Twisted N = 2 supersymmetry and Donaldson invariants∗ Domenico Monaco June 8, 2012

1 The SUSY action In his article [Wi88], Witten proposed a SUSY Lagrangian (actually the action functional) for a gauge theory on a general 4-manifold M , and derived an expression for the Donaldson invariants [DK90] of M in terms of path integrals. These are polynomial invariants on the homology of M . The starting point for Witten’s work is a 3-dimensional Hamiltonian theory proposed by Atiyah which gave the Floer homology groups of the 3d base manifold as ground states. Due to the connection between Floer theory and Donaldson invariant for 4manifolds with boundary, there were founded expectations that a relativistic generalisation of Atiyah’s model could indeed yield to Donaldson theory. I will not talk about this 3-dimensional model, but present directly its Lorentz-invariant counterpart. In the following, M is a smooth, closed, oriented, 4-dimensional Riemannian manifold. Notice in particular that in this setting there is no natural notion of “time translations”, so we cannot formulate the theory within the Hamiltonian formalism. We let E → M be a complex vector bundle, whose sections are the matter fields of the relativistic theory; we assume for simplicity that the structure group of E is G = SU (N ) (i.e. the bundle is endowed with a metric and an orientation). We are interested in gauge fields: these will be viewed as functions or more generally differential forms on M . They will also have an additive quantum number U , sometimes called the “ghost number” or “R charge”, that in the 3-dimensional picture would correspond to the de Rham degree of the fields viewed as differential forms on the space of all gauge connections for E. The fermion fields are given by: • a one-form ψ with U = +1; ∗

In partial fulfillment of the exam “Introduction to Topological Field Theories”. The lectures were given by prof. Alessandro Tanzini during the period November 2011 – January 2012.

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1 • a self-dual two-form χ (thus χαβ = −χβα = εαβγδ g γρ g δσ χρσ ) with U = −1; 2 • a zero-form η with U = −1. The boson fields are given by: • a one-form A with U = 0 (the gauge connection); • a scalar field φ with U = +2; • a scalar field λ with U = −2. We will denote by F the curvature of A, i.e. the field strength, and by D = d + A the associated covariant derivative. The action is Z h1 1 S0 = d volg Tr Fαβ F αβ + φDα Dα λ − iηDα ψ α + i(Dα ψβ )χαβ − 4 2 M i i i − φ[χαβ , χαβ ] − [ψα , ψ α ] . (1) 8 2 It is invariant under the symmetry δAα = iεψα , 1 δη = ε[φ, λ], 2

δφ = 0, δψα = −εDα φ,

δλ = 2iεη,   1 γδ δχαβ = ε Fαβ + εαβγδ F , 2

(2)

where ε is an anticommuting parameter. Notice that δε δε0 − δε0 δε is zero only up to gauge transformations, and that to prove this one uses the equations of motion for χ. Define the SUSY operator Q (or rather the linear transformation {Q, ·}) by the relation δO = −iε{Q, O}. The above observations show that if O is gauge-invariant and does not contain explicitly 1 the field χ, then {Q, O} is Q-invariant. This forces O to be of the form O = Tr([φ, λ]η), 4 and thus we may add to the action (1) the term Z Z hi i 1 (3) S1 = − d volg {Q, O} = − d volg Tr φ[η, η] + [φ, λ]2 . 2 8 M M We denote by S = S0 + S1 . (The sign of the kinetic (φ, λ) term may be corrected by setting φ = −λ, i.e. considering φ and λ as two complex conjugate fields rather than two independent real fields. However, we prefer the action to be real-valued so as to have the reality of Donaldson invariants for free.) Remark. The above construction may be considered rather ad hoc. However, consider the case of M = R4 with flat metric and the usual N = 2 supersymmetric Yang-Mills gauge theory. This has a symmetry SU (2)L × SU (2)R × SU (2)I × U (1)U

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where SU (2)L × SU (2)R is the (universal cover of the) rotation group of R4 , SU (2)I are internal symmetries and U (1)U is the symmetry corresponding to the charge U . Replacing SU (2)L × SU (2)R with SU (2)L × SU (2)0R , where SU (2)0R is the diagonal sum of SU (2)R and SU (2)I (an “exotic” action of the rotation group), gives exactly the symmetries obeyed by our fields A, φ, λ, ψ, χ, η. Thus the action S is that of a twisted N = 2 SUSY gauge theory, at least on flat R4 . Its advantage is that it will remain supersymmetric on any 4-manifold M . A key observation for what will come next is the following. Recall that the energymomentum tensor of the theory is defined in terms of the variation of the action S under a change of the metric gαβ of our base manifold M , by the relation Z 1 d volg Tαβ δg αβ . (4) δg S = 2 M It thus plays a fundamental rˆole in checking that some quantities do not depend on the metric of M , but only on its differential-topological structure. One finds that Tαβ = {Q, λαβ }

(5)

is a “BRST commutator”. This will turn out to be very useful in future computations. Another valuable remark is the following. Define 1 Feαβ = εαβγδ Fγδ . 2 Then the first Pontryagin class of the vector bundle E can be written as Z Z 1 1 Tr F ∧ F = 2 d volg Fαβ Feαβ . p1 (E) = 2 8π M 4π M It is a topological invariant, hence it is undisturbed by all possibile infinitesimal transformations that we considered earlier. Now set Z 1 0 S =S+ d volg Fαβ Feαβ ; 4 M then one finds that also S 0 is a BRST commutator, S 0 = {Q, V }.

(6)

Thus S 0 is a more convenient choice for the action rather than S.

2 Path integral representation of Donaldson invariants We will now consider some path integrals constructed via the action functional S 0 . We will abbreviate DX = DA Dφ Dλ Dψ Dχ Dη

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for the “measure” on the space of fields. We want to compute expressions of the form Z
Z(O) = DX exp −S 0 /e2 O where e is the gauge coupling constant (a real parameter). This integral also represents the expectation value of the observable O in the vacuum, and is usually also denoted by hOi. Some basic properties of these expressions are: (P1) if O = {Q, O0 } is a BRST commutator, then hOi = 0: this is due to the action S 0 being supersymmetric; (P2) if {Q, O} = 0, then hO{Q, O0 }i = 0 for any O0 . We first focus on the partition function Z
Z = DX exp −S 0 /e2 = h1i. This is a topological invariant (i.e. it does not depend on the metric g on M ) because using (4), (5) and (P1) we get   Z
δg S 0 0 2 = δg Z = DX exp −S /e − 2 e  Z  Z
1 0 2 αβ = − 2 DX exp −S /e Q, d volg λαβ δg = 2e M  Z  1 αβ Q, d volg λαβ δg = 0. =− 2 2e M Another important feature is that Z is independent of the gauge coupling constant e. Indeed, Equations (6) and (P1) yield to   Z
1 0 2 δe Z = DX exp −S /e δe − 2 S 0 = e Z 
1 DX exp −S 0 /e2 {Q, V } = = δe − 2 e   1 = δe − 2 h{Q, V }i = 0. e Thus we can evaluate the integral Z in the UV limit, namely when e2 → 0: in this limit, the path integral is dominated by classical minima. Considering the gauge field terms in S 0 Z   1Z    1 αβ αβ e d volg Tr Fαβ F + Fαβ F = d volg Tr Fαβ + Feαβ F αβ + Feαβ 4 M 8 M we see that these are minimal if and only if Fαβ = −Feαβ

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i.e. when F is anti-self-dual. Therefore, the evaluation of Z is done expanding around (anti)instantons. Denote by M the moduli space of instantons: it depends on the base manifold M and on the bundle E. This space has a “virtual dimension” dim(M) which can be computed starting from some topological numbers of M and E; for example, when G = SU (2) the formula is 3 dim(M) = 8p1 (E) − [χ(M ) + σ(M )] 2 where χ(M ) is the Euler characteristic of M and σ(M ) its signature. We will always assume that the manifold and the bundle are such that this “virtual dimension” equals the actual dimension of the moduli space: this is true provided the generic instanton is an irreducible connection, i.e. is not invariant under any proper subgroup of G. Under these assumptions, we easily see that dim(M) equals the number of ψ zero modes, because ψ ∝ δA is a direction tangent to the moduli space at the generic instanton A. In general, the number of ψ zero modes minus the number of (η, χ) zero modes equals dim(M); but as ψ has U -charge equal to +1 while (η, χ) have U -charge equal to −1, this is also equal to ∆U , the net violation of U . Thus, if we want U to be conserved, we must have ∆U = dim(M) = 0 i.e. the moduli space consists only of isolated, irreducible instantons. Expanding around these instantons the path integral Z, one needs only to keep track of the quadratic terms of the action in the bosons Φ = (A, φ, λ) and the fermions Ψ = (ψ, χ, η). These are of the general form Z 0 S(2) = d volg (Φ∆B Φ + iΨDF Ψ) M

where ∆B is an elliptic second-order operator (acting on bosons) while DF is a real, skew-symmetric operator (acting on fermions). The standard Gaussian integration gives Pfaff(DF ) Z=p . det(∆B ) Due to SUSY, the eigenvalues of DF and ∆B are related: more precisely, if µ 6= 0 is an eigenvalue of DF , iDF Ψ = µΨ, µ 6= 0, (these µ’s come in complex conjugate pairs, because DF is skew-symmetric) then there is a corresponding eigenvalue for ∆B , ∆B Φ = µ2 Φ. Thus

Y Pfaff(DF ) =± Z=p det(∆B ) µ∈spec(∆

B

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µ p . 2 |µ| )\{0}

The sign in front of the product depends on the choice of an orientation. More precisely, we require the Pfaffian line bundle over the space of gauge orbits of connections to be orientable. This is established by Donaldson’s work on the orientability of the moduli space M [DK90]. Thus we can simply choose an instanton A0 and declare that Pfaff(DF ) A0 > 0; then the sign of Pfaff(DF ) on any other instanton A is evaluated counting the number nA of times that Pfaff(DF ) has a zero eigenvalue at At = tA0 + (1 − t)A (this integer will not depend on the choice of the path due to the above-mentioned orientability). Hence we obtain the first Donaldson invariant X Z= (−1)nA A∈M

in the case when dim(M) = 0. In order to evaluate Z(O) when dim(M) > 0, to begin with we need to require that O has a quantum number U equal to dim(M), so as to compensate for the fermion zero modes (otherwise Z(O) vanishes). In order for Z(O) to be a topological invariant, we will need to impose further conditions on O. We calculate   Z
δg S 0 0 2 δg Z(O) = DX exp −S /e − 2 O + δg O = e   Z   Z
1 0 2 αβ = DX exp −S /e d volg λαβ δg O + δg O . − 2 Q, 2e M The first summand in the brackets of the second line of the above expression vanishes whenever {Q, O} = 0, due to (P2). The second vanishes if O does not depend on the metric g (or more generally if δg O = {Q, O0 }, but actually δg O = 0 for the observables we will be interested in). Moreover, we know by (P1) that if O = {Q, O0 } is a BRST commutator then Z(O) = 0. Thus, “meaningful” topological invariants will be produced from gauge-invariant metric-independent observables O such that {Q, O} = 0, modulo those of the form O = {Q, O0 }; that is, from the Q-cohomology. We notice from (2) that the scalar field φ obeys these conditions, apart from gauge invariance: but any invariant polynomial in φ will satisfy our criteria. The number of such polynomials is equal to the rank of G; for G = SU (N ) this is N − 1. In particular, for G = SU (2) we have only the invariant 1 Tr φ2 (p), p ∈ M. (7) 2 Notice in particular that the U -charge of W0 is 4. We now show that W0 does not depend on the choice of the point p ∈ M (rather, it depends on its homology class [p] ∈ H0 (M, Q)). Indeed, one has O = W0 (p) =

∂ 1 ∂ W0 = Tr φ2 = Tr φDα φ = i{Q, Tr φψα } α ∂x 2 ∂xα by (2). This yields to  Z p  Z p ∂ α 0 W0 dx = i Q, W1 W0 (p) − W0 (p ) = α p0 p0 ∂x

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where we have introduced the operator-valued 1-form W1 = Tr(φψ). Hence we have   Z p 0 =0 hW0 (p) − W0 (p )i = Q, i W1 p0

by (P1), as we wanted. We may now use a “descent” argument, namely define inductively 0 = i{Q, W0 }, dW2 = i{Q, W3 },

dW0 = i{Q, W1 }, dW3 = i{Q, W4 },

dW1 = i{Q, W2 }, dW4 = 0

(8)

where Wk is an operator-valued k-form on M , and has U = 4 − k. One finds explicitly, for W0 as in (7), 1 W0 = Tr φ2 , W1 = Tr(φψ), 2   1 W2 = Tr (9) ψ ∧ ψ + iφF , 2 1 W3 =i Tr(ψ ∧ F ), W4 = − Tr(F ∧ F ). 2 The topological invariants are now defined as follows. Let γk be a k-dimensional cycle in M , and set Z I(γk ) = Wk . (10) γk

This is BRST invariant (i.e. a Q-cycle), because Z Z {Q, I(γk )} = {Q, Wk } = −i dWk−1 = 0. γk

γk

Moreover, its Q-cohomology class depends only on the homology class [γk ] ∈ Hk (M, Q), since if γk = ∂βk then   Z Z Z Z I(∂βk ) = Wk = dWk = i {Q, Wk+1 } = Q, i Wk+1 . ∂βk

βk

βk

βk

We can finally recover Donaldson polynomial invariants. Indeed, pick cycles γ1 , . . . , γs in M of dimension k1 , . . . , ks , respectively, such that dim(M) =

s X

(4 − ki ),

i=1

so that Wk1 · · · Wks has U = dim(M). Then Z Z(γ1 , . . . , γs ) =

s
Y DX exp −S 0 /e2 i=1

Z W ki = γi

* s Z Y i=1

+ W ki

(11)

γi

gives a topological invariant which coincides with one of the polynomial invariants introduced by Donaldson.

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3 Donaldson invariants as differential forms on M Recall that under our assumptions on the moduli space of instantons M, there are exactly n = dim(M) fermionic ψ zero modes (and no (η, χ) zero modes), and that these are related by SUSY to bosonic A zero modes (and there are no (φ, λ) zero modes). The ψ zero modes can be identified with tangent directions to M. Consider an observable O having U = dim(M); it may contain zero as well as non-zero modes, but we can always integrate out the non-zero modes to get an effective functional (0) (0) O0 . If we denote by a1 , . . . , an (respectively ψ1 , . . . , ψn ) the bosonic (respectively fermionic) zero modes, then O0 will be of the general form (0)

(0)

O0 = Φi1 ,...,in (a1 , . . . , an )ψi1 · · · ψin . Due to the fact that the ψ’s are anticommuting, we can assume that Φ is a skewsymmetric tensor with n indices, that is, a differential n-form on M. Hence we obtain that Z Z (0) (0) (0) (0) i1 ,...,in Z(O) = da1 · · · dan dψ1 · · · dψn Φ (a1 , . . . , an )ψi1 · · · ψin = Φ. TM

M

Moreover, if O = O1 · · · Os is a product of s observables, then at least at the first order in e2 (which is valid since we are computing topological invariants in the UV limit) Φ = Φ 1 ∧ · · · ∧ Φs i.e. the n-form Φ corresponding to O is the wedge product of the forms Φi corresponding to Oi . Consequently Z Z(O1 · · · Os ) =

Φ1 ∧ · · · ∧ Φs . M

We want to apply the above considerations to the observables (10), for Wk as in (9). We want to integrate out the non-zero modes out of the W ’s. This is done as follows: • whenever F appears, approximate it by the classical instanton; • whenever ψ appears, replace it with the zero-mode wave functions ψ (0) ; • whenever φ appears, replace it by Z   i a (x) = −i d volg Gij (x, y) ψα(0) (y), ψ α,(0) (y) j , M

where Gij (x, y) is the Green function of the Laplacian ∆ = Dα Dα , i.e. the unique solution to ∆Gij (x, y) = δ ij δ 4 (x − y). If we call Φγk the differential forms associated to I(γk ) (here γk is a k-cycle in M ), then we get the “explicit formula” for the Donaldson invariants (11), namely Z Φγ1 ∧ · · · ∧ Φγs . Z(γ1 , . . . , γs ) = M

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4 Seiberg-Witten solution of N = 2 SUSY gauge theory In subsequent works on N = 2 SUSY gauge theories (see e.g. [Se88]), the superspace formalism was adopted: M is thus considered as a supermanifold with coordinates (xα , θα ). This allowed Seiberg and Witten to give an exact solution to the N = 2 SUSY Yang-Mills theory presented in [SW94]. The idea is as follows [Ta04]. For notational simplicity, we will consider the case of gauge group equal to SU (2). We know that in this case the only gauge invariant SUSY observable is W0 = 1/2 Tr φ2 as in (7). Thus, when the parameter u = hTr φ2 i is nonzero, the gauge group is broken to the U (1) commuting with φ. Thus it is important to understand the theory of this U (1) gauge theory. We first turn our attention to the N = 1 SUSY theory. Consider in general a U (1)n gauge theory. The fields can be arranged in U (1) vector fields V i = (Ai , η i ) and neutral chiral multiplets Ai = (φi , ψ i ) (here we regard each φ as a complex scalar, so that in the previous notations φ = −λ), for i = 1, . . . , n. The generic Lagrange density which respects N = 1 SUSY can be written as Z Z Z n τ o j ij 2 2 i 2 2 i 2 i j i d θd θK(A , A ) + d θd θκi V + d θ W W + U (A ) + h.c. 16πi where W i is the coefficient in the superspace θ-expansion of the anti-self-dual part F−i of the gauge field strength, and Θij 4π +i 2 2π e is the complexified coupling constant. Here Θij is the coupling parameter from QCD: the “complexified” super Yang-Mills density reads Z Z Z Z Θij iτij 1 iτij j i j i j i F ∧ ∗F + i 2 F ∧ F = F+ ∧ F+ + F−i ∧ F−j . 2 e 8π 4π 4π τij =

If one wants to impose a second SUSY, then the ψ’s and the η’s should appear symmetrically in the Lagrangian. This implies that • the potential U can be at most linear in the Ai , U = ζi Ai ; • for the kinetic terms of the ψ’s and η’s to be equal, one should have ∂2K 1 (τij − τ ij ) = gij = j. 8πi ∂Ai ∂A The latter can be solved as a partial differential equation for K if    1  D i j i K Ai , A = Ai A − AD A i 8πi where AD i is an holomorphic function of A, and moreover ∂AD ∂AD j i τij = = . j ∂A ∂Ai

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This is integrable due to the symmetry of τij , which implies that AD i =

∂F ∂Ai

for some homolorphic function F of the A’s, called the prepotential. Thus this function controls the N = 2 SUSY action, which discarding the Fayet-Iliopoulos terms κV reads Z  Z 1 ∂2F i 2 2 ∂F 2 i j S= = d θd θ i A + d θ i j W W . 4π ∂A ∂A ∂A An important feature of this action is that it exhibits the so-called electromagnetic duality. Indeed, we have found that the metric can be expressed as  1  1  D j j g= = τij (A)dAi dA = = dAj dA . 4π 4π This formula is symmetric in AD and A, so that we may use also AD as a local parameter. The expression for the metric as in the first equality (with A replaced by AD ) changes accordingly, with another harmonic function replacing =τ . It can be shown that this function is related to τ by the relation τ D (AD ) = −

1 . τ (A)

The transformation τ → −1/τ is the above-mentioned duality. This corresponds to the transformation F → ∗F in Maxwell’s equation (without sources). Moreover, as the function τ always appears in terms of its imaginary part, also the shift transformation τ → τ + 1 leaves the action invariant. It should be noted, however, that duality maps the description of the system via an action S into the the description in terms of another action S ∗ , while the shift is a symmetry of the system. Together, these two transformations generate the group Sp(2n, Z). Coming back to the SU (2) gauge theory, we have seen (compare Equation (3)) that the classical potential is V (φ) ∝ Tr[φ, φ]2 . This implies that the classical teory has a family of vacuum states, for φ commuting with φ; with this hypothesis, φ can be diagonalised, say φ = diag(a, −a), so that u = 2a2 . The prepotential for the unbroken U (1) gauge theory is classically 1 Fcl (a) = τcl a2 2 Θ 4π with τcl = + i 2 . In the quantum theory, instead, the gauge coupling is regulated 2π e by the so-called renormalisation group equation, which yields τΛ1 = τΛ0 +

2i Λ1 log . π Λ0

This immediately implies that the renormalisation-group invariant parameter is Λ4 = Λ40 exp (2πiτΛ0 ) .

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The one-loop correction to the prepotential in the large-a range is Fpert =

1 2 a2 a log 2 . 2π Λ

(12)

No higher-order perturbative corrections are present. In the non-perturbative regime (a ' 0), instead, problems arise. Indeed, an harmonic function like =τ (a) cannot have a minimum if it is defined for all a, so the coupling constant e2 may become negative. Here we exploit the electromagnetic duality, which reverses the coupling constant. This implies that other corrections to the prepotential come from instantons. More precisely, it can be shown that F = Fpert + Finst where Fpert is as in Equation (12) while Finst

∞ X Λ4k = Fk 2πik k=0

(13)

where the sum is over the instanton charge k. In the paper [MW97], Moore and Witten used this description to study again the Donaldson invariant of a 4-manifold.

5 Instanton counting The evaluation of the terms in (13) was limited to the first values of k up until when Nekrasov [Ne03] proposed a way to compute Finst via localisation formulæ. The proof of Nekrasov’s conjecture was achieved by Nakajima and Yoshioka [NY05, NY03]. The construction is based on equivariant integration on the moduli space Mk of framed k-instantons on the complex projective space CP2 . The term “framed” means that we have fixed a trivialisation of the bundle on the line at infinity  `∞ = [z0 : z1 : z2 ] ∈ CP2 : z0 = 0 . We consider the action of the torus (C∗ )2 on the base CP2 given by (t1 , t2 ) · [z0 : z1 : z2 ] = [z0 : t1 z1 : t2 z2 ] and the obvious action of the N -dimensional maximal torus T of GL(N, C), given by diagonal matrices, on the framing at infinity. We denote by (~ε, ~a) ≡ (ε1 , ε2 , a1 , . . . , ar ) the generators of the Lie algebra of the torus (C∗ )2 ×T and define the generating partition function Z ∞ X 4k Z(~ε, ~a; Λ) = Λ 1 k=0

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Mk

where integration is given by applying the localisation formula to the equivariant coho∗ ∗ mology class 1 ∈ H(C ε, ~a) = H(C ∗ )2 ×T (Mk ), and is thus a rational function in C(~ ∗ )2 ×T ({pt}). Nekrasov conjecture states that Finst (~ε, ~a; Λ) = ε1 ε2 log Z(~ε, ~a; Λ) is analytic around ε1 , ε2 = 0, and that Finst (~0, ~a; Λ) is the instanton part of the SeibergWitten prepotential, as in (13). The proof of this fact is achieved considering also d2 and certain differential equations (the blowup and contact instantons on the blowup CP equations) satisfied by Finst .

References [Wi88]

Edward Witten, Topological quantum field theory. Commun. Math. Phys. 117, 353–386 (1988).

[Se88]

Nathan Seiberg, Supersymmetry and non-perturbative beta functions. Physics Letters B 206, no. 1, 75 – 80 (1988).

[DK90] S.K. Donaldson and P.B. Kronheimer, The Geometry of Four-Manifolds. Oxford University Press, 1990. [SW94] Nathan Seiberg and Edward Witten, Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. arXiv:hep-th/9407087v1. [MW97] Gregory Moore and Edward Witten, Integration over the u-plane in Donaldson theory. arXiv:hep-th/9709193v2. [Ne03]

Nikita A. Nekrasov, Seiberg-Witten prepotential from instanton counting. arxiv:hep-th/0306211v1.

[NY03]

Hiraku Nakajima and K¯ota Yoshioka, Lectures on instanton counting. arXiv:math/0311058v1.

[NY05]

Hiraku Nakajima and K¯ota Yoshioka, Instanton counting on blowup. I: 4dimensional pure gauge theory. arXiv:math/0306198v2.

[Ta04]

Yuji Tachikawa, Seiberg-Witten theory and instanton counting. Master Thesis.

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