chern) representation of the Donaldson invariant. We begin by recalling that our derivation of the Atiyah-Jeffrey formula (5.53) assumes the existence of a Riemannian metric on the principal bundle space (in our case, Aˆ ) for which the group ( Gˆ ) acts by isometries. Such a metric is easy to produce. Since Aˆ is open in A (the space of all connections on P ) and A is an affine space modeled on Ω1 ( B, ad P ), T! ( Aˆ ) ∼ = Ω1 ( B , ad P ) ˆ Now, all of the spaces Ωk ( B, ad P ) have natural L2 -inner for each ω ∈ A. products arising from the metric g on B (and the corresponding Hodge star ∗ ) and an invariant inner product on the Lie algebra. Taking the inner product on su( 2 ) to be ( A, B ) = −tr( AB ) this is given by Z h α , β ik = − tr ( α ∧ ∗ β ) . (5.54) B

81

ˆ Since In particular, this is true for T! ( Aˆ ) and this defines a metric on A. the inner product is invariant under the action of Gˆ (pointwise conjugation by ˆ This metric defines an element of P ×Ad SU ( 2 ) ), Gˆ acts by isometries on A. a connection on Gˆ ,→ Aˆ → Bˆ whose horizontal spaces are the orthogonal complements to the gauge orbits. Indeed, we already know these horizontal spaces since (4.11) gives the orthogonal decomposition T! ( Aˆ ) ∼ = T! ( ω · Gˆ ) ⊕ ker ( δ ! ) = Im ( d! ) ⊕ ker ( δ ! ) ,

(5.55)

where d! : Ω0 ( B, ad P ) → Ω1 ( B, ad P ) is the covariant exterior derivative and δ ! : Ω1 ( B, ad P ) → Ω0 ( B, ad P ) is its formal adjoint relative to the natural inner products (5.54) on the spaces of forms. The first term we must contend with in the exponent of the Atiyah-Jeffrey formula (5.53) is − 12 k S( ω ) k2 , where S = F + : Aˆ → Ω2+ ( B, ad P ) and the norm is computed in the natural inner product h , i2 on Ω2 ( B, ad P ) at ˆ Thus, each ω ∈ A. Z Z °2 ¡ + ∗ +¢ 1 ¡ ¢ 1 1° 1 2 ° − kS (ω )k = − °F+ tr F ∧ F = tr F + ∧ F+ . = ω ω ω ω ω 2 2 2 B 2 B Using the orthogonality of the Hodge decomposition one finds that this can be written as Z Z 1 1 1 2 − kS(ω)k = tr ( F ω ∧ ∗ F ω ) + tr ( F ω ∧ F ω ) . (5.56) 2 4 B 4 B The first term is of the typical Yang-Mills variety for a classical gauge theory, whereas the second Witten [46] calls a topological term because it is, up to a constant, the Chern number of the underlying SU ( 2 )-bundle. Remark: Witten [46] employs the notation more common in physics whereby everything is written in such a way as to appear local. We will not attempt to translate all that we do into this language, but will illustrate with (5.56). Let { Ta } be an orthonormal basis for su( 2 ) relative to ( A, B ) = −tr( AB ), e.g., Ta = − √12 i σa , a = 1, 2, 3, and σ1, σ2, σ3 are the Pauli spin matrices. Write a Fαβ dxα ∧ dxβ , where Fαβ = Fαβ Ta and ∗F ω = 12 F˜αβ dxα ∧ dxβ , 0 0 a where F˜αβ = F˜αβ Ta . Raise indices with g to get F αβ = g αα g ββ Fα0 β 0 and 0 0 F˜ αβ = g αα g ββ F˜ 0 0 . A quick computation shows that 1 tr( F ω ∧ ∗ F ω ) =

Fω =

1 2

α β

4

tr( Fαβ F )volg and 41 tr( F ω ∧F ω ) = 14 tr( Fαβ F˜ αβ )volg . Writing volg √ as g d4 x one obtains the two terms corresponding to (5.56) in (2.41) of [46]: ¶ µ Z 1 1 1 √ 2 (5.57) − kS(ω)k = g d4 x tr Fαβ F αβ + Fαβ F˜ αβ . 2 4 4 B 1 4

αβ

To proceed we must sort out the appropriate analogues, in the Donaldson theory context, of the maps C, C ∗ and R. At each point ω in the prinˆ C! is the map from the Lie algebra of G, ˆ which we cipal bundle space A, 82

have seen can be identified with Ω0 ( B, ad P ), to the tangent space T! ( Aˆ ) ∼ = Ω1 ( B, ad P ) defined, for each ξ ∈ Ω0 ( B, ad P ) by C! ( ξ ) =

¯ d ( ω · exp ( tξ ) ) ¯t=0 . dt

Computing this derivative locally gives C! ( ξ ) = d ! ξ .

(5.58)

∗ is the formal adjoint Consequently, C!

∗ C! = δ ! : Ω1 ( B , ad P ) −→ Ω0 ( B , ad P )

(5.59)

of d! relative to the natural inner products on the spaces of forms and so ∗ 0 0 R! = C! ◦ C! = ∆! 0 : Ω ( B , ad P ) −→ Ω ( B , ad P )

(5.60)

is the scalar Laplacian corresponding to ω. With this information in hand we consider the term −i( φ, R! λ ) in (5.53). Both φ and λ are in the Lie algebra so we introduce two “bosonic” fields φ , λ ∈ Ω0 ( B , ad P ) and interpret ( , ) as the natural inner product h , i0 on Ω0 ( B, ad P ). Remark: We apply the adjectives “bosonic” and “fermionic” to the fields we introduce only because of the type of integral these variables correspond to in (5.53). We do not claim to have justified any physical connotations associated with the terms. Thus, the term −i( φ, R! λ ) is to be interpreted as Z ¡ ¢ −i ( φ, R! λ ) = −i h φ, ∆! λ i = i tr φ ∧ ∗( ∆! 0 0 λ) 0 B

Z =i

tr B

¡



¢

( φ ∆! . 0 λ)

(5.61)

Remark: As Atiyah and Jeffrey [3] point out, the real field φ must be replaced by iφ and λ must be replaced by 12 λ to conform to Witten’s notation. In physics notation, the corresponding term in [46] is ¶ µ Z 1 √ φ Dα D α λ . g d4 x tr 2 B ∗ ∗ Next we consider the term ( C! χ, η ) in (5.53). Since C! maps Ω1 ( B, ad P ) 0 to Ω ( B, ad P ) we will need two fermionic fields

η ∈ Ω0 ( B , ad P ) 83

and

χ ∈ Ω1 ( B , ad P )

and, as above, ( , ) = h , i0 . Thus, we find that ∗ ( C! χ , η ) = h δ ! χ , η i0 = h χ , d! η i1 = −

Z

tr ( χ ∧ ∗ d! η ) . (5.62)

B

The fermionic variable ψ in the Mathai-Quillen formalism arises from the odd generators of the exterior algebra of the fiber vector space V . In our case this vector space is Ω2+ ( B, ad P ) so we introduce a fermionic field ψ ∈ Ω2+ ( B , ad P ) . Now consider the term i ψ T dS! ( χ ) in (5.53). S is the self-dual curvature map S = F + : Aˆ → Ω2+ ( B, ad P ) and we noted in Section 4 that the derivative of this map at ω ∈ Aˆ is identified with 1 2 d! + : Ω ( B , ad P ) −→ Ω+ ( B , ad P )

so

dS! ( χ ) = d! +χ

(5.63)

for each χ ∈ Ω1 ( B, ad P ). We will interpret finite-dimensional expressions such as  1  B  ¡ 1 ¢ T r  .. 1 1 r r A B = A ··· A  .   = A B + ··· + A B Br in terms of the appropriate field-theoretic inner product so that i ψ T dS! ( χ ) becomes ® ­ ! i ψ, d! + χ 2 = i h ψ, d χ i2 ( because ψ is self-dual and the Hodge decomposition is orthogonal ) = i h d! χ, ψ i2 Z = −i tr ( d! χ ∧ ψ ) B

Z i ψ T dS! ( χ ) = −i

tr ( d! χ ∧ ψ ) .

(5.64)

B

Next we consider the term 12 ψ T ( ρ∗( φ ) )ψ in (5.53). We know already that φ ∈ Ω0 ( B, ad P ) and ψ ∈ Ω2+ ( B, ad P ). In the Mathai-Quillen form, ρ 84

corresponds to the action of G on V that gives rise to the associated vector bundle. In our case, Gˆ (regarded as sections of the nonlinear adjoint bundle) acts on Ω2+ ( B, ad P ) pointwise by conjugation. At each point this is just the ordinary adjoint action of SU ( 2 ) on its Lie algebra for which the infinitesimal action is just bracket. Thus, for each φ ∈ Ω0 ( B, ad P ), ρ∗( φ ) acts on ψ ∈ Ω2+ ( B, ad P ) by ρ∗ ( φ ) ψ = [ φ , ψ ] so 12 ψ T ( ρ∗( φ ) )ψ is interpreted as we rearrange as follows:

1 2

ψ T ( ρ∗( φ ) )ψ =

1 2

h ψ, [ φ, ψ ] i2 which

1 T 1 1 ψ ( ρ∗ ( φ ) ) ψ = h ψ, [ φ , ψ ] i2 = h [ φ , ψ ] , ψ i2 2 2 2 Z 1 tr ( [ φ , ψ ] ∧ ∗ ψ ) =− 2 B Z 1 tr ( [ φ , ψ ] ∧ ψ ) =− 2 B because ψ is self-dual. Now, any ad-invariant inner product ( , ) on any Lie algebra satisfies ( x, [ y, z ] ) = ( [ x, y ], z ) so in this last integral we may replace [ φ, ψ ] ∧ ψ by φ[ ψ, ψ ]. Moreover, tr( AB ) = tr( BA ) so we conclude that Z 1 T 1 ψ ( ρ∗ ( φ ) ) ψ = − tr ( [ ψ , ψ ] φ ) . (5.65) 2 2 B ∗ The only remaining term in (5.53) is i( dC! ( χ, χ ), λ ) and this requires a ∗ ˆ bit more work. C is a 1-form on A with values in Ω0 ( B, ad P ). We compute ˆ Since A is an affine space and dC ∗ at ω ∈ Aˆ as follows. Fix χ1 , χ2 ∈ T! A. ˆ Aˆ is open in A we may regard χ1 and χ2 as constant vector fields on A. Thus, ∗ dC! ( χ1 , χ2 ) = χ1 ( C ∗ χ2 ) − χ2 ( C ∗ χ1 ) − C ∗ ( [ χ1 , χ2 ] )

= χ1 ( C ∗ χ2 ) − χ2 ( C ∗ χ1 ) , where C ∗ χi is the function on θ → C∗ χi = δ  χi for i = 1, 2. Now, ( χ1 ( C ∗ χ2 ) ) ( ω ) = χ1 ( ω ) ( C ∗ χ2 ) = χ1 ( C ∗ χ2 ) = ( χ1 ( C ∗ χ2 ) ) ( ω ) =

¯ d ¯ ∗ C! +t1 ( χ2 ) ¯ dt t=0 ¯ d !+t1 ( χ2 ) ¯ t=0 . δ dt

We compute δ !+t1 as follows: For any λ ∈ Ω0 ( B, ad P ), d!+t1 ( λ ) = d! λ + t [ χ1 , λ ] = d! λ + t B1 ( λ ) 85

(5.66)

where B1 : Ω0 ( B, ad P ) → Ω1 ( B, ad P ) is given by B1 ( λ ) = [ χ1 , λ ]. Thus, ∗ δ !+t1 ( χ2 ) = δ ! χ2 + B ( χ2 ) 1 ∗ where B : Ω1 ( B, ad P ) → Ω0 ( B, ad P ) is the adjoint of B1 . We claim 1 that ∗ B ( χ2 ) = − ∗ [ χ1 , ∗ χ2 ] . (5.67) 1

Indeed, for any λ ∈ Ω0 ( B, ad P ), Z D E B1 ( λ ) , χ2 = h [ χ1 , λ ] , χ2 i1 = − 1

B

Z = B

tr ( [ λ , χ1 ] ∧ ∗ χ2 ) =

Z =

B

tr ( [ χ1 , λ ] ∧ ∗ χ2 ) Z B

tr ( λ ∧ [ χ1 , ∗ χ2 ] )

tr ( λ ∧ ∗∗ [ χ1 , ∗ χ2 ] ) = − h λ , ∗ [ χ1 , ∗ χ2 ] i0

= h λ , − ∗ [ χ1 , ∗ χ2 ] i0 which establishes (5.67). Thus, δ !+t1 ( χ2 ) = δ ! χ2 − t ∗[ χ1 , ∗ χ2 ] and computing the derivative at t = 0 gives, from (5.66), ( χ1 ( C ∗ χ2 ) ) ( ω ) = − ∗ [ χ1 , ∗ χ2 ] . Interchanging χ1 and χ2 gives ( χ2 ( C ∗ χ1 ) ) ( ω ) = − ∗ [ χ2 , ∗ χ1 ] . Since these are independent of ω we have χ1 ( C ∗ χ2 ) = − ∗ [ χ1 , ∗ χ2 ] and χ2 ( C ∗ χ1 ) = − ∗ [ χ2 , ∗ χ1 ]. One can verify that, for any α, β ∈ Ω1 ( B, ad P ), ∗ [ β , ∗ α ] = − ∗ [ α , ∗ β ] so we may write ∗ [ χ2 , ∗ χ1 ] = − ∗ [ χ1 , ∗ χ2 ] and thereby obtain ∗ dC! ( χ1 , χ2 ) = − 2 ∗ [ χ1 , ∗ χ2 ] (5.68) ˆ Thus, for any ω ∈ A. ∗ ∗ ( χ1 , χ2 ) i0 ( χ1 , χ2 ) , λ i0 = i h λ, dC! i h dC! Z ¡ ¢ = −i tr λ ∧ ∗ ( − 2 ∗ [ χ1 , ∗ χ2 ] )

Z

B

= 2i Z

B

= 2i B

tr ( λ ∧ [ χ1 , ∗ χ2 ] ) tr ( [ χ1 , ∗ χ2 ] λ )

and finally

Z ∗

i ( dC! ( χ , χ ) , λ ) = 2 i 86

tr ( [ χ , ∗ χ ] λ ) . B

(5.69)

With this we have identified all of the terms in the exponent in (5.53). Each is the integral over B of a trace and so we may collect them all together into µ Z 1 1 1 tr F ω ∧∗ F ω + F ω ∧ F ω − [ψ, ψ] φ − id! χ ∧ ψ 4 4 2 B (5.70) ¶ ! ∗ ∗ ∗ ! + 2i [χ, χ] λ + i (φ ∆0 λ) − χ ∧ d η . Now we introduce some of the terminology used in physics. Each fixed ˆ φ, λ ∈ Ω0 ( B, ad P ) ) and three fermionic choice of the three bosonic ( ω ∈ A, 0 1 ( η ∈ Ω ( B, ad P ), χ ∈ Ω ( B, ad P ), ψ ∈ Ω2+ ( B, ad P ) ) fields will be called a field configuration and will be denoted Φ = (ω, φ, λ, η, χ, ψ) . For each such choice the expression (5.70) is a number so this integral can be regarded as a function of Φ. Minus this function is the Donaldson-Witten action functional µ Z 1 1 1 SDW [Φ] = tr − F ω ∧∗ F ω − F ω ∧ F ω + [ψ, ψ] φ + id! χ ∧ ψ 4 4 2 B (5.71) ´ ! ∗ ! ∗ ∗ − 2i [χ, χ] λ − i (φ ∆0 λ ) + χ ∧ d η . Thus, in our present infinite-dimensional context the integral in (5.53) can be written Z e−SDW [ Φ ] D Φ , (5.72) where we have abbreviated Dχ Dη Dψ dλ dφ dω as simply DΦ. It is only in this last expression that we leave the world of mathematically well-defined objects and proceed “formally”. In (5.72) we have omitted the constant (2π)−n (2π)−k in (5.53) since, in our present circumstances, both n and k would be infinite. In the physics literature one often sees the integral (5.72) normalized with a factor of 1/vol ( G ), where vol ( G ) is intended to represent the “volume” of the gauge group G. About this we will have nothing further to say, but, for certain remarks we wish to make here and in Section 7, we point out that the physicists often include in the exponent in (5.72), or directly in the action (5.71), a factor of 1/e2 , where e is a so-called coupling constant. Mathematically, one can view the inclusion of such a factor in SDW [ Φ ] as simply a different choice of invariant inner product on the Lie algebra su( 2 ), which is determined only up to a positive constant multiple. Classically, one can rescale and give this factor any convenient value. However, upon quantization the different values of the coupling constant give rise to an entire one-parameter family of quantum field theories and the computability (“renormalizability”) of the theory generally depends on this value. Since this 87

dependence on the coupling constant (or, rather, a lack thereof in the cases of topological interest) is relevant to a few comments we will make here and somewhat later, we record the following alternative to (5.72). Z 2 e−SDW [ Φ ] / e D Φ . (5.73) This integral represents the partition function of the quantum field theory constructed by Witten in [46]. Of course, Witten arrived at the action (5.71) and therefore the partition function by quite a different route than the one we have followed. We began with the 0-dimensional Donaldson invariant, regarded it as an “Euler number” and massaged the Mathai-Quillen integral representation of this Euler number until it could be formally applied in our infinite-dimensional context to yield (5.71) and (5.72). Witten’s arguments leading to SDW [ Φ ] were physical, but the objective was to describe a quantum field theory in which the Donaldson invariants appeared as expectation values of certain observables and, in particular, the 0-dimensional invariant was the partition function. How then did Witten uncover the Donaldson invariants in the field theory with action SDW [ Φ ]?. Witten chose the field content Φ = ( ω, φ, λ, η, χ, ψ ) and action SDW [ Φ ] in order to ensure the presence of certain symmetries (gauge invariance and “BRST-like” symmetries). The BRST symmetries are expressed in terms of a certain operator Q on the fields which squares to zero ( Q ◦ Q = 0 ) and so determines cohomology classes that are taken to represent the physical states of the theory. The energy-momentum tensor of the theory (defined in terms of the variation of the action under an infinitesimal change in the Riemannian metric g of the underlying 4-manifold B ) turns out to be Q-exact (and so cohomologically trivial). With this, certain formal manipulations with functional integrals imply that the partition function of the theory is independent of both the metric g and the coupling constant e in the sense that its infinitesimal variation with respect to either g or e is zero. Note: These are hallmarks of what are today called cohomological field theories. In such field theories the expectation values of observables are also “independent of g ” in the same sense. This has led the physicists to refer to such field theories as “topological quantum field theories” and the expectation values as “topological invariants”. However, these are very different uses of the terms “independent of g ” and “topological invariant” than one would encounter in mathematics. For example, we have seen in Section 4 that the 0-dimensional Donaldson invariant γ0 ( B ) is only independent of a generic choice of g and even this is true only when b+ 2 ( B ) > 1. Even granting this, γ0 ( B ) is an invariant of the differentiable structure of B and certainly not of its topology. The fact that the partition function is independent of the coupling constant e is particularly significant since one is then free to compute it in the limit of either small ( e → 0 ) or large ( e → ∞ ) values, whichever is most conve-

88

nient or most informative. For small values of e, physicists employ a technique known as semiclassical approximation which, again because of the symmetries of SDW [ Φ ], one can show is actually exact in our case. This phenomenon is an infinite-dimensional analogue of a well-known finite-dimensional theorem on the exactness of the stationary phase approximation due to Duistermaat and Heckman [13]. As we shall see in the next section this theorem is most properly understood within the context of equivariant cohomology and the localization of certain integals of equivariant differential forms to the fixed point set of the group action. Similarly, the BRST operator Q can be viewed as the equivariant exterior derivative in a model of the G-equivariant cohomology of A and Witten shows that, for certain well-chosen observables in his field theory, the path integral representations of their expectation values localize to the (finitedimensional) moduli spaces of anti-self-dual connections thus yielding integral formulas for the Donaldson invariants. For the particular case we have under consideration, the partition function (which, being invariant, descends to the moduli space of fields) localizes to a sum over the 0-dimensional moduli space of ASD connections yielding γ0 ( B ). In Part II of this survey we will take up our story at this point with a rather detailed discussion of the simplest of the “Equivariant Localization” theorems and its relation to the theorem of Duistermaat and Heckman on exact stationary phase approximation. This done, we will turn to the question of what can be learned by examining the partition function in the limit e → ∞ of large coupling constants in a section on “Duality and Seiberg-Witten”. Finally then we will consider “The Witten Conjecture” itself.

References [1] Atiyah M.F. and Bott R., The Moment Map and Equivariant Cohomology, Topology, 23(1984), 1-28. [2] Atiyah M.F., Hitchin N.J. and Singer I.M., Self-Duality in FourDimensional Riemannian Geometry, Proc. Roy. Soc. Lond., A 362(1978), 425-461. [3] Atiyah M.F. and Jeffrey L., Topological Lagrangians and Cohomology, J. Geo. Phys., 7(1990), 119-136. [4] Belavin A., Polyakov A., Schwarz A. and Tyupkin Y., Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett., 59 B(1975), 85-87. [5] Berline N., Getzler E. and Vergne M., Heat Kernels and Dirac Operators, Springer-Verlag, New York, Berlin, 1996. [6] Blau M., The Mathai-Quillen Formalism and Topological Field Theory, J. Geo. Phys., 11(1993), 95-127. [7] Blau M. and Thompson G., Localization and Diagonalization, J. Math. Phys., 36(1995), 2192-2236. 89

[8] Cordes S., Moore G. and Ramgoolam S., Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theory, arXiv: hepth/9402107. [9] Dirac P.A.M., Quantised Singularities in the Electromagnetic Field , Proc. Roy. Soc., A 133(1931), 60-72. [10] Donaldson S.K., An Application of Gauge Theory to Four-Dimensional Topology, J. Diff. Geo., 18(1983), 279-315. [11] Donaldson S.K., Polynomial Invariants for Smooth 4-Manifolds, Topology, 29(1990), 257-315. [12] Donaldson S.K. and Kronheimer P., The Geometry of Four-Manifolds, Oxford University Press, Oxford, 1990. [13] Duistermaat J.J. and Heckman G., On the Variation in the Cohomology of the Symplectic Form of the Reduced Phase Space, Invent. Math., 69(1982), 250-268. Addendum, 72(1983), 153-158. [14] Feehan, P.M.N. and Leness T.G., On Donaldson and Seiberg-Witten Invariants, arXiv: math.DG/0106221. [15] Freed D. and Uhlenbeck K.K., Instantons and Four-Manifolds, SpringerVerlag, New York, Berlin, 1984. [16] Freedman M., The Topology of Four-Dimensional Manifolds, J. Diff. Geo., 17(1982), 357-454. [17] Friedman R. and Morgan J., Smooth Four-Manifolds and Complex Surfaces, Springer-Verlag, New York, Berlin, 1994. [18] Guillemin V.W. and Sternberg S., Geometric Asymptotics, Amer. Math. Soc., Providence, RI, 1977. [19] Guillemin V.W. and Sternberg S., Symplectic Techniques in Physics, Cambridge University Press, Cambridge, England, 1984. [20] Guillemin V.W. and Sternberg S., Supersymmetry and Equivariant De Rham Theory, Springer-Verlag, New York, Berlin, 1999. [21] Hitchin N.J., The Geometry and Topology of Moduli Spaces. In: Global Geometry and Mathematical Physics, Springer Lecture Notes in Mathematics 1451, Springer-Verlag, New York, Berlin, 1990, 1-48. [22] Jaffe A. and Taubes C.H., Vortices and Monopoles, Birkhauser, Boston, MA, 1980. [23] Kalkman J., BRST Model for Equivariant Cohomology and Representatives for the Equivariant Thom Class, Comm. Math. Phys., 153(1993), 447-463.

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[24] Kobayashi S. and Nomizu K., Foundations of Differential Geometry, Vols. I, II , Wiley Interscience, 1996. [25] Kronheimer P.B. and Mrowka T.S., Recurrence Relations and Asymptotics for Four-Manifold Invariants, Bull. Amer. Math. Soc., 30(1994), 215-221. [26] Kronheimer P.B. and Mrowka T.S., Embedded Surfaces and the Structure of Donaldson’s Polynomial Invariants, J. Diff. Geo., 3(1995), 573-734. [27] Labastida J.M.F. and Marino M., A Topological Lagrangian for Monopoles on Four-Manifolds, Phys. Lett. B 351(1995), 146-152. [28] Labastida J.M.F. and Lozano C., Lectures on Topological Quantum Field Theory, arXiv: hep-th/9709192. [29] Lang S., Introduction to Differentiable Manifolds, John Wiley & Sons, New York, London, 1962. [30] Lawson H.B., The Theory of Gauge Fields in Four Dimensions, Amer. Math. Soc., Providence, RI, 1985. [31] Lawson H.B. and Michelsohn M-L., Spin Geometry, Princeton University Press, Princeton, NJ, 1989. [32] Mathai V. and Quillen D., Superconnections, Thom Classes and Equivariant Differential Forms, Topology, 25(1986), 85-110. [33] Naber G., Topological Methods in Euclidean Spaces, Dover Publications, Mineola, NY, 2000. [34] Naber G., Topology, Geometry, and Gauge Fields: Foundations, SpringerVerlag, New York, Berlin, 1997. [35] Naber G., Topology, Geometry, and Gauge Fields: Interactions, SpringerVerlag, New York, Berlin, 2000. [36] O’Grady K.G., Donaldson’s Polynomials for K3 Surfaces, J. Diff. Geo., 35(1992), 415-427. [37] Parker T.H., Gauge Theories on Four-Dimensional Riemannian Manifolds, Comm. Math. Phys., 85(1982), 563-602. [38] Pidstrigach V. and Tyurin A., Localization of Donaldson’s Invariants along Seiberg-Witten Classes, arXiv: dg-ga/9507004. [39] Seiberg N. and Witten E., Monopoles, Duality and Chiral Symmetry Breaking in N = 2 Supersymmetric QCD, Nucl. Phys. B 431(1994), 581-640. [40] Szabo R.J., Equivariant Localization of Path Integrals, arXiv: th/9608068.

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[41] Taubes, C.H., Self-Dual Connections on Non-Self-Dual 4-Manifolds, J. Diff. Geo., 17(1982), 139-170. [42] Taubes, C.H., The Seiberg-Witten Invariants (Videotape), AMS Selected Lectures in Mathematics, Amer. Math. Soc., Providence, RI, 1995. [43] Uhlenbeck K., Removable Singularities in Yang-Mills Fields, Comm. Math. Phys., 83(1982), 11-30. [44] Vajiac A., A Generalization of Witten’s Conjecture Relating Donaldson and Seiberg-Witten Invariants, arXiv: hep-th/0003214. [45] Warner F.W., Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, New York, Berlin, 1983. [46] Witten E., Topological Quantum Field Theory, Comm. Math. Phys., 117(1988), 353-386. [47] Witten E., Two Dimensional Gauge Theories Revisited , J. Geom. Phys., 9(1992), 303-368. [48] Witten E., Monopoles and Four-Manifolds, Math. Res. Lett., 1(1994), 769796. [49] Yang C.N. and Mills R.L., Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev., 96(1954), 191-195.

92

TOPOLOGY, GEOMETRY AND PHYSICS: BACKGROUND FOR THE WITTEN CONJECTURE PART II Gregory L. Naber Department of Mathematics, California State University Chico, CA 95929-0525, U.S.A.

Abstract This is a continuation of the article “Topology, Geometry and Physics: Background for the Witten Conjecture, Part I” which appeared in the previous issue of this journal. Although much of the material presented in this paper can be read independently of its predecessor we will nevertheless continue the numbering of sections, equations and results from the point at which the previous paper ended.

6

Equivariant Localization

Motivated by our discussion of the Witten Lagrangian, its equivariant symmetries, and the localization of the corresponding partition function to the antiself-dual moduli space to yield the Donaldson invariant, we return now to the finite-dimensional context and consider the general phenomenon of “equivariant localization.” In order to provide a relatively complete treatment and because it describes the finite-dimensional analogue of Witten’s partition function (the 0-dimensional Donaldson invariant) we will restrict our attention to the simplest (“discrete”) equivariant localization theorem. We consider a compact, oriented, smooth manifold M of dimension n = 2k and denote by ν a volume form on M . Suppose H : M → IR is a Morse function on M , i.e., a smooth function whose critical points p (dH(p) = 0) are all nondegenerate (this means that the Hessian Hp : Tp (M ) × Tp (M ) → IR , defined by Hp ( Vp , Wp ) = Vp ( W (H) ), where Vp , Wp ∈ Tp ( M ) and W is a vector field on M with W (p) = Wp , is a nondegenerate bilinear form). Finally, 1

let T denote some real parameter. We consider the integral Z ei T H ν

(6.1)

M

and are especially interested in its asymptotic behavior as T → ∞. The Stationary Phase Theorem (Chapter I of [22] ) asserts roughly that, for large T , the dominant contributions to such an integral come from the critical points of H. More precisely, one has Z ei T H ν = M

¯ X µ 2π ¶k ¡ ¢ ¯¯ − 21 i T H (p) ¯ eπi (Sgn Hp )/ 4 ¯ det Hp ei , ej ¯ e T

(6.2)

p∈M

dH (p)=0

³ ´ + O T −(k+1) ,

where Sgn Hp is the signature (number of positive eigenvalues minus the number of negative eigenvalues) of any matrix representing Hp , { e1 , . . . , e2k } is a basis for Tp (M ) with ν p ( e1 , . . . , e2k ) = 1, and O ( T −(k+1) ) stands for terms which, in modulus, are bounded by C / T k+1 for some constant C and all T outside some compact set in IR . The terms preceding O ( T −(k+1) ) on the right-hand side of (6.2) constitute the stationary phase approximation of the integral. These terms arise in the proof of (6.2) from writing H near p as a quadratic function in some coordinates (that this is possible is the content of the Morse Lemma) and computing directly the resulting Gaussian integral. It follows from the Morse Lemma that the critical points of a Morse function are isolated. Since M is compact, H can have only finitely many critical points so the sum in (6.2) is necessarily finite. Let us write out a simple example used by Witten [53] to illustrate the phenomenon we wish to study. We take M to be the 2-sphere S 2 in IR 3 and let ν be the usual metric volume form on S 2 (this is the restriction to S 2 of the 2-form x dy ∧ dz − y dx ∧ dz + z dx ∧ dy on IR 3 ). Let H : S 2 → IR be the “height function” ( H( x, y, z ) = z for any ( x, y, z ) ∈ S 2 ). We claim that the critical points of H are the north and south poles, i.e., N = ( 0, 0, 1 ) and S = ( 0, 0, −1 ), and that both are nondegenerate so H is a Morse function on S 2 . For example, on z > 0 in S 2 , ( x, y, z ) → ( x, y ) is a chart with inverse ( x, y ) → 1

1

( x, y, ( 1 − x2 − y 2 ) 2 ) and, in these coordinates, H( x, y ) = ( 1 − x2 − y 2 ) 2 so 1 dH( x, y ) = −( 1 − x2 − y 2 )− 2 ( x dx + y dy ). Thus, the only critical point in z > 0 occurs when ( x, y ) = ( 0, 0 ), i.e., at N ( 0, 0, 1 ). Furthermore, the Hessian (which, in any coordinate system, is represented by the matrix of second order

2

partial derivatives) is given by ¡

2

− 1−x −y

¢ 3 2 −2

Ã

!

1 − y2

xy

xy

1 − x2

µ Thus, at ( x, y ) = ( 0, 0 ) we obtain HN =

−1

0

0

−1

. ¶ and this is, indeed,

nonsingular. The region z < 0 on S 2 is, of course, handled in the same way and projecting onto other coordinate planes shows that there are no critical points with z = 0. Now we shall write out the stationary phase approximation for the integral Z e i Tz ν . (6.3) S2

From (6.2) with k = 1 and p = N, S, this is ¶ µ ¯ ¡ ¢ ¯¯ − 21 i T z ( N ) 2π ¯ e + eπi ( Sgn HN )/4 ¯ det HN ei , ej ¯ T µ

2π T

(6.4)

¶ eπi ( Sgn HS )/4 µ

Now, Sgn HN = Sgn

−1

0

0

−1

¯ ¢ ¯¯ − 21 i T z ( S ) ¡ ¯ e . ¯ det HS ei , ej ¯ ¶ = −2. Next note that evaluating x dy∧dz −

∂ ∂ ∂ x , ∂ y ) gives z ∂ ∂ ∂ x (N ), ∂ y (N ) }

∂ ∂x

(N ),

∂ ∂y

y dx ∧ dz + z dx ∧ dy at (

so, at N , ν N (

(N ) ) =

1. Thus, { e1 , e2 } = {

is a basis of the required type for

2

TN ( S ) so ¯ à ¯ ¯ ¡ ¢ ¯¯ − 12 −1 ¯ ¯ = ¯ det ¯ det HN ei , ej ¯ ¯ 0

0 −1

! ¯ − 12 ¯ ¯ = 1. ¯ ¯

1

Similarly, Sgn HS = 2 and | det HS ( ei , ej ) | − 2 = 1. Substituting all of this into (6.4) gives, for the stationary phase approximation to (6.3), ¶ µ ¢ 2π i ¡ −i T sin T iT . (6.5) e − e = 4π T T Next we observe that the integral (6.3) is actually easy to compute exactly. Let ι : S 2 ,→ IR 3 be the inclusion map so that ν = ι∗ ( x dy∧dz−y dx∧dz+z dx∧ dy ). Define an orientation preserving diffeomorphism ϕ of ( 0, π ) × ( −π, π ) 3

into S 2 by ( ι ◦ ϕ ) ( φ , θ ) = ( sin φ cos θ , sin φ sin θ , cos φ ) . The image of this map covers all of S 2 except a set of measure zero. A simple computation shows that ϕ∗ ν = sin φ dφ ∧ dθ and so ¡ ¢ ϕ∗ ei T z ν = ei T cos φ sin φ dφ ∧ dθ . Denoting by dm Lebesgue measure on the plane we therefore have Z Z e i Tz ν = ei T cos φ sin φ dφ ∧ dθ S2

( 0,π )×( −π,π )

Z ei T cos φ sin φ dm

= [ 0,π ]×[ −π,π ]

Z

π

Z

π

= −π

=− =

ei T cos φ sin φ dφ dθ

0

2π £ i T cos φ ¤π e 0 iT

¤ 2π i £ −i T e − ei T = 4π T

µ

sin T T

¶ .

We find then that, in this particular case, the stationary phase approximation (6.5) to the integral (6.3) actually gives the exact value of the integral. Our goal now is to uncover the underlying features of this example which account for this exactness of the stationary phase approximation. We begin with a few observations on the preceding example. Note that the volume form ν on S 2 is also a symplectic form, i.e., a closed, nondegenerate 2-form. Indeed, any volume form ν on any orientable surface is a symplectic form (it is closed because it is a 2-form on a 2-dimensional manifold and nondegenerate because, at each point, an oriented basis { e1 , e2 } for the tangent space satisfies ν( e1 , e2 ) > 0 so if v = v 1e1 + v 2e2 6= 0 (say, v 1 6= 0 ), then ν( v , e2 ) = v 1 ν( e1 , e2 ) 6= 0 ). When thinking of ν as a symplectic form on S 2 we will denote it ω. Now, the height function H, like any smooth, real-valued function on the symplectic manifold ( S 2 , ω ), determines a corresponding Hamiltonian vector field VH on S 2 . This is defined to be the unique vector field on S 2 satisfying dH = ιV ω ,

(6.6)

H

where ιV

H

is interior multiplication by VH (so that, for any vector field W on

2

S , dH( W ) = ω( VH , W ) ). We claim that if 4

∂ ∂θ

is the θ-coordinate velocity

field of the spherical coordinate chart on S 2 (taken to be zero at N and S ), then ∂ VH = . ∂θ First note that (6.6) and the nondegeneracy of ω imply that VH must vanish at the critical points N and S of H and so it agrees with ∂∂θ there. At any other point, H( φ, θ ) = cos φ so dH( φ, θ ) = − sin φ dφ. Since ω = sin φ dφ ∧ dθ (here and henceforth we adopt the time-honored custom of omitting references to the diffeomorphism ϕ whenever it is convenient to do so), we have ι

∂ ∂ θ

ω=ι

( sin φ dφ ⊗ dθ − sin φ dθ ⊗ dφ )

∂ ∂ θ

µ = sin φ

µ dφ

∂ ∂θ

¶¶

µ dθ − sin φ

µ dθ

∂ ∂θ

¶¶ dφ

= − sin φ dφ = dH as required. The integral curves of VH =

∂ ∂θ

are then easily found. They are

points at N and S and elsewhere they are “horizontal” circles traversed at speed one. The unique one through p = ϕ( φ, θ ) at time t = 0 is αp ( t ) = ( sin φ cos ( θ + t ), sin φ sin ( θ + t ), cos φ ) (we shall also omit references to the inclusion ι : S 2 ,→ IR 3 ). These integral curves are therefore periodic with period 2π. The flow α : S 2 × IR −→ S 2 α ( p , t ) = αp ( t ) is therefore also periodic in t. Finally, recall that any symplectic manifold ( M 2k , ω ) has a canonical orientation (volume form) ν ω called the Liouville form and defined by νω =

1 1 k k ω ∧ ··· ∧ ω = ω . k! k!

For k = 1 this is just ω so, in our example on S 2 , ν, ω and ν ω are all the same. Duistermaat and Heckman [16] have shown that the exactness of the staR tionary phase approximation of S 2 ei T z ν is a consequence of the fact that the Hamiltonian vector field of the height function on S 2 has a periodic flow. More generally, we have

5

Theorem 6.1 (Duistermaat-Heckman) Let M be a compact manifold of dimension n = 2k with symplectic form ω and oriented by the corresponding Liouville form ν ω = k1! ω k . Let H ∈ C ∞ ( M ) be a Morse function on M and VH its Hamiltonian vector field ( dH = ιV ω ) . If the flow of VH is H periodic, then, for any real number T > 0, Z ei T H ν ω = M

¯ X µ 2π ¶k ¡ ¢ ¯¯ − 21 i T H (p) ¯ eπi (Sgn Hp )/ 4 ¯ det Hp ei , ej ¯ e , T

p∈M

dH (p)=0

where Hp : Tp ( M )×Tp ( M ) → IR is the Hessian of H at p and { e1 , . . . , e2k } is a basis for Tp (M ) with ν ω ( e1 , . . . , e2k ) = 1. Note that the set of critical points must be nonempty since M is compact and so H must achieve maximum and minimum values. We intend to provide a complete proof of this result, but will proceed toward it in a rather roundabout fashion. First we return to our example on S 2 and isolate a group action which suggests a more general perspective on the Duistermaat-Heckman Theorem. We formulate this new perspective as a Generalized Duistermaat-Heckman Theorem that concerns itself with Hamiltonian actions on symplectic manifolds and show that this new result implies our Theorem 6.1. Still our perspective is not broad enough, however, and we focus our attention on general group actions on manifolds and their associated equivariant cohomologies. In this context we prove the simplest of the so-called Equivariant Localization Theorems and find that it has as a simple consequence the Generalized Duistermaat-Heckman Theorem and therefore also Theorem 6.1. Let us then consider again the height function H on the symplectic manifold S 2 . Since the Hamiltonian vector field VH has a periodic flow it gives rise to an obvious action of S 1 on S 2 (rotate points of S 2 around the integral curves containing them). Specifically, if g = ei T ∈ S 1 and p = ϕ( φ, θ ) = ( sin φ cos θ, sin φ sin θ, cos φ ) ∈ S 2 , then we define g · p = ei T · ( sin φ cos θ , sin φ sin θ , cos φ ) = ( sin φ cos ( θ + T ), sin φ sin ( θ + T ), cos φ ) (if p = N or S we define g · p = p for all g ∈ S 1 ). This clearly defines a (left) action of S 1 on S 2 . As usual, we identify the Lie algebra of S 1 with i IR . Each ξ = i a in the Lie algebra gives rise to an associated vector field ξ # on S 2 defined by d ξ# ( p ) = ( exp ( − tξ ) · p ) | t=0 . (6.7) dt 6

It is a simple matter to compute ξ # ( p ) explicitly. At p = N, S it is zero and, otherwise, ξ# ( p ) =

d ( exp ( − tξ ) · p ) | t=0 dt

=

¢¯ d ¡ − i at e · p ¯ t=0 dt

=

d ( sin φ cos ( θ − at ), sin φ sin ( θ − at ), cos φ ) | t=0 dt

= − a ( − sin φ sin θ , sin φ cos θ , 0 ) = −a

∂ (p) ∂θ

= − a VH ( p ) = V−a H ( p ) (the last equality is easy to check by verifying that d ( − aH ) = ι− a V ω ). H Thus, ξ = i a =⇒ ξ # = V− a H . In particular, every ξ # is the Hamiltonian vector field of some smooth function on S 2 . We can therefore define a map ¡ ¢ ¡ ¢ µ : Lie S 1 = i IR −→ C ∞ S 2 by µ(ξ ) = µ(ia) = −aH which has the following properties: 1. µ is linear. 2. ξ # is the Hamiltonian vector field on S 2 determined by µ ( ξ ). 3. µ is equivariant with the respect to the natural actions of S 1 on Lie ( S 1 ) and C ∞ ( S 2 ), i.e., µ(g · ξ ) = g · µ(ξ ) . Remarks: Regarding (3), the natural action of S 1 on Lie ( S 1 ) is the adjoint action ( g · ξ = gξ g −1 ) which, in this case, is trivial since S 1 is abelian. Thus, µ( g · ξ ) = µ( ξ ). The action of S 1 on C ∞ ( S 2 ) is defined by ( g · ψ )( p ) = ψ( g −1·p ) so ( g·µ( ξ ) )( p ) = µ( ξ )( g −1·p ) = ( −aH )( g −1·p ) = ( −aH )( p ) = µ( ξ )( p ) because H is constant on the orbits. Thus, g·µ( ξ ) = µ( ξ ) = µ( g·ξ ). Now we abstract these properties of our example and formulate general definitions. Let ( M, ω ) be a compact symplectic manifold of dimension n = 2k 7

and G a compact Lie group (with Lie algebra G ) that acts smoothly on M on the left (we will write such an action as σ : G × M → M with σ( g, p ) = g · p = σg ( p ) = σp ( g ) ). The action is said to be Hamiltonian if there is a map µ : G −→ C ∞ ( M ) such that 1. µ is linear. 2. For each ξ ∈ G the vector field ξ # on M (defined by (6.7) ) is the Hamiltonian vector field associated with µ ( ξ ), i.e., dµ ( ξ ) = ιξ# ω .

(6.8)

3. µ is equivariant, i.e., µ(g · ξ ) = g · µ(ξ ) . The function µ( ξ ) is called the symplectic moment of ξ and one defines the associated moment map Φ : M −→ G ∗ (where G ∗ is the dual of the vector space G ) by (Φ(p) )(ξ ) = (µ(ξ ) )(p) . Although they will play no role in our story here, these moment maps have many striking and beautiful properties (see, e.g., [2] and [23] ). Notice that it follows at once from (2) and the nondegeneracy of ω that the critical points of µ( ξ ) coincide with the zeros of ξ # . Moreover, every fixed point of the G-action is, by (6.7), a zero of every ξ # (and so, a critical point of every µ( ξ ) ). If ξ ∈ G has the property that ξ # vanishes only at the fixed points of the G-action, then ξ is said to be nondegenerate and, in this case, one can show that µ( ξ ) is necessarily a Morse function ( see [23] ). We assume that some invariant Riemannian metric h , iG on M has been selected and note that any vector field ξ # defined by (6.7) for some ξ ∈ G is then necessarily a Killing vector field for h , iG , i.e., Lξ# h , iG = 0 , where Lξ# denotes the Lie derivative with respect to ξ # . This last condition can be written equivalently as ­ £ ¤® ­ £ # ¤ ® (6.9) ξ # ( h V, W iG ) = ξ , V , W G + V, ξ # , W G for all vector fields V and W on M .

8

Now, fix a ξ ∈ G. We denote by Z( ξ # ) the set of zeros of the vector field ξ # . For each p ∈ Z( ξ # ) we define a linear transformation Lp ( ξ ) : Tp ( M ) −→ Tp ( M ) by

´ ¡ ¢ ³ £ ¤ L p ( ξ ) Vp = L ξ # V = ξ# , V p ,

(6.10)

p

where V is any vector field on M with V ( p ) = Vp . By writing out the definition of the Lie derivative explicitly one obtains the following alternative expression for Lp ( ξ )( Vp ). ´ ¡ ¢ ¡ ¢ d ³ L p ( ξ ) Vp = − σexp (−tξ) Vp . dt ∗p

(6.11)

Note that, since ξ # ( p ) = 0, σexp (−tξ) ( p ) = p for every t so ( σexp (−tξ) )∗p : Tp ( M ) → Tp ( M ) and the derivative in (6.11) is computed in the single tangent space Tp ( M ). We claim that Lp ( ξ ) is skew-symmetric with respect to the inner product on Tp ( M ) supplied by h , iG , i.e., that ­

D ¡ ¢ ® ¡ ¢E L p ( ξ ) Vp , W p G = − Vp , L p ( ξ ) W p

(6.12)

G

for all Vp , Wp ∈ Tp ( M ). To see this one simply evaluates (6.9) at p and uses the fact that ξ # ( p ) = 0 and the definition (6.10) of Lp ( ξ ). Next we will require a lemma which follows from a simple manipulation of well-known identities from differential geometry, but, since we use the result several times, we supply a proof. Lemma 6.2 Let H be an arbitrary smooth function on the symplectic manifold ( M, ω ) and VH its Hamiltonian vector field ( d H = ιV ω ). Suppose p ∈ M H and VH ( p ) = 0. Then, for any Vp , Wp ∈ Tp ( M ), ¡

¢

µ ³

Hp Vp , Wp = − ω

LVH V

where Hp is the Hessian of H at p, LV

H



´ p

, Wp

,

(6.13)

is the Lie derivative with respect to

VH and V is any vector field on M with V ( p ) = Vp . Proof: By definition, Hp ( Vp , Wp ) = Vp ( W ( H ) ) and ³ W ( H ) = LW H = d H ( W ) =

´ ´ ³ (ω) . ιV ω ( W ) = ιW ◦ ιV H

9

H

Now compute V ( W ( H ) ) = LV ( W ( H ) ) = LV ◦ ι W ◦ ι V ( ω ) H

³ =

´ ι[ V,W ] + ιW ◦ LV

◦ ιV ( ω ) H

= ι[ V,W ] ◦ ιV ( ω ) + ιW ◦ LV ◦ ιV ( ω ) H

H

³ = ι[ V,W ] ◦ ιV ( ω ) + ιW ◦ H

´ ι[ V,V

H

+ ιV

]

H

◦ LV

(ω)

= ω ( VH , [ V, W ] ) − ω ( [ VH , V ] , W ) + ( LV ω ) ( VH , W ) . Now, evaluate at p and use V ( p ) = 0 to obtain ³ ´ Vp ( W ( H ) ) = 0 − ω [ VH , V ] p , W p + 0 , i.e.,

¡ ¢ Hp Vp , Wp = − ω

µ ³



´ LVH V

p

, Wp

as required. Proposition 6.3 Let ( M, ω ) be a symplectic manifold with a Hamiltonian action of the compact Lie group G. Let ξ ∈ G be nondegenerate and p ∈ Z ( ξ # ). Then Lp ( ξ ) : Tp ( M ) → Tp ( M ) is nonsingular. Proof: We apply Lemma 6.2 to H = µ ( ξ ). Then VH = Vµ( ξ ) = ξ # . For any Vp ∈ Tp ( M ) we select a vector field V on M with V ( p ) = Vp . Then ( LV V )p = ( Lξ# V )p = Lp ( ξ )( Vp ). Since ξ is nondegenerate, µ ( ξ ) is a H Morse function so its Hessian Hp is nondegenerate. Thus, the equality ¡ ¢ ¡ ¡ ¢ ¢ Hp Vp , Wp = − ω Lp ( ξ ) Vp , Wp implies that Lp ( ξ )( Vp ) cannot be zero unless Vp = 0. Remark: We will improve this result shortly by showing that if G is any compact Lie group acting on any (not necessarily symplectic) manifold M and if ξ ∈ G has the property that ξ # ( defined by (6.7) ) has only isolated zeros, then each Lp ( ξ ) ( defined by (6.10) ) is nonsingular. Now, let us assume that ξ ∈ G is nondegenerate and p ∈ Z ( ξ # ). Then Lp ( ξ ) is nonsingular and skew-symmetric with respect to h , iG . Thus, we can find a basis { e1 , . . . , e2k } for Tp (M ) that is orthonormal with respect to h , iG and oriented (with respect to the Liouville form) and relative to which

10

the matrix of Lp ( ξ ) is of the form           

0 − λ1 0 0 .. .

λ1 0 0 0 .. .

0 0 0 − λ2 .. .

0 0 λ2 0 .. .

··· ··· ··· ···

0 0 0 0 .. .

0 0 0 0 .. .

0 0

0 0

0 0

0 0

··· ···

0 − λk

λk 0

          

(6.14)

with λj ∈ IR −{ 0 } for j = 1, . . . , k. We define a square root of the determinant of Lp ( ξ ) by taking h ¡ ¢ i 12 det Lp ( ξ ) = λ1 λ2 · · · λk . (6.15) Remark: This is, in fact, the Pfaffian of the matrix (6.14). Although this observation will play no role in what we do here it is crucial in formulating more general localization theorems than the one we will prove since these involve the so-called equivariant Euler class of a certain (equivariant) vector bundle and this is constructed, a la Chern-Weil, from the Pfaffian. With this we are prepared to formulate what we will call the Generalized Duistermaat-Heckman Theorem. Theorem 6.4 Let ( M, ω ) be a compact, symplectic manifold of dimension n = 2k with a Hamiltonian action of a compact Lie group G (and corresponding symplectic moments given by µ : G → C ∞ ( M ) ). Orient M with the Liouville form ν ω = k1! ω k (and assume that a G-invariant Riemannian metric h , iG on M has been chosen). If ξ ∈ G is nondegenerate, then Z ei µ ( ξ ) ν ω = M

X

( 2π i )

k

h

¡ ¢ i− 12 i µ ( ξ )( p ) det Lp ( ξ ) e .

(6.16)

p∈M ξ # (p)=0

Remarks: Since ξ is nondegenerate, µ ( ξ ) is Morse and so has (at least two and at most) finitely many critical points. These critical points coincide with the zeros of ξ # so the sum in (6.16) is (nonvacuous and) finite. Furthermore, Lp ( ξ ) is nonsingular so det ( Lp ( ξ ) ) 6= 0 and the right-hand side of (6.16) is meaningful. As we mentioned earlier we shall eventually derive Theorem 6.4 as a consequence of our equivariant localization theorem. Our task for the present is simply to show that Theorem 6.4 implies Theorem 6.1. Thus, we begin with a 11

compact, symplectic manifold ( M, ω ) of dimension n = 2k and oriented by the the Liouville form ν ω . We let H ∈ C ∞ ( M ) be a Morse function and VH the corresponding Hamiltonian vector field on M . The assumption of Theorem 6.1 is that the flow of VH is periodic. By rescaling we may assume that the period is 2π. Now, just as for our example on S 2 , this gives rise to a circle action on M with the property that ¡ ¢ ξ = i a ∈ Lie S 1 =⇒ ξ # = V− aH . (6.17) In somewhat more detail, the action moves p ∈ M along the integral curve of VH that begins at p so ´¯ ¢¯ d ³ −t( −i ) d ¡ ti ¯ # (i( − 1) ) (p) = e · p ¯ = e · p ¯ t=0 = VH ( p ) . dt dt t=0 Moreover, V−aH = −a VH because d ( −aH ) = −adH = −a ιV ω = ι−aV ω H H so #

ξ # ( p ) = ( ia )

(p) = −a( i( − 1) )

#

( p ) = − a VH ( p ) = V−aH ( p ) .

Consequently, the S 1 -action is Hamiltonian with symplectic moments given by µ ( ξ ) = µ ( i a ) = − aH (equivariance is proved in the same way as for the S 2 example). Next we record a simple, but crucial fact about S 1 -actions in general. Lemma 6.5 Let M be a smooth manifold and suppose S 1 acts smoothly on M on the left. Then, for any nonzero ξ in the Lie algebra of S 1 , the zero set Z ( ξ # ) of the vector field ξ # coincides with the fixed point set of the S 1 -action. Proof: Since ξ 6= 0 and S 1 is 1-dimensional, ξ spans the Lie algebra of S 1 , i.e., Lie( S 1 ) = { −tξ : t ∈ IR }. The exponential map of Lie( S 1 ) to S 1 is onto so the orbit of any p ∈ M coincides with { exp( −tξ ) · p : t ∈ IR }, i.e., with the integral curve of ξ # through p. If ξ # ( p ) = 0, then this integral curve is a point and therefore the orbit of p is a point, i.e., p is a fixed point for the S 1 -action. Since a fixed point is obviously a zero of any ξ # , the result follows. Returning to the derivation of Theorem 6.1 from Theorem 6.4 we now have that any nonzero ξ in the Lie algebra of S 1 is nondegenerate. In particular, for any T > 0, we may apply Theorem 6.4 to ξ = i ( −T ) to obtain Z h X ¡ ¢ i− 12 i T H ( p ) k ei T H ν ω = ( 2π i ) det Lp ( − i T ) e M

p∈M ξ # (p)=0

=

h X µ 2π ¶k ¡ ¢ i− 12 i T H ( p ) k (iT ) det Lp ( − i T ) e . T

p∈M

dH(p)=0

Comparing this with the conclusion of Theorem 6.1 we find that we need only show 12

k

¯ ¡ ¢ i− 12 ¡ ¢ ¯¯− 21 ¯ det Lp ( − i T ) = eπi ( Sgn Hp )/ 4 ¯ det Hp ei, ej ¯

h

(i T)

(6.18)

to complete the proof ( here { e1 , . . . , e2k } is a basis for Tp ( M ) that satisfies k

1 k!

( ω ∧ · · · ∧ ω ) ( e1 , . . . , e2k ) = 1 ). While largely computational, the proof of (6.18) relies on one nontrivial result so we shall go through it in some detail. First note that ´ ³ ´ ¡ ¢ ³ Lp ( − i T ) Vp = L( −i T )# V = T L( i( −1 ) )# V ³ =T

p

´ LV H V

p

p

.

Thus, (6.13) can be written ¡ ¢ ¡ ¡ ¢ ¢ T H p Vp , W p = − ω L p ( − i T ) Vp , W p .

(6.19)

In particular, if { e1 , . . . , e2k } is any basis for Tp ( M ), ¡ ¢ ¡ ¢ T Hp ei , ej = − ω Lp ( − i T ) ( ei ) , ej and if we write Lp ( −i T )( ei ) = Lli el , then ¢ ¢ ¡ ¡ T Hp ei , ej = − Lli ω el , ej for all i, j = 1, . . . , 2k. As a matrix product this is 

T Hp ( e1 , e1 ) · · ·  ..  . T Hp ( e2k , e1 ) · · ·   

− L11

.. . − L12k

··· ···

− L2k 1



 T Hp ( e1 , e2k )  ..  = . T Hp ( e2k , e2k )

ω ( e1 , e1 ) · · ·  .. ..  . . ω ( e − L2k 2k , e1 ) · · · 2k



(6.20)

ω ( e1 , e2k )  .. . . ω ( e2k , e2k )

Now we will make a particular choice of basis. The classical Darboux Theorem guarantees the existence of an oriented, orthonormal basis for Tp ( M ) relative to which   0 1 0 0 ··· 0 0  −1 0 0 0 · · · 0 0   ³ ¡ ¢´   .. ..  =  ... ... ... ... ω ei , e j (6.21) . .     0 0 0 0 ··· 0 1  0 0 0 0 · · · −1 0 and

1 k!

k

( ω ∧ · · · ∧ ω ) ( e1 , . . . , e2k ) = 1. We know also that we can find 13

an oriented, orthonormal basis in which the matrix of Lp (−i T) has the form (6.14). It so happens that for circle actions (and, more generally, for torus actions) it is possible to do all of this simultaneously, i.e., to find one oriented, orthonormal basis { e1 , . . . , e2k } for Tp ( M ) in which (6.14), (6.21) and ν ω( e1 , . . . , e2k ) = 1 are all satisfied ( see Section 32 of [23] ). Making such a choice of basis, substituting (6.14) and (6.21) into (6.20) and taking determinants gives ³ ¡ ¢´ T 2k det Hp ei , ej = λ21 · · · λ2k . Thus,

¯ ³ ¡ ¢ ´ ¯¯ 21 ¯ T k ¯ det Hp ei , ej ¯ = Sign ( λ1 · · · λk ) λ1 · · · λk h = Sign ( λ1 · · · λk )

( where Sign ( λ1 · · · λk ) = 1 if λ1 · · · λk > λ1 · · · λk < 0 ) and so h ¡ ¢ i − 12 = Sign ( λ1 · · · λk) T k det Lp ( − iT )

¡ ¢ i 12 det Lp ( − i T )

0 and Sign ( λ1 · · · λk ) = −1 if ¯ ³ ¢ ´ ¯¯ − 21 ¡ ¯ . (6.22) ¯ det Hp ei , ej ¯

Comparing (6.22) and (6.18) we see that all that remains is to prove k

Sign ( λ1 · · · λk ) = ( − i ) e

πi ( Sgn Hp ( ei, ej ) ) / 4

.

(6.23)

This will follow easily by induction if we can show that it is true for k = 1. In this case, (6.20) gives à ! à !à ! T Hp ( e1 , e1 ) T Hp ( e1 , e2 ) 0 − λ1 0 1 = T Hp ( e2 , e1 ) T Hp ( e2 , e2 ) λ1 0 −1 0 and so

Ã

H p ( e1 , e 1 )

H p ( e1 , e 2 )

H p ( e2 , e 1 )

H p ( e2 , e 2 )

!

à =

λ1 / T

0

0

λ1 / T

! .

Now, if λ1 > 0 ( Sign ( λ1 ) = 1 ), then Sgn ( Hp ( ei, ej ) ) = 2 so 1

( −i) e

πi ( Sgn Hp ( ei, ej ) )/ 4

= − i eπi / 2 = 1 = Sign ( λ1 )

and, similarly, if λ1 < 0 ( Sign ( λ1 ) = −1 ), then Sgn ( Hp ( ei, ej ) ) = −2 so 1

( −i) e

πi ( Sgn Hp ( ei, ej ) )/ 4

= − i e− πi / 2 = − 1 = Sign ( λ1 ) .

Leaving the induction to the reader this completes the proof of (6.23) and therefore the derivation of the Duistermaat-Heckman Theorem 6.1 from the Generalized Duistermaat-Heckman Theorem 6.4. We shall find that Theorem 6.4 itself is a simple consequence of a beautiful localization theorem in equivariant cohomology.

14

The basic philosophical principle behind all of the equivariant localization theorems is that, in some sense, “G-equivariant cohomology is determined by the fixed point set of the G-action”. Our first lemma is an initial indication of what this means and why it is true. Roughly, it says that if α ∈ Ω∗G ( M ) is G-equivariantly closed, then, for each ξ ∈ G, α( ξ )[ n ] ( n = dim M ) is cohomologically trivial away from the zero set Z( ξ # ) (which contains the fixed point set of the G-action and, for S 1 -actions and ξ 6= 0, coincides with it by Lemma 6.5). Lemma 6.6 Let M be a smooth n-manifold and G a compact Lie group that acts smoothly on M on the left. Let α ∈ Ω∗G ( M ) be G-equivariantly closed. Then, for each ξ ∈ G, α( ξ )[ n ] is ( de Rham ) exact on M − Z( ξ # ) . Proof:

Fix a nonzero ξ ∈ G ( the result is vacuous if ξ = 0 ).

Remarks: We will actually prove more than is asserted in the lemma. Since the additional strength will be required in the derivation of the DuistermaatHeckman Theorem we will elaborate. Note that, with ξ ∈ G held fixed and for any α ∈ Ω∗G ( M ), ³ ´ ( dG α ) ( ξ ) = d − ιξ# ( α ( ξ ) ) . Define dξ# = d − ιξ# .

(6.24)

Then dξ# acts on Ω∗ ( M ) , and, by (5.9), dξ# ◦ dξ# = − Lξ# . Consequently, on the subspace n o Ω∗ξ# ( M ) = ϕ ∈ Ω∗ ( M ) : Lξ# ϕ = 0

(6.25)

of ξ # -invariant forms dξ # ◦ dξ # = 0 .

(6.26)

Applying analogous formulas for d and ιξ# one obtains the Leibnitz Rule ³ dξ# ( ω ∧ η ) =

i ´ h n dξ# ω ∧η+ ω [ 0 ] − ω [ 1 ] + · · · + ( − 1 ) ω [ n ] ∧dξ# η (6.27)

for any ω, η ∈ Ω∗ ( M ) . The proof of Lemma 6.6 will rely only on the fact that dξ# ( α( ξ ) ) = 0 and the properties of dξ# just described. In particular, the conclusion will also be true of any Ω∗ ( M )-valued map ξ → α( ξ ) on G even if it is not polynomial in ξ, provided only that dξ# ( α( ξ ) ) = 0 . 15

Now we return to the proof of Lemma 6.6. Using the G-invariant Riemannian metric h , iG on M we construct a 1-form θ on M dual to ξ # , i.e., we define ­ ® θ ( V ) = ξ# , V G (6.28) for each vector field V on M . Claim

#

1:

θ is ξ # -invariant, i.e., Lξ# θ = 0.

To see this we fix a p ∈ M and Vp ∈ Tp ( M ) and show that ( Lξ# θ )p ( Vp ) = 0 . By definition, ´¯ d ³ ¯ ∗ Lξ # θ = σexp ( −tξ θ ¯ ) dt t=0 so µ ³ ³ ´ ¡ ¢ ´ ¡ ¢ ¶ ¯¯ d ¯ Lξ # θ Vp = σexp ( −tξ∗) θ Vp ¯ dt p p t=0 ¶¶¯ µ µ ³ ´ ¯ ¡ ¢ d ¯ Vp θ σexp ( −tξ ) = ¯ dt ∗p t=0 ¿ ³ ´ ¡ ¢ À ¯¯ ¡ ¢ d ¯ Vp = ξ # exp ( − tξ ) · p , σexp ( −tξ ) . ¯ dt ∗p G t=0 We leave it to the reader to verify that ³ ´ ξ # ( exp ( − tξ ) · p ) = σexp ( −tξ )

¡ ∗p

ξ# ( p )

¢

which gives ³ Lξ # θ

´ ¡ p

Vp

¢

d = dt

¿³

´ σexp (−tξ )

¡

∗p

´ ¢ ³ ξ ( p ) , σexp ( −tξ )

¡

#

∗p

Vp

¢

À ¯ ¯ ¯ ¯

G t=0

® ¯¯ d ­ # = ξ ( p ) , Vp G ¯ dt t=0 =0 because h , iG is invariant under the G-action. This proves Claim #1 and from it and (6.26) we conclude that ³ ´ dξ# dξ# θ = 0 . (6.29) Now notice that ° °2 ­ ® dξ# θ = dθ − ιξ# θ = dθ − ξ # , ξ # G = − ° ξ # ° G + dθ .

(6.30)

This is a (nonhomogeneous) element of Ω∗ ( M ) whose scalar ( i.e., Ω0 ( M ) ) 16

part is −k ξ # k 2G and this scalar part is nonzero on M − Z ( ξ # ) . Remark: A nonhomogeneous element of Ω∗ ( M ) with nonzero scalar part always has a multiplicative inverse ( relative to ∧ ) obtained from the geometric series. Indeed, if we write such an element as a + α with a ∈ Ω0 ( M ), a 6= 0, and α ∈ Ω∗ ( M ) with α[ 0 ] = 0 and define (a + α)

−1

=

∞ ´ 1 X³ α k − a a k=0

(a finite sum), then it is easy to verify that ( a + α )−1 ∧ ( a + α ) = ( a + α ) ∧ ( a + α )−1 = 1 ∈ Ω0 ( M ) . We conclude that, on M − Z ( ξ # ), dξ# θ = −k ξ # k 2G + dθ is invertible and ³ dξ # θ

´−1

´ ° ° −2 ³ ° ° −2 1 + ° ξ # ° G dθ + · · · . = − ° ξ# ° G

(6.31)

Thus, on M − Z ( ξ # ), we can define an element β of Ω∗ ( M ) by ³ β = θ ∧ Claim

#

2:

dξ # θ

´−1

.

(6.32)

On M − Z ( ξ # ), dξ# β = 1 and Lξ# β = 0.

To prove this we first compute µ ³ ´−1 ¶ dξ# β = dξ# θ ∧ dξ# θ ³ = dξ # θ ∧

dξ # θ

´−1

µ ³ − θ ∧ dξ#

µ ³

dξ# θ

´−1 ¶

´−1 ¶

= 1 − θ ∧ dξ#

dξ# θ

µ

´−1 ¶

and Lξ# β = Lξ#

³ θ ∧

dξ# θ

³ = Lξ# θ ∧

dξ # θ

´−1

µ ³ + θ ∧ Lξ #

µ ³ = 0 + θ ∧ Lξ #

dξ# θ

´−1 ¶

dξ # θ

´−1 ¶

.

Now we show that dξ# ( ( dξ# θ )−1 ) and Lξ# ( ( dξ# θ )−1 ) are both zero. 17

Beginning with

³

´ dξ # θ

³ ∧

dξ # θ

´−1

= 1

we compute dξ# of both sides to obtain ³

´

dξ# dξ# θ

³ ∧

dξ# θ

´−1

h +

µ ³ i ´−1 ¶ ° # °2 ° ° − ξ + dθ ∧ dξ# dξ# θ = 0 G

so, by (6.29) and (6.30), µ ³ ³ ´ ´−1 ¶ dξ# θ ∧ dξ# = 0. dξ # θ Now multiply on both sides by ( dξ# θ )−1 . The proof for Lξ# ( ( dξ# θ )−1 ) is the same so this proves Claim #2. Finally, we define λ ∈ Ω∗ ( M ) by µ ³ ´−1 ¶ λ = β ∧ α ( ξ ) = θ ∧ dξ# θ ∧ α(ξ ) and compute d ξ # λ = dξ # ( β ∧ α ( ξ ) ) ³ ´ h i = dξ# β ∧ α ( ξ ) + β [ 0 ] − β [ 1 ] + · · · ∧ dξ# ( α ( ξ ) ) = 1 ∧ α(ξ ) + 0 = α(ξ ) . Thus, dλ − ιξ# λ = α ( ξ ) . Now look at the top ( nth ) degree parts. ιξ# λ has none and ( dλ )[ n ] = d ( λ[ n−1 ] ) so

³ ´ α ( ξ )[ n ] = d λ[ n−1 ]

and this completes the proof of Lemma 6.6. For future reference we summarize what we have just proved. ­ ® dξ# ( α ( ξ ) ) = 0 and θ = ξ # , · G =⇒ µ µ ¶ ³ ´−1 ¶ α ( ξ ) = dξ # θ ∧ dξ# θ ∧ α(ξ ) and

à µ µ

α ( ξ )[ n ] = d ¡

on M − Z ξ

¢ #

³ θ ∧

dξ# θ

. 18

´−1 ¶

¶ ∧ α(ξ ) [ n−1 ]

!

In order to proceed further we must understand more about the structure of the set Z ( ξ # ) = { p ∈ M : ξ # ( p ) = 0 } of zeros of ξ # . Notice that it is clear from the definition (6.7) of ξ # that any fixed point of the G-action on M is a zero of every ξ # so every Z ( ξ # ) contains the fixed point set MG = { p ∈ M : g · p = p ∀ g ∈ G } . If ξ ∈ G has the property that Z ( ξ # ) = M G (i.e., ξ # vanishes only at fixed points of the G-action), then ξ is said to be nondegenerate (for G = S 1 actions, every ξ in the Lie algebra of S 1 is nondegenerate by Lemma 6.5). Notice that, in general, one can define, for any ξ ∈ G, the subgroup Tξ = closureG { exp ( − tξ ) : t ∈ IR } of G. Then Z ( ξ # ) clearly coincides with the fixed point set of the action on M of Tξ and, being compact, connected, and abelian, Tξ is a torus. Thus, the zero set of ξ # is always the fixed point set of a torus action on M . As for Hamiltonian actions, we define, for each p ∈ Z ( ξ # ) a linear transformation Lp ( ξ ) : Tp ( M ) → Tp ( M ) by (6.10) and note that it is skewsymmetric with respect to the G-invariant Riemannian metric h , iG on M . Now we let expp be the (metric) exponential map on Tp ( M ) corresponding to h , iG . This carries a Vp ∈ Tp ( M ) onto γVp ( 1 ) , where γVp is the geodesic of h , iG with γV0p ( 0 ) = Vp and it is a local diffeomorphism of some neigh-

borhood of 0 in Tp ( M ) onto some neighborhood of p in M . The G-invariance of h , iG implies that, on some neighborhood of 0 in Tp ( M ) , ³ ¡ ¢´ ¡ ¢ expp Vp + t Lp ( ξ ) Vp = exp ( − tξ ) · expp Vp . (6.33) Thus, if Lp ( ξ ) is the vector field on Tp ( M ) corresponding to Lp ( ξ ) , i.e., ¡ ¢i d h , Vp + t L p ( ξ ) Vp Lp ( ξ ) = dt t=0 then

ξ# =

¡

expp

¢ ¡ ∗

Lp ( ξ )

¢

on some neighborhood of p in M . In particular, integral curves of Lp ( ξ ) are (locally) mapped by expp to integral curves of ξ # Now, suppose Vp ∈ ker( Lp ( ξ ) ). Then Lp ( ξ ) ( Vp ) = 0 and this is the case if and only if the integral curve of Lp ( ξ ) through Vp is a point, i.e., the integral curve of ξ # through expp ( Vp ) is a point. Since this is the case if and only if ξ # ( expp ( Vp ) ) = 0 we conclude that, on some neighborhood of 0 in Tp ( M ) ¡ ¢ ¡ ¢ ¡ ¢ Vp ∈ ker Lp ( ξ ) ⇐⇒ expp Vp ∈ Z ξ # . Now ker( Lp ( ξ ) ) is a linear subspace (and therefore a submanifold) of Tp ( M ) 19

so the restriction of expp to some open set in ker( Lp ( ξ ) ) maps diffeomorphically onto a neighborhood of p in Z( ξ # ). Thus, Z( ξ # ) has a local manifold structure near each of its points p whose dimension is dim ( ker( Lp ( ξ ) ) ). This dimension need not be the same at each p ∈ Z( ξ # ), but is constant on the connected components of Z( ξ # ). Thus, we find that Z( ξ # ) is a disjoint union of submanifolds of M each of which has dimension dim ( ker( Lp ( ξ ) ) ), where p is any point in the submanifold. In particular, we have the promised generalization of Proposition 6.3. Proposition 6.7 Let M be a smooth manifold, G a compact Lie group acting smoothly on M on the left, ξ an element of the Lie algebra G of G and p ∈ Z( ξ # ) a zero of ξ # . Then p is an isolated point of Z( ξ # ) if and only if Lp ( ξ ) : Tp ( M ) → Tp ( M ) is invertible. Henceforth we assume that p is an isolated zero of ξ # . Then Lp ( ξ ) is invertible and skew-symmetric with respect to h , iG . It follows that the dimension of M must be even, say, n = 2k and that there exists an oriented, orthonormal basis { e1 , . . . , e2k } for Tp ( M ) relative to which the matrix of Lp ( ξ ) has the form (6.14) with λj ∈ IR − { 0 } for j = 1, . . . , k. As before we define a square root of the determinant of Lp ( ξ ) by (6.15). Now, if Vp ∈ Tp ( M ) and we write Vp = Vpi ei (summation convention), then ¢ ¢ ¡ ¡ ¢ ¡ Lp ( ξ ) Vp = λ1 Vp2 e1 − Vp1 e2 + · · · + λk Vp2k e2k−1 − Vp2k−1 e2k . If, as before, we identify Lp ( ξ ) with a vector field Lp ( ξ ) on Tp ( M ) and recall that, on some neighborhood of p in M , ξ # agrees with ( expp )∗ ( Lp ( ξ ) ), then, in normal coordinates x1 , . . . , x2k on that neighborhood determined by expp and { e1 , . . . , e2k }, we have µ ¶ µ ¶ ∂ ∂ ∂ 1 ∂ 2k 2k−1 ξ # = λ1 x 2 − x + · · · + λ x − x . (6.34) k ∂ x1 ∂ x2 ∂ x2k−1 ∂ x2k Note that if p happens to be a fixed point of the G-action (e.g., if ξ is nondegenerate), then this neighborhood can be chosen G-invariant (restrict to some ²-ball relative to h , iG ). With this we are finally prepared to prove our major result. Theorem 6.8 (Equivariant Localization Theorem) Let M be a compact,

20

oriented manifold of dimension n = 2k and G a compact Lie group acting smoothly on M on the left. Let α be a G-equivariantly closed differential form on M . Then, for any nondegenerate ξ ∈ G for which ξ # has only isolated zeros, Z h i− 12 X k (6.35) α ( ξ )[ 0 ] ( p ) . α(ξ ) = ( − 2π ) det Lp ( ξ ) M

p∈M ξ # (p)=0

Remarks: 1. Since M is compact and Z ( ξ # ) is discrete, the sum in (6.35) is finite. If Z ( ξ # ) happens to be empty, then Lemma 6.6 implies that α ( ξ )[ n ] R is exact on all of M so Stokes’ Theorem gives M α ( ξ ) = 0 and (6.35) is vacuously satisfied. 2. For S 1 -actions Lemma 6.5 implies that the nondegeneracy assumption in Theorem 6.8 is unnecessary. 3. As was the case for Lemma 6.6 our proof of Theorem 6.8 will not use the full strength of the assumption that α is a G-equivariantly closed differential form on M , but only that dξ# ( α ( ξ ) ) = 0 for the particular ξ ∈ G referred to in the Theorem. Proof: By the first remark above we may assume Z ( ξ # ) 6= ∅. Let p ∈ Z ( ξ # ). We have shown that we can find a G-invariant neighborhood Up of p and (normal) coordinates x1 , . . . , x2k on Up such that ξ # | Up is given by 1 (6.34), where [ det( Lp ( ξ ) ) ] 2 = λ1 · · · λ2k 6= 0. On Up we define a 1-form θ p by ¡ 2 1 ¢ ¡ 2k 2k−1 ¢ θ p = λ−1 x dx − x1 dx2 + · · · + λ−1 x dx − x2k−1 dx2k . (6.36) 1 k Then a few simple computations show ¡ ¢ ¡ ¢2 ¡ ¢2 θ p ξ # = x1 + · · · + x2k .

(6.37)

³ ´ d ιξ# θ p = 2x1 dx1 + · · · + 2x2k dx2k

(6.38)

ιξ# ( dθ p ) = −2x1 dx1 − · · · − 2x2k dx2k

(6.39)

and, from the last two of these, ³ ´ Lξ# θ p = d ◦ ιξ# + ιξ# ◦ d ( θ p ) = 0 .

(6.40)

Now, each of the sets Up , p ∈ Z ( ξ # ), is G-invariant by construction and M − Z ( ξ # ) is G-invariant because ξ is assumed nondegenerate (so Z ( ξ # ) = 21

M G which is surely G-invariant). Thus © ª © ¡ ¢ª Up p∈Z( ξ# ) ∪ M − Z ξ # is a G-invariant open cover of M . By choosing a partition of unity subordinate to this cover and averaging each of its elements over G (as we did to produce h , iG and the map I in Section 5) one can produce a G-invariant partition of unity subordinate to the cover. With this and the 1-forms θ p on Up and (as in the proof of Lemma 6.6) θ 0 = h ξ # , · iG on M − Z ( ξ # ), one can piece together a 1-form θ on all of M with the following properties: 1. θ agrees with θ p on some neighborhood of p. 2. Lξ# θ = 0 . 3. dξ# θ is invertible on M − Z ( ξ # ). Exactly as in the proof of Lemma 6.6, properties (2) and (3) together with dξ# ( α ( ξ ) ) = 0 imply that µ µ α ( ξ ) = dξ #

³ θ∧

dξ # θ

´−1 ¶

¶ ∧ α(ξ )

¡ ¢ on M − Z ξ # .

(6.41)

Now we compute the integral on the left-hand side of (6.35). For each p ∈ Z ( ξ # ) and ² > 0 sufficiently small we let n o ¡ ¢ ¡ ¢2 ¡ ¢2 2 B² ( p ) = x = x1 , . . . , x2k : k x kG = x1 + · · · + x2k ≤ ² ⊆ Up and

n S² ( p ) =

o

2

x : k x kG = ²

and give both their usual orientations. Since Z ( ξ # ) is a finite set, Z Z Z α(ξ ) = α ( ξ ) = lim α(ξ ) M

M −Z( ξ # )

²→0

M −∪

µ µ

Z = lim

²→0

M −∪

p∈Z( ξ# )

B² ( p )

dξ#

d

= lim

M −∪

p∈Z( ξ# )

B² ( p )

B² ( p )

³ θ ∧

µ µ

Z ²→0

p∈Z( ξ# )

dξ # θ

³ θ ∧

dξ# θ

´−1 ¶

´−1 ¶

¶ ∧ α(ξ ) ¶

∧ α(ξ )

³

´ because the ιξ# term can have no top degree part

22

 = lim  − ²→0

µ

Z

X

³ θ ∧

p∈Z( ξ # )

S² ( p )

dξ# θ

´−1 ¶

 ∧ α(ξ ) 

( the minus sign being due to the switch from boundary to standard orientations ) ! Ã µ Z ³ ´−1 ¶ X ∧ α(ξ ) . θ ∧ dξ# θ lim −

=

p∈M ξ

#

²→0

S² ( p )

(p)=0

Comparing this with (6.35) we see that it remains only to prove that, for each p ∈ Z ( ξ # ), Ã ! µ Z ³ ´−1 ¶ lim − θ ∧ dξ# θ ∧ α(ξ ) = ²→0

S² ( p )

k

( − 2π )

£

det Lp ( ξ )

¤− 21

(6.42) α ( ξ )[ 0 ] ( p ) .

Thus, we fix a p ∈ Z ( ξ # ). For each ² > 0 sufficiently small, θ = θ p on S² ( p ). For such an ² > 0 we introduce a change of coordinates on Up √ by each xi by a factor of ² , i.e., we replace xi everywhere with √ rescaling i ² x , i = 1, . . . , 2k: √ xi −→ ² xi , i = 1 , . . . , 2k . (6.43) In the new coordinates, S² ( p ) becomes the unit sphere S1 ( p ). Write α² ( ξ ) for α ( ξ ) written in these new coordinates, i.e., ¡√ ¢ √ α² ( ξ ) ( x , dx ) = α ( ξ ) ² x , ² dx . Notice that, as ² → 0, all of the α² ( ξ )[ i ] with i > 0 approach 0, whereas α² ( ξ )[ 0 ] → α ( ξ )[ 0 ] ( p ) since p = ( 0, . . . , 0 ). Now we consider the effect of this substitution on θ ∧ ( dξ# θ )−1 . Near p, θ = θ p is given by (6.36) so (6.43) introduces an extra factor of ² . On the other hand, ¢ ¡ 2 −1 1 2 2k−1 dξ# θ = dθ − ιξ# θ = − 2 λ−1 ∧ dx2k − k x k G 1 dx ∧ dx + · · · + λk dx so this also picks up a factor of ² . Consequently, ( dξ# θ )−1 acquires a new factor of 1² and, as a result, θ ∧( dξ# θ )−1 is unaffected by the rescaling (6.43). Thus, µ µ Z Z ³ ´−1 ¶ ³ ´−1 ¶ θ ∧ dξ # θ ∧ α(ξ ) = θ ∧ dξ# θ ∧ α² ( ξ ) S² ( p )

S1 ( p )

23

and so

à lim

²→0

µ

Z −

θ ∧

dξ# θ

µ

³

S² ( p )

Ã

´−1 ¶

³

Z

θ ∧



dξ# θ

S1 ( p )

We therefore compute Z Z ³ ´−1 − θ ∧ dξ # θ =− S1 ( p )

! ∧ α(ξ )

´−1 ¶

= (6.44)

! α ( ξ )[ 0 ] ( p ) .

θ ∧ ( dθ − 1 )

−1

S1 ( p )

³

´

2

k x k G = 1 on S1 ( p ) Z =

θ ∧ ( 1 − dθ )

−1

S1 ( p )

Z

³

=

2

θ∧

k−1

1 + dθ + ( dθ ) + · · · + ( dθ )

S1 ( p )

³

k

´

+ ( dθ )

´

2

where ( dθ ) = dθ ∧ dθ , etc. Z =

θ ∧ ( dθ )

k−1

S1 ( p )

Z

( since dim S1 ( p ) = 2k − 1 ) k

=

( dθ ) B1 ( p )

³

³ ´ k−1 by Stokes’ Theorem since d θ ∧ ( dθ ) = ³ ´ k−1 k−1 dθ ∧ ( dθ ) − θ ∧ d ( dθ ) = ´ k ( dθ ) − 0

Z

¡

= B1 ( p )

¡ ¢ ¢k −1 1 2 2k−1 ( − 2 ) λ−1 ∧ dx2k 1 dx ∧ dx + · · · + λk dx Z

k

−1 = ( − 2 ) k! λ−1 1 · · · λk

dx1 ∧ · · · ∧ dx2k B1 ( p )

µ k

−1

= ( − 2 ) k! ( λ1 · · · λk ) k

= ( − 2π )

h det Lp ( ξ ) 24

i− 21

πk k! .



Substituting this into (6.44) yields (6.42) and so completes the proof of Theorem 6.8. Finally we will derive the Generalized Duistermaat-Heckman Theorem 6.4 from our Localization Theorem 6.8. Recall that the scenario is as follows. We have a compact, symplectic manifold ( M, ω ) of dimension 2k and oriented by the Liouville form ν ω = k1! ω k . There is a Hamiltonian action of a compact Lie group G on M with corresponding equivariant moments given by µ : G → C ∞ ( M ). Finally, we have a ξ ∈ G which is nondegenerate. Notice that, because the action is Hamiltonian, nondegeneracy of ξ implies that ξ # has isolated zeros ( µ ( ξ ) is a Morse function and the zeros of ξ # coincide with the critical points of µ ( ξ ) ). Our objective is to prove (6.16). We consider a map G → Ω∗ ( M ) called the G - equivariant symplectic form ω G defined by ωG = µ + ω , i.e., ωG ( ξ ) = µ ( ξ ) + ω for every ξ ∈ G. We claim that dξ# ( ω G ( ξ ) ) = 0

(6.45)

for every ξ ∈ G. Indeed, ³ dξ# ( ω G ( ξ ) ) =

´ d − ιξ #

( µ(ξ ) + ω )

= d ( µ ( ξ ) + ω ) − ιξ # ( µ ( ξ ) + ω ) = dµ ( ξ ) + 0 − 0 − ιξ# ω =0

( by ( 6.8 ) ) .

Now consider the element ei ωG ( ξ ) ∈ Ω∗ ( M ): ei ωG ( ξ ) = 1 + i ω G ( ξ ) −

1 ω ( ξ ) ∧ ωG ( ξ ) + · · · 2 G

(a finite sum). Since (6.45) and the Leibnitz Rule (6.27) imply that dξ# ( ω G ( ξ ) ∧ · · · ∧ ω G ( ξ ) ) = 0 , we conclude that

³ dξ#

ei ωG ( ξ )

´ = 0.

(6.46)

Remark (3) following Theorem 6.8 implies that (6.46) is sufficient to apply the

25

Localization Theorem to ei ωG ( ξ ) . Since ei ωG ( ξ ) = ei (µ ( ξ )+ω ) = ei µ ( ξ ) ei ω µ ¶ 1 = ei µ ( ξ ) 1 + i ω − ω 2 + · · · 2 we have

³

ei ω G ( ξ )

´ [0]

= ei µ ( ξ ) .

Thus, (6.35) gives X

k

( − 2π )

h det Lp ( ξ )

i− 12

Z iµ(ξ)(p)

e

ei ω G ( ξ )

= M

p∈M ξ # (p)=0

Z ei µ ( ξ ) ei ω

= M

µ

Z ei µ ( ξ )

= M

1 k k i ω k!



Z

= ik M

ei µ ( ξ ) ν ω

which is (6.16).

7

Duality and Seiberg-Witten

In Section 5 we briefly described an argument which led Witten [52] to identify the partition function Z 2 ZDW = e− SDW [ Φ ] / e D Φ (7.1) of Donaldson-Witten theory with the 0-dimensional Donaldson invariant. Even more briefly, the idea is this: The symmetries built into the action SDW [ Φ ] ensure that ZDW is independent of the coupling constant e so that it can be computed in the limit e → 0. For these small values of the coupling constant one has available the semiclassical (stationary phase) approximation which, again by virtue of the symmetries, one can show (formally) must be exact. In finite dimensions at least we saw in Section 6 that exactness of the stationary phase approximation is tantamount to the localization of the integral. Since SDW [ Φ ] is gauge invariant, the integration in ZDW can be understood over the space of field configurations modulo gauge transformations and, thought of in this way, the integral in (7.1) is found to localize to the moduli space of instantons, thus yielding Donaldson’s definition of the 0-dimensional invariant. 26

Remarkable as it is that the subtle differential-topological invariants of Donaldson can be recast in these quantum field-theoretic terms, Witten’s achievement would perhaps amount to no more than a fascinating novelty were it not for the fact that quantum field theory immediately suggests a new line of investigation. The partition function ZDW is independent of e and all that we said above was obtained by analyzing it perturbatively in the so-called weak coupling limit e → 0. Perhaps an entirely new perspective on ZDW (i.e., on the Donaldson invariant) could be gleaned from an analysis in the strong coupling regime e → ∞. Until relatively recently, however, such an analysis was out of the question because the strongly coupled, nonperturbative behavior of such quantum field theories was not well understood. In 1994, Seiberg and Witten [45] made an astonishing breakthrough in understanding this behavior for a certain class of quantum field theories (“N = 2 supersymmetric Yang-Mills theories”). Two key ingredients led to the unraveling of this behavior. The first was that an important part of the action depended holomorphically on the coupling constants. This puts severe restrictions on how the quantum theories can change as the coupling constants are varied. We will, however, have nothing further to say about this part of the story and will turn instead to that aspect of the Seiberg-Witten discovery that is more directly relevant to Witten’s conjecture regarding the Donaldson invariants. Since the earliest days of quantum mechanics it has been understood that a given classical system can admit more that one “quantization” (if for no other reason than the ordering ambiguity one encounters in replacing classical observables, which commute, with operators, which do not). More surprizing, and a more recent discovery, is the fact that a given quantum system can result from the quantization of two quite distinct classical systems. Perhaps the best known example of this phenomenon is the quantization of the Sine-Gordon and Thirring Models in two spacetime dimensions (see, for example, [21] ). When such a situation occurs the two classical models are said to be dual and one is presented with the possibility of adopting two, possibly quite different views of the quantum theory. For example, for a theory (such as SU ( 2 ) Yang-MillsHiggs) in which magnetic monopoles appear as excitations of the fundamental fields (i.e., soliton-like solutions to the field equations), a “dual” description of the theory might have the monopoles themselves as the fundamental fields with the previously fundamental fields arising as excitations of these. The hallmark of such duality transformations is a symmetry that interchanges “electric” and “magnetic” aspects of the theory and therefore also (as we shall see) strong and weak coupling. When such a duality exists one can translate a question that is inherently nonperturbative (and therefore intractible) into a “dual” question in the weak coupling regime where there is some hope of finding an answer. Seiberg and Witten [45] applied this strategy to the physical problem of confinement and Witten [54] used it to sort out the dual version of Donaldson theory. Duality symmetries of this sort are exceedingly subtle and deep and we would not presume to offer an exegesis (see [ 1 ] for a rather detailed outline and

27

Volume 2 of [11] for more details). However, we will pause to show that this notion of duality actually has its roots in classical electromagnetic theory and how a natural symmetry might interchange strong and weak coupling. As in Section 3 we will let X denote some open submanifold of Minkowski spacetime IR 1,3 with its usual orientation and semi-Riemannian metric and standard coordinates x0, x1, x2, and x3. We identify an electromagnetic field on X with a U ( 1 )-gauge field strength on X satisfying Maxwell’s equations. π In more detail, we consider a principal U ( 1 )-bundle U ( 1 ) ,→ P −→ X over X and a connection ω on it with curvature Ω = dω ( U ( 1 ) is abelian). Identifying the Lie algebra u( 1 ) of U ( 1 ) with iIR and letting s : V → P denote a section of the bundle we write the corresponding gauge potential A and field strength F as A = s∗ ω = − i A and

F = s∗ Ω = d A = − i d A = − i F ,

where A and F are the usual real-valued forms describing the potential and field in physics. Writing these out in standard coordinates gives A = Aα dxα = −i Aα dxα and F =

1 1 F dxα ∧ dxβ = − i Fαβ dxα ∧ dxβ . 2 αβ 2

Assuming (for the moment) that X contains none of the sources of the electromagnetic field, F satisfies the (source free) Maxwell equations dF = 0

(7.2)

d ∗F = 0 ,

(7.3)

where ∗ is the Hodge star on X determined by the Minkowski metric and the usual orientation. The first appearance of electromagnetic duality in classical physics is the obvious invariance of (7.2) and (7.3) under the symmetry F → ∗ F . To make contact with the notation commonly used in physics we define functions E 1 , E 2 , E 3 and B 1 , B 2 , B 3 (thought of as components of the ordinary spatial ~ and magnetic B ~ field vectors) by electric E Fio = E i

and Fij = εijk B k ,

i, j, k = 1, 2, 3 .

(7.4)

Thus, ¢ ¡ F = E 1 dx1 + E 2 dx2 + E 3 dx3 ∧ dx0 + B 3 dx1 ∧ dx2 − B 2 dx1 ∧ dx3 + B 1 dx2 ∧ dx3

28

(7.5)

and a computation of the Hodge dual gives ¡ ¢ ∗ F = − B 1 dx1 − B 2 dx2 − B 3 dx3 ∧ dx0 + E 3 dx1 ∧ dx2 − E 2 dx1 ∧ dx3 + E 1 dx2 ∧ dx3 .

(7.6)

Computing d F and d ∗F one finds that Maxwell’s equations (7.2) and (7.3) assume their more familiar forms ~ · B ~ = 0 5

and

~ ~ × E ~ + ∂ B = ~0 5 ∂ x0

(dF = 0)

(7.7)

~ · E ~ = 0 5

and

~ ~ × B ~ − ∂ E = ~0 5 ∂ x0

( d ∗F = 0 ) ,

(7.8)

~ =( where 5

∂ ∂ x1

,

∂ ∂ x2

,

∂ ∂ x3

) is the usual spatial gradient operator and the

· and × refer to the dot and cross products on IR 3 . Notice also that the ~ →B ~ symmetry F → ∗ F of (7.2) and (7.3) becomes, from (7.5) and (7.6), E ~ → −E ~ so that, up to a sign, it interchanges electric and magnetic fields. and B ~ and B ~ into E ~ + iB, ~ F → ∗ F can be written Consequently, if we combine E ³ ´ ³ ´ ~ + iB ~ −→ B ~ − iE ~ = −i E ~ + iB ~ = e− π2 i E ~ + iB ~ . E (7.9) Seen in this light one finds that the equations (7.7) and (7.8) actually admit a U ( 1 ) symmetry ³ ´ ³ ´ ³ ´ ~ +iB ~ −→ eφ i E ~ + iB ~ = cos φE ~ − sin φB ~ +i sin φE ~ + cos φB ~ E (7.10) ~ 0 = cos φE ~ − sin φB ~ and where eφ i ∈ U ( 1 ) is arbitrary (just substitute E 0 ~ = sin φE+cos ~ ~ into (7.7) and (7.8) ). Finally note that, with this notation, B φB the four equations in (7.7) and (7.8) can be written as ³ ´ ~ · E ~ + iB ~ = 0 5 (7.11) and

´ ³ ´ ∂ ³~ ~ × E ~ + i5 ~ + iB ~ = ~0 . E + iB 0 ∂x

(7.12)

For regions X in which sources for the electromagnetic field are present ~ × ~ ·E ~ = 0 and ∂ E~0 − 5 equations (7.7) and (7.8) are modified by replacing 5 ∂x ~ ·E ~ = ~0 by 5 ~ = q and B

~ ∂E ∂ x0

~ ×B ~ = ~je , respectively, where q is the − 5

electric charge density and ~je is the electric current density. Thus, (7.11) and

29

(7.12) become ~ · 5

³

~ + iB ~ E

´ = q

´ ³ ´ ∂ ³~ ~ × E ~ + i5 ~ + iB ~ = ~je . E + i B ∂ x0

(7.13) (7.14)

These equations, of course, no longer admit even the special case (7.9) of the U ( 1 ) symmetry (7.10). We will see that the full U ( 1 ) symmetry can be restored if one is willing to hypothesize the presence of analogous magnetic charge g and current ~jm densities. Since such magnetic densities would presumably arise, as in the electric case, from a fundamental magnetically charged particle and since such a particle has never been observed in nature, we should pause to briefly discuss such “magnetic monopoles.” Dirac [12] was the first to take seriously the implications, especially for quantum mechanics, of the possible existence of magnetic charges, which are ~ ·B ~ = 0 ) by the traditional form (7.13) and (7.14) of explicitly forbidden ( 5 Maxwell’s equations. Such a magnetic charge (or monopole) is a point object assumed to create the “magnetic analogue” of a Coulomb field and we wish to consider one of “magnetic charge” g at rest at the origin. Since our situation is entirely static we may restrict the analysis to a single spatial cross-section of Minkowski spacetime (say, x0 = 0 ) which we will denote simply IR 3 and in which we will write standard rectangular and spherical coordinates ( x, y, z ) and ( ρ, φ, θ ) respectively (for the record, our φ is measured from the positive z-axis and takes values in 0 ≤ φ ≤ π ). The field we wish to consider then is ~ = ~0 and defined on IR 3 − { ( 0, 0, 0 ) } and is given there by E ¡ ¢ g ~ = g e = B x ex + y ey + z ez , (7.15) ρ 2 2 2 2 3/2 ρ (x +y +z ) where eρ is the outward unit radial field and ex , ey and ez are unit vectors in the positive x, y and z directions, respectively. One can write the field strength 2-form F in either rectangular ¡ ¢−3/2 F = g x2 + y 2 + z 2 ( x dy ∧ dz − y dx ∧ dz + z dx ∧ dy ) (7.16) or spherical F = g sin φ dφ ∧ dθ

(7.17)

coordinates. Notice that (7.17) is independent of ρ and so may be regarded as a 2-form on the unit sphere S 2 in IR 3 (we will return to this point shortly). It is easy to see that F is not exact on IR 3 − { ( 0, 0, 0 ) }, i.e., that there does not exist a smooth 1-form A on IR 3 −{ ( 0, 0, 0 ) } for which dA = F there ( (7.17) R is easily integrated over S 2 ⊆ IR 3 to give S 2 F = 4πg, but the existence of such an A would, by Stokes’ Theorem, give a value of zero for the integral). Dirac was well aware of this, of course, and also knew (although perhaps not in these terms) that if one deletes from IR 3 not only the location of the monopole, but also some ray extending from it to infinity (a so-called Dirac string), then 30

the result is a submanifold of IR 3 whose 2nd de Rham cohomology is trivial so that every 2-form on it (e.g., F ) is exact. Taking the Dirac string to be the nonnegative z-axis one can explicitly write down a 1-form AS on IR 3 − { ( 0, 0, z ) : z ≥ 0 } with dAS = F there. One such is g AS = ( y dx − x dy ) = −g ( 1 + cos φ ) dθ . (7.18) ρ(ρ−z) Similarly, on IR 3 − { ( 0, 0, z ) : z ≤ 0 }, g AN = ( y dx − x dy ) = −g ( 1 − cos φ ) dθ ρ(ρ+z)

(7.19)

satisfies dAN = F . Together the domains of AS and AN cover all of U . Thus, although F has no globally defined potential on IR 3 − { ( 0, 0, 0 ) }, it does have two locally defined potentials whose domains exhaust the domain of F . Based on just what we have seen to this point, Dirac made a remarkable discovery. Roughly, his argument can be phrased in the following way. On the intersection of their domains the potentials AN and AS are related by AN = AS + d ( 2 g θ ) .

(7.20)

Now consider an electric charge q in the field of the monopole. Each potential AN and AS gives rise (via the Schr¨odinger equation) to an expression for the wavefunction of q. Denote these ψN and ψS , respectively. It is a basic property of the Schr¨odinger equation that, because AN and AS differ by d( 2gθ ), ψN and ψS are related by ψN = e 2qgθ i ψS . (7.21) Now, the circle ρ = 1, φ = π2 , is in the intersection of the domains of AN and AS so both ψN and ψS are defined (and single-valued) at points of this circle. Thus, for such points, θ → θ + 2π must leave both ψN and ψS unchanged. But (7.21) then implies that e 2qg( θ+2π )i = e 2qgθ i . This, in turn, implies that e 4πqg i = 1 so 4πqg = 2nπ for some n ∈ ZZ , i.e., 1 n (7.22) 2 for some integer n. This is known as the Dirac Quantization Condition and it is interpreted to mean that if a magnetic monopole of charge g exists (or did at one time exist) somewhere in the universe, then all electric charges 1 q are quantized in units of 2g . Since no other plausible “explanation” for the quantization of charge has ever been put forward, this is often regarded as persuasive evidence that monopoles do (or did) exist. Of course, if we have a number of purely electric charges qi and a number of purely magnetic charges gj , then any pair of them will satisfy qg =

qi gj =

31

1 n 2 ij

(7.23)

for some nij ∈ ZZ . If we are now prepared to accept the existence of magnetically charged particles, then one can define magnetic charge density g and magnetic current density ~jm just as in the electric case and introduce them into (7.13) and (7.14) to obtain ³ ´ ~ · E ~ + iB ~ = q + ig 5 (7.24) ´ ³ ´ ∂ ³~ ~ × E ~ + i5 ~ + iB ~ = ~j + i ~j . E + iB e m 0 ∂x

(7.25)

For these equations the full U ( 1 ) symmetry (7.10) is restored, provided we also rotate the electric and magnetic charges q + ig −→ e φi ( q + ig ) = ( q cos φ − g sin φ ) + i ( q sin φ + g cos φ ) , (7.26) (just substitute (7.10) and q 0 = q cos φ − g sin φ, g 0 = q sin φ + g cos φ into (7.24) and (7.25) ). ~ →B ~ and Now notice that, in the special case φ = − π2 , corresponding to E ~ → −E, ~ the charges q → g and g → −q are interchanged. But these charges B measure the strengths of the interactions, i.e., they play the role of coupling constants and, according to the Dirac Quantization Condition (7.22), they are inversely proportional. In particular, if one is “large”, then the other is “small” so that our symmetry interchanges strong and weak coupling. We conclude our discussion of the Dirac monopole with one last observation. If we measure electric charge in multiples of the charge of the electron we may take q = 1 in (7.22) so that, for any magnetic monopole, 2g = n is an integer. Rewriting the potential (7.19) as AN =

1 n ( 1 − cos φ ) d θ 2

and regarding this now as a form on the 2-sphere ρ = 1 we obtain, except for the Lie algebra factor −i, the gauge potential An for the connection on U ( 1 ) ,→ Pn → S 2 described in the Remark following (1.24). The duality symmetry of Donaldson-Witten theory is, of course, much more subtle and complex than the classical electromagnetic duality we have described and its consequences are much more profound. It has led, in particular, to a dual version of the Donaldson invariants, known as the Seiberg-Witten invariants, and these have precipitatied a revolution in the differential topology of 4-manifolds. Fundamentally, these are defined in a manner quite analogous to the Donaldson invariants from moduli spaces of classical fields, but, whereas in Donaldson theory these are pure SU ( 2 ) Yang-Mills fields, the dual version has a spinor field coupled to a U ( 1 ) connection. The relative simplicity of SeibergWitten over Donaldson theory resides in the shift from nonabelian ( SU ( 2 ) ) to Abelian ( U ( 1 ) ) and the fact that, quite unlike Donaldson theory, the moduli 32

spaces of Seiberg-Witten theory are “usually” finite and always compact. The price one must pay for the eventual simplicity of the theory, however, is an initial expenditure of time and energy to assemble the rather substantial algebraic machinery required to write down the relevant equations. Our final objective is to lay this algebraic foundation and then describe, as we did for Donaldson theory in Section 4, the construction of the 0-dimensional Seiberg-Witten invariant. Much of the algebraic background we require is most conveniently phrased in the language of Clifford algebras. We recall that any finite dimensional, real vector space V with an inner product h , i has a Clifford algebra Cl( V ) which can be described abstractly as the quotient of the tensor algebra J ( V ) by the 2-sided ideal I( V ) generated by elements of the form v ⊗ v + h , i1 with v ∈ V . More concretely, if { e1, . . . , en } is an orthonormal basis for V , then Cl( V ) is the real associative algebra with unit 1 generated by { e1, . . . , en } and subject to the relations ­ ® ei ej + ej ei = −2 ei , ej 1 , i, j = 1, . . . , n . (7.27) We intend to be even more concrete and construct an explicit matrix model for the Clifford algebra Cl( 4 ) = Cl( IR 4 ) of IR 4 with its usual positive definite inner product. The procedure will be to identify IR 4 with a real linear subspace if a matrix algebra, find an orthonormal basis for this copy of IR 4 satisfying the defining conditions (7.27), where the product is matrix multiplication and 1 is the identity matrix, and form the subalgebra it generates. One can, of course, identify IR 4 with the algebra IH of quaternions q = 1 q + q 2 i + q 3 j + q 4 k , but we wish to embed this into the real, associative algebra IH 2×2 of 2 × 2 quaternionic matrices: (à ! ) q11 q12 2×2 IH = : qij ∈ IH , i, j = 1, 2 . q21 q22 Specifically, we identify IR 4 with the real linear subspace of IH 2×2 consisting of all elements of the form à ! 0 q x = , q ∈ IH (7.28) − q¯ 0 (this is, of course, not a subalgebra of IH 2×2 ). Notice that det x = k q k2 so, defining a norm on the set of x given by (7.28) by 2

k x k = det x

(7.29)

and an inner product by polarization ( h x, y i = 14 ( k x + y k2 − k x − y k2 ) we find that the subspace of IH 2×2 consisting of all x of the form (7.28) is isomorphic to IR 4 as an inner product space. One easily checks that { e1, e2, e3, e4 } 33

given by à e1 =

0

1

−1

0

!

à , e2 =

0

!

i

Ã

0

, e3 =

i 0

j

!

à , e4 =

j 0

0 k k

!

0

is an orthonormal basis and, moreover, satisfies ­ ® ei ej + ej ei = −2 ei , ej 11 , i, j = 1, 2, 3, 4 ,

(7.30)

(7.31)

where we use 11 generically for the identity matrix of any size (2 × 2 in this case). Note that it follows from (7.31) that x, y ∈ IR 4 .

x y + y x = −2 h x , y i ,

(7.32)

The real subalgebra of IH 2×2 generated by { e1, e2, e3, e4 } is the real Clifford algebra of IR 4 and is denoted Cl( 4 ). Writing out products of basis vectors and using (7.31) to eliminate linear dependencies gives the following basis for Cl( 4 ) : Ã ! 1 0 e0 = = 11 0 1 Ã e1 =

0

1

−1

0 Ã

e1e2 = Ã e2e3 =

!

à e2 =

i

0

0

−i

k

i

i

0

!

!

Ã

!

à e2e4 =

0 k à e1e2e3 = Ã

0

k

−k

0

0

i

Ã

0

e3 =

e1e3 =

0

e1e3e4 =

0

j

0

0

−j

−j

0

0

−j

à e1e4 =

! e3e4 = 0

−j

j

0

0

−1

−1

0

à e2e3e4 = −1 0

0 k k

k

0

0

−k

Ã

e1e2e4 =

!

e1e2e3e4 =

à e4 =

!

Ã

Ã

!

j 0

!

−i 0

j

i

0

0

i

!

0 !

! (7.33)

!

!

!

0 1

Thus, dim Cl( 4 ) = 16. Since IH 2×2 itself has real dimension 16 we conclude that, in fact, Cl ( 4 ) = IH 2×2 . (7.34) Notice that the basis (7.33) gives Cl( 4 ) a natural ZZ 2 -grading Cl ( 4 ) = Cl0 ( 4 ) ⊕ Cl1 ( 4 ) ,

34

(7.35)

where Cl0 ( 4 ) is spanned by e0, e1e2, e1e3, e1e4, e2e3, e2e4, e3e4 and e1e2e3e4 and Cl1 ( 4 ) is spanned by e1, e2, e3, e4, e1e2e3, e1e2e4, e1e3e4 and e2e3e4 . The elements of Cl0 ( 4 ) are said to be even, while those of Cl1 ( 4 ) are odd. Regarding ZZ 2 as { 0, 1 } with addition modulo 2, ¡ ¢ (7.36) ( Cli ( 4 ) ) Clj ( 4 ) ⊆ Cli+j ( 4 ) for i, j = 0, 1, so Cl( 4 ) is a ZZ 2 -graded algebra, i.e., a superalgebra. From (7.33) it is clear that the decomposition (7.34) corresponds simply to à ! à ! à ! q11 q12 q11 0 0 q12 = + . q21 q22 q21 0 0 q22 Lemma 7.1 The center Z( Cl( 4 ) ) of Cl( 4 ) is Span{ e0 } ∼ = IR . Proof: Since e0 = 11 it commutes with everything in Cl( 4 ), Span{ e0 } ⊆ Z( Cl( 4 ) ) is clear. To complete the proof it will suffice to show that every eI = ei1 · · · ei , k

1 ≤ k ≤ 4,

1 ≤ i1 < · · · < ik ≤ 4

fails to commute with something in Cl( 4 ). For k = 1 this is clear since eiej = −ejei for i 6= j. For k = 4, eI = e1e2e3e4 so e1eI = ( e1e1 )e2e3e4 = −e2e3e4, whereas eI e1 = ( e1e2e3e4 )e1 = ( −1 )3 ( e1e1 )e2e3e4 = e2e3e4. Now suppose 1 < k < 4. Then eI ei1 = ( −1 )k−1 ei1 eI and, if el is not among ei1 , . . . , ei , k

eI el = ( −1 )k el eI . Thus, eI cannot commute with both ei1 and el . Lemma 7.2 If x ∈ IR 4 ⊆ Cl( 4 ) and k x k = 1, then x is a unit in Cl( 4 ) (i.e., is invertible) and x−1 = −x. Proof:

h x, x i = 1 and xx + xx = −2h x, x i11 imply xx = −11.

We denote by Cl× ( 4 ) the multiplicative group of units in Cl( 4 ) and by Pin( 4 ) the subgroup of Cl× ( 4 ) generated by all of the x ∈ IR 4 with k x k = 1 (see Lemma 7.2). Now, an x of the form (7.28) has k x k = 1 if and only if q ∈ Sp( 1 ) (the Lie group of unit quaternions) and the set of all such is closed under inversion ( x−1 = −x ). Thus, Pin( 4 ) is just the set of all products of such elements. The even elements of Pin( 4 ) are just its diagonal elements and they form a subgroup denoted Spin ( 4 ) = Pin ( 4 ) ∩ Cl0 ( 4 ) ! ) ( Ã u1 0 : u1 , u2 ∈ Sp ( 1 ) = 0 u2

(7.37)

∼ = Sp ( 1 ) × Sp ( 1 ) . The topology and differentiable structure Spin( 4 ) inherits from IH 2×2 ∼ = = IH 4 ∼ 35

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