TOPOLOGICAL QUANTUM FIELD THEORIES PAVEL SAFRONOV Abstract. These are lecture notes from a class on TQFTs taught at the University of Zurich in fall 2017.
Contents 1. TFTs and categories 1.1. Functorial field theories from physics 1.2. Monoidal categories 1.3. Bordism categories 2. One-dimensional TFTs 2.1. 0d cobordism hypothesis 2.2. Dualizable objects 2.3. Operations with duals 2.4. Uniqueness of duality data 2.5. 1d cobordism hypothesis 2.6. Morse and Cerf theory 3. Two-dimensional TFTs 3.1. Frobenius algebras 3.2. Examples 3.3. Oriented 2d TFTs 3.4. Unoriented, spin and framed 2d TFTs 3.5. Dijkgraaf–Witten theory 4. Extended two-dimensional TFTs 4.1. Bicategories 4.2. Symmetric monoidal bicategories 4.3. Fully dualizable objects 4.4. Examples 4.5. Bordism bicategory 4.6. 2d cobordism hypothesis 4.7. Extracting non-extended theories 5. Three-dimensional TFTs 5.1. Oriented 3d TFTs 5.2. Fusion categories 5.3. Ribbon categories 5.4. Modular tensor categories 5.5. Classification of 3-2-1 TFTs 5.6. Knot invariants 1
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5.7. Knot invariants from TFTs 6. Higher categories 6.1. ∞-categories 6.2. Higher bordism categories 6.3. Cobordism hypothesis 6.4. Even higher categories References
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1. TFTs and categories 1.1. Functorial field theories from physics. The notion of a topological quantum field theory in mathematics arose as an attempt by Atiyah [Ati88] (following Segal’s formalization [Seg04] of conformal field theories) to formalize a special class of quantum field theories arising in physics inspired by works [Wit89] [Wit88] of Witten. In this section we will give a schematic description of what these theories are without going into details. A quantum field theory is given by a big collection of data and we will only extract a small piece that will be relevant for us. Let M be an oriented n-dimensional manifold without boundary. A topological n-dimensional quantum field theory has the following data: • The space of fields FM which is a manifold equipped with a measure. Typically FM is given by the space of smooth maps from M to a fixed manifold or, more generally, by the space of sections of a fiber bundle. • The action functional SM : FM → C. Remark 1.1. One can drop the adjective “topological” if we moreover equip the manifolds M we consider with a Riemannian metric. Given such data we obtain the partition function Z eiSM Z(M ) = FM
which is a certain number associated to the manifold M . If the manifold M has a boundary ∂M , the action functional may not be well-defined (like it happens in the Chern–Simons theory). Even if it is a well-defined number, we can consider fields with a certain boundary condition along ∂M . To formalize this situation, we consider the following additional data: • The space of boundary conditions F∂M which is a manifold. • A complex line bundle L∂M on F∂M . Given such data we obtain the space of states Z(∂M ) = Γ(F∂M , L∂M ), the space of sections of L∂M → F∂M . The action functional SM in this case is no longer a well-defined function, but its exponential exp(iSM ) can be considered as a section of the line bundle res∗ L∂M , where res : FM → F∂M extracts the boundary condition. Then the partition function is not a number, but instead is a section of L∂M , i.e. a vector in Z(∂M ).
TOPOLOGICAL QUANTUM FIELD THEORIES
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To summarize, an n-dimensional topological quantum field theory gives the following assignment: • To an (n − 1)-manifold N without boundary it assigns a vector space Z(N ). • To an n-manifold M with boundary ∂M it assigns a vector Z(M ) ∈ Z(N ). Note that these assignments satisfy various additional properties, such as: • When N = ∅ is`the empty manifold, we have Z(N ) ∼ = C. • When N = N1 `N2 is a disjoint union of manifolds, we have Z(N ) ∼ = Z(N1 )⊗Z(N2 ). • When M = M1 N M2 is a manifold glued out of two manifolds M1 , M2 along the common boundary N . Then Z(M ) can be obtained by pairing the vectors Z(M1 ) and Z(M2 ). We will succinctly formulate these data and these conditions by saying that Z defines a symmetric monoidal functor from a certain category of manifolds (the category of cobordisms) to the category of vector spaces. Before we proceed, let us make the following observation. The partition function Z(M ) for M an n-manifold is given by summing the numbers exp(iSM (φ)) for every φ ∈ F∂M . In a similar manner, the space of states Z(N ) for N an (n − 1)-manifold is given by adding the vector spaces LN,φ , the fibers of LN at φ ∈ FN . Thus, a natural question is what happens when N itself has a boundary. The idea of categorification gives a sequence of objects: • Complex numbers C, • C-vector spaces, • C-linear categories, • ... Thus, if N is a manifold with boundary ∂N , we may expect that we have a space of codimension 2 boundary conditions F∂N which carries a family of C-linear categories G∂N . The appropriate notion of a family of C-linear categories in our context is that of a gerbe. We will not use gerbes in these lectures, but the reader is referred to [Fre94] for a more detailed explanation of these ideas. Let us just point out that just as the space of sections of a line bundle is a vector space, the space of sections of a gerbe is a category. Therefore, we can define a C-linear category Z(∂N ) = Γ(F∂N , G∂N ) associated to an (n − 2)-manifold ∂N . As before, the vector space Z(N ) is no longer welldefined when N has a boundary, but should be considered as an object of the category Z(∂N ). This perspective gave rise to the theory of extended topological quantum field theories. One can continue the procedure and consider lower-dimensional manifolds. A fully extended n-dimensional topological quantum field theory is one where we can consider manifolds of arbitrary dimension ≤ n. Then, for instance, the value Z(pt) will be a certain C-linear (n − 1)-category. The cobordism hypothesis (a theorem in certain cases) states that a fully extended ndimensional topological quantum field theory is completely determined by its value Z(pt) on the point. It turns out that not every linear (n − 1)-categories arise in this way, but only fully dualizable ones. We will define this notion later, but let us just mention now that it is a certain finiteness condition.
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1.2. Monoidal categories. We will repeatedly use the language of category theory throughout these notes and it is assumed the reader has some familiarity with it. In this section we recall the necessary facts about monoidal categories. Recall that a monoid is a set equipped with a distinguished object (the unit) and a binary operation (multiplication) which is associative and unital. A monoidal category is a similar notion where one replaces sets with categories. Correspondingly, the associativity and unitality equations become specified natural transformations which have to satisfy higher coherences. Definition 1.2. A monoidal category is given by the following data: • A category C. • A functor ⊗ : C × C → C called the tensor product. • An object 1 ∈ C. • A natural isomorphism (associator ) ∼
αx,y,z : (x ⊗ y) ⊗ z − → x ⊗ (y ⊗ z) between the two functors C × C × C → C. • Natural isomorphisms (unitors) ∼
∼
λx : 1 ⊗ x − → x,
ρx : x ⊗ 1 − →x
between of functors C → C. These have to satisfy the following axioms: • (Triangle axiom) The diagram αx,1,y
(x ⊗ 1) ⊗ y
/
∼ ∼
ρx ⊗idy
x ⊗ (1 ⊗ y)
∼
&
x
idx ⊗λy
x⊗y commutes for every pair of objects x, y ∈ C. • (Pentagon axiom) The diagram ((x ⊗ y) ⊗ z) ⊗ w αx,y,z ⊗idw
u
αx⊗y,z,w
∼
∼
)
(x ⊗ (y ⊗ z)) ⊗ w
(x ⊗ y) ⊗ (z ⊗ w)
αx,y⊗z,w ∼
x ⊗ ((y ⊗ z) ⊗ w)
∼ αx,y,z⊗w ∼ idx ⊗αy,z,w
/
x ⊗ (y ⊗ (z ⊗ w))
commutes for every quadruple of objects x, y, z, w ∈ C. Given two monoidal categories C, D we can talk about functors C → D preserving the monoidal structure. Definition 1.3. A monoidal functor F : C → D between monoidal categories (C, ⊗, 1C , α, λ, ρ), (D, ⊗, 1D , α, λ, ρ) is given by the following data: • A functor F : C → D.
TOPOLOGICAL QUANTUM FIELD THEORIES
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• A natural isomorphism ∼
Jx,y : F (x) ⊗ F (y) − → F (x ⊗ y) of functors C × C → D. • An isomorphism
∼
: 1D − → F (1C ). These have to saitsfy the following axioms: • The diagram idF (x) ⊗Jy,z
F (x) ⊗ (F (y) ⊗ F (z)) αF (x),F (y),F (z)
4
/
∼
F (x) ⊗ F (y ⊗ z) Jx,y⊗z ∼
∼
(F (x) ⊗ F (y)) ⊗ F (z)
(
F (x ⊗ (y ⊗ z)) ∼
Jx,y ⊗idF (z)
∼
*
6
F (αx,y,z ) ∼
F (x ⊗ y) ⊗ F (z)
/
Jx⊗y,z
F ((x ⊗ y) ⊗ z)
commutes for every triple of objects x, y, z ∈ C. • The diagrams 1 ⊗ F (x)
⊗idF (x)
/
∼
F (1) ⊗ F (x) ∼ J1,x
λF (x) ∼
∼
F (x) o
F (1 ⊗ x)
F (λx )
and F (x) ⊗ 1
idF (x) ⊗ ∼
/
ρF (x) ∼
F (x) ⊗ F (1) ∼ Jx,1
o
F (x)
∼
F (x ⊗ 1)
F (ρx )
commute for every object x ∈ C. Definition 1.4. A monoidal functor F : C → D is a monoidal equivalence if the underlying functor of categories is an equivalence. Definition 1.5. Suppose F1 , F2 : C → D are two monoidal functors. A monoidal natural transformation F1 ⇒ F2 is given by a natural transformation η : F1 ⇒ F2 satisfying the following axioms: • The diagram F1 (x) ⊗ F1 (y)
ηx ⊗ηy
/
F2 (x) ⊗ F2 (y)
1 ∼ Jx,y
2 ∼ Jx,y
F1 (x ⊗ y) commutes for every x, y ∈ C.
ηx⊗y
/
F2 (x ⊗ y)
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PAVEL SAFRONOV
• The diagram 1 1
}
2 ∼
∼
F1 (1)
/
η1
!
F2 (1)
commutes. For ordinary monoids we can ask if the multiplication is commutative. For categories such a commutativity becomes additional data. Moreover, there are now two notions: braided monoidal and symmetric monoidal categories depending whether the commutativity data is given by a natural transformation or a natural isomorphism. Definition 1.6. A braided monoidal category is given by the following data: • A monoidal category (C, ⊗, 1, α, λ, ρ). • A natural isomorphism (braiding ) ∼
σx,y : x ⊗ y − →y⊗x between the two functors C × C → C. These have to satisfy the following axioms: • (Hexagon axioms) The diagrams (x ⊗ y) ⊗ z
αx,y,z
/
x ⊗ (y ⊗ z)
∼
σx,y ⊗idz
σx,y⊗z
∼
w
∼
(y ⊗ x) ⊗ z
'
(y ⊗ z) ⊗ x
∼ αy,x,z
'
y ⊗ (x ⊗ z)
∼
/
w
∼ αy,z,x
y ⊗ (z ⊗ x)
idy ⊗σx,z
and x ⊗ (y ⊗ z)
α−1 x,y,z
/
(x ⊗ y) ⊗ z
∼
idx ⊗σy,z
w
σx⊗y,z
∼
∼
x ⊗ (z ⊗ y)
'
z ⊗ (x ⊗ y) ∼
α−1 x,z,y
∼
'
(x ⊗ z) ⊗ y
∼
/
w
(z ⊗ x) ⊗ y
σx,z ⊗idy
commute for every triple of objects x, y, z ∈ C.
α−1 z,x,y
TOPOLOGICAL QUANTUM FIELD THEORIES
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Definition 1.7. A braided monoidal functor F : C → D between braided monoidal categories C and D is a monoidal functor F : C → D such that the diagram F (x) ⊗ F (y)
σF (x),F (y)
/
F (y) ⊗ F (x)
Jx,y ∼
∼ Jy,x
/
F (x ⊗ y)
F (σx,y )
F (y ⊗ x)
Definition 1.8. A symmetric monoidal category is a braided monoidal category (C, ⊗, 1, α, λ, ρ, σ) for which the composite σx,y
σy,x
x ⊗ y −−→ y ⊗ x −−→ x ⊗ y is the identity for every pair of objects x, y ∈ C. Example 1.9. Consider the category Vect of vector spaces over a field k. It has a natural symmetric monoidal structure given by the tensor product of vector spaces V ⊗k W . Depending on the precise construction, Vect is usually not a strict symmetric monoidal category. g ⊂ Vect be the full subcategory of Its strictification can be constructed as follows. Let Vect ⊕α vector spaces of the form k for some cardinal α. It is an example of a skeletal category: g are isomorphic iff they are equal. Moreover, every vector space admits two objects of Vect g ⊂ Vect is essentially surjective, hence is an equivalence of a basis, so the inclusion Vect g has an obvious strict symmetric monoidal structure given on categories. The category Vect objects by the addition of cardinals. Example 1.10. Another important example of a symmetric monioidal category is VectZ , the category of Z-graded vector spaces. Given two Z-graded vector spaces V and W we define σ : V ⊗ W → W ⊗ V to be σ(v ⊗ w) = (−1)|w||v| w ⊗ v on homogeneous elements v ∈ V and w ∈ W of degrees |v| and |w| respectively. When dealing with monoidal categories, one often suppresses the associators and unitors in the diagrams. The justification is given by coherence theorems that we will recall. Definition 1.11. A monoidal category (C, ⊗, 1, α, λ, ρ) is strict if the natural transformations α, λ, ρ are identities. Theorem 1.12 (MacLane). (1) For every monoidal category C there is a strict monoidal category str(C) together with a monoidal equivalence str(C) ∼ = C. (2) The free monoidal category on a category C is equivalent to the free strict monoidal category on C. The same coherence results hold verbatim for symmetric monoidal categories, see [JS93].
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1.3. Bordism categories. Our next goal is to formalize the notion of a topological quantum field theory from section 1.1. The key definition will be that of a bordism which is an nmanifold describing a time evolution of an (n − 1)-manifold. Definition 1.13. A manifold M is closed if it is compact and has no boundary. The following definition is taken from [Fre12]. Definition 1.14. Let N0 and N1 be closed (n − 1)-manifolds. A bordism from N0 to N1 is given by the following data: • A compact n-manifold M with boundary. • A continuous ` map ∂M → {0, 1} giving rise to a decomposition of the boundary ∂M = (∂M )0 (∂M )out . • Embeddings θ0 : [0, 1) × N0 −→ M θ1 : (−1, 0] × N1 −→ M such that θi ({0} × Ni ) = (∂M )i for i = 0, 1. Remark 1.15. The words “bordism” and “cobordism” are usually used interchangeably. N1 N0
M
Figure 1. “Pair of pants” bordism. Examples. (1) Let N be a closed (n − 1)-manifold and consider the cylinder M = [0, 1] × N . It can be considered as a bordism from N to N if we let θ0 and θ1 to be the standard inclusions θ0 : [0, 1) × N → [0, 1] × N given by t 7→ t/2 on the interval and θ1 : (−1, 0] × N → [0, 1] × N given by t 7→ t/2 + 1 on the interval. (2) Suppose N is a closed (n − 1)-manifold and f : N → N is a diffeomorphism. We can consider the manifold M = [0, 1] × N as in the previous example. We let θ0 be as before and θ1 be the map θ1 : (−1, 0] × N → [0, 1] × N given by t 7→ t/2 + 1 on the interval and f on N . This is known as the mapping cylinder . ` (3) A typical bordism is shown in fig. 1 known as a pair of pants bordism from S 1 S 1 to S 1 . The blue shaded region denotes the image of θ0 and the red shaded region denotes the image of θ1 . Definition 1.16. Let (M, θ0 , θ1 ) be a bordism from N0 to N1 . A diffeomorphism of the bordism is a diffeomorphism f : M → M making the diagrams [0, 1) × N0 θ0
M
y
θ0 f
/
%
M
TOPOLOGICAL QUANTUM FIELD THEORIES
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and (−1, 0] × N1 θ1
M
y
θ1
%
f
/
M
commute. The diffeomorphisms θi in the definition of a bordism give rise to the collarings of the boundary. The reason we have included the collars is to be able to glue two bordisms into a smooth manifold. Indeed, given two smooth functions f : [0, 1) → R and g : (−1, 0] → R such that f (0) = g(0), they can be glued to a continuous function (−1, 1) → R which is not in general smooth. Similarly, if M1 , M `2 are two manifolds with a diffeomorphism of the boundaries ∂M1 ∼ = ∂M2 , their union M1 ∂M1 M2 does not have a canonical structure of a smooth manifold. Definition 1.17. Let N0 , N1 , N2 be closed (n − 1)-manifolds. Suppose (M 0 , θ00 , θ10 ) is a bordism from N0 to N1 and (M 00 , θ000 , θ100 ) is a bordism from N1 to N2 . Their composition is the bordism with bound` from N0 to N2 constructed as follows. As a manifold ` ary, it is M = M 0 N1 M 00 . Its boundary is split as ∂M = (∂M 0 )0 (∂M 00 )1 and we let (∂M )0 = (∂M 0 )0 and (∂M )1 = (∂M 00 )1 . We let θ0 = θ00 and θ1 = θ100 .
M0
◦
M 00
= M
Figure 2. An example of composition. Remark 1.18. The only nontrivial step in the previous definition is to define the structure of ` 0 00 a smooth manifold on M N1 M . The key idea is to glue θ10 and θ000 into a diffeomorphism of a neighborhood of N1 ⊂ M and (−1, 1) × N1 (see the red shaded area on fig. 2); Lemma 1.19. Composition of bordisms is compatible with diffeomorphisms and hence it descends to a composition of diffeomorphism classes of bordisms. Definition 1.20. The category Cobn is defined as follows: • Its objects are closed (n − 1)-manifolds. • A morphism N0 → N1 is a diffeomorphism class of bordisms from N0 to N1 . • The identity id : N → N is given by the bordism M = [0, 1] × N from N to N with θ0 : [0, 1) × N → M is given by the obvious inclusion and θ1 : (−1, 0] × N → M induced from the map t + 1 : (−1, 0] → [0, 1]. • The composition of morphisms is defined to be the composition of bordisms. The category Cobn has a symmetric monoidal structure given by the disjoint union of manifolds.
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Example 1.21. Let us work out the case n = 0. There is a single (−1)-dimensional manifold which is the empty set. Bordisms from the empty set to the empty set are given by finite sets of points. An isomorphism class of a finite set is determined by its cardinality, so we get an equivalence Cob0 ∼ = ∗/N, where N is the monoid of natural numbers under addition. The symmetric monoidal structure on Cob0 is given by the disjoint union of bordisms, i.e. the functor ∗/N × ∗/N −→ ∗/N is induced by the addition N × N → N. We can modify the definition of the bordism category Cobn as follows. Let Cobor n be the category defined as before where objects and bordisms are equipped with orientations and all diffeomorphisms are required to be orientation-preserving. One might also want to consider spin bordisms or framed bordisms, so let us present a general theory of tangential structures incorporating these examples. Definition 1.22. Let M be an n-dimensional manifold (possibly with boundary) and V → M a real vector bundle of rank k. Its frame bundle F(V ) is the principal GL(k, R)-bundle whose fiber at m ∈ M is the set of frames in Vm , i.e. the set of isomorphisms of real vector spaces Vm ∼ = Rk . We denote FM = F(TM ) called the frame bundle of M . Example 1.23. Suppose n = 1. Then GL(1, R) = R× is the group of nonzero real numbers under multiplication. The frame bundle FM as a manifold can be obtained as TM − M , the tangent bundle minus the zero section. Definition 1.24. Let H → G be a homomorphism of groups and P → M a principal Gbundle over a manifold M . An H-reduction of the bundle P is an H-bundle PH → M together with an isomorphism of G-bundles PH ×H G ∼ = P. Remark 1.25. One can give a homotopy-theoretic interpretation of H-reductions as follows. One can identify principal G-bundles with their classifying maps M → BG. An H-reduction is then a homotopy lift of the classifying map along BH → BG. That is, it is given by a map M → BH together with a homotopy between the two composites in the diagram BH <
PH
M
P
/
BG
Definition 1.26. Let M be an n-dimensional manifold (possibly with boundary) and ρ : G → GL(n, R) a homomorphism of groups. A G-structure on M is a G-reduction of the frame bundle FM . Examples. (1) Suppose G = GL(n, R). Then a G-structure on M carries no extra information.
TOPOLOGICAL QUANTUM FIELD THEORIES
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(2) The group GL(n, R) has two connected components and let’s denote by G = GL+ (n, R) ⊂ GL(n, R)
(3) (4) (5) (6)
the connected component of the identity. A G-structure on M is the same as an orientation. Let G = O(n) ⊂ GL(n, R). Then a G-structure on M is the same as a Riemannian metric. Let G = Spin(n) → GL(n, R). Then a G-structure on M is known as the spin structure. Suppose n = 2m is even and let G = GL(m, C) ⊂ GL(2m.R). Then a G-structure on M is an almost complex structure. Let G = {e} ⊂ GL(n, R) be the trivial subgroup. Then a G-structure on M is the same as a framing.
Remark 1.27. Note that any two O(n)-structures on M are isomorphic since GL(n, R)/O(n, R) is contractible. Suppose M is a manifold with boundary. The restriction of the tangent bundle of M to the boundary naturally fits into an exact sequence 0 −→ T(∂M ) −→ TM |∂M −→ R −→ 0, where R is the trivial one-dimensional vector bundle over the boundary. A collar of the boundary gives rise to a splitting of the above sequence, so that TM |∂M ∼ = R⊕T(∂M ). Using this observation we are now going to define the category of bordisms with a G-structure. Definition 1.28. Let N0 and N1 be closed (n − 1)-manifolds equipped with G-reductions PGi of the frame bundles F(R ⊕ TNi ) for i = 0, 1. A bordism from N0 to N1 equipped with a G-structure is a bordism (M, θ0 , θ1 ) from N0 to N1 such that M is equipped with a G-structure PG and the embeddings θi intertwine the G-structures. That is, we are given an isomorphism of the G-structures [0, 1) × PG0 → [0, 1) × N0 and θ0∗ PG over [0, 1) × N0 and similarly for θ1 . We can also talk about diffeomorphisms of bordisms equipped with G-structures which interwine the G-structures. Definition 1.29. Let ρ : G → GL(n, R) be a homomorphism. The category CobG n is defined as follows: • Its objects are closed (n − 1)-manifolds N equipped with a G-reduction of the frame bundle F(R ⊕ TN ). • A morphism N0 → N1 is an diffeomorphism class of bordisms from N0 to N1 equipped with a G-structure. +
{e} fr (n,R) For instance, CobGL = Cobor n n is the category of oriented bordisms and Cobn = Cobn is the category of framed bordisms. We will now show that the bordism category only depends on the homotopy type of G.
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Proposition 1.30. Consider a diagram G1
4
*
GL(n, R)
G2 of Lie groups and suppose that G1 → G2 is a weak equivalence, i.e. an equivalence on G2 1 homotopy groups. Then the forgetful functor CobG n → Cobn on bordism categories is an equivalence. Proof. The fibrations Gi → EGi → BGi for i = 0, 1 induce long exact sequences of homotopy groups · · · −→ πm (Gi ) −→ πm (EGi ) −→ πm (BGi ) −→ πm−1 (Gi ) −→ . . . Since EGi are contractible, we get isomorphisms πm (BGi ) ∼ = πm−1 (Gi ). Therefore, BG1 → BG2 is a weak equivalence as well. Next, recall that for G a Lie group, the set of isomorphism classes of principal G-bundles over a manifold M is isomorphic to the set [M, BG] of homotopy classes of maps M → BG. Since BG1 → BG2 is a weak equivalence, the map [M, BG1 ] → [M, BG2 ] is an isomorphism. That is, the set of isomorphism classes of G1 -bundles on M is isomorphic to that for G2 . G2 1 The forgetful functor CobG n → Cobn is therefore essentially surjective since by the above argument every G2 -reduction can be refined to a G1 -reduction. The same argument shows that the functor is fully faithful and hence is an equivalence by [Mac71, Theorem IV.4.1]. For example, the inclusion O(n) ⊂ GL(n, R) is a homotopy equivalence (its inverse can be constructed using the QR decomposition) and hence the forgetful functor CobO(n) → Cobn n is an equivalence. We are now ready to define the notion of a topological quantum field theory. Definition 1.31. Let C be a symmetric monoidal category. An n-dimensional topological field theory is a symmetric monoidal functor Z : Cobn −→ C. Definition 1.32. An n-dimensional topological quantum field theory is a symmetric monoidal functor Cobn → Vect, where Vect is the category of vector spaces. Given a homomorphism G → GL(n, R) we can also consider symmetric monoidal functors CobG n → C which we call n-dimensional topological field theories on manifolds with a Gstructure. For instance, for Cobor n we talk about oriented topological field theories and for fr Cobn we talk about framed topological field theories. It is easy to see that this definition formalizes the path integral for topological quantum field theories from section 1.1: • A closed (n − 1)-manifold N is an object of Cobn , so Z(N ) is a vector space. This encodes the space of states. • The empty (n − 1)-manifold is the unit for the symmetric monoidal structure, so Z(∅) ∼ = k.
TOPOLOGICAL QUANTUM FIELD THEORIES
13
• A closed n-manifold M is a bordism from the empty manifold to itself. Therefore, Z(M ) is an endomorphism of k in Vect, i.e. a number. This encodes the partition function. Since we are using the language of category theory, it is natural to ask what topological field theories form. Functors Cobn → C organize into a category Fun(Cobn , C) whose objects are functors and morphisms are natural transformations. Similarly, symmetric monoidal functors form a category Fun⊗ (Cobn , C) whose objects are symmetric monoidal functors and morphisms are symmetric monoidal (equivalently, monoidal) natural transformations. Thus, topological field theories form a category. We will show later (see corollary 2.29) that this category is a groupoid, i.e. every morphism of topological field theories is an isomorphism. 2. One-dimensional TFTs 2.1. 0d cobordism hypothesis. Our goal in this course will be to formulate the cobordism hypothesis. As a warm-up we are now going to present its various versions in the toy case of 0-dimensional manifolds. Let Bord00 be the set of diffeomorphism classes of compact 0-dimensional manifolds. The set Bord00 has a natural structure of a commutative monoid given by disjoint sum. A compact 0-dimensional manifold is a finite set of points, so Bord00 ∼ = N where N is considered as a commutative monoid under addition. We denote by CMon the category of commutative monoids. Theorem 2.1 (0d cobordism hypothesis, version 1). Let X be a commutative monoid. The map HomCMon (Bord00 , X) → X given by evaluating the map on a single point pt is an isomorphism. Proof. Under the isomorphism Bord00 ∼ = N the point goes to 1 ∈ N and the statement becomes that N is freely generated as a commutative monoid by 1. The set Bord00 we have considered consisted of diffeomorphism classes of manifolds. We can also consider the groupoid Bord10 of compact 0-dimensional manifolds with morphisms given by diffeomorphisms. Given an arbitrary category C we denote by π0 (C) the set of isomorphism classes of objects. Then we get by definition Bord00 = π0 (Bord10 ). The groupoid Bord10 has a natural symmetric monoidal structure given by the disjoint union of manifolds. Theorem 2.2 (0d cobordism hypothesis, version 2). Let X be a symmetric monoidal groupoid. The functor Fun⊗ (Bord10 , X) → X given by evaluating the functor on pt is an equivalence of groupoids. Proof. The groupoid Bord10 has isomorphism classes of objects given by standard finite sets on n objects. The isomorphism group of such a finite set is Sn . Thus, we get that a Bord1 ∼ ∗/Sn . = 0
n∈N
14
PAVEL SAFRONOV
Note that the groupoid on the right-hand side has a strict symmetric monoidal structure. Moreover, it is obviously the free strict symmetric monoidal groupoid on the one-object category ∗. By the coherence theorem (symmetric monoidal analog of theorem 1.12) it is equivalent to the free symmetric monoidal groupoid on ∗. Since it is free, we get equivalences ∼
→ Fun(∗, X) ∼ Fun⊗ (Bord10 , X) − = X. Let us observe that theorem 2.1 follows from theorem 2.2. Indeed, any commutative monoid X can be considered as a symmetric monoidal groupoid with only identity morphisms. Moreover, for any symmetric monoidal groupoid G the set of isomorphism classes of objects π0 (G) is a commutative monoid. Thus, if X is a commutative monoid, we get equivalences Fun⊗ (Bord10 , X) ∼ = HomCMon (π0 (Bord10 ), X) = HomCMon (Bord00 , X). 2.2. Dualizable objects. In quantum field theory we consider integrals over non-compact manifolds and a delicate analysis is needed to prove their convergence. Similarly, in topological field theories the objects that appear (partition function, spaces of states, ...) satisfy a certain finiteness condition. The simplest form of finiteness is that of dualizability that we will now define. Definition 2.3. Let C be a monoidal category and x ∈ C an object. A left dual to x is given by the following data: • An object x∨ ∈ C. • A morphism ev : x ⊗ x∨ → 1 called the evaluation. • A morphism coev : 1 → x∨ ⊗ x called the coevaluation. These have to satisfy the following axioms: (1) The composite ∼
x
/
x⊗1
idx ⊗coev
/
x ⊗ x∨ ⊗ x
ev⊗idx
/
1⊗x
∼
/
x
is equal to idx . (2) The composite x∨
∼
/
1 ⊗ x∨
coev⊗idx∨
/
x∨ ⊗ x ⊗ x∨
idx∨ ⊗ev
/
x∨ ⊗ 1
∼
/
x∨
is equal to idx∨ . Definition 2.4. An object x ∈ C is left dualizable if it admits a left dual. One similarly defines right duals. In a braided monoidal category a left dual can be turned into a right dual using the braiding. Thus, in a braided monoidal category we will simply speak of dualizable objects. Definition 2.5. A monoidal category is left rigid (right rigid ) if every object admits a left (right) dual. We say it is rigid if every object admits both left and right duals.
TOPOLOGICAL QUANTUM FIELD THEORIES
x
f
15
y
Figure 3. Morphism f : x → y. The axioms of duality are called the snake relations (in common parlance, mark of Zorro axioms) because of their pictorial representation that we are now going to describe. Suppose C is a category and f : x → y is a morphism. We may draw it as shown in fig. 3. Concatenating two morphisms we get compositions as shown in fig. 4. The identity morphism is drawn without any decorations. x
f
y
z = x
g
g◦f
z
Figure 4. Composition of morphisms. Suppose morever C is a monoidal category. Then we draw tensor products by stacking morphisms vertically as shown in fig. 5. z x
g f
w y
Figure 5. Morphism f ⊗ g : x ⊗ z → y ⊗ w. Note that by convention we are reading morphisms from left to right and reading tensor products from bottom to top. Remark 2.6. To read off a morphism from its pictorial representation, we may have to insert associators and unitors in some way. To see that the result is unambiguous, replace the monoidal category C by the equivalent strictly monoidal category. Now fix a pair of objects x, x∨ ∈ C together with morphisms ev : x ⊗ x∨ → 1 and coev : 1 → x∨ ⊗ x. We will draw x with an outgoing arrow and x∨ with an incoming arrow, see fig. 6.
Figure 6. Morphisms ev : x ⊗ x∨ → 1 and coev : 1 → x∨ ⊗ x. Using this notaion the axioms of duality in definition 2.3 can be drawn pictorially as follows:
16
PAVEL SAFRONOV
= (1)
= (2) Lemma 2.7. Let C be a monoidal category and x, y ∈ C left dualizable objects. Then x ⊗ y is left dualizable with the dual y ∨ ⊗ x∨ . Proof. We can construct the evaluation map as id⊗evy ⊗id
ev
x x ⊗ y ⊗ y ∨ ⊗ x∨ −−−−−−→ x ⊗ x∨ −−→ 1
and the coevaluation map as coevy
id⊗coev ⊗id
x 1 −−−→ y ∨ ⊗ y −−−−−− −→ y ∨ ⊗ x∨ ⊗ x ⊗ y.
The duality axioms for x ⊗ y follow from the duality axioms for x and y.
Let us move on to examples of dualizable objects. Consider the category Vect of vector spaces over a field k. If V is a finite-dimensional vector space, we can construct its dual as V ∨ = Hom(V, k). We have an obvious evaluation map V ⊗ V ∨ → k. To construct the coevaluation, pick a basis {ei } of V and let {e∨i } be the corresponding dual basis of V ∨ . The coevaluation k → V ∨ ⊗ V is the same as an element coev(1) ∈ V ∨ ⊗ V which we let to be X coev(1) = e∨i ⊗ ei . i
Lemma 2.8. The element coev(1) is independent of the choice of the basis of V . Proof. Consider the map V ∨ ⊗ V → End(V ) given by ! X i
φi ⊗ vi 7→
w 7→
X
φi (w)vi
.
i
By picking a basis it is easy to see that the map is injective. Moreover, it is an isomorphism since both sides have the same dimension. Under this map coev(1) is send to the identity morphism id ∈ End(V ) which is independent of a basis. Now we are going to show that these are all dualizable objects of Vect.
TOPOLOGICAL QUANTUM FIELD THEORIES
17
Proposition 2.9. A vector space V ∈ Vect is dualizable iff it is finite-dimensional. Proof. We have already shown that a finite-dimensional vector space is dualizable. Conversely, suppose V ∈ Vect is dualizable. Given φ ∈ V ∨ and v ∈ V we denote φ(v) = ev(v, φ). Let coev(1) = i fi ⊗ ei for some vectors fi ∈ V ∨ and ei ∈ V where we note that the sum is finite. The first duality axiom implies that for any vector v ∈ V we have a decomposition X v= fi (v)ei . P
i
In other words, {ei } span V and since there are finitely many of ei , V is finite-dimensional. More generally, consider a commutative ring R and let ModR be the category of R-modules. It has a symmetric monoidal structure defined by M ⊗R N = coeq(M ⊗ R ⊗ N ⇒ M ⊗ N ), where the two maps are the actions of R on M and on N . Proposition 2.10. An R-module M ∈ ModR is dualizable iff it is finitely generated and projective. This statement generalizes to noncommutative rings as follows. Definition 2.11. Suppose A and B are two rings. An (A, B)-bimodule is an abelian group M equipped with a left A-module structure, a right B-module structure such that they commute: for all a ∈ A, m ∈ M and b ∈ B we have (am)b = a(mb). Let R be a ring and LModR the category of left R-modules. It no longer carries a symmetric monoidal structure. Given a right R-module (equivalently, a left Rop -module) N we have an abelian group N ⊗R M and an (R, Rop )-bimodule M ⊗ N . We say that M ∈ LModR is dualizable if there is a right R-module N and maps coev : Z → N ⊗R M,
ev : M ⊗ N → R
where coev is a morphism of abelian groups while ev is a morphism of (R, R)-bimodules. Moreover, they have to satisfy the duality axioms. For instance, the composite id⊗coev
ev⊗id
M −−−−→ M ⊗ (N ⊗R M ) −−−→ M is the identity. Proposition 2.12. The bordism category Cobn is rigid. Proof. Recall that the unit of the symmetric monoidal structure on Cobn is given by the empty (n − 1)-manifold. Now suppose M ∈ Cobn is a closed (n − 1)-manifold. We define M ∨`= M . The evaluation is given by the manifold M × [0, 1] considered as a bordism from M M to ∅. The coevaluation is given by reversing the arrows. The duality axioms are checked by applying M × (−) to the pictures (1) and (2).
18
PAVEL SAFRONOV
Note that the claim is also true for the bordism categories CobG n of manifolds with a G-structure. One also has the following trivial observation. Lemma 2.13. Suppose F : C → D is a monoidal functor and x ∈ C is left dualizable. Then F (x) is left dualizable. Proof. Given a duality data (x∨ , ev, coev) for x in C we define the duality for F (x) in D by applying F . For instance, the dual object is given by F (x∨ ). Corollary 2.14. Suppose Z is an n-dimensional topological field theory valued in C. Then Z(M ) ∈ C is dualizable for any closed (n − 1)-manifold M . For instance, the spaces of states in a topological quantum field theory are all finitedimensional. This should be contrasted with the case of ordinary quantum field theories where the spaces of states are infinite-dimensional. For instance, the space of states in quantum mechanics of a particle on a line is L2 (R). A related notion is that of an invertible object. Definition 2.15. Suppose C is a symmetric monoidal category. We say that x ∈ C is invertible if it admits a duality data (x∨ , ev, coev) with the evaluation and coevaluation maps being isomorphisms. Example 2.16. A vector space V ∈ Vect is invertible iff it is one-dimensional. Indeed, since the evaluation map V ⊗ V ∨ → k is an isomorphism, both sides have the same dimension, so (dim V )2 = 1. Remark 2.17. If C is a symmetric monoidal groupoid, an object is dualizable iff it is invertible. 2.3. Operations with duals. Fix a monoidal category C. Definition 2.18. Let f : y → x be a morphism of left dualizable objects. Its dual f ∨ : x∨ → y ∨ is defined to be the composite x∨
∼
/
1 ⊗ x∨
coevy
/ y∨
id⊗f ⊗id ∨ /
⊗ y ⊗ x∨
y ⊗ x ⊗ x∨
id⊗evx
/ y∨
⊗1
∼
/ y∨.
Note that in the case of vector spaces this definition recovers the definition of the dual map. Lemma 2.19. Suppose f : y → x is a morphism of left dualizable objects. Then the diagram f ⊗id
x9 ⊗ x∨
evx
"
y ⊗ x∨ id⊗f ∨
%
y⊗y commutes.
evy ∨
<
1
TOPOLOGICAL QUANTUM FIELD THEORIES
19
Proof. Expanding f ∨ by definition we get a commutative diagram f ⊗id
y ⊗ x∨ id⊗coevy ⊗id
/ 4
evy ⊗f ⊗id
x ⊗O x∨ evx
y ⊗ y ∨ ⊗ y ⊗ x∨
)
evy ⊗id⊗id
evy
*
id⊗id⊗f ⊗id
1O
y ⊗ y ∨ ⊗ x ⊗ x∨
/ id⊗id⊗evx
y ⊗ y∨
where the top left triangle commutes by the first duality axiom. Now let us assume that C is a symmetric monoidal category.
Definition 2.20. Let x ∈ C be a left dualizable object and f : x → x an endomorphism. Its trace tr(f ) ∈ EndC (1) is the composite 1
coev
/
x∨ ⊗ x
σx∨ ,x
/
x ⊗ x∨
f ⊗id
/
x ⊗ x∨
ev
/
1.
Definition 2.21. Suppose x ∈ C is a left dualizable object. Its dimension is dim(V ) = tr(id) ∈ EndC (1). As an example, consider a finite-dimensional vector space V ∈ Vect with an endomorphism f : V → V . Fix a basis {ei } of V and let {e∨i } be the dual basis of V ∨ . Then tr(f ) ∈ EndVect (k) ∼ = k is given by X tr(f ) = he∨i , f (ei )i i
which coincides with the trace of the matrix of V in this basis. For instance, the categorical dimension of V is given by the image under the natural map Z → k of its dimension as a vector space. Note that, for instance, the categorical dimension of a p-dimensional vector space over a field of characteristic p is zero. By lemma 2.7 the dual of x ⊗ x∨ can be identified with itself. Thus, the dual of evaluation is a map ev∨ : 1 → x ⊗ x∨ . Proposition 2.22. Let x ∈ C be a dualizable object. Then ev∨ : 1 −→ x ⊗ x∨ coincides with
coev
σ
1 −−→ x∨ ⊗ x → − x ⊗ x∨ . Proof. We can identify σ ◦ ev∨ with the composite coev
coev
id⊗ev⊗id
1 −−→ x∨ ⊗ x −−→ x∨ ⊗ x ⊗ x∨ ⊗ x −−−−−→ x∨ ⊗ x. By the duality axiom the composite of the last two morphisms is equal to id, so the whole composite is coev : 1 → x∨ ⊗ x.
20
PAVEL SAFRONOV
2.4. Uniqueness of duality data. Let C again be a monoidal category. This discussion is inspired by [Lur17, Section 4.6.1]. Our first observation is that it is enough to have the evaluation map to define duals. Let DDatf ull (C) be the following category: • Its objects are (x, x∨ , ev, coev) where x and x∨ are objects of C and ev : x ⊗ x∨ → 1 and coev : 1 → x∨ ⊗ x are morphisms which satisfy the duality axioms. • Morphisms (x, x∨ , evx , coevx ) → (y, y ∨ , evy , coevy ) are given by f : x → y and g : x∨ → y ∨ making the diagrams x∨; ⊗ x coevx
1
g⊗f coevy
"
y∨ ⊗ y and x ⊗ x∨ evx
<
f ⊗g
y⊗y
#
1
evy ∨
commute. Let DDat(C) be the following category: • Its objects are (x, x∨ , ev) where x and x∨ are objects of C and ev : x ⊗ x∨ → 1 is a morphism such that for every objects z, w ∈ C the composite HomC (z, w ⊗ x)
(−)⊗x∨
/
HomC (z ⊗ x∨ , w ⊗ x ⊗ x∨ )
ev
/
HomC (z ⊗ x∨ , w)
is an isomorphism. • Morphisms (x, x∨ , evx ) → (y, y ∨ , evy ) are given by f : x → y and g : x∨ → y ∨ compatible with the evaluation maps as before. Definition 2.23. Suppose x, x∨ ∈ C are objects and ev : x ⊗ x∨ → 1 is a morphism. We say that ev is a nondegenerate pairing if (x, x∨ , ev) ∈ DDat(C). Proposition 2.24. Let C be a closed monoidal category, i.e. there is an internal Hom functor Hom : Cop × C → C together with a natural isomorphism HomC (z, Hom(x, y)) ∼ = HomC (z ⊗ x, y). Then a pairing ev : x ⊗ x∨ → 1 is nondegenerate iff the morphism y ⊗ x −→ Hom(x∨ , y) represented by idy ⊗ evx : y ⊗ x ⊗ x∨ → y is an isomorphism.
TOPOLOGICAL QUANTUM FIELD THEORIES
21
Proof. By the Yoneda lemma y ⊗ x → Hom(x∨ , y) is an isomorphism iff for all z ∈ C HomC (z, y ⊗ x) −→ HomC (z, Hom(x∨ , y)) is an isomorphism. Applying the definition of the internal Hom this morphism is equal to ev
HomC (z, y ⊗ x) −→ HomC (z ⊗ x∨ , y ⊗ x ⊗ x∨ ) − → HomC (z ⊗ x∨ , y) which is an isomorphism iff ev : x ⊗ x∨ → 1 is nondegenerate. f ull
Lemma 2.25. Both DDat
(C) and DDat(C) are groupoids.
Proof. Suppose f : x → y and g : x∨ → y ∨ determine a morphism (x, x∨ , evx , coevx ) → (y, y ∨ , evy , coevy ) in DDatf ull (C). Then f ∨ : y ∨ → x∨ and we claim it is inverse to g. Indeed, the equality g ◦ f ∨ = idy∨ is given by the commutative diagram id⊗f ⊗id
x∨ ⊗ O x ⊗ y
/
coevx ⊗id coevy ⊗id
y∨
id⊗evy
x∨ ⊗ y ⊗ y ∨ / y∨
/
x∨
g⊗id⊗id
⊗ y ⊗ y∨
id⊗evy
5/
g
y∨
idy∨
where the top composite is g ◦ f ∨ . The equality f ∨ ◦ g = idx∨ is given by the commutative diagram y ∨O
coevx ⊗id
/
g
x∨
x∨ ⊗ xO ⊗ y ∨
id⊗f ⊗id
/
x∨ ⊗ y ⊗ y ∨
id⊗id⊗g coevx ⊗id
/
x∨ ⊗ x ⊗ x∨
id⊗evx
/5
id⊗evy
x∨
idx∨
The claim in DDat(C) is proved similarly.
There is a natural forgetful functor DDatf ull (C) → DDat(C) given by forgetting the coevaluation. Proposition 2.26. The forgetful functor DDatf ull (C) → DDat(C) is an equivalence. Proof. First, we have to show that the functor is well-defined, i.e. for any duality data (x, x∨ , ev, coev) ∈ DDatf ull (C) the morphism Hom(z, w ⊗ x)
(−)⊗x∨
/
Hom(z ⊗ x∨ , w ⊗ x ⊗ x∨ )
ev
/
Hom(z ⊗ x∨ , w)
is an isomorphism for all z, w ∈ C. Indeed, its inverse is given by the composite HomC (z ⊗ x∨ , w)
(−)⊗x
/
HomC (z ⊗ x∨ ⊗ x, w ⊗ x)
coev
/
HomC (z, w ⊗ x).
Next, we have to show that it is fully faithful. In other words, give a pair of morphisms f : x → y, g : x∨ → y ∨ compatible with the evaluations, they are also compatible with the
22
PAVEL SAFRONOV
coevaluations. Since we have to check that two morphisms 1 → y ∨ ⊗ y, namely coevy and (g ⊗ f ) ◦ coevx , are equal, we can check it in Hom(y ∨ , y ∨ ) since Hom(1, y ∨ ⊗ y)
(−)⊗y ∨
/
Hom(y ∨ , y ∨ ⊗ y ⊗ y ∨ )
evy
/
Hom(y ∨ , y ∨ )
is an isomorphism. The image of coevy under the above map is idy∨ . (g ⊗ f ) ◦ coevx is given by the composite id⊗evy
g⊗f ⊗id
coev
The image of
x x∨ ⊗ x ⊗ y ∨ −−−−→ y ∨ ⊗ y ⊗ y ∨ −−−−→ y ∨ y ∨ −−−→
which after rearranging becomes g ◦ f ∨ . But by lemma 2.25 g ◦ f ∨ = idy∨ . Finally, we have to show that it is essentially surjective, i.e. given (x, x∨ , ev) ∈ DDat(C) we have to construct the coevaluation satisfying the duality axioms. Let coev : 1 → x∨ ⊗ x ∼ be the preimage of idx∨ under the isomorphism HomC (1, x∨ ⊗ x) − → HomC (x∨ , x∨ ). By construction the second axiom of duality is satisfied. To check the first axiom of duality, observe that the image of id⊗coev ev⊗id x −−−−→ x ⊗ x∨ ⊗ x −−−→ x ∼ under Hom(x, x) − → Hom(x ⊗ x∨ , 1) is given by the composite id⊗coev⊗id
ev⊗id
ev
x ⊗ x∨ −−−−−−→ x ⊗ x∨ ⊗ x ⊗ x∨ −−−→ x ⊗ x∨ − → 1. By the second axiom of duality it coincides with ev : x ⊗ x∨ → 1 which is also the image of ∼ id under Hom(x, x) − → Hom(x ⊗ x∨ , 1). Thus, we see that the coevaluation is not necessary to define the duality data and one can instead just consider nondegenerate evaluation maps, i.e. those that induce an isomorphism Hom(z, w ⊗ x) ∼ = Hom(z ⊗ x∨ , w). Next, we show that even the evaluation is not necessary. Let Cld ⊂ C be the full subcategory of left dualizable objects. Let (Cld )∼ its underlying groupoid where we throw away non-invertible morphisms. Proposition 2.27. The forgetful functor DDat(C) → (Cld )∼ is an equivalence. Proof. By construction the functor is essentially surjective. Note that a morphism in Cld is a morphism f : x → y while a morphism in DDat(C) is a pair of morphisms f : x → y and g : x∨ → y ∨ compatible with the evaluation. We already know that g = (f ∨ )−1 , so we have to show that this choice of g is automatically compatible with the evaluation map. But this is exactly the content of lemma 2.19. To summarize, the two forgetful functors DDatf ull (C) −→ DDat(C) −→ (Cld )∼ are equivalences which shows that dualizability is indeed a property of an object and not extra data. Proposition 2.28. Let C and D be symmetric monoidal categories and assume that D is rigid. The category Fun⊗ (D, C) of symmetric monoidal functors from D to C is a groupoid. Proof. Suppose η : F1 ⇒ F2 is a symmetric monoidal natural transformation between symmetric monoidal functors F1 , F2 : D → C. To show that it is an isomorphism, we have to show that for every object x ∈ D the induced map F1 (x) → F2 (x) is an isomorphism. But as in lemma 2.25 one can check that the dual of F1 (x∨ ) → F2 (x∨ ) is its inverse.
TOPOLOGICAL QUANTUM FIELD THEORIES
23
Corollary 2.29. The category Fun⊗ (Cobn , C) of n-dimensional topological field theories is a groupoid. Proof. The claim follows from proposition 2.12 and proposition 2.28.
2.5. 1d cobordism hypothesis. We are now ready to state and prove the one-dimensional version of the cobordism hypothesis. Its formulation will be modeled after theorem 2.2. Moreover, we will consider manifolds with a G-structure. In dimension 1 we have to pick a group G with a homomorphism G → O(1) ∼ = Z/2. There are two such interesting homomorphisms: 1 → Z/2 and id : Z/2 → Z/2. These correspond to the oriented bordism category Cobor 1 and the unoriented bordism category Cob1 respectively. The category Cobor 1 has objects given by compact oriented 0-manifolds. Every compact 0-manifold is a finite collection of points. An orientation on a point is the same as an orientation of the line R; there are two such orientations and we denote pt+ and pt− the points equipped with the corresponding orientations. Note that (pt )∨ ∼ = pt . +
−
Let Cd ⊂ C be the full subcategory of dualizable objects. Theorem 2.30 (1d cobordism hypothesis). The functor d ∼ Fun⊗ (Cobor 1 , C) −→ (C )
given by evaluation Z 7→ Z(pt+ ) is an equivalence. Before we give a proof, let us state the following basic theorem whose proof can be found in [Mil65]. Theorem 2.31. Every connected compact 1-manifold is diffeomorphic to the circle S 1 or the interval [0, 1]. Proof of theorem 2.30. Let us describe the category Cobor 1 . Its objects are finite sets of points each of which is either pt+ or pt− . There are 4 connected bordisms: (1) The identity id : pt+ → pt+ . (2) The identity id : pt− → pt− . ` (3) The evaluation bordism ev : pt+ pt− → ∅. ` (4) The coevaluation bordism coev : ∅ → pt− pt+ . Note that the functor Fun⊗ (Cob1 , C) → (Cd )∼ factors as F
Fun⊗ (Cob1 , C) − → DDatf ull (C) −→ (Cd )∼ where the functor F is given by sending Z to the duality data (Z(pt+ ), Z(pt− ), Z(ev), Z(coev)). By proposition 2.26 and proposition 2.27 it is enough to show that F is an equivalence. Let us begin by showing that F is fully faithful. A natural transformation Z1 ⇒ Z2 is given by morphisms Z1 (X) → Z2 (X) for every finite set of oriented points X which are natural under bordisms. Since we consider symmetric monoidal natural transformation, this reduces to the data of maps Z1 (pt+ ) → Z2 (pt+ ) and Z1 (pt− ) → Z2 (pt− ). The naturality axioms can be checked on connected bordisms and using the previous classification of connected bordisms we exactly get relations in the category DDatf ull (C).
24
PAVEL SAFRONOV
Now suppose (x, x∨ , evx , coevx ) ∈ DDatf ull (C) is a duality data. We can construct a topological field theory as follows. We let a aa O Z( pt+ pt− ) = (⊗i x) (⊗j x∨ ) i
j
on objects. We set Z(ev) = evx and Z(coev) = coevx and extend it to disconnected bordisms using the tensor products. To see that this is well-defined, let us check that Z is indeed a functor, i.e. it respects compositions. The only nontrivial relations are exactly the snake relations for the duality data. So far we have classified all oriented one-dimensional topological field theories. Let us also give a statement in the unoriented case. Definition 2.32. Suppose G is a discrete group and C is a category. An action of G on C is the data of a monoidal functor G → Fun(C, C). Definition 2.33. If C and D are categories equipped with a G-action, a G-equivariant functor from C to D is a functor F : C → D together with a natural isomorphism of the two composites in the diagram Fun(C, C) :
F ◦(−)
'
Fun(C, D)
G
7
$
(−)◦F
Fun(D, D) Let us unpack this definition. A data of a G-action on C is given by a collection of functors Fg : C → C together with natural isomorphisms Jg,h : Fh ◦ Fg ∼ = Fgh and : id ∼ = Fe satisfying the following properties: (1) The diagram of natural isomorphisms Fi ◦ Fh ◦ Fg
Jg,h
/
Fi ◦ Fgh
Jh,i
Fhi ◦ Fg
Jg,hi
/
Jgh,i
Fghi
is commutative. (2) The composite
Je,g
Jg,e
Fg ◦ id → − Fg ◦ Fe −−→ Fg is the identity. (3) The composite id ◦ Fg → − Fe ◦ Fg −−→ Fg is the identity.
TOPOLOGICAL QUANTUM FIELD THEORIES
25
Definition 2.34. Suppose C is a category equipped with a G-action. The category of homotopy fixed points ChG is defined as follows: ∼ • Its objects are pairs of an object x ∈ C and a collection of isomorphisms tg : Fg (x) − →x for all g ∈ G such that te = x and the diagram tgh
Fgh (x)
/
O
Jg,h
xO
th
Fh (Fg (x))
Fh (tg )
/
Fh (x)
commutes. • Morphisms are given by morphisms f : x → y in C compatible with the isomorphisms {tg }. Observe that we have an O(1)-action on Cobor 1 given by changing the orientation of the d ∼ manifold. We also have an action of O(1) on (C ) given by sending x 7→ x∨ . Remark 2.35. The O(1)-action on the groupoid (Cd )∼ does not come from an O(1)-action on the category Cd : given a morphism f : x → y of dualizable objects, its dual is a morphism f ∨ : y ∨ → x∨ but there is no way of producing a morphism x∨ → y ∨ unless f is an isomorphism. d ∼ Observe that the functor Fun⊗ (Cobor is equivariant with respect to O(1)1 , C) → (C ) ∨ ∼ ∨ ∼ actions since Z(pt− ) = Z(pt− ) = Z(pt+ ).
Lemma 2.36. The groupoid of homotopy fixed points ((Cd )∼ )hO(1) is the groupoid of objects x ∈ C equipped with a symmetric nondegenerate pairing ev : x ⊗ x → 1. Proof. By definition the groupoid of homotopy fixed points ((Cd )∼ )hO(1) is identified with the following groupoid: ∼ • Its objects are dualizable objects x together with an isomorphism f : x − → x∗ such ∼ that f ∗ : x − → x∗ is equal to f . • Its morphisms are isomorphisms x → y in C compatible with the isomorphisms f . We have an isomorphism Hom(x, x∗ ) ∼ = Hom(x ⊗ x, 1) under which isomorphisms x → x∗ correspond to nondegenerate pairings. Finally, the condition f ∗ = f is equivalent to the symmetry of the pairing. Observe now that the functor Fun⊗ (Cob1 , C) −→ (Cd )∼ given by Z 7→ Z(pt) factors as Fun⊗ (Cob1 , C) −→ ((Cd )∼ )hO(1) −→ (Cd )∼ . Indeed, if we let Z(pt) = x, the value on the cap gives a nondegenerate evaluation pairing ev : x ⊗ x → 1. Moreover, it is symmetric since we have a diffeomorphism of bordisms shown in fig. 7. Theorem 2.37 (1d cobordism hypothesis for unoriented manifolds). The functor Fun⊗ (Cob1 , C) −→ ((Cd )∼ )hO(1) given by Z 7→ Z(pt) is an equivalence. Proof. The proof is a slight variation on the proof of theorem 2.30, so we omit it.
26
PAVEL SAFRONOV
∼ = Figure 7. Symmetry of the pairing. 2.6. Morse and Cerf theory. We are going to give another proof of the one-dimensional cobordism hypothesis which uses Morse theory. Similar ideas are used to prove the higherdimensional case as well. The main idea is to use Morse functions to present manifolds combinatorially via a handle decomposition. We will state all theorems here without proofs following the notes [Fre12, Lecture 23]. We refer to [Mil63] for details. Definition 2.38. Suppose f : M → R is a smooth function. We say m ∈ M is a critical point of f if the differential dfm : Tm M → R is zero. We refer to f (m) as the critical value. Suppose m ∈ M is a critical point. Then we have its matrix of second derivatives called the Hessian d2 fm ∈ Sym2 (T∗m M ). Definition 2.39. A critical point m ∈ M of f : M → R is nondegenerate if the Hessian d2 fm ∈ Sym2 (T∗m M ) is a nondegenerate bilinear form. The index of a nondegenerate critical point is the number of negative eigenvalues of the Hessian d2 fm . A function f is Morse if all its critical points are nondegenerate. Suppose f : M → R is a Morse function and consider f −1 (a) as we vary a ∈ R. Consider two values a0 < a00 , none of which are critical. If there are no critical values in [a0 , a00 ], then f −1 ([a0 , a00 ]) is diffeomorphic to the product of [a0 , a00 ] and f −1 (a0 ). If there is a single critical point of index q, then f −1 ([a0 , a00 ]) is obtained from f −1 (a0 ) by attaching a q-handle in the following sense. ˜ be n-manifolds with boundary together with a map Definition 2.40. Let M and M g : S q−1 × Dn−q → M. ˜ is obtained by attaching a q-handle to M if M ˜ is homeomorphic to We say that M a M (Dq × Dn−q ). g
Example 2.41. Consider the height function on the 2-torus as a Morse function shown in fig. 8. It has four critical points whose indices are shown on the picture. We have the following transitions: • The manifold f −1 (x) is empty if x < a. • The manifold f −1 ([a − , a + ]) is diffeomorphic to the disk D2 , i.e. it is obtained by attaching a 0-handle to the empty manifold. • The manifold f −1 ([a−, b+]) is obtained by attaching a 1-handle to f −1 ([a−, b−]) ∼ = D2 as shown in fig. 9. • The manifold f −1 ([a − , c + ]), the 2-torus minus a disk, is obtained by attaching a 1-handle to f −1 ([a − , c − ]) as shown in fig. 10.
TOPOLOGICAL QUANTUM FIELD THEORIES
27
2 d
f:
1
−→
1
c b
a 0 Figure 8. Morse function on the torus.
∼ =
Figure 9. Passing through critical value b.
∼ =
Figure 10. Passing through critical value c.
∼ =
Figure 11. Passing through critical value d. • The manifold f −1 ([a − , d + ]), i.e. the full 2-torus, is given by attaching a 2-handle to f −1 ([a − , d − ]) as shown in fig. 11.
28
PAVEL SAFRONOV
We will be interested in studying Morse functions on bordisms, so it will be convenient to consider Morse functions compatible with the structure of the bordism. Definition 2.42. Let M be a bordism from N0 to N1 . An excellent function f : M → R is a Morse function satisfying the following conditions: (1) f takes the constant value a0 on N0 . (2) f takes the constant value a1 on N1 . (3) The critical points m1 , . . . , mN have distinct values c1 . . . , cN which satisfy a0 < c1 < · · · < cN < a1 . Proposition 2.43. The space of excellent functions on a given bordism C ∞ (M ) is open and dense in C ∞ (M ). An excellent function on a bordism allows one to write a bordism as a composition of some elementary ones. Definition 2.44. A bordism M from N0 to N1 equipped with an excellent Morse function f : M → R is elementary if f has a unique critical point. A presentation of a given bordism in terms of a composition of elementary bordisms is not unique since the space of excellent functions is not connected. Instead, we will embed excellent functions into a connected space. Definition 2.45. A function f : M → R has a birth-death singularity at m ∈ M if there are local coordinates x1 , . . . , xn near m such that f (x) = x31 + x22 + · · · + x2r − x2r+1 − · · · − x2n + f (m). We refer to fig. 12 for an example of a family of functions ft : [0, 1] → R which are Morse away from a single value t = t0 at which ft0 has a birth-death singularity.
Figure 12. A family of excellent functions passing through a birth-death singularity. Definition 2.46. Let M be a bordism from N0 to N1 . We say that a function f : M → R is good if one of the two things happen: (1) f is excellent away from a single point which is a birth-death singularity. (2) f is excellent away from two nondegenerate critical points m1 , m2 ∈ M such that f (m1 ) = f (m2 ). Theorem 2.47 (Cerf). Let M be a bordism from N0 to N1 . The space of good functions on M is connected.
TOPOLOGICAL QUANTUM FIELD THEORIES
29
Thus, elementary bordisms give generators for the bordism category and relations are obtained either by having a bordism with a birth-death singularity or two critical points with the same critical value. We will now use it to give a Morse-theoretic proof of the one-dimensional cobordism hypothesis. Proof of theorem 2.30. As before, we know that the objects of Cobor 1 are pt+ and pt− . The only elementary oriented one-dimensional bordisms are shown in fig. 13.
Figure 13. Elementary one-dimensional bordisms. A natural transformation of topological field theories Z1 ⇒ Z2 is given by morphisms Z1 (pt+ ) → Z2 (pt+ ) and Z1 (pt− ) → Z2 (pt+ ) which are compatible with the elementary f ull bordisms. Therefore, the functor Fun⊗ (Cobor (C) is fully faithful. 1 , C) → DDat To prove essential surjectivity, consider a dualizable object x ∈ C and set Z(pt+ ) = x and Z(pt− ) = x∨ . Given a one-dimensional bordism M we can choose an excellent function to present it as a composition of elementary bordisms to which we assign ev and coev. To see that the result is independent of the choice of the excellent function, we have to show that families of good functions give equalities of the compositions. In one dimension a good function necessarily has a birth-death singularity. Passing through a birth-death singularity gives a relation between an excellent function with two critical points and that with no critical points, so the only option is to have a family of the form shown in fig. 12. This gives two relations depending on the orientation of the interval. But these relations are exactly the duality axioms. 3. Two-dimensional TFTs 3.1. Frobenius algebras. We are now going to study 2-dimensional topological field theories. We will see that oriented 2d TFTs are the same as commutative Frobenius algebras. In this section we introduce the necessary algebraic definitions. Let C be a monoidal category. Definition 3.1. A Frobenius algebra in C is a unital algebra A ∈ C equipped with a trace θ θ : A → 1 such that A ⊗ A → A → − 1 is a nondegenerate pairing. In particular, we see that a Frobenius algebra is necessarily dualizable. Passing to the dual, we see that A is also a coalgebra. Remark 3.2. However, A is not a bialgebra. In fact, to define a bialgebra, C has to be braided while Frobenius algebras can be defined in any monoidal category. Proposition 3.3. A Frobenius algebra in C is equivalent to the following data: an object A ∈ C equipped with an associative unital multiplication m : A ⊗ A → A and a coassociative counital comultiplication ∆ : A → A ⊗ A such that the three maps ∆⊗id
id⊗m
A ⊗ A −−−→ A ⊗ A ⊗ A −−−→ A ⊗ A,
30
PAVEL SAFRONOV m
∆
A⊗A− →A− →A⊗A and id⊗∆
m⊗id
A ⊗ A −−−→ A ⊗ A ⊗ A −−−→ A ⊗ A are equal. A morphism of Frobenius algebras is an algebra morphism A1 → A2 which preserves the trace. We denote by FrobAlg(C) the category of Frobenius algebras in C. Lemma 3.4. The category of Frobenus algebras FrobAlg(C) is a groupoid. Proof. Indeed, given a morphism f : A1 → A2 of Frobenius algebras, it preserves the traces and hence the pairings A1 ⊗ A1 → 1 and A2 ⊗ A2 → 1. But we have shown in lemma 2.25 that such a morphism admits an inverse given by (f ∗ )−1 : A2 → A1 . Now suppose C is a braided monoidal category. Definition 3.5. A Frobenius algebra A ∈ C is symmetric if θ(ab) = θ(ba) for all a, b ∈ A, i.e. the composites σ m θ A⊗A→ − A⊗A− →A→ − A and m θ A⊗A− →A→ − A are equal. Given an algebra object A ∈ C we have its opposite algebra Aop ∈ C defined by applying the braiding to the multiplication. Consider the enveloping algebra Ae = A⊗Aop with the componentwise multiplication. Then A becomes an Ae -module via the left and right action on itself. If A ∈ C is dualizable, the dual A∗ is also an Ae -module. Lemma 3.6. Let A ∈ C be an algebra object. A Frobenius structure on A is the same as an ∼ isomorphism A − → A∗ of A-modules. A symmetric Frobenius structure on A is the same as ∼ an isomorphism A − → A∗ of Ae -modules. Definition 3.7. An algebra object A ∈ C is called separable if A is dualizable as an Ae -module. In this case we denote the dual Ae -module by A! = HomAe (A, Ae ). Consider the multiplication map µ : Ae → A of Ae -modules. We have the following result from [DI71, Proposition 1.1]. Proposition 3.8. An algebra A in Vect is separable iff it admits a separability idemponent p ∈ Ae such that µ(p) = 1 and (a ⊗ 1) · p = (1 ⊗ a)p for any a ∈ A. Proof. The map Ae → A is surjective, so A is finitely-generated as an Ae -module. Therefore, by proposition 2.10 it is separable iff A is projective as an Ae -module. Consider an exact sequence of Ae -modules µ 0 −→ J −→ Ae → − A −→ 0, where J is defined to be the kernel of µ. Note that J is generated by elements a ⊗ 1 − 1 ⊗ a for a ∈ A. Then A is projective iff the morphism µ splits. Such a splitting is uniquely determined on 1 ∈ A which gives the required element p ∈ Ae .
TOPOLOGICAL QUANTUM FIELD THEORIES
31
Remark 3.9. Note that p ∈ Ae is indeed idempotent: p2 − p = (p − 1 ⊗ 1)p, but p − 1 ⊗ 1 lies in J and hence (p − 1 ⊗ 1) · p = 0. The following is [DI71, Proposition 2.1]. Proposition 3.10. A separable k-algebra is finite dimensional. Example 3.11. Let A = Matn (k) be the algebra of n × n-matrices over a field k. Let {eij } be the basis of A consisting of matrices with 1 in the (i, j) entry and zero otherwise. Then X pj = eij ⊗ eji i
for any j is a separability idempotent and hence A is separable. Theorem 3.12 (Artin–Wedderburn). Suppose k is an algebraically closed field. Then a separable k-algebra A is isomorphic to a direct sum of matrix algebras. 3.2. Examples. (1) The simplest example of a Frobenius algebra is A = Matn (k), the algebra of n × nmatrices over a field k. As we have previously mentioned, it is separable. The trace tr : Matn (k) → k endows A with a symmetric Frobenius structure. Moreover, one can show that any symmetric Frobenius structure on A is proportional to the trace. (2) Let G be a finite group and consider the group algebra A = k[G]. It admits a natural trace θ : A → k given by extracting the coefficient of the unit e ∈ G. Thus, it becomes a symmetric Frobenius algebra. If the order |G| does not divide char k, we have a separability idempotent given by 1 X p= g ⊗ g −1 |G| g∈G and hence in this case A is separable. (3) Consider the ring of class functions A = C[G]G , i.e. elements of C[G] invariant with respect to the adjoint action. We have a trace θ : A → C given by the formula X 1 θ(f ) = f ([g]), |CG (g)| [g]∈G/G
where the sum goes over conjugacy classes in G. The algebra A is separable, in fact we can split M A= C · χi , where χi are the characters of irreducible representations. By the Schur orthogonality relations θ(χi ) = 1 and so θ is nondegenerate. (4) By theorem 3.12 every commutative separable algebra over C is isomorphic to a direct sum of the trivial algebra C. Let us denote by Pi ∈ A the projectors, so that M A= C · Pi . i
A Frobenius structure θ : A → C is given by specifying nonzero numbers θi = θ(Pi ). The coproduct is given by ∆(Pi ) = θi−1 Pi ⊗ Pi .
32
PAVEL SAFRONOV
(5) Let M be a compact oriented manifold of even dimension and consider the gradedcommutative algebra A = H• (M ; k). It defines a commutative algebra in the category of Z/2-graded vector spaces VectZ/2 . Moreover, the integration pairing defines a map H • (M ; k) → k which is nondegenerate by Poincaré duality. In other words, we obtain a commutative Frobenius algebra in VectZ/2 . Note that this Frobenius algebra is almost never separable. (6) Let X be a compact Calabi–Yau manifold of complex dimension n with the complex volume form Ω. Consider the graded-commutative algebra M A= Hp (X, ∧q TX ), p,q
where the (p, q) piece is in degree p + q. It has a natural trace map given by the composite Ω → Hn (X, OX ) −→ C, A −→ Hn (X, ∧n TX ) − where the last map is given by Serre duality. This is again a commutative Frobenius algebra in VectZ/2 . ∂f (7) Let f : Cn → C be a polynomial function and denote by fi (z) = ∂z its derivatives. i Consider the Jacobian algebra of f . A = C[z1 , . . . , zn ] (f1 , . . . , fn ) . Suppose that f has at most one singularity at the origin and dim A < ∞. Let B be the real n-ball defined by fi (z) = for some small . Then we have a trace θ : A → C given by Z g(z) · dz1 ∧ · · · ∧ dzn 1 . θ(g) = (2πi)2n B f1 (z) . . . fn (z) Then A equipped with the trace θ is a Frobenius algebra. 3.3. Oriented 2d TFTs. Now we are going to relate oriented 2-dimensional topological field theories to commutative Frobenius algebras. We begin by constructing a functor Fun⊗ (Cobor 2 , C) −→ CFrobAlg(C). An object of Cobor 2 is given by a finite collection of oriented circles. Note that the antipodal map identifies the opposite orientations on S 1 , so we only have to consider a single orientation. Since we restrict to symmetric monoidal functors, we see that symmetric monoidal functors are determined by their value on S 1 equipped with some extra structure. 1 Given a symmetric monoidal functor Z : Cobor 2 → C we consider the object A = Z(S ) ∈ C. It has multiplication m : A ⊗ A → A, unit e : 1 → A and trace θ : A → 1 given by evaluating Z on bordisms fig. 14, fig. 15 and fig. 16 respectively. Proposition 3.13. The data (A, m, e, θ) defines a commutative Frobenius algebra structure on A. Proof. The associativity of the multiplication follows from the diffeomorphism of bordisms shown in fig. 17. The commutativity of the multiplication follows from the diffeomorphism of bordisms shown in fig. 18. The unitality of the multiplication follows from the diffeomorphism
TOPOLOGICAL QUANTUM FIELD THEORIES
33
Figure 14. Multiplication m : A ⊗ A → A.
Figure 15. Unit e : 1 → A.
Figure 16. Trace θ : A → 1.
∼ =
Figure 17. Associativity of the multiplication.
∼ =
Figure 18. Commutativity of the multiplication.
∼ =
Figure 19. Unitality of the multiplication. of bordisms shown in fig. 19. Let us note that by comutativity of the multiplication we only m θ need to check one unitality axiom. The nondegeneracy of the pairing A ⊗ A − → A → − 1 e ∆ follows from the existence of the coevaluation 1 → − A− → A ⊗ A which satisfies the duality axiom by fig. 20. The previous proposition asserts that we have a well-defined functor Fun⊗ (Cobor 2 , C) −→ CFrobAlg(C). The following theorem has been a folklore (going back to a thesis of Dijkgraaf) and the first careful proof is given by [Abr96] using the classification of 2d surfaces. We will instead use Morse-theoretic arguments given in [MS06, Section A.1].
34
PAVEL SAFRONOV
∼ =
Figure 20. Nondegeneracy of the pairing. Theorem 3.14 (Classification of oriented 2d TFTs). The functor Fun⊗ (Cobor 2 , C) −→ CFrobAlg(C) given by Z 7→ Z(S 1 ) is an equivalence. Proof. As we have already mentioned, objects of Cobor 2 are given by finite disjoint unions of circles. The elementary two-dimensional bordisms are shown in fig. 21 which give generators for the bordism category Cobor 2 .
Figure 21. Elementary two-dimensional bordisms. Thus, morphisms Z1 ⇒ Z2 of 2d TFTs are morphisms Z1 (S 1 ) → Z2 (S 1 ) which preserve multiplication m, comultiplication ∆, unit e and trace θ. In the definition of CFrobAlg(C) we merely ask morphisms to preserve m, e and θ, so to prove full faithfulness, we need to show that any morphism in CFrobAlg(C) automatically preserves the comultiplication as well. Indeed, a morphism of commutative Frobenius algebras f : A1 → A2 preserves the multiplication and the trace, so it intertwines the evaluation pairings A1 ⊗A1 → 1 and A2 ⊗A2 → 1. Given an evaluation pairing we can uniquely reconstruct the coevaluation by proposition 2.26, so the morphism f automatically preserves the coevaluation pairing as well. But the comultiplication can be written in terms of the multiplication and the coevaluation as shown in fig. 22. To prove essential surjectivity, we have to understand relations between elementary cobordisms. Such relations are given by one-parameter families of good functions which are excellent away from a single value of the parameter. To reduce the number of computations, observe that if a relation is satisfied, so is its time-reversal. We begin with the case of a birth-death singularity. It creates two critical points of adjacent indices, so it gives a relation between bordisms shown in fig. 23. Such a relation corresponds to unitality of the multiplication (see fig. 19). Next, we have to analyze the case of two critical points having the same critical value c. If f −1 ([c − , c + ]) is disconnected, the corresponding relation follows from naturality of the
TOPOLOGICAL QUANTUM FIELD THEORIES
35
∼ =
Figure 22. Reconstruction of the comultiplication from the coevaluation.
∼ =
Figure 23. Passing through a birth-death singularity. tensor product. Thus, it is enough to consider connected bordisms with two critical points having the same critical value. It is easy to see that both critical points have to have index 1. Using Morse theory one can show that the Euler characteristic of such a bordism is −2. But it is also equal to 2 − 2g − n, where g is the genus and n is the number of boundary components. Therefore, the two cases are given by the torus with two boundary components and the sphere with four boundary components. In each elementary bordism the number of outgoing circles is equal to the number of incoming circles ±1. Therefore, we have the following three cases: (1) We consider a bordism from S 1 to S 1 of genus 1 shown in fig. 24.
Figure 24. Case (1). There are many ways to cut it, but all of them give rise to the same composition ∆
m
A− →A⊗A− →A so there is nothing to prove. ` ` (2) We consider a bordism from S 1 S 1 to S 1 S 1 of genus 0 shown in fig. 25 It gives rise to three compositions m
∆
A⊗A− →A− → A ⊗ A, ∆⊗id
id⊗m
id⊗∆
m⊗id
A ⊗ A −−−→ A ⊗ A ⊗ A −−−→ A ⊗ A and
A ⊗ A −−−→ A ⊗ A ⊗ A −−−→ A ⊗ A.
36
PAVEL SAFRONOV
Figure 25. Case (2). They are equal by proposition`3.3. ` (3) We consider a bordism from S 1 S 1 S 1 to S 1 of genus 0 shown in fig. 26.
Figure 26. Case (3). This gives rise to the two compositions m⊗id
m
id⊗m
m
A ⊗ A ⊗ A −−−→ A ⊗ A − →A and A ⊗ A ⊗ A −−−→ A ⊗ A − →A which are equal by associativity of the multiplication. Under the equivalence between oriented 2d TFTs and Frobenius algebras, we have the following examples of oriented 2d TFTs based on examples of Frobenius algebras from section 3.2: (1) Suppose G is a finite group and A = C[G]G is the algebra of class functions. Then it defines the (untwisted) 2d Dijkgraaf–Witten theory associated to G. (2) Let X be a compact Calabi–Yau manifold and A = H• (X, ∧• TX ). Then it defines the B-model associated to X. (3) Let f : Cn → C be a polynomial with an isolated singularity at the origin and A the Jacobian algebra of f . Then it defines the Landau–Ginzburg B-model associated to f . Let us also work out the complete description of 2d TFT associated to a separable Frobenius C-algebra M A= C · Pi , i 2
where (Pi ) = Pi and θ(Pi ) = θi .
TOPOLOGICAL QUANTUM FIELD THEORIES
37
Proposition 3.15. Let Σg be the closed oriented Riemann surface of genus g and Z the oriented 2d TFT associated to the commutative Frobenius aglebra A. Then X 1−g Z(Σg ) = θi . i
Proof. Dualizing the product, we obtain the coproduct ∆(Pi ) = θi−1 Pi ⊗ Pi . Σg can be written as a composition of a cup, g “genus–changing operators” given by fig. 24 and a cap. The value of the 2d TFT on the genus-changing operator is the morphism A → A given by Pi 7→ θi−1 Pi . Therefore, X 1−g Z(Σg ) = θi . i
3.4. Unoriented, spin and framed 2d TFTs. In this section we explain the classification of 2d TFTs on unoriented, spin and framed manifolds. We begin with the unoriented case. The functor Fun⊗ (Cobor 2 , C) → CFrobAlg(C) did not use orientations on the manifolds, so it still makes sense in the unoriented case. However, the corresponding commutative Frobenius algebra has extra structure. Definition 3.16. An unoriented commutative Frobenius algebra is a commutative Frobenius algebra (A, m, θ) in C together with an involution φ : A → A and an element c : 1 → A which satisfy the following equations: (1) The composites c⊗id m A −−→ A ⊗ A − →A and φ c⊗id m A −−→ A ⊗ A − →A→ − A are equal. (2) The composites c⊗c m 1 −−→ A ⊗ A − →A and φ⊗id e ∆ m 1→ − A− → A ⊗ A −−−→ A ⊗ A − →A are equal. Remark 3.17. These were introduced under the name “extended Frobenius algebras” in [TT06]. Let CFrobAlgunor (C) be the category of unoriented commutative Frobenius algebras in C. We are now going to construct a functor Fun⊗ (Cob2 , C) −→ CFrobAlgunor (C). Given a symmetric monoidal functor Z : Cob2 → C, we have a commutative Frobenius algebra A = Z(S 1 ). Let r : S 1 → S 1 be an orientation-reversing diffeomorphism and let
38
PAVEL SAFRONOV
Cylr be the corresponding mapping cylinder which is a bordism from S 1 to S 1 . Note that since π0 (Diff(S 1 )) = Z/2, it is uniquely defined. The value of Z on Cylr defines a morphism φ : A → A. Now consider the Möbius band RP2 − D2 which is a bordism from ∅ to S 1 . Then Z(RP2 − D2 ) gives a morphism c : 1 → A. Note that RP2 is not orientable, so the value of c is not expressible from the data of a commutative Frobenius algebra. Theorem 3.18 ([TT06]). The functor Fun⊗ (Cob2 , C) −→ CFrobAlgunor (C) is an equivalence. Next, we move on to the spin case. Let Σ be a compact oriented surface (possibly with boundary). We have the following description of spin structures on Σ. Proposition 3.19. Pick an arbitrary complex structure on Σ. A spin structure on Σ is given by choosing a complex line bundle L on Σ together with an isomorphism of complex line bundles L⊗2 ∼ = TΣ . Proof. Let GL+ (2, R) be the connected component of the identity in GL(2, R). The inclusion GL(1, C) → GL+ (2, R) is a weak equivalence. Indeed, both groups retract onto U(1) → SO(2) which is an isomorphism. Spin(2) → SO(2) is equivalent to the homomorphism U(1) → U(1) given by z 7→ z 2 . A z2
reduction of the structure group of a complex line bundle along U(1) − → U(1) is the same as a choice of a square root. In particular, we see that spin structures form a torsor over H1 (Σ, Z/2). If Σ is closed, one also has the following alternative description of spin structures. Definition 3.20. Let V be a Z/2-vector space and h−, −i : V ⊗ V → Z/2 a nondegenerate pairing. Its quadratic refinement is a quadratic form q : V → Z/2 such that q(x + y) − q(x) − q(y) = hx, yi,
∀x, y ∈ V.
We have an intersection form H1 (Σ, Z/2) ⊗ H1 (Σ, Z/2) → Z/2 which is nondegenerate by Poincaré duality. Proposition 3.21 ([Joh80]). Let Σ be a closed Riemann surface. A choice of a spin structure on Σ is the same as a choice of a quadratic refinement of the intersection pairing. Two quadratic refinements of the intersection pairing differ by a linear form on H1 (Σ, Z/2) which is the same as a class in H1 (Σ, Z/2). Example 3.22. The cylinder Σ = S 1 × [0, 1] has two spin structures since H1 (Σ, Z/2) ∼ = Z/2. 2 We can embed Σ ⊂ D as an annulus and the induced spin structure on Σ is known as the Neveu–Schwarz (or bounding) spin structure. The other spin structure is known as the Ramond (or non-bounding) spin structure. Note that spin structures on S 1 are simply Z/2-covers. Using the obvious embedding Spin(1) → Spin(2) a spin structure on S 1 induces one on S 1 × [0, 1]. For example, the trivial spin structure on S 1 corresponds to the Ramond spin structure while the nontrivial spin structure on S 1 corresponds to the Neveu–Schwarz spin structure.
TOPOLOGICAL QUANTUM FIELD THEORIES
39
Definition 3.23. A spin commutative Frobenius algebra is given by the following data: • A commutative Frobenius algebra AN S ∈ C. • An AN S -module AR together with a multiplication map AR ⊗ AR −→ AN S . • An involution T : AR → AR . These have to satisfy the following axioms: • The pairing θ AR ⊗ AR −→ AN S → − 1 is nondegenerate. • The diagram coevR
1
/
AR ⊗ AR
coevN S
/
m
AN S ⊗ AN S
m
AN S
is commutative. • The diagram AR ⊗ AR
σ
/
AR ⊗ AR
T ⊗id
AR ⊗ AR
/
m
m
AN S
is commutative. • The diagram AN S ⊗ AR
/
m
id⊗T
AR
AN S ⊗ AR
m
/
T
AR
is commutative. Note that applying θ to the second axiom we deduce that dim(AN S ) = dim(AR ) ∈ EndC (1). Let CFrobAlgspin (C) be the category of spin commutative Frobenius algebras. We are now going to construct a functor Fun⊗ (Cobspin , C) −→ CFrobAlgspin (C). 2 spin 1 Let SN be the circle equipped with the Neveu–Schwarz spin structure and S ∈ Cob2 1 similarly for SR . Given a symmetric monoidal functor Z : Cobspin → C we let 2 1 AN S = Z(SN S ),
AR = Z(SR1 ).
S Given the generator g ∈ Z/2 we obtain an automorphism of a spin structure. Let CylN g 1 1 and CylR g be the corresponding mapping cylinders on SN S and SR . One can check that S S ∼ CylN while CylR = CylN g e g gives rise to an involution T : AR → AR . The disk has a unique spin structure which gives rise to the unit and trace maps
e : 1 → AN S ,
θ : AN S → 1.
40
PAVEL SAFRONOV
Next, we consider the pair of pants. It admits a complex structure with the trivial complex tangent bundle. Therefore, spin structures correspond to Z/2-bundles. The holonomy of the corresponding Z/2-bundles on the neighborhood of the boundary circles is multiplicative under the pair of pants cobordism, so we have cobordisms a a a 1 1 1 1 1 1 1 1 SN S → S , S S → S , S SR1 → SN S NS NS NS R R R S. These give rise to the multiplications AN S ⊗ AN S → AN S ,
AR ⊗ AR → AN S
and the action map AN S ⊗ AR → AR . The proof of the following statement is sketched in [MS06, Section 2.6] Theorem 3.24. The functor Fun⊗ (Cobspin , C) −→ CFrobAlgspin (C) 2 is an equivalence. Finally, we discuss the framed case. Definition 3.25. Let M be an m-manifold. An n-framing on M for n ≥ m is the trivialization of Rn−m ⊕ TM . The objects of Cobf2 r are equipped with the 2-framing, so we have to understand the set of 2-framings on the circle. Given any two framings on the circle, get the induced automorphism of the trivial rank 2 vector bundle, so the set of isomorphism classes of framings is a torsor over π0 (Map(S 1 , GL(2, R))) ∼ = Z/2 × Z. Here the Z/2-factor comes from the two orientations on the circle. However, there is an orientation-reversing diffeomorphism S 1 → S 1 which gives rise to an isomorphism between the corresponding objects. Thus, we can parametrize isomorphism classes of (connected) objects of Cobf2 r by Z, so we get circles Sn1 for n ∈ Z. We can fix the basepoint as follows. Any 1-framing on S 1 induces a 2-framing on S 1 , but there is a unique 1-framing on S 1 compatible with the given orientation. We call the induced 2-framing S11 . The disk gives rise to bordisms ∅ → S01 ,
S21 → ∅.
One can see that the pair of pants admits framings which give rise to bordisms a 1 1 Sn1 Sm → Sn+m . For instance, an incoming disk from the identity Sn1 → Sn1 to obtain the pair `we1 can remove 1 1 of pants Sn S0 → Sn . Remark 3.26. A 2-framing on a surface is the same as an orientation and a nonvanishing vector field v. Indeed, we may arbitrarily pick a complex structure J compatible with the given orientation and define the 2-framing to be given by the pair of vector fields (v, Jv). Such a description is useful for pictorial representation of framed bordisms.
TOPOLOGICAL QUANTUM FIELD THEORIES
41
We can summarize all the structure in the following definition. Definition 3.27. L A framed commutative Frobenius algebra is a Z-graded commutative algebra A = n An together with a trace θ : A2 → 1 satisfying the following conditions: • The induced pairings θ
An ⊗ A2−n −→ A2 → − 1 are nondegenerate. • The morphisms coev
1 −−→ An ⊗ A2−n −→ A2 are independent of n. Remark 3.28. Note that the commutativity on the multiplication does not involve signs, i.e. we require that the composite σ
An ⊗ Am → − Am ⊗ An −→ Am+n is equal to the multiplication An ⊗ Am → An+m . Let CFrobAlgf r (C) be the category of framed commutative Frobenius algebras. We have a functor Fun⊗ (Cobf2 r , C) −→ CFrobAlgf r (C) L which sends Z 7→ n Z(Sn1 ). Theorem 3.29. The functor Fun⊗ (Cobf2 r , C) −→ CFrobAlgf r (C) is an equivalence. 3.5. Dijkgraaf–Witten theory. Our next goal will be to understand extended 2d topological field theories. The relevant formalism will be introduced in the next sections. In this section we will motivate extended topological field theories by considering a special example which is the Dijkgraaf–Witten theory introduced in [DW90]. This will be an example where the path integral formalism of section 1.1 can be made precise. An interesting modification from the formalism of section 1.1 will be that our spaces will have symmetries, i.e. they will be groupoids. Let us introduce the necessary formalism. Definition 3.30. Let G be a groupoid. A function f : G → C is given by a function f : Ob G → C on the set of objects such that f (x) = f (y) for any isomorphic objects x, y ∈ G. We denote by O(G) the commutative algebra of complex-valued functions on G. Note that it is isomorphic to the algebra of functions O(π0 (G)) on the set of isomorphism classes of objects. Definition 3.31. Given a morphism of groupoids Φ : G1 → G2 , we define the pullback map Φ∗ : O(G2 ) −→ O(G1 ) by declaring (Φ∗ f )(x) = f (Φ(x)) for every f ∈ O(G2 ) and x ∈ G1 .
42
PAVEL SAFRONOV
Lemma 3.32. Let Φ : G1 → G2 and Ψ : G2 → G3 be morphisms of groupoids. Then Φ∗ Ψ∗ = (Ψ ◦ Φ)∗ : O(G3 ) → O(G1 ). Proof. Given f : G3 → C we have ((Ψ ◦ Φ)∗ f )(x) = f (Ψ(Φ(x)), (Ψ∗ f )(y) = f (Ψ(y)) and (Φ∗ Ψ∗ f )(x) = f (Ψ(Φ(x))). Definition 3.33. Let Φ : G1 → G and Ψ : G2 → G be morphisms of groupoids. Their homotopy pullback G1 ×hG G2 is given by the following groupoid: • Its objects are triples consisting of objects x ∈ G1 and y ∈ G2 together with an ∼ isomorphism h : Φ(x) − → Ψ(y). • Its morphisms are morphisms f : x1 → x2 in G1 and g : y1 → y2 in G2 such that the diagram Φ(x1 )
Φ(f )
/
Φ(x2 )
h1
Ψ(y1 )
Ψ(g)
/
h2
Ψ(y2 )
commutes. ˜ : G1 ×h G2 → G2 and Ψ ˜ : G1 ×h G2 → G1 which make the We have natural projections Φ G G diagram of groupoids G1 ×hG G2
˜ Ψ
/
G1
˜ Φ
/
Ψ
G2
Φ
G
commute up to a natural isomorphism. Definition 3.34. A diagram of groupoids G0
˜ Ψ
/
˜ Φ
G1
G2
Ψ
/
Φ
G
commuting up to a natural isomorphism is homotopy Cartesian if the natural morphism G0 → G1 ×hG G2 is an equivalence of groupoids. Definition 3.35. Let Φ : G1 → G2 be a morphism of groupoids and x ∈ G2 an object. The homotopy fiber Φ−1 (x) is the homotopy pullback of Φ and x : ∗ → G2 . Explicitly, the homotopy fiber Φ−1 (x) is given by the following groupoid:
TOPOLOGICAL QUANTUM FIELD THEORIES
43
• Its objects are pairs of an object z ∈ G1 and an isomorphism g : Φ(z) → x. • Morphisms (z1 , g1 ) → (z2 , g2 ) are given by morphisms h : z1 → z2 in G1 such that g2 ◦ Φ(h) = g1 . Definition 3.36. A morphism of groupoids Φ : G1 → G2 is finite if for every object x ∈ G2 the groupoid Φ−1 (x) has finitely many isomorphism classes of objects and the automorphism group of every object in Φ−1 (x) is finite. We say that a groupoid G is finite if the natural map G → ∗ is finite. Lemma 3.37. Let Φ : G1 → G2 be a morphism of finite groupoids. Then Φ is finite. Definition 3.38. Given a finite morphism of groupoids Φ : G1 → G2 we define the pushforward map Φ∗ : O(G1 ) −→ O(G2 ) by declaring X f (z) (Φ∗ f )(x) = |AutΦ−1 (x) (z)| −1 [z]∈π0 (Φ
(x))
for every f ∈ O(G1 ) and x ∈ G2 . The reason we quotient by the order of the automorphism group is that we want the following statement to hold. Proposition 3.39. Let Φ : G1 → G2 and Ψ : G2 → G3 be finite morphisms of groupoids. Then Ψ∗ Φ∗ = (ΨΦ)∗ : O(G1 ) → O(G3 ). Proof. Given f : G1 → C we have (Φ∗ f )(y) =
X [x]∈π0 (Φ−1 (y))
f (x) |AutΦ−1 (y) (x)|
.
Given g : G2 → C we have (Ψ∗ g)(z) =
X [y]∈π0 (Ψ−1 (z))
g(y) |AutΨ−1 (z) (y)|
.
Therefore, (Ψ∗ Φ∗ f )(z) =
X
X
[y]∈π0 (Ψ−1 (z)) [x]∈π0 (Φ−1 (y))
f (x) . |AutΨ−1 (z) (y)||AutΦ−1 (y) (x)|
We have a morphism of groupoids (ΨΦ)−1 (z) → Ψ−1 (z) given by sending (x, h1 : ΨΦ(x) → z) to (Φ(x), h1 ). Its homotopy fiber over y ∈ Ψ−1 (z) can be identified with Φ−1 (y). Therefore, (ΨΦ)∗ f is given by the same formula. Φ
Ψ
Example 3.40. Let G be a finite group and consider the morphisms ∗ − → ∗/G − → ∗. The algebras of functions on each groupoid is C. The pushforward (Φ)∗ is given by multiplication by |G| while the pushforward (Ψ)∗ is given by multiplication by 1/|G|. The composite ΨΦ is the identity and so is Ψ∗ Φ∗ .
44
PAVEL SAFRONOV
Proposition 3.41 (Base change formula). Let Φ : G1 → G and Ψ : G2 → G be morphisms of ˜ : G1 ×G G2 → G2 is also finite and we have an equality groupoids where Φ is finite. Then Φ ˜ ∗Ψ ˜∗ Ψ∗ Φ∗ = Φ of maps O(G1 ) → O(G2 ). ˜ −1 (y) is the following groupoid: Proof. Let y ∈ G2 be an object. Then Φ h
η
• Its objects are x ∈ G1 , y˜ ∈ G2 equipped with isomorphisms Φ(x) → − Ψ(˜ y ) and y˜ → − y. • Morphisms (x1 , y˜1 , h1 , η1 ) → (x2 , y˜2 , h2 , η2 ) are given by morphisms f : x1 → x2 and g : y˜1 → y˜2 such that the diagrams g
y˜1 η1
y
/ y˜2 η2
and h1
Φ(x)
/
Ψ(˜ y1 )
Φ(f ) h2
Φ(x2 )
/
Ψ(g)
Ψ(˜ y2 )
commute. ˜ −1 (y) which sends (x ∈ G1 , h : Φ(x) → Ψ(y)) There is an obvious functor Φ−1 (Ψ(y)) → Φ ˜ to (x, y˜ = y, h) which can be shown to be fully faithful and essentially surjective. Thus, Φ is finite since Φ is so. Now consider a function f : G1 → C. On the one hand, we have X f (x) (Φ∗ f )(z) = |AutΦ−1 (z) (x)| −1 [x]∈π0 (Φ
(z))
and (Ψ∗ Φ∗ f )(y) =
f (x)
X [x]∈π0 (Φ−1 (Ψ(y)))
|AutΦ−1 (Ψ(y)) (x)|
.
On the other hand, we have ˜ ∗ f )(x, y, h) = f (x) (Ψ ˜ −1 (y), and, using the equivalence Φ−1 (Ψ(y)) ∼ =Φ X f (x) ˜ ∗ f )(y) = (Φ∗ Ψ . |AutΦ−1 (Ψ(y)) (x)| −1 [x]∈π0 (Φ
(Ψ(y))
Now we fix a finite group G. We are going to construct the Dijkgraaf–Witten theory in dimension n as a symmetric monoidal functor F : Cobn −→ Vect.
TOPOLOGICAL QUANTUM FIELD THEORIES
45
Let M be a compact manifold. We denote by LocG (M ) the groupoid of principal Gbundles. Since G is finite and M is compact, the groupoid LocG (M ) is finite. Given a closed (n − 1)-manifold N ∈ Cobn we let Z(N ) = O(LocG (N )). Given a bordism M from N1 to N2 we have a correspondence LocG (M ) resin
resout
x
&
LocG (N1 )
LocG (N2 )
We let Z(M ) : Z(N1 ) → Z(N2 ) be the map (res2 )∗ res∗1 : O(LocG (N1 )) −→ O(LocG (N2 )). To check that it defines a functor, we have to show that compositions are sent to compositions. Given a bordism M1 from N1 to N2 and M2 from N2 to N3 we can identify LocG (M2 ◦ M1 ) with the homotopy pullback LocG (M1 ) ×hLocG (N2 ) LocG (M2 ). Therefore, we get a diagram LocG (M2 ◦ M1 ) p1
p2
v
(
LocG (M1 )
LocG (M2 )
res1in
res2in
res1out
w
(
LocG (N1 )
res2out
v
LocG (N2 )
'
LocG (N2 )
We have Z(M2 ) ◦ Z(M1 ) = (res2out )∗ (res2in )∗ (res1out )∗ (res1in )∗ : Z(N1 ) −→ Z(N3 ). By the base change formula (res2in )∗ (res1out )∗ = (p2 )∗ p∗1 . Therefore, Z(M2 ) ◦ Z(M1 ) = (res2out )∗ (p2 )∗ p∗1 (res1in )∗ = (res2out ◦ p2 )∗ (res1in ◦ p1 )∗ = Z(M2 ◦ M1 ). The symmetric monoidal structure comes from natural isomorphisms a O(LocG (N1 N2 )) ∼ = O(LocG (N1 ) × LocG (N2 )) ∼ = O(LocG (N1 )) ⊗ O(LocG (N2 )). We can summarize it as follows. Proposition 3.42. The assignment N 7→ O(LocG (N )) and M 7→ (resout )∗ res∗in defines a symmetric monoidal functor Z : Cob2 −→ Vect. Remark 3.43. Given a class c ∈ Hn (G, C× ) in group cohomology one can consider the twisted Dijkgraaf–Witten theory. Namely, such a class defines a complex line bundle on LocG (N ) for a closed (n − 1)-manifold and one defines Zc (N ) to be the space of sections of this line bundle.
46
PAVEL SAFRONOV
Let us temporarily restrict to the case n = 2. By theorem 3.18 we get the structure of an unoriented commutative Frobenius algebra on Z(S 1 ) = O(LocG (S 1 )). We can identify the groupoid LocG (S 1 ) with the groupoid whose objects are given by elements of g and morphisms are given by conjugation: for every h ∈ G there is a morphism g → hgh−1 . We denote this groupoid by G/G. Then Z(S 1 ) ∼ = O(G/G) = O(G)G . Let us compute the unoriented Frobenius structure. • The unit is given by evaluating Z on the disk with the outgoing circle. We can identify LocG (D) ∼ = pt/G, so that the correspondence LocG (∅) ← LocG (D) → LocG (S 1 ) is identified with pt/G
∗
# ~
G/G
The pullback O(∗) → O(pt/G) is the identity map C → C and the pushforward O(pt/G) → O(G/G) sends a number x to the function xδe . In other words, δe is the unit. • The multiplication is given by evaluation ` Z on the pair of pants bordism MP oP . We 1 can identify the correspondence LocG (S S 1 ) ← LocG (MP oP ) → LocG (S 1 ) with (G × G)/G p1 ×p2
m
w
&
G/G × G/G
G/G,
where (G × G)/G denotes the groupoid whose objects are (g1 , g2 ) ∈ G × G with morphisms (g1 , g2 ) → (hg1 h−1 , hg2 h−1 ) for every h ∈ G. Then Z(MP oP ) = ∗ : O(G)G ⊗ O(G)G → O(G)G is given by X (f1 ∗ f2 )(x) = f1 (g1 )f2 (g2 ) g1 ,g2 , g1 g2 =x
which is known as the convolution product which in fact defines an associative product on all of O(G). We have an isomorphism C[G] ∼ = O(G) given by sending g ∈ C[G] to the function that takes value 1 on g and zero otherwise. Under this isomorphism the standard product on C[G] corresponds to the convolution product on O(G). We can identify X (f2 ∗ f1 )(x) = f1 (g2 g1 g2−1 )f2 (g2 ), g1 ,g2 , g1 g2 =x
so this product is commutative on the subspace O(G)G ⊂ O(G) of conjugationinvariant functions.
TOPOLOGICAL QUANTUM FIELD THEORIES
47
• The trace is given by evaluating Z on the disk with the incoming circle. Again we can identify the correspondence LocG (S 1 ) ← LocG (D) → LocG (∅) with pt/G {
∗
G/G
The pullback O(G/G) → O(pt/G) ∼ = C is given by evaluating function on the unit conjugacy class and the pushforward O(pt/G) → C is given by multiplication by 1/|G|. Therefore, the trace is given by θ(f ) =
f (e) . |G|
• The diffeomorphism S 1 → S 1 given by antipodal involution gives rise to an isomorphism LocG (S 1 ) ∼ = LocG (S 1 ) of groupoids. Explicitly, it is the isomorphism G/G ∼ = G/G given by g 7→ g −1 . Therefore, the involution φ : O(G)G → O(G)G is given by φ(f )(g) = f (g −1 ). • The element c ∈ Z(S 1 ) is given by evaluating Z on the Möbius band RP2 −D2 which is a bordism from ∅ to S 1 . Instead of describing c, we will describe the functional (c, −) : Z(S 1 ) → C where (−, −) is the natural pairing on Z(S 1 ). The correspondence LocG (∅) ← LocG (RP2 − D2 ) → LocG (S 1 ) is equivalent to the correspondence G/G g7→g 2
∗
# ~
G/G
Therefore, we get (c, f ) =
1 X f (g 2 ) |G| g∈G
known as the Frobenius–Schur indicator . Let {Vi } be the collection of irreducible complex representations of G and denote by χi ∈ O(G)G the character of Vi . For the following, see e.g. [Isa76, Theorem 2.8, Theorem 2.13]. Proposition 3.44. χi ∗ χj = δij
|G| χi . dim Vi
The morphism M given by Vi 7→ χi is an isomorphism.
CVi −→ O(G)G
48
PAVEL SAFRONOV
Therefore, the algebra O(G)G admits idempotents Pi =
dim Vi χi |G|
so that θi = θ(Pi ) =
(dim Vi )2 . |G|2
Let |Hom(π1 (Σg ), G)| be the cardinality of the set of homomorphisms π1 (Σ) → G from the fundamental group of Σg to G. The following statement is known as Mednyh’s formula proved by [Med78]. Proposition 3.45. We have |Hom(π1 (Σg ), G)| X = |G| i
dim Vi |G|
2−2g .
Proof. By proposition 3.15 we obtain Z(Σg ) =
X dim Vi 2−2g |G|
i
.
More directly, by construction Z(Σg ) is given by applying pullback and pushforward of functions along the correspondence LocG (Σg ) x
&
LocG (∅)
LocG (∅).
This is equivalent to the number of elements of LocG (Σg ) weighted by their automorphisms. Fixing a point x ∈ Σg we get a restriction morphism LocG (Σg ) → LocG (pt) ∼ = pt/G. The homotopy fiber of the unique object of pt/G is equivalent by the set Hom(π1 (Σg , x), G). By proposition 3.39 we can compute the pushforward along LocG (Σg ) → pt as a composite of pushforwards along LocG (Σg ) −→ pt/G −→ pt, so Z(Σg ) =
|Hom(π1 (Σg , x), G)| . |G|
Considering the case g = 0 we obtain the following corollary. Corollary 3.46. Let G be a finite group and {Vi } the collection of complex irreducible representations. Then X |G| = (dim Vi )2 . i
TOPOLOGICAL QUANTUM FIELD THEORIES
49
We are now going to sketch how to construct the Dijkgraaf–Witten theory as an extended topological field theory. Namely, given a closed (n−2)-manifold P we are supposed to assign to it a linear category Z(P ). To an (n − 1)-dimensional bordism N from P1 to P2 we are supposed to assign a linear functor Z(N ) : Z(P1 ) → Z(P2 ). Definition 3.47. Let G be a groupoid. The category of vector bundles on G is Vect(G) = Fun(G, Vect). For a morphism of groupoids Φ : G1 → G2 we have a pullback functor Φ∗ : Vect(G2 ) → Vect(G1 ) given by precomposition. Lemma 3.48. Let Φ : G1 → G2 be a morphism of groupoids. The pullback functor Φ∗ : Vect(G2 ) −→ Vect(G1 ) admits a right adjoint Φ∗ : Vect(G1 ) −→ Vect(G2 ). Example 3.49. Consider pt/G for a finite group G. Then Vect(pt/G) is equivalent to the category Rep G of representations of G. The pullback functor p∗ : Vect(pt) → Vect(pt/G) associated to the natural projection p : pt/G → pt gives the functor Vect → Rep G which sends a vector space V to itself equipped with the trivial action of G. Its right adjoint p∗ : Rep G → Vect is the functor of G-invariants. Given a pair of morphisms Φ : G1 → G and Ψ : G2 → G of groupoids, consider the homotopy pullback G1 ×hG G2
˜ Ψ
/
˜ Φ
G1
G2
Ψ
/
Φ
G
˜ ∗Ψ ˜ ∗ obtained via the composite We have a natural transformation of functors Ψ∗ Φ∗ → Φ ˜ ∗Φ ˜ ∗ Ψ∗ Φ∗ ∼ ˜ ∗Ψ ˜ ∗ Φ∗ Φ∗ → Φ ˜ ∗Ψ ˜∗ Ψ∗ Φ∗ → Φ =Φ ˜∗ a Φ ˜ ∗ and the last map is given where the first map is given by the unit of the adjunction Φ by the counit of the adjunction Φ∗ a Φ∗ . Proposition 3.50 (Base change formula). Let Φ : G1 → G and Ψ : G2 → G be morphisms of groupoids. Then the natural transformation ˜ ∗Ψ ˜∗ Ψ∗ Φ∗ −→ Φ is an isomorphism. Proof sketch. The claim is invariant under equivalences of groupoids, so we may assume that ` all groupoids are skeletal, i.e. are equal to i pt/Gi for some groups Gi . Next, the general statement follows from the case when G1 is connected. In this case only the connected component of the image of Φ : G1 → G affects the calculation, so we may
50
PAVEL SAFRONOV
assume G is connected as well. Since G1 ×hG (−) preserves coproducts, we may assume G2 is connected as well. Thus, we consider the homotopy pullback ˜ Φ
pt/G ×hpt/H pt/K
/
pt/G
˜ Ψ
/
Ψ
pt/K
Φ
pt/H
We can identify the groupoid pt/G with the groupoid H/(G × H) whose objects are given by H and morphisms by G × H. Thus, we are reduced to proving the claim for the diagram H/(G × K)
˜ Ψ
/
H/(G × H)
˜ Φ
pt/K
Ψ
/
Φ
pt/H
for which it is obvious.
Let Ho Catk be the category whose objects are linear categories and morphisms are given by natural isomorphism classes of functors. Then proceeding as before, from the base change formula one deduces that the assignment P 7→ Vect(LocG (P )) gives rise to a functor Cobn−1 −→ Ho Catk . So, at the moment we have constructed the n-dimensional Dijkgraaf-Witten theory on n and (n−1)-manifolds and on (n−1) and (n−2)-manifolds separately. One can also construct the theory uniformly for manifolds of dimension n, (n − 1) and (n − 2), but for this we need to unify Cobn−1 and Cobn into a single bicategory that we will denote by Bord[n−2,n] . 4. Extended two-dimensional TFTs In this section we will discuss extended two-dimensional topological field theories, i.e. we allow 1-manifolds to have boundaries. This increases the categorical level: the bordism category becomes the bordism bicategory. This significantly increases the complexity of checking all axioms; in these notes we will merely explain general ideas without giving complete proofs. 4.1. Bicategories. Let X be a topological space. We can construct a category π≤1 (X) known as the fundamental groupoid as follows: • Objects of π≤1 (X) are points of X. • Morphisms in π≤1 (X) from x ∈ X to y ∈ X are given by homotopy classes of paths from x to y. • The composition is given by concatenation of paths. Note that composition is not associative on the level of paths, but it becomes associative when we pass to homotopy classes. A prototypical bicategory will be the fundamental 2-groupoid of a topological space X: • Objects of π≤2 (X) are points of X.
TOPOLOGICAL QUANTUM FIELD THEORIES
51
• 1-morphisms from x ∈ X to y ∈ X are given by paths γ : x → y. • 2-morphisms from γ1 to γ2 are given by homotopy equivalence classes of homotopies h : γ1 ⇒ γ2 that fix endpoints that we can pictorially draw as γ1 h
x
y
γ2
Note that as we have already mentioned, the composition of 1-morphisms in π≤2 (X) is not strictly associative, i.e. the corresponding paths are not equal, but it is associative up to a 2-isomorphism, i.e. the corresponding paths are homotopic. Let us also recall that if M is a monoid, we have the category ∗/M . Similarly, if C is a monoidal category, ∗/C will be a bicategory. Thus, axioms of a bicategory will be “multiobject” versions of axioms of a monoidal category. Definition 4.1. A bicategory C is given by the following data: • A set of objects Ob C. • Given two objects x, y ∈ Ob C we get a category HomC (x, y). Its objects γ are 1morphisms from x to y. Morphisms in HomC (x, y) from γ1 to γ2 are 2-morphisms from γ1 to γ2 . • Composition functors cx,y,z : HomC (y, z) × HomC (x, y) −→ HomC (x, z) for all x, y, z ∈ C. For two objects g ∈ HomC (y, z) and f ∈ HomC (x, y), i.e. 1-morphisms in C, we denote cx,y,z (g, f ) = g ◦ f. For two morphisms β ∈ HomC (y, z) and α ∈ HomC (x, y), i.e. 2-morphisms in C, we denote cx,y,z (β, α) = β ∗ α. • Objects idx ∈ HomC (x, x). • Natural isomorphisms (associators) ∼
αh,g,f : (h ◦ g) ◦ f − → h ◦ (g ◦ f ) for every objects x, y, z, w ∈ C and 1-morphisms f : x → y, g : y → z, h : z → w. • Natural isomorphisms (unitors) ∼
∼
λf : idy ◦ f − → f,
ρf : f ◦ idx − → f.
These have to satisfy the following axioms: • (Triangle axiom) The diagram αg,idy ,f
(g ◦ idy ) ◦ f ρg ∗idf
&
g◦f
x
/
g ◦ (idy ◦ f )
idg ∗λf
commutes for all objects x, y, z ∈ C and 1-morphisms f : x → y and g : y → z.
52
PAVEL SAFRONOV
• (Pentagon axiom) The diagram ((k ◦ h) ◦ g) ◦ f αk,h,g ∗idf
αk◦h,g,f
u
)
(k ◦ (h ◦ g)) ◦ f αk,h◦g,f
(k ◦ h) ◦ (g ◦ f )
k ◦ ((h ◦ g) ◦ f )
/
idk ∗αh,g,f
αk,h,g◦f
k ◦ (h ◦ (g ◦ f ))
commutes for all objects x, y, z, w, u ∈ C and morphisms f : x → y, g : y → z, h : z → w and k : w → u. Definition 4.2. A bicategory C is a strict 2-category if the associators and unitors are identities. It is immediate that for any object x ∈ C of a bicategory the category HomC (x, x) becomes a monoidal category under composition by comparing the axioms of a bicategory with that of a monoidal category (see definition 1.2). Moreover, if C is a strict 2-category, HomC (x, x) is a strict monoidal category. Recall that given two sets, there is a notion of a morphism between them. Given two categories, there is a notion of a functor between them and given two functors F, G : C → D there is a notion of a natural transformation. Thus, there is a strict 2-category Cat of categories, functors and natural transformations. Our next goal is to study maps between bicategories. Definition 4.3. A functor of bicategories F : C → D is given by the following data: • A morphism of sets F : Ob C → Ob D. • For every pair of objects x, y ∈ C a functor Fx,y : HomC (x, y) → HomD (F (x), F (y)). • Natural isomorphisms ∼
φg,f : F (g) ◦ F (f ) − → F (g ◦ f ) for all objects x, y, z ∈ C and morphisms f : x → y and g : y → z. • Isomorphisms ∼ φx : idF (x) − → F (idx ) for every object x ∈ C. These have to satisfy the following axioms: • The diagram idF (h) ∗φg,f
/
F (h) ◦ (F (g) ◦ F (f )) αF (h),F (g),F (f )
4
F (h) ◦ F (g ◦ f ) φh,g◦f
(
(F (h) ◦ F (g)) ◦ F (f ) φh,g ∗idF (f )
F (h ◦ (g ◦ f )) 6
*
F (h ◦ g) ◦ F (f )
/ φh◦g,f
F (αh,g,f )
F ((h ◦ g) ◦ f )
TOPOLOGICAL QUANTUM FIELD THEORIES
53
commutes for all objects x, y, z, w ∈ C and morphisms f : x → y, g : y → z and h : z → w. • The diagrams φy ∗idF (f )
idF (y) ◦ F (f ) λF (f )
/
F (idy ) ◦ F (f )
φidy ,f
F (f ) o
F (idy ◦ f )
F (λf )
and idF (f ) ∗φx
F (f ) ◦ idF (x) ρF (f )
/
F (f ) ◦ F (idx )
F (f ) o
φf,idx
F (f ◦ idx )
F (ρf )
commute for all objects x, y ∈ C and morphisms f : x → y. If M1 , M2 are monoidal categories and ∗/Mi denotes the associated one-object bicategories, then a functor of bicategories ∗/M1 → ∗/M2 is the same as a monoidal functor M1 → M2 . Similarly to the case of categories, given two functors of bicategories F, G : C → D there is a notion of a natural transformation η : F ⇒ G. It is given by the data of morphisms ηx : F (x) → G(x) for every object x ∈ C but the naturality diagram F (x)
F (f )
ηx
/
G(x)
F (f )
/
F (y)
ηy
G(y)
commutes up to a specified 2-isomorphism. Moreover, these data have to be compatible with the bicategorical structure. Remark 4.4. Note that while functors of bicategories F, G : ∗ /M1 → ∗/M2 are the same as monoidal functors M1 → M2 , natural transformations F → G of functors are not the same as monoidal natural transformations of the monoidal functors M1 → M2 . Instead, one has to consider ∗/Mi as pointed bicategories and require functors, natural transformations etc to preserve the pointing. Since natural transformations between functors of bicategories involve the data of morphisms ηx : F (x) → F (y), we have a notion of modifications between natural transformations η1 , η2 which are given by 2-morphisms σx : η1,x ⇒ η2,x which satisfy certain axioms. We refer to [Lei98] for details on natural transformations and modifications. Definition 4.5. Let C be a bicategory. Its homotopy category Ho C is defined as follows: • Objects of Ho C are objects of C. • HomHo C (x, y) = π0 (HomC (x, y)), i.e. the isomorphism classes of 1-morphisms. Given functors F, G : C → D between bicategories, they induce functors F, G : Ho C → Ho D on the underlying homotopy categories. Similarly, natural transformations η : F ⇒ G induce natural transformations between the induced functors on the homotopy categories.
54
PAVEL SAFRONOV
4.2. Symmetric monoidal bicategories. Recall that a monoid is a pointed set equipped with a binary operation which satisfies the associativity and unitality axioms. A monoid is commutative if it satisfies an additional axiom that x · y = y · x. Going up in the categorical level, a monoidal category is a pointed category equipped with a bifunctor where the associativity and unitality are witnessed by specified natural isomorphisms which in addition satisfy some coherences (pentagon and triangle axioms). A braided monoidal category is one where the commutativity axiom is witnessed by a specified ∼ natural isomorphism (braiding) σ : x ⊗ y − → y ⊗ x which satisfies some coherences (hexagon axiom). However, a braided monoidal category is not completely commutative and one can σ σ ask if the isomorphism x ⊗ y → − y ⊗x → − x ⊗ y is the identity which is the axiom of symmetric monoidal categories. Note that while commutativity for sets had two levels (associative and commutative monoids), commutativity for categories had three levels (monoidal, braided monoidal and symmetric monoidal categories). Now we are going to sketch the analogous notions for bicategories. This time commutativity will have four levels: monoidal, braided monoidal, sylleptic monoidal and symmetric monoidal bicategories. We refer to [Sch09, Appendix C] for detailed definitions of these notions, while we will merely sketch the data without specifying the axioms they have to satisfy. Definition 4.6. A monoidal bicategory consists of the following structure: • • • • •
A bicategory C. The tensor product functor ⊗ : C × C → C. An object 1 ∈ C. ∼ Natural isomorphisms (associators) αx,y,z : (x ⊗ y) ⊗ z − → x ⊗ (y ⊗ z) for x, y, z ∈ C. The invertible modification π (pentagonator ) which witnesses commutativity of the diagram ((x ⊗ y) ⊗ z) ⊗ w αx,y,z ⊗idw
αx⊗y,z,w
u
)
(x ⊗ (y ⊗ z)) ⊗ w αx,y⊗z,w
(x ⊗ y) ⊗ (z ⊗ w)
x ⊗ ((y ⊗ z) ⊗ w)
/
idx ⊗αy,z,w
αx,y,z⊗w
x ⊗ (y ⊗ (z ⊗ w))
∼
∼
• Natural isomorphisms (unitors) λx : 1 ⊗ x − → x and ρx : x ⊗ 1 − → x. • Invertible modifications (2-unitors) which witness commutativity of the diagrams αx,1,y
(x ⊗ 1) ⊗ y ρx ⊗idy
&
x
/
x⊗y
x ⊗ (1 ⊗ y)
idx ⊗λy
TOPOLOGICAL QUANTUM FIELD THEORIES α1,x,y
(1 ⊗ x) ⊗ y λx ⊗idy
/
&
55
1 ⊗ (x ⊗ y)
λx⊗y
x
x⊗y and αx,y,1
(x ⊗ y) ⊗ 1 ρx⊗y
/
idx ⊗ρy
x
&
x ⊗ (y ⊗ 1)
x⊗y Definition 4.7. A braided monoidal bicategory consists of the following structure: • A monoidal bicategory C. ∼ • A natural isomorphism (braiding ) σx,y : x ⊗ y − → y ⊗ x. • Invertible modifications R and S which witness commutativity of the diagrams σx,y⊗z
x⊗y⊗z σx,y ⊗idz
/
y7 ⊗ z ⊗ x
idy ⊗σx,z
'
y⊗x⊗z and σx⊗y,z
x⊗y⊗z idx ⊗σy,z
'
/
z7 ⊗ x ⊗ y
σx,z ⊗idy
x⊗z⊗y where we omit associators. Definition 4.8. A sylleptic monoidal bicategory consists of the following structure: • A braided monoidal bicategory C. • An invertible modification (syllepsis) which witnesses commutativity of the diagram x⊗y
σx,y
/
y⊗x
σy,x
/
x9 ⊗ y
idx⊗y
Finally, a symmetric monoidal bicategory is a sylleptic monoidal bicategory where a certain equation on syllepsis is satisfied. The following statement is clear from the definitions. Proposition 4.9. • Let C be a monoidal bicategory. Then the homotopy category Ho C is a monoidal category. • Let C be a braided monoidal bicategory. Then Ho C is a braided monoidal category. • Let C be a sylleptic monoidal bicategory. Then Ho C is a symmetric monoidal category.
56
PAVEL SAFRONOV
Using the previous observation, we can define dualizable objects in monoidal bicategories as follows. We will restrict to the case of symmetric monoidal bicategories, but the general case is identical. Definition 4.10. Let C be a symmetric monoidal bicategory. An object x ∈ C is dualizable if it is dualizable as an object of the symmetric monoidal category Ho C. Explicitly, it means the following. A dualizable object x ∈ C is given by the following data: • An object x∨ ∈ C. • The evaluation map ev : x ⊗ x∨ → 1. • The coevaluation map coev : 1 → x∨ ⊗ x. • Cusp 2-isomorphisms τ1 and τ2 which witness commutativity of the diagrams idx ⊗coev
x
/
x ⊗ x∨ ⊗ x
ev⊗idx
/6
x
idx
and x∨
coev⊗idx∨
/
x∨ ⊗ x ⊗ x∨
idx∨ ⊗ev
6/
x∨
idx∨
Remark 4.11. Note that we do not require the cusp 2-isomorphisms to satisfy any additional axioms. There is a notion of a coherent duality data (see e.g. [Pst14, Section 2.1]) where the cusp 2-isomorphisms satisfy the so-called swallowtail axioms. In fact, as shown in [Pst14, Theorem 2.7] one can modify one of the cusp 2-isomorphisms so that the swallowtail axioms are satisfied. Recall from definition 2.23 the notion of a nondegenerate pairing in a 1-category. We can similarly define this notion in a bicategory. Definition 4.12. Let C be a symmetric monoidal bicategory and ev : x⊗y → 1 a morphism. It is a nondegenerate pairing if for all objects z, w ∈ C the functor ⊗y
ev
HomC (z, w ⊗ x) −→ HomC (z ⊗ y, w ⊗ x ⊗ y) − → HomC (z ⊗ y, w) is an equivalence. Proposition 4.13. Let C be a symmetric monoidal bicategory. A morphism ev : x ⊗ y → 1 is a nondegenerate pairing in C iff it is so in the homotopy category Ho C. Proof. If ev is a nondegenerate pairing in C, the functor ⊗y
ev
F : HomC (z, w ⊗ x) −→ HomC (z ⊗ y, w ⊗ x ⊗ y) − → HomC (z ⊗ y, w) is an equivalence, hence ⊗y
ev
π0 (HomC (z, w ⊗ x)) −→ π0 (HomC (z ⊗ y, w ⊗ x ⊗ y)) − → π0 (HomC (z ⊗ y, w)) is an isomorphism, so ev is nondegenerate in Ho C.
TOPOLOGICAL QUANTUM FIELD THEORIES
57
Conversely, suppose ev is a nondegenerate pairing in Ho C. By proposition 2.26 we can pick the coevaluation coev : 1 → y ⊗ x in Ho C which realizes y as the dual of x. We get the functor ⊗x
coev
G : HomC (z ⊗ y, w) −→ HomC (z ⊗ y ⊗ x, w ⊗ x) −−→ HomC (z, w ⊗ x) and the cusp 2-isomorphisms establish natural isomorphisms F G ∼ = id and GF ∼ = id.
4.3. Fully dualizable objects. Recall that dualizability in a symmetric monoidal category was a certain finiteness condition inherent in 1-dimensional topological field theories. We will see in the next sections that 2-dimensional topological field theories are finite in an even stronger sense and they correspond to fully dualizable objects that we will define shortly. Definition 4.14. Let C be a bicategory with objects x, y ∈ C and a 1-morphism f : x → y. A right adjoint to f is given by the following data: • A 1-morphism g : y → x. • A 2-morphism : f ◦ g ⇒ idy called the counit. • A 2-morphism η : idx ⇒ g ◦ f called the unit. These have to satisfy the following axioms: (1) The composite 2-morphism idf ∗η
∗idf
η∗idg
idg ∗
f −−−→ f ◦ g ◦ f −−−→ f is equal to idf . (2) The composite 2-morphism g −−−→ g ◦ f ◦ g −−−→ g is equal to idg . In this situation we also say that f is left adjoint to g. Example 4.15. Let C = Cat be the bicategory of categories, functors and natural transformations. Then adjoints for morphisms in Cat coincide with the notion of adjoint functors. Example 4.16. Let M be a monoidal category and C = pt/M be the one-object bicategory. A 1-morphism in C is the same as an object x of M. Its right adjoint in C is the same as a right dual to x in M. Example 4.17. Let C be a category considered as a bicategory with only identity 2-morphisms. Then a 1-morphism has a left or right adjoint iff it is invertible. Indeed, in this case η and are identities, i.e. f ◦ g = idy and g ◦ f = idx . As for dualizability, adjointness is a property and can be formulated purely in terms of the unit of the adjunction (equivalently, in terms of the counit). Proposition 4.18. Let C be a bicategory with objects x, y ∈ C and 1-morphisms f : x → y and g : y → x. A 2-morphism η : idx → g ◦ f is a unit of the adjunction between f and g iff the morphism of sets ◦f
η
HomHomC (y,y) (h ◦ g, k) −→ HomHomC (x,y) (h ◦ g ◦ f, k ◦ f ) → − HomHomC (x,y) (h, k ◦ f ) is an isomorphism for all 1-morphisms k : y → y and h : x → y. In this case the choice of the counit is unique.
58
PAVEL SAFRONOV
Proof. The proof is similar to the proof of proposition 2.26, so we omit it.
Let us mention that adjoints in symmetric monoidal bicategories interact in an expected way with duality. Proposition 4.19. Suppose f : x → y is a 1-morphism in a bicategory C and g : y → x its right adjoint. Moreover, suppose x and y are dualizable. Then f ∨ : y ∨ → x∨ is right adjoint to g ∨ : x∨ → y ∨ . Proof. Let : f ◦ g ⇒ idy be the counit of the adjunction and η : idx ⇒ g ◦ f be the unit. We have a 2-isomorphism (f ◦ g)∨ ∼ = g ∨ ◦ f ∨ . The 1-morphism (f ◦ g)∨ by definition is given by coevy ⊗id
id⊗(f ◦g)⊗id
id⊗evy
y ∨ −−−−−→ y ∨ ⊗ y ⊗ y ∨ −−−−−−−→ y ∨ ⊗ y ⊗ y ∨ −−−−→ y ∨ . In particular, this composite has a natural 2-morphism 0 : (f ◦ g)∨ ⇒ (idy )∨ ∼ = idy∨ . In the same way one can construct the counit of the adjunction. The adjunction axioms for g ∨ a f ∨ follow from those for f a g. Definition 4.20. A bicategory C has adjoints if every 1-morphism admits a left and a right adjoint. Given a bicategory C, there is an initial bicategory Cadj whih has adjoints and which admits a functor Cadj → C. It can be constructed as follows. Let Cad ⊂ C be the bicategory which has the same objects as C, 1-morphisms in Cad are 1-morphisms in C which have adjoints in C and 2-morphisms between them are the same in both bicategories. We can inductively define C0 = C, Cn = (Cn−1 )ad . Then we get an inverse system Cn of bicategories and we define Cadj = lim Cn . n
In other words, 1-morphisms in C are given by those 1-morphisms in C which admit an infinite string of left and right adjoints. adj
Definition 4.21. Let C be a symmetric monoidal bicategory. An object x ∈ C is fully dualizable if it is dualizable in Cadj . In other words, a fully dualizable object x ∈ C admits the following data: • Its dual x∨ ∈ C. • The evaluation and coevaluation maps ev : x ⊗ x∨ → 1,
coev : 1 → x∨ ⊗ x.
• The iterated left adjoints evL , (evL )L , . . . to evaluation and iterated left adjoints coevL , (coevL )L , . . . to coevaluation. • The iterated right adjoints evR , (evR )R , . . . and coevR , (coevR )R , . . . . Definition 4.22. Let x ∈ C be a dualizable object such that the evaluation map admits a right adjoint evR . The Serre map of x is id ⊗evR
id⊗σ
ev⊗id
x S : x −−x−−−→ x ⊗ x ⊗ x∨ −−−→ x ⊗ x∨ ⊗ x −−−−→ x.
TOPOLOGICAL QUANTUM FIELD THEORIES
59
The following is shown in [Pst14, Proposition 3.5]. Proposition 4.23. Let x ∈ C be a dualizable object. Suppose that the evaluation and coevaluation maps admit right adjoints evR and coevR . Then the Serre map is invertible. Proof. We construct the inverse Serre map as coev⊗id
id⊗coevR
σ⊗id
T : x −−−−→ x∨ ⊗ x ⊗ x −−−→ x ⊗ x∨ ⊗ x −−−−−→ x. Denote coev g = σ ◦ coev : 1 → x ⊗ x∨ and ev e = ev ◦ σ : x∨ ⊗ x → 1 which realize x∨ as a right dual to x. Then it is easy to see that evR ∼ g coevR ∼ e ◦ (idx∨ ⊗ T ). = (S ⊗ idx∨ ) ◦ coev, = ev Passing to right adjoints in the duality axiom we see that the identity idx is isomorphic to the composite evR ⊗id
id⊗coevR
x −−−−→ x ⊗ x∨ ⊗ x −−−−−→ x. Substituting the above expressions for evR and coevR we obtain T ∼ = S Using the duality axiom for (coev, g ev) e we can simplify the composite on the left to ST : x → x. Therefore, ST ∼ id . = x Passing to right adjoints in the other duality axiom for (coev, ev) we deduce that
S
T
∼ =
Passing to the dual we get that T S ∼ = idx . Therefore T is the inverse to S and hence S is invertible. It might appear that to show that an object is fully dualizable requires a lot of checks since we need to exhibit an infinite number of adjoints. We will now show that we can significantly reduce the number of adjoints we need to exhibit (see [Lur09, Proposition 4.2.3] and [Pst14, Theorem 3.9] for related statements). Proposition 4.24. Let C be a symmetric monoidal bicategory. The following properties of x ∈ C are equivalent: (1) The object x ∈ C is fully dualizable. (2) There is a nondegenerate pairing ev : x ⊗ x∨ → 1 which admits right and left adjoints evR , evL . (3) x ∈ C is dualizable and ev and coev admit right adjoints evR and coevR . Proof.
60
PAVEL SAFRONOV
• (1) ⇒ (2) is obvious. • (2) ⇒ (3). By proposition 2.22 the evaluation map ev : x ⊗ x∨ → 1 is dual to coevaluation coev : 1 → x∨ ⊗ x. Therefore, by proposition 4.19 the evaluation map ev has a left adjoint iff coev has a right adjoint. • (3) ⇒ (1). Let T : x → x be the inverse to S. Then T and S are both sided adjoints of each other. We have evR ∼ g = (S ⊗ idx∨ ) ◦ coev. Therefore, for any integer n ev e ◦ (id ⊗ S n ) a (id ⊗ S 1−n ) ◦ coev. We also have coevR ∼ e ◦ (idx∨ ⊗ T ). = ev Therefore, for any integer m (id ⊗ S m ) ◦ coev g a ev ◦ (S −1−m ⊗ idx∨ ). This shows that ev and coev have an infinite chain of right and left adjoints. Let Cf d ⊂ C be the bicategory whose objects are fully dualizable objects of C while 1morphisms and 2-morphisms are all morphisms in C. As before, we denote by (Cf d )∼ its underlying 2-groupoid where we only keep invertible 1-morphisms and invertible 2-morphisms. As we are going to show now, the Serre automorphism has an interpretation in terms of an SO(2)-action on (Cf d )∼ . Recall from definition 2.32 the notion of an action of a discrete group on a category. This can be generalized to actions of topological groups as follows. Let G be a topological group. Then its fundamental 2-groupoid π≤2 (G) is a monoidal bicategory. Definition 4.25. Let G be a topological group and C a bicategory. A G-action on C is a monoidal functor π≤2 (G) → Fun(C, C). Remark 4.26. Monoidal functors between bicategories form a bicategory: 1-morphisms are given by monoidal transformations and 2-morphisms are given by monoidal modifications. Thus, the collection of G-actions on a given bicategory organizes itself into a 2-category. As a space we can identify SO(2) ∼ = S 1 . In particular, π≤2 (SO(2)) ∼ = π≤1 (SO(2)) ∼ = pt/Z, where we regard Z as a monoid, hence pt/Z is a monoidal category, i.e. a monoidal bicategory where all 2-morphisms are identities. A functor pt/Z → Fun(C, C) is the same as a choice of an endofunctor F : C → C together with a natural isomorphism F ⇒ F . Since we are looking for monoidal functors, the unique object of pt/Z is the monoidal unit while the monoidal unit of Fun(C, C) is the identity functor id : C → C. Thus, we expect that an SO(2)-action corresponds to a natural isomorphism S : id ⇒ id of the identity functor which satisfies some further conditions. More precisely, we have the following statement shown in [Sch14, Section 18].
TOPOLOGICAL QUANTUM FIELD THEORIES
61
Proposition 4.27. Let C be a bicategory. The bicategory of SO(2)-actions on C is equivalent to the following category: • Objects are natural isomorphisms S : id ∼ = id of the identity functor id : C → C satisfying the following property. Let T be its inverse and consider the composite 2morphism in the diagram idx
7
x
Sx
Tx
' 7 Ex
x Tx
'
Sx
x idx
where the 2-morphism at the top is given by T S ∼ = id, the 2-morphism in the middle square is given by the naturality of S with respect to Tx and the 2-morphism at the bottom is given by T S ∼ = id. The required property is that the composite 2-morphism idx → idx is the identity 2-morphism for all x ∈ C. • 1-morphisms from S 1 to S 2 are given by modifications S 1 ⇒ S 2 of natural isomorphisms idx ⇒ idx . • 2-morphisms are the identities. To show that the Serre automorphism can be lifted to give an SO(2)-action on (Cf d )∼ , observe that the bicategory of G-actions on a bicategory C1 is equivalent to the bicategory of G-actions on a bicategory C2 if C1 ∼ = C2 . Proposition 4.28. Let C be a symmetric monoidal bicategory. Let FDDat(C) be the following bicategory: • Its objects are objects x, x∨ ∈ C equipped with a nondegenerate pairing ev : x⊗x∨ → 1, a morphism evR : 1 → x ⊗ x∨ and the unit of the adjunction η : id ⇒ evR ◦ ev. Moreover we assume ev has a left adjoint. ∨ R • 1-morphisms (x, x∨ , evx , evR x , ηx ) → (y, y , evy , evy , ηy ) are given by 1-morphisms f : x → y and g : x∨ → y ∨ in C together with 2-isomorphisms witnessing commutativity of the diagrams x ⊗ x∨
evR x
evx
&
f ⊗g
y ⊗ y∨
evy
9
1
x8 ⊗ x∨
1
f ⊗g evR y
%
y ⊗ y∨
which are compatible with the units ηx and ηy . • 2-morphisms (f1 , g1 ) ⇒ (f2 , g2 ) are given by 2-morphisms f1 ⇒ f2 and g1 ⇒ g2 in C making the obvious diagrams commute.
62
PAVEL SAFRONOV
The forgetful functor FDDat(C) −→ (Cf d )∼ which sends (x, x∨ , ev, evR , η) to x is an equivalence of bicategories. Proposition 4.29. The Serre automorphism lifts to an SO(2)-action on FDDat(C). Similarly to the case of discrete group actions on categories, given a G-action on a bicategory one can define the bicategory ChG of homotopy fixed points. For instance, ChSO(2) has objects given by a pair of object x ∈ C and a 2-isomorphism Sx ∼ = idx . 4.4. Examples. Given two k-algebras A and B let us recall that an (A, B)-module is a k-vector space equipped with commuting left A-module and right B-module structures. Definition 4.30. The Morita bicategory Mork is defined as follows: • Its objects are k-algebras. • 1-morphisms from A to B are given by (A, B)-bimodules. • 2-morphisms from an (A, B)-bimodule M to an (A, B)-bimodule N are given by intertwiners, i.e. morphisms M → N compatible with the actions of A and B. • The identity 1-morphism id : A → A is given by A viewed as an (A, A)-bimodule. • Given an (A, B)-bimodule M and a (B, C)-bimodule N , their composition is the (A, C)-bimodule M ⊗B N . The Morita bicategory Mork has a natural symmetric monoidal structure given by the tensor product of vector spaces. Proposition 4.31. Every object of Mork is dualizable. Proof. Let A ∈ Mork be a k-algebra. We define its dual to be the opposite k-algebra Aop . Let Ae = A ⊗ Aop be the enveloping algebra. A is naturally an (A, A)-bimodule via the left and right actions on itself. We define the evaluation to be A viewed as an (Ae , k)-bimodule. We define the coevaluation to be A viewed as an (k, Ae )-bimodule. The composite id⊗coev
ev⊗id
A −−−−→ A ⊗ Aop ⊗ A −−−→ A gives the (A, A)-bimodule (A ⊗ A)A⊗Aop ⊗A (A ⊗ A) ∼ = A ⊗Aop A ∼ = A, so the first duality axiom is satisfied. The second duality axiom is checked in the same manner. Recall that every dualizable object x in a symmetric monoidal (bi)category C has a notion of a dimension which is an object dim(x) ∈ EndC (1). We can easily compute it for algebras. Definition 4.32. Let A be a k-algebra. Its zeroth Hochschild homology is the vector space A . HH0 (A) = [A, A] Proposition 4.33. Let A ∈ Mork be a k-algbra. Its dimension dim(A) ∈ EndMork (k) ∼ = Vect is isomorphic to HH0 (A).
TOPOLOGICAL QUANTUM FIELD THEORIES
63
Proof. The dimension is defined to be the composite coev
ev
1 −−→ A∨ ⊗ A − → 1. Using proposition 4.31 we see that the dimension dim(A) is given by the (k, k)-bimodule, i.e. a vector space, dim(A) ∼ = A ⊗A⊗Aop A. In other words, it consists of elements x ⊗ y modulo relations • ax ⊗ y ∼ x ⊗ ya for all a ∈ A. • xb ⊗ y ∼ x ⊗ by for all b ∈ A. Using the first relation every pure tensor x ⊗ y can be uniquely written as 1 ⊗ y. Using the first relation b ⊗ y ∼ 1 ⊗ yb. Using the second relation b ⊗ y ∼ 1 ⊗ by. Therefore, we get an isomorphism A . dim(A) ∼ = [A, A] Proposition 4.34. Let A ∈ Mork be a k-algebra. (1) The evaluation ev admits a right adjoint iff A is dualizable as a k-module, i.e. A is finite-dimensional. (2) The coevaluation coev admits a right adjoint iff A is dualizable as an Ae -module, i.e. A is separable. Proof. (1) The right adjoint evR to the evaluation bimodule is an (k, Ae )-bimodule A∗ together with a map η : Ae → A ⊗ A∗ of Ae -bimodules and a map : A∗ ⊗Ae A → k of vector spaces which satisfy the adjunction axioms. In particular, we get maps η
k −→ Ae → − A ⊗ A∗ ,
A∗ ⊗ A −→ A∗ ⊗Ae A → − k
which satisfy the duality axioms. Therefore, A∗ is the dual of A as a k-module. Conversely, suppose A is dualizable as a k-module and let A∗ = Homk (A, k) be the dual. In particular, it carries a canonical right Ae -module structure such that the evaluation map A ⊗ A∗ → k factors as A ⊗ A∗ → A ⊗Ae A∗ → k which defines the counit of the adjunction. A map Ae → A∗ ⊗ A ∼ = Homk (A, A) is a map of Ae -bimodules iff the image of 1 ∈ Ae is a morphism A → A which preserves the Ae -module structure. We let the unit of the adjunction be the identity morphism A → A. It is then straightforward to check the adjunction axioms. (2) The right adjoint coevR to the coevaluation bimodule is an (Ae , k)-bimodule A! together with a map of vector spaces η : A ⊗Ae A! → k and a map η : Ae → A! ⊗ A of Ae -bimodules which satisfy the adjunction axioms. But this is precisely the definition of the dual of A as a left Ae -module. In particular, by definition A is separable and A! = HomAe (A, Ae ).
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Thus, (Morfk d )∼ is identified with the 2-groupoid of separable finite-dimensional algebras. Let us now describe SO(2)-homotopy fixed points on (Morfk d )∼ . From the proof of proposition 4.34 we know that the right adjoint evR to the evaluation is given by the (k, Ae )bimodule A∗ = Homk (A, k). Therefore, the Serre map S : A → A is given by A∗ viewed as an (A, A)-bimodule. An SO(2)-homotopy fixed point on (Morfk d )∼ is given by a separable finite-dimensional algebra A together with an isomorphism A ∼ = A∗ of (A, A)-bimodules. By lemma 3.6 this is the same as the data of a symmetric Frobenius algebra on A. Therefore, we have the following statement. Proposition 4.35. The 2-groupoid ((Morfk d )∼ )hSO(2) of SO(2)-homotopy fixed points is equivalent to the 2-groupoid of separable symmetric Frobenius algebras. Here is another important symmetric monoidal bicategory. Definition 4.36. The bicategory 2Vectk of 2-vector spaces is defined as follows: • Its objects are small k-linear additive categories which are idempotent-complete, i.e. every idempotent has a kernel. • 1-morphisms from C1 to C2 are given by k-linear functors. • 2-morphisms are given by natural transformations. Given two k-linear categories C1 , C2 we can define a tensor product C1 ⊗ C2 whose objects x1 x2 are given by pairs of objects x1 ∈ C1 and x2 ∈ C2 such that HomC1 ⊗C2 (x1 x2 , y1 y2 ) = HomC1 (x1 , y1 ) ⊗ HomC2 (x2 , y2 ). Given two categories C1 , C2 ∈ 2Vectk , their usual tensor product C1 ⊗ C2 is usually not an b 2 ∈ 2Vectk by adding object of 2Vectk . Instead, one can define the Cauchy-completion C1 ⊗C the necessary biproducts and splitting idempotents. This defines a symmetric monoidal structure on 2Vectk whose unit is the category Veck of finite-dimensional vector spaces. g Example 4.37. Suppose A, B are two k-algebras and let LModproj,f be the category of A proj,f g proj,f g projective finitely generated modules. Then LModA , LModB ∈ 2Vectk and proj,f g proj,f g ∼ b LModproj,f g ⊗LMod . = LMod A
B
A⊗B
Let us now give a classification of dualizable and fully dualizable objects of 2Vectk . Definition 4.38. Suppose C is an abelian category. An object x ∈ C is simple if the only subobjects of x are 0 and x. Lemma 4.39 (Schur). Suppose C is a k-linear abelian category and x ∈ C a simple object. Then EndC (x) is a division algebra over k. For instance, suppose k is algebraically closed and EndC (x) is finite-dimensional. Then EndC (x) = k. Definition 4.40. An abelian category C is semisimple if any object is a finite direct sum of simple objects. g Proposition 4.41. Suppose A is a separable k-algebra. Then the category LModproj,f = LModfAd A of finite-dimensional A-modules is a semisimple category.
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Example 4.42. Suppose G is a finite group such that char k does not divide |G|. Then we have shown in section 3.2 that the group algebra k[G] is separable. Therefore, the category d Repf d (G) = LModfk[G] of finite-dimensional k[G]-modules is semisimple. Definition 4.43. A k-linear category C is proper if HomC (x, y) is finite-dimensional for every x, y ∈ C. The following is [Til98, Theorem 2.5]. Proposition 4.44. A category C ∈ 2Vectk is dualizable iff it is a proper semisimple category with finitely many isomorphism classes of simple objects. Proof. We explain one direction. Suppose C is a proper semisimple category with x1 , . . . , xn ∈ C b op → Veck the non-isomorphic simple objects. The dual is defined to be Cop . We define ev : C⊗C op b to be the Hom functor. We define coev : Veck → C ⊗C to be the functor n M k 7→ xi xi . i=1
The composite id⊗coev
ev⊗id
b op ⊗C b −−−→ C C −−−−→ C⊗C is the functor y 7→
n M
y xi xi 7→
i=1
n M
HomC (xi , y) ⊗ xi .
i=1
To prove that it is equivalent to the identity, it is enough to consider the case where y is simple where it is obvious. Definition 4.45. Let C be a semisimple k-linear category. Its Grothendieck group K(C) is the free abelian group generated by isomorphism classes of simple objects of C. Proposition 4.46. Suppose C ∈ 2Vectk is a semisimple k-linear category such that EndC (x) = k for every simple object x ∈ C. Then dim(C) ∈ Hom2Vectk (Vec, Vec) ∼ = Vec is equivalent to K(C) ⊗Z k. Proof. Since every object is a finite sum of simple objects which have finite-dimensional endomorphism algebras, the category C is proper. In particular, it is dualizable by proposition 4.44. Therefore, the dimension of C is well-defined. dim(C) is given by the composite ev ◦ coev which is the functor Veck → Veck which sends n n M M ∼ k 7→ HomC (xi , xi ) = k i=1
and hence dim(C) ∼ = K(C) ⊗Z k.
i=1
We are now going to show that any dualizable object of 2Vectk is automatically fully dualizable. Lemma 4.47. Let MorLk be the bicategory Mork where all 1-morphisms have right adjoints in Mork . Then we have a symmetric monoidal functor MorLk −→ 2Vectk
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p given on objects by A 7→ LModproj,f . A
Proposition 4.48. A dualizable category C ∈ 2Vectk is fully dualizable. Proof. Let C be a semisimple category with non-isomorphic simple objects x1 , . . . , xn ∈ C. Then we have an equivalence of categories p C −→ RModfEnd(x 1 ⊕···⊕xn )
where the functor is given by y 7→ Hom(x1 ⊕ · · · ⊕ xn , y). In particular, since C is dualizable, the algebra A = End(x1 ⊕ · · · ⊕ xn ) is separable. Therefore, A ∈ Mork is fully dualizable by proposition 4.34. But then C is fully dualizable as well since by lemma 4.47 the functor MorLk −→ 2Vectk is symmetric monoidal. 4.5. Bordism bicategory. We are now going to define the bordism bicategory Bord[n−2,n] whose objects will be closed (n − 2)-manifolds, 1-morphisms will be (n − 1)-dimensional bordisms and 2-morphisms will be “bordisms between bordisms”. The construction will have the following two properties: • HomBord[n−2,n] (∅, ∅) = Cobn . • There is a functor Cobn−1 → Ho Bord[n−2,n] . We will closely follow [Sch09, Section 3.1]. To define the notion of a bordism between bordisms, we need to consider manifolds with corners. Let us recall that an n-manifold is covered by charts which look like open subsets of Rn . An n-manifold with boundary is covered by charts which look like open subsets of Rn−1 × R+ , where R+ is the set of nonnegative real numbers. Points on the boundary correspond to points lying on Rn−1 × {0} in a chart. Definition 4.49. Let M be a topological manifold. A chart at m ∈ M is given by a continuous map φ : U → Rn−p × Rp+ which is a homeomorphism onto its image where U 3 p is an open set. Two charts φ1 : U1 → Rn−p × Rp+ and φ2 : U2 → Rn−p are compatible if n−p φ2 ◦ φ−1 × Rp+ is smooth. A manifold with corners is a topological 1 : φ1 (U1 ) → R manifold M equipped with a compatible family of charts for every m ∈ M . Definition 4.50. Let M be a manifold with corners. Suppose m ∈ M is a point with a chart φ : U → Rn−p × Rp+ . The index of m ∈ M is the number of last p coordinates which are zero. Note that the index of a point is independent of the chart. We denote by ∂M ⊂ M the subset of points of index ≥ 1. Example 4.51. A manifold with boundary is a manifold with corners. A point in the interior has index 0 and a point on the boundary has index 1. Definition 4.52. Let M be a manifold with corners. A connected face of M is the closure of a component in {m ∈ M | index(m) = 1}. We will refer to a disjoint union of connected faces as a face. Definition 4.53. A h2i-manifold is a manifold with corners M together with a pair of faces ∂0 M, ∂1 M such that ∂M = ∂0 M ∪ ∂1 M and ∂0 M ∩ ∂1 M is a (possibly empty) face of ∂0 M and ∂1 M .
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Definition 4.54. Let K0 and K1 be closed (n − 2)-manifolds and (Ni , θ0i , θ1i ) for i = 0, 1 be two (n − 1)-dimensional bordisms from K0 to K1 as in definition 1.14. A 2-bordism from N0 to N1 is given by the following data: (1) A compact n-dimensional h2i-manifold M . ` (2) A continuous map ∂0 M → {0, 1} giving rise to a decomposition ∂0 M = ∂0,0 M ∂0,1 M . (3) Embeddings θ˜0 : [0, 1] × [0, 1) × K0 → M θ˜1 : [0, 1] × (−1, 0] × K1 → M such that θ˜i ([0, 1] × {0} × Ki ) = ∂0,i M for i = 0, 1. ` (4) A continuous map ∂1 M → {0, 1} giving rise to a decomposition ∂1 M = ∂1,0 M ∂1,1 M . (5) Embeddings φ0 : [0, 1) × N0 → M φ1 : (−1, 0] × N1 → M such that φi ({0} × Ni ) = ∂1,i M for i = 0, 1. Moreover, we require the diagrams [0, 31 ) × [0, 1) × K0
[0,
1 ) 3
id×θ00
θ˜0
8
/
M
[0,
( 32 , 1] × [0, 1) × K0 (− 13 , 0]
(−1)×θ0
× N1
id×θ10
φ0
× N0
[0, 31 ) × (−1, 0) × K1
φ1
θ˜0
8
/
M
1 ) 3
θ˜1
/7
M
φ0
× N0
( 23 , 1] × (−1, 0] × K1
(− 31 , 0]
(−1)×θ1
θ˜1
/7
M
φ1
× N1
to commute. In other words, a 2-bordism is a manifold with corners M whose boundary is split into two pieces ∂0 M and ∂1 M where ∂1 M has boundaries given by N0 and N1 while ∂0 M has boundaries given by the identity bordisms [0, 1] × K0 and [0, 1] × K1 . Moreover, we choose a collar near the boundary of M which is compatible with the existing collars in N0 and N1 . ` Example 4.55.` Let K0 = K1 =` pt pt. Consider the coevaluation and evaluation bordisms coev : ∅ → pt pt and ev : pt pt → ∅ and let N0 = idpt ` pt , N1 = coev ◦ ev. A 2-bordism from N0 to N1 is shown in fig. 27. The shaded areas are as follows: • θ˜0 is blue. • θ˜1 is red. • φ0 is purple. • φ1 is green. Definition 4.56. Let K0 and K1 be closed (n−2)-manifolds. The category HomBord[n−2,n] (K0 , K1 ) is defined as follows:
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N1 [0, 1] × K1 [0, 1] × K0
N0 Figure 27. Example of a 2-bordism. • Its objects are bordisms from K0 to K1 . • Morphisms from N0 to N1 , two bordisms from K0 to K1 , are given by diffeomorphism classes of 2-bordisms from N0 to N1 . • The identity morphism from N to N is given by the manifold with corners [0, 1] × N considered as a 2-bordism in a trivial way. • The composition of morphisms is given by stacking 2-bordisms on top of each other as shown in fig. 28. Such a composition uses the data of the collars φi to endow the composition with a structure of a manifold with corners.
=
Figure 28. Vertical composition of 2-bordisms. These categories also admit a horizontal composition HomBord[n−2,n] (K1 , K2 ) × HomBord[n−2,n] (K0 , K1 ) −→ HomBord[n−2,n] (K0 , K2 ) which is given on objects by the composition of bordisms (see definition 1.17) and on morphisms given by the horizontal composition of 2-bordisms as shown in fig. 29 which uses the collars θ˜i to endow the composition with a structure of a manifold with corners. Definition 4.57. The bordism bicategory Bord[n−2,n] is defined as follows:
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69
=
Figure 29. Horizontal composition of 2-brodisms. • Its objects are closed (n − 2)-manifolds. • HomBord[n−2,n] (K0 , K1 ) for two closed (n − 2)-manifolds K0 and K1 is given by definition 4.56. • The identities idK0 ∈ HomBord[n−2,n] (K0 , K0 ) are given by the identity bordisms [0, 1] × K0 . One can endow Bord[n−2,n] with a symmetric monoidal structure given by disjoint union of manifolds. We refer to [Sch09, Section 3.1.4] for more details. Example 4.58. Let K be a closed (n−2)-manifold. Recall that we have the mapping cylinder construction which gives a group homomorphism π0 (Diff(K)) −→ HomCobn−1 (K, K). Similarly, we have a homomorphism of monoidal categories π≤1 (Diff(K)) −→ HomBord[n−2,n] (K, K) which on objects is given by the mapping cylinder. Let ft : K → K be a one-parameter family of diffeomorphisms of K. Then we can construct a 2-bordism which as a manifold with corners is [0, 1] × [0, 1] × K. We define θ˜0 to be the trivial collared inclusion of [0, 1] × K. We define θ˜1 to be trivial collared inclusion of [0, 1]×K precomposed with the diffeomorphism ft on the second factor. Example 4.59. Let N be a bordism from K0 to K1 , two closed (n−2)-manifolds and consider a diffeomorphism f : N → N which fixes the collars of K0 and K1 . Then we can construct a 2-bordism which as a manifold with corners is [0, 1] × N . The collar φ0 is given by the obvious inclusion and the collar φ1 is given by precomposing the obvious inclusion by f . We will also call this 2-bordism the mapping cylinder. By construction we have • HomBord[n−2,n] (∅, ∅) = Cobn . • The homotopy category Ho Bord[n−2,n] has objects given by closed (n − 2)-manifolds and morphisms given by 2-isomorphism classes of bordisms. In particular, mapping cylinders of bordisms provide invertible 2-bordisms, so diffeomorphic bordisms are 2-isomorphic. Therefore, we get a natural full functor Cobn−1 −→ Ho Bord[n−2,n] . Definition 4.60. Let C be a symmetric monoidal bicategory. An extended n-dimensional topological field theory is a symmetric monoidal functor Bord[n−2,n] → C.
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As for ordinary topological field theories, extended n-dimensional topological field theories form a bicategory Fun⊗ (Bord[n−2,n] , C). Definition 4.61. Let C be a symmetric monoidal bicategory. Its loop space is the symmetric monoidal category ΩC = HomC (1, 1). Example 4.62. As we have observed before, ΩBord[n−2,n] = Cobn . Example 4.63. The unit in Mork is the trivial k-algebra k. The category HomMork (k, k) consists of (k, k)-bimodules, i.e. k-vector spaces. Therefore, ΩMork ∼ = Vectk . Let Z : Bord[n−2,n] → C be an extended n-dimensional topological field theory. Then we can construct the non-extended theories out of it as follows: (1) The composition Z Cobn = HomBord[n−2,n] (∅, ∅) − → HomC (Z(∅), Z(∅)) ∼ = HomC (1, 1)
gives rise to an n-dimensional topological field theory valued in the symmetric monoidal category ΩC. (2) The composition Z
Cobn−1 −→ Ho Bord[n−2,n] − → Ho C gives rise to an (n − 1)-dimensional topological field theory valued in the symmetric monoidal category Ho C. We will now study dualizability properties of objects of the extended bordism category. Proposition 4.64. Every object K ∈ Bord[n−2,n] is fully dualizable. Proof. We will prove that K = pt ∈ Bord[0,2] is fully dualizable. The general case is obtained by applying (−) × K to the duality data of the point. Recall that pt ∈ Cob1 is dualizable with the duality data given as in fig. 6. Using the functor Cob1 → Ho Bord[0,2] this gives that pt ∈ Bord[0,2] is dualizable. The composite ev ◦ ev is the circle. Therefore, the disk provides a 2-bordism : ev ◦ coev ⇒ id∅ . We also let the 2-bordism idpt ` pt ⇒ coev ◦ ev be the one shown in fig. 27. The commutativity of the diagram ev
idev ∗η
+3
ev ◦ coev ◦ ev
∗idev
+3 4<
ev
id
follows from the diffeomorphism of 2-bordisms shown in fig. 30.
TOPOLOGICAL QUANTUM FIELD THEORIES
71
∼ =
Figure 30. Proof of the adjunction ev a coev. The commutativity of the diagram coev
η∗idcoev
+3
coev ◦ ev ◦ coev
idev ∗
+3
;3 coev
id
reading the diagram fig. 30 from right to left. The proof of the adjunction coev a ev is completely anologous and we omit it.
fr We can similarly consider the symmetric monoidal bicategories Bordor [n−2,n] and Bord[n−2,n] of oriented and framed bordisms. More generally, for a homomorphism of topological groups G → GL(n, R) we have a symmetric monoidal bicategory BordG [n−2,n] , so that SO(n)
Bordor [n−2,n] = Bord[n−2,n] ,
r Bordf[n−2,n] = Bord∗[n−2,n] .
Similarly to proposition 4.64, every object of BordG [n−2,n] is fully dualizable. As before, it G is enough to reduce to the case of Bord[0,2] and since a point admits a 2-framing, it is enough r . The proof is similar to the proof of proposition 4.64, but now we to prove it for Bordf[0,2] have to be careful with framings on the 2-bordisms. We refer to [CS15, Section 9.3] for a detailed proof. Proposition 4.65. Every object of BordG [n−2,n] is fully dualizable. 4.6. 2d cobordism hypothesis. We will now restrict to the case n = 2, so define G BordG 2 = Bord[0,2] .
Since the bicategory Bord2 contains all manifolds of dimension at most 2, we refer to it as a fully extended bordism bicategory. Definition 4.66. Let C be a symmetric monoidal bicategory. A fully extended twodimensional topological field theory for G-manifolds is a symmetric monoidal functor BordG 2 → C. By proposition 4.65 the point pt ∈ Bordf2 r is fully dualizable, so for every symmetric monoidal bicategory C we have a restriction functor Fun⊗ (Bordf2 r , C) −→ (Cf d )∼ given by sending Z : Bordf2 r −→ C to Z(pt) ∈ C.
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Theorem 4.67 (2d cobordism hypothesis). The restriction functor Fun⊗ (Bordf2 r , C) −→ (Cf d )∼ is an equivalence of bicategories. This statement follows from a more general cobordism hypothesis proved by Lurie in [Lur09]. In this form it was proved in [Pst14] following the important work of [Sch09] on the oriented case. Example 4.68. By proposition 4.34 we know that (Morfk d )∼ is the 2-groupoid of finitedimensional separable k-algebras which gives a complete classification of framed fully extended two-dimensional topological field theories Bordf2 r → Mork . The objects, 1-morphisms and 2-morphisms in Bordf2 r are equipped with a 2-framing. In particular, there is a GL(2, R)-action on Bordf2 r given by changing the 2-framing. Lemma 4.69. The evaluation functor Fun⊗ (Bordf2 r , C) −→ (Cf d )∼ given by Z 7→ Z(pt) is GL(2, R)-equivariant. Theorem 4.70 (2d cobordism hypothesis for manifolds with G-structure). Let G → GL(2, R) be a homomorphism of topological groups. There is an equivalence of bicategories ∼
Fun⊗ (BordG → ((Cf d )∼ )hG 2 , C) − given on objects by Z 7→ Z(pt). SO(2)
Example 4.71. Consider the case G = SO(2), so that Bord2 = Bordor 2 is the bicategory of oriented bordisms. By proposition 4.35 we can identify the 2-groupoid of SO(2)-homotopy fixed points ((Morfk d )∼ )hSO(2) with the 2-groupoid of separable symmetric Frobenius algebras. Thus, the 2-groupoid of oriented fully extended two-dimensional topological field theories with the 2-groupoid of separable symmetric Frobenius algebras. 4.7. Extracting non-extended theories. Let Z : Bordf2 r −→ C be a fully extended twodimensional topological field theory. As we have explained in section 4.5, we can restrict it to a framed two-dimensional topological field theory Cobf2 r = HomBordf r (∅, ∅) −→ ΩC. 2
Since a fully extended TFT is completely determined by its value x = Z(pt) ∈ C on the point, one can completely describe the 2d TFT from the data of x. Recall from section 3.4 that the framed bordism category Cobf2 r is generated by the 2framed circles Sn1 for n ∈ Z which can be decomposed as Sn1 = ev ◦ (S n−1 ⊗ id) ◦ coev. From this description we immediately obtain the following statement. Lemma 4.72. Let Z : Bordf2 r → C be a symmetric monoidal functor with Z(pt) = x ∈ C. Then Z(S11 ) ∼ = dim(x) ∈ ΩC.
TOPOLOGICAL QUANTUM FIELD THEORIES
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Let us now consider the case C = Mork . Fully dualizable objects in Mork are given by finite-idmensional separable k-algebras. To such an algebra we can associate its center Z(A) and its zeroth Hochschild homology HH0 (A) = A/[A, A]. The center Z(A) is naturally a commutative algebra which acts on HH0 (A) by multiplication. Proposition 4.73. Let Z : Bordf2 r → Mork be a symmetric monoidal functor with Z(pt) = A. Then: (1) Z(S01 ) ∼ = Z(A). (2) Z(S11 ) ∼ = HH0 (A). (3) The commutative product Z(S01 )⊗Z(S01 ) → Z(S01 ) is the natural product on the center Z(A). (4) The action map Z(S01 ) ⊗ Z(S11 ) → Z(S11 ) is the natural action map. Proof. We will give a proof of the first two statements leaving the last two as an exercise. Recall from the proof of proposition 4.34 that the Serre (A, A)-bimodule S is given by ∗ A = Homk (A, k) while its inverse T is given by A! = HomAe (A, Ae ). Using the decomposition S01 ∼ = ev ◦ (T ⊗ id) ◦ coev we can identify Z(S 1 ) ∼ = A! ⊗Ae A. 0
By proposition 2.24 since A is dualizable over Ae , we can identify A! ⊗Ae A ∼ = HomAe (A, A). Let f : A → A be a morphism of Ae -modules. Using the left A-module structure, we get f (a) = af (1). Using the right A-module structure we get f (a) = f (1)a. Therefore, such a morphism is equivalent to the data of f (1) ∈ A in the center Z(A). The second statement is the content of proposition 4.33. Example 4.74. Let G be a finite group and consider the group algebra A = C[G]. Since A is separable and finite-dimensional, it is fully dualizable. Moreover, A admits a symmetric Frobenius structure θ : A → C given by extracting the coefficient of e ∈ G. In this way we obtain an oriented 2d TFT which coincides with the Dijkgraaf–Witten theory constructed in section 3.5. Given a fully extended 2-dimensional topological field theory, we have extracted a (nonextended) 2-dimensional topological field theory. It is natural to ask whether any (nonextended) 2-dimensional topological field theory can be fully extended, i.e. whether the functor G ⊗ Fun⊗ (BordG 2 , C) −→ Fun (Cob2 , ΩC) is essentially surjective. It turns out not to be the case. For instance, consider the case G = SO(2), so we are considering oriented bordisms. Then Fun⊗ (Bordor 2 , Mork ) is equivalent to the 2-groupoid of separable symmetric Frobenius alge⊗ bras. Fun (Cobor 2 , Vectk ) is equivalent to the groupoid of commutative Frobenius algebras. The functor from the former to the latter is given by taking the center of a separable symmetric Frobenius algebra, which is a commutative Frobenius algebra. But the center of a separable algebra is separable, hence a non-separable commutative Frobenius algebra, such as k[x]/xn , provides a counterexample.
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5. Three-dimensional TFTs 5.1. Oriented 3d TFTs. theorem 3.14 gives a complete classification of oriented twodimensional TFTs which is given in terms of commutative Frobenius algebras. This was proved by observing that the 2-dimensional bordism category Cobor 2 has a unique generating object given by S 1 and 4 generating morphisms giving the unit, trace, product and coproduct. Let us try to repeat the same in 3 dimensions. Then we have generating objects for Cobor 3 given by Σg for g ∈ Z≥0 , closed Riemann surfaces of genus g. We have the following generating morphisms: ` • Bordism from the disjoint union Σg1 Σg2 to the connected sum Σg1 ]Σg2 ∼ = Σg1 +g `2 . • The time-reversed versions of the previous bordisms: bordisms from Σg1 +g2 to Σg1 Σg2 . • The 3-ball considered as a bordism from ∅ to S 2 = Σ0 . • The 3-ball considered as a bordism from Σ0 to ∅. • The mapping cylinders providing a representation of the mapping class group: π0 (Diff or (Σg )) −→ HomCobor (Σg , Σg ). 3 • Bordism from Σg to Σg+1 given by attaching a handle to Σg . • The time-reversed version of the previous bordism which is a bordism from Σg+1 to Σg . Such a structure subject to certain axioms is called a J-algebra in [Juh14, Definition 5.13]. For a symmetric monoidal category C we denote by JAlg(C) be the category of J-algebras in C. The following is given by [Juh14, Theorem 1.10]. Theorem 5.1 (Classification of oriented 3d TFTs). Let C be a symmetric monoidal category. The functor Fun⊗ (Cobor 3 , C) −→ JAlg(C) L given by Z 7→ g≥0 Z(Σg ) is an equivalence. As we see, this is a fairly involved description of three-dimensional theories. Recall that while the classification of framed two-dimensional field theories (see theorem 3.29) was also quite complicated in that one has to specify an infinite number of objects with certain maps, the classification of framed extended two-dimensional field theories (see theorem 4.67) was much easier and was given in terms of a single object satisfying some properties. Similarly, as we shall see, the classification of extended three-dimensional theories will also be easier than the the classification of non-extended ones; the trade-off is that we have to work with bicategories. 5.2. Fusion categories. From now on we assume that the ground field k is algebraically closed. This is not essential, but will simplify the exposition. Definition 5.2. A multi-fusion category is a semisimple rigid monoidal category C with finitely many isomorphism classes of simple objects. A multi-fusion category is fusion if the unit object is simple. The rank of a fusion category is the number of isomorphism classes of simple objects. For instance, any category of rank 1 is equivalent to Veck , the category of finite-dimensional vector spaces.
TOPOLOGICAL QUANTUM FIELD THEORIES
75
Proposition 5.3. Suppose C is a rigid monoidal category. The tensor product functor (−) ⊗ x : C → C and x ⊗ (−) : C → C preserves all limits and colimits. Proof. Let x∨ , ∨ x ∈ C be the right and left dual objects. Then we have natural isomorphisms HomC (z, w ⊗ x) ∼ = HomC (z ⊗ x∨ , w) and HomC (z ⊗ x, w) ∼ = HomC (z, w ⊗ ∨ x) for any z, w ∈ C. Therefore, the functor (−) ⊗ x∨ is left adjoint to (−) ⊗ x and the functor (−) ⊗ ∨ x is right adjoint to (−) ⊗ x. Since (−) ⊗ x admits both adjoints, it commutes with all limits and colimits. The proof for x ⊗ (−) is identical. Lemma 5.4. Let C be a fusion category and suppose x ∈ C is simple. Then its right and left duals x∨ and ∨ x are simple. Proof. Suppose x is simple and decompose x∨ ∼ =
M
Vi ⊗ xi .
i
Therefore, x ∼ ⊗ xi . Since x is simple, Vi = 0 for all i except for one we denote i0 . = ∼ Moreover, Vi1 = k. Therefore, x ∼ = xi0 . The proof for ∨ x is identical. = ∨ xi0 and hence x∨ ∼ ⊕i Vi∗
∨
Fusion categories can be completely described by a finite amount of data. Let 1 = x0 , x1 , . . . , xn ∈ C be the representatives of isomorphism classes Ln of simple objects. We have a natural functor ⊕n Veck −→ C which sends (V1 , . . . , Vn ) 7→ i=0 Vn ⊗ xn . Since xi are simple, this functor is fully faithful and since C is semisimple, it is essentially surjective. Thus, we may assume C is equivalent as a category to Vec⊕n k . By semisimplicity we can identify xi ⊗ xj ∼ =
(3)
n M
l
k ⊕Nij ⊗ xl
l=0
for some nonnegative integers Nijl . The only remaining data is the associator. It is uniquely determined by its value Φabc : (xa ⊗ xb ) ⊗ xc → xa ⊗ (xb ⊗ xc ) on the simple objects. Using the identification (3) we see that the associator is given by an invertible linear map M M j j i i ∼ Φabc : k ⊕Nab ⊗ k ⊕Nic − → k ⊕Naj ⊗ k ⊕Nbc i,j
i,j
satisfying the pentagon axiom. Moreover, we have to check that each simple object has a dual.
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Example 5.5. Let G be a finite group and suppose ω ∈ Z3 (G, k × ) is a 3-cocycle, i.e. for every a, b, c ∈ G an element ω(a, b, c) ∈ k × satisfying for a, b, c, d ∈ G the cocycle identity 0 = ω(b, c, d) − ω(ab, c, d) + ω(a, bc, d) − ω(a, b, cd) + ω(a, b, c). To this data we are going to associate a fusion category VecG,ω . As a plain category, VecG,ω is the category of G-graded vector spaces. It has |G| simple objects given by ka for a ∈ G with the tensor product ka ⊗ kb = kab . The associator ∼
α : (ka ⊗ kb ) ⊗ kc − → ka ⊗ (kb ⊗ kc ) ∼
is the isomorphism kabc − → kabc given by multiplication by ω(a, b, c). The right and left duals of ka are given by ka−1 , so the category VecG,ω is fusion. It is easy to see that cohomologous 3-cocycles give rise to monoidally equivalent categories, so such fusion categories are parametrized by ω ∈ H3 (G, k × ). As we have mentioned above, fusion categories of rank 1 are equivalent to Veck . One also has the following classification of fusion categories of rank 2. The following is a result of Ostrik [Ost03]. Theorem 5.6. Suppose k has characteristic 0. Let C be a k-linear fusion category of rank 2 with simple objects 1 and x. Then it is equivalent to one of the following 4 categories: • VecZ/2,ω for ω ∈ H3 (Z/2, k × ) ∼ = Z/2. • Two fusion categories with x ⊗ x ∼ = 1 ⊕ x (Fibonacci and Yang–Lee categories). Lemma 5.7. Suppose C is a monoidal category. Then EndC (1) is a commutative monoid. Proof. The composition of endomorphisms induces a monoid structure ◦ : EndC (1) × EndC (1) −→ EndC (1). The tensor product functor C × C → C induces a different monoid structure • : EndC (1) × EndC (1) −→ EndC (1 ⊗ 1) ∼ = EndC (1). Moreover, • : EndC (1) × EndC (1) → EndC (1) is a morphism of monoids with respect to the ◦ product. Using the Eckmann–Hilton argument the two multiplications must coincide and, moreover, they have to be commutative. If C is a monoidal category, it is canonically a EndC (1)-linear category. Indeed, we can identify HomC (x, y) ∼ = HomC (x, y ⊗ 1) which carries a natural action by EndC (1). Note, however, that the EndC (1)-linearity structure is not compatible with the monoidal structure in general. Lemma 5.8. Suppose C is a multi-fusion category. Then the unit object 1 is a direct sum of non-isomorphic simple objects. Ln Proof. By semi-simplicity we have 1 ∼ = i=1 Vi ⊗ 1i for some simple objects 1i ∈ C. Our goal will be to prove that LnVi are at most one-dimensional. We have EndC (1) ∼ = i=1 End(Vi ). By lemma 5.7, EndC (1) is commutative, but End(Vi ) is commutative only if dim(Vi ) ≤ 1.
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77
We will now give an intrinsic characterization of multi-fusion categories. Let C be a 2b → C a monoidal structure. Suppose ∆ admits vector space (see definition 4.36) and ∆ : C⊗C R b a right adjoint ∆ : C → C⊗C. Then there is a canonical morphism ∆R (x)⊗(1y) → ∆R (x⊗y) constructed as follows: (4) ∆R (x) ⊗ (1 y) −→ ∆R ∆(∆R (x) ⊗ (1 y)) ∼ = ∆R (∆∆R (x) ⊗ y) −→ ∆R (x ⊗ y). One similarly constructs a morphism (x 1) ⊗ ∆R (y) → ∆R (x ⊗ y). The following discussion is inspired by [Gai15, Appendix D.1]. Definition 5.9. Let C be a monoidal 2-vector space. It satisfies property (R) if the following conditions are satisfied: (1) The unit functor e : Vec → C admits a right adjoint eR : C → Vec. b → C admits a right adjoint ∆R : C → C⊗C. b (2) The tensor product functor ∆ : C⊗C R R R R (3) The morphisms (x 1) ⊗ ∆ (y) → ∆ (x ⊗ y) and ∆ (x) ⊗ (1 y) → ∆ (x ⊗ y) are isomorphisms. Remark 5.10. The equation (4) shows that the diagram b C⊗C
(5)
id⊗∆R
/
b ⊗C b C⊗C
∆ ∆R
C
/
∆⊗id
b C⊗C
commutes up to a natural isomorphism. Therefore, by proposition 3.3 the trace eR : C → Vec defines a Frobenius algebra structure on C ∈ Ho 2Vectk . Proposition 5.11. Let C be a multi-fusion category. Then it satisfies property (R). Proof. The unit e : Vec → C admits a right adjoint eR : C → Vec given by Hom(1, −) which is well-defined since CL is proper. ∼ Suppose xi ⊗ xj = nk=1 Vijk ⊗ xk . Then it is easy to see that M ∆R (xk ) = (Vijk )∗ ⊗ (xi xj ) i,j
defines a right adjoint to ∆. Recall that (−) ⊗ ∨ x admits a right adjoint (−) ⊗ x. Therefore, we get a sequence of isomorphisms for every x, y, z, w ∈ C Hom b (z w, ∆R (x ⊗ y)) ∼ = HomC (z ⊗ w, x ⊗ y) C⊗C
∼ = HomC ((z ⊗ w) ⊗ ∨ y, x) ∼ = Hom b (z (w ⊗ ∨ y), ∆R (x)) C⊗C
R ∼ = HomC⊗C b (z w, ∆ (x) ⊗ (1 y))
which are natural in z, w ∈ C. Therefore, we get an isomorphism ∆R (x)⊗(1y) ∼ = ∆R (x⊗y) which can be shown to coincide with (4). The proof that (x 1) ⊗ ∆R (y) → ∆R (x ⊗ y) is an isomorphism is identical and we omit it.
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Proposition 5.12. Suppose C is a monoidal 2-vector space satisfying property (R). Then it is a self-dual object of 2Vectk . Proof. We construct the evaluation as the composite eR
∆
b − ev : C⊗C → C −→ Vec and the coevaluation as the composite ∆R
e
b coev : Vec → − C −−→ C⊗C. We have a diagram of 2-vector spaces C
id⊗e
/
b C⊗C
id⊗∆R
∆
/
b ⊗C b C⊗C /'
id
∆⊗id
/
b 7 C⊗C
eR ⊗id
/9
C
∆R
C
id
which commutes up to a natural isomorphism. Here the natural isomorphisms in the outer triangles are given by the unitors and the natural isomorphism in the middle triangle is given by (5). This proves the first duality axiom. The second duality axiom is proved by considering a diagram of 2-vector spaces C
e⊗id
/
b C⊗C
∆R ⊗id
∆ id
/
b ⊗C b C⊗C /'
id⊗∆
/ 7
b C⊗C
id⊗eR
/9
C
∆R
C
id
which again commutes up to a natural isomorphism.
Proposition 5.13. Suppose C is a monoidal 2-vector space satisfying property (R). Then every object of C admits left and right duals. Proof. By proposition 5.12 C is dualizable, so by proposition 4.44 it is proper. In particular, for x ∈ C we have a well-defined Hom functor Hom(x, −) : C −→ Vec. b → C the functor given by Hom(x, −) on the For x ∈ C we denote by Hom1 (x, −) : C⊗C first factor. Let x∨ = Hom1 (x, ∆R (1)) ∈ C. We are going to prove that x∨ is the right dual of x. The natural morphism x ⊗ Hom(x, y) → y for x, y ∈ C gives rise to a morphism x x∨ −→ ∆R (1). Let η : id → ∆R ∆ and : ∆∆R → id be the unit and counit for the ∆ a ∆R adjunction. We define the evaluation to be the composite
ev : x ⊗ x∨ = ∆(x x∨ ) −→ ∆∆R (1) → − 1.
TOPOLOGICAL QUANTUM FIELD THEORIES
79
We define the coevaluation coev to be the composite 1 −→ Hom1 (x, x 1) η
→ − Hom1 (x, ∆R (x)) ∼
← − Hom1 (x, ∆R (1) ⊗ (1 x)) ∼ = Hom1 (x, ∆R (1)) ⊗ x = x∨ ⊗ x where the first morphism is given by the inclusion of the identity morphism and the backwards isomorphism follows from the last axiom of property (R). Let us denote the composite x 1 −→ x Hom1 (x, x 1) η
→ − x Hom1 (x, ∆R (x)) ∼
← − x Hom1 (x, ∆R (1) ⊗ (1 x)) ∼ = (x Hom1 (x, ∆R (1))) ⊗ (1 x) ∼ = ∆R (1) ⊗ (1 x) ∼
− → ∆R (x) by f : x 1 → ∆R (1). Then the composite id⊗coev
ev⊗id
x −−−−→ x ⊗ x∨ ⊗ x −−−→ x is given by x ◦ ∆(f ). Let us now simplify f . Interchanging evaluation and ∆R (1) ⊗ (1 x) → ∆R (x) we get that f is equal to the composite x 1 −→ x Hom1 (x, x 1) η
→ − x Hom1 (x, ∆R (x)) ∼
← − x Hom1 (x, ∆R (1) ⊗ (1 x)) ∼
− → x Hom1 (x, ∆R (x)) −→ ∆R (x). The two isomorphisms are inverses to each other, so f is equal to the composite η
x 1 −→ x Hom1 (x, x 1) → − x Hom1 (x, ∆R (x)) −→ ∆R (x). Finally, we can interchange η and evaluation to see that f is equal to the composite η
x1 x 1 −→ x Hom1 (x, x 1) −→ x 1 −− → ∆R (1).
The first two morphisms compose to the identity, so f = ηx1 . Therefore, the first duality axiom follows since x ◦ ∆(ηx1 ) = idx by the adjunction axioms. The second duality axiom is proved similarly. We define ∨ x = Hom2 (x, ∆R (1)). The fact that it is a left dual is proved similarly to x∨ , so we omit the details.
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We refer to [Bar+15, Proposition 4.8] for an alternative proof of proposition 5.13. We can summarize the previous three statements in the following assertion. Theorem 5.14. Let C be a monoidal 2-vector space. It satisfies property (R) iff it is a multi-fusion category. Proof. Suppose C satisfies property (R). By proposition 5.12 it is dualizable in 2Vectk . Therefore, by proposition 4.44 we conclude that it is a proper semisimple category with finitely many simple objects. Moreover, by proposition 5.13 every object admits both duals, so C is rigid. Therefore, C is a multi-fusion category. The convese is given by proposition 5.11. 5.3. Ribbon categories. Definition 5.15. A balanced monoidal category is a braided monoidal category C to∼ gether with a natural isomorphism θx : x − → x for each x ∈ C called a balancing which satisfies the following equations: (1) θx⊗y = (θx ⊗ θy ) ◦ σy,x ◦ σx,y for all x, y ∈ C. (2) θ1 = id1 . Example 5.16. Suppose C is a symmetric monoidal category. Then θx = idx defines a balancing. Recall that in a monoidal category we can draw morphisms in terms of strings, so that parallel strings correspond to tensor products. In a braided monoidal category we can draw the braiding σx,y : x⊗y → y ⊗x as shown in fig. 31. We will draw the balancing θx : x → x as a twist shown in fig. 32. Note that the twist is isotopic to the identity, but such an isotopy is not compatible with the natural framings. Such framings can be drawn in the picture using ribbons which we will not do. y x x Figure 32. Balancing.
Figure 31. Braiding. The relation
θx⊗y = (θx ⊗ θy ) ◦ σy,x ◦ σx,y corresponds to the isotopy shown in fig. 33.
=
=
Figure 33. Balancing relation. Definition 5.17. A ribbon category is a rigid balanced monoidal category C satisfying (θx )∗ = θx∗ ,
∀x ∈ C.
TOPOLOGICAL QUANTUM FIELD THEORIES
81
Example 5.18. Let g be a semisimple Lie algebra. Then the category Repf d g of finitedimensional representations is a rigid symmetric monoidal category. In particular, by example 5.16 it has a ribbon structure. It is also semisimple with a simple unit, but it has infinitely many non-isomorphic simple objects. Therefore, it is not fusion. Example 5.19. Suppose g is a simple Lie algebra. Then the category of finite-dimensional representations of the quantum group Repfq d g is a Q(q 1/2 )-linear ribbon category. One has an equivalence of plain categories Repfq d g ∼ = Repf d g. In a fusion category C with simple objects 1 = x0 , . . . , xn and the fusion rules n M xi ⊗ xj ∼ Vijk ⊗ xk , = k=0
Vijk ∼ the braiding is given by isomorphisms = Vjik and the balancing is given by numbers θi = θxi ∈ EndC (xi ) = k. The balancing equation is then Bijk :
k (Bji ◦ Bijk )θi θj = θk .
Example 5.20. Consider the category C = VecZ/2,ω for ω ∈ H3 (Z/2, k × ) ∼ = µ2 , the group of second roots of unity. We may assume ω is represented by a normalized cocycle, i.e. ω : (Z/2)×3 → k × is nontrivial only for ω(1, 1, 1) where 1 ∈ Z/2 is the generator. The 0 braiding and balancing are given by nonzero numbers Bxx , θx ∈ k × . The hexagon relation gives 0 2 ω(1, 1, 1) = (Bxx ) while the balancing equation gives 0 (Bxx θx )2 = 1. Thus, a braided monoidal structure on C is given by µ4 , the group of fourth roots of unity and a balanced monoidal structure is given by µ4 × µ2 . Since x ⊗ x = 1, x is self-dual. Therefore, we automatically have the relation θx∗ = (θx )∗ . Therefore, for all elements of µ4 × µ2 , C is a ribbon fusion category. Example 5.21. Consider the other example from theorem 5.6. Namely, it is the category with two simple objects 1, x with the tensor product functor x ⊗ x = 1 ⊕ x. We can identify (x ⊗ x) ⊗ x ∼ = (1 ⊕ x) ⊗ x ∼ = 1 ⊕ x⊕2 and similarly x ⊗ (x ⊗ x) ∼ = 1 ⊕ x⊕2 . The associator (x ⊗ x) ⊗ x → x ⊗ (x ⊗ x) is given by the identity 1 → 1 and a 2 × 2-matrix a 1 a −a giving a map x⊕2 → x⊕2 , where a is a root of a2 + a − 1 = 0. 0 x The braiding and balancing are given by nonzero numbers Bxx , Bxx , θx ∈ k × . The hexagon relation gives 0 x 2 0 x +1=0 Bxx = (Bxx ), Bxx + aBxx while the balancing equation gives x 2 θx (Bxx ) = 1.
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PAVEL SAFRONOV
x = e±3πi/5 while if a So, if a is the positive root (the Fibonacci fusion category) we get Bxx x is the negative root (the Yang-Lee fusion category) we get Bxx = e±πi/5 . Since x is self-dual, θx∗ = (θx )∗ .
Let x ∈ C be an object of a rigid monoidal category. Then the right dual x∨ admits a left dual given by x itself. However, the double right dual (x∨ )∨ in general does not have to be isomorphic to x. See [BK01, Section 2.2] for the following statement. Proposition 5.22. Let C be a rigid braided monoidal category. The ribbon structure is equivalent to the data of natural isomorphisms ∼
δx : x − → (x∨ )∨ satisfying (1) δx⊗y = δx ⊗ δy . (2) δ1 = id1 . (3) δx∗ = (δx∗ )−1 . Proof. Let us define a collection of morphisms ψx : (x∨ )∨ → x as follows: id⊗σ −1
coev
id⊗ev
ψx : (x∨ )∨ −−→ x ⊗ x∨ ⊗ (x∨ )∨ −−−−→ x ⊗ (x∨ )∨ ⊗ x∨ −−−→ x. Since (ev ◦ σ, σ ◦ coev) realizes x∨ as a left dual of x if (ev, coev) realize x∨ as a right dual of x, it is easy to see that ψx is an isomorphism. The correspondence between δ and θ is given by θx = ψx ◦ δx . Let C be a ribbon category. Given an object x ∈ C together with a right dual x∨ we will draw the associated morphisms as shown in figs. 34 to 37.
Figure 34. Morphism ev : x∨ ⊗ x → 1.
Figure 36. Morphism ev ∨ δx ⊗id x ⊗ x −−−→ (x∨ )∨ ⊗ x∨ − → 1.
Figure 35. Morphism coev : 1 → x ⊗ x∨ .
37. Morphism
Figure coev
∨
−1 ∨ ∨ id⊗δx
1 −−→ x ⊗ (x ) −−−−→ x∨ ⊗ x.
In a ribbon category we can draw the twist in two different ways as shown in fig. 38.
TOPOLOGICAL QUANTUM FIELD THEORIES
83
= Figure 38. Twist in a ribbon category. Proposition 5.23. Let C be a ribbon category and x ∈ C an object. Then the composite σx,x ⊗id
id⊗coev
id⊗δ ⊗id
id⊗ev
x x −−−−→ x ⊗ x ⊗ x∨ −−−−→ x ⊗ x ⊗ x∨ −−−− −→ x ⊗ (x∨ )∨ ⊗ x∨ −−−→ x
is equal to θx . Definition 5.24. Let C be a ribbon tensor category and x ∈ C an object. Its dimension dim(x) ∈ EndC (1) is given by δ ⊗id
coev
ev
x 1 −−→ x ⊗ x∨ −− −→ (x∨ )∨ ⊗ x∨ − → 1.
Remark 5.25. When C is a rigid symmetric monoidal category equipped with the canonical ribbon structure, the previous definition reduces to definition 2.21.
x Figure 39. Dimension dim(x). We may draw the dimension as shown in fig. 39. The direction of the arrow in fact does not matter (see e.g. [TV17, Corollary 3.4]). Lemma 5.26. Let C be a ribbon category. Then the left and right dimensions of objects coincide, i.e. = x
x
Proof. This follows from the following isotopy shown in fig. 40. =
=
=
x
x Figure 40. Left and right dimensions coincide.
Here in the first equality we have used id = σ ◦ σ −1 , in the second equality we have used the relation between δ and θ (see the proof of proposition 5.22) and in the last equality we have used id = θ ◦ θ−1 .
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PAVEL SAFRONOV
Lemma 5.27. Suppose C is a semisimple ribbon category with a simple unit. Then every simple object y ∈ C has nonzero dimension dim(y) ∈ EndC (1) ∼ = k. Proof. By semisimplicity
y ⊗ y∨ ∼ = (V0 ⊗ 1) ⊕ x, where x does not have 1 as a direct summand. We can identify V0 ∼ = HomC (1, y ⊗ y ∨ ) ∼ = HomC (y, y) ∼ = k, so V0 is one-dimensional. The coevaluation 1 → 1 ⊕ x is given by a map 1 → 1 which is necessarily nonzero by the duality axioms. Similarly, the evaluation 1 ⊕ x ∼ = y ∨ ⊗ y → 1 is given by 1 → 1 which is again nonzero by the duality axioms. Therefore, the dimension coev
δ⊗id
ev
dim(x) : 1 −−→ y ⊗ y ∨ −−→ (y ∨ )∨ ⊗ y ∨ − →1 is given by a product of nonzero elements of EndC (1) ∼ = k, hence it is nonzero. Definition 5.28. Let C be a ribbon category and f : x → x a morphism. tr(f ) ∈ EndC (1) is given by (δx ◦f )⊗id
coev
Its trace
ev
1 −−→ x ⊗ x∨ −−−−−→ (x∨ )∨ ⊗ x∨ − → 1. By [Eti+15, Proposition 4.7.3 (4)] we have the following. Proposition 5.29. Let C be a ribbon category and f, g : x → x two endomorphisms. Then tr(gf ) = tr(f g) ∈ EndC (1). 5.4. Modular tensor categories. Definition 5.30. Let C be a k-linear category. Its zeroth Hochschild cohomology HH0 (C) is the k-algebra of natural endomorphisms of the identity functor EndFun(C,C) (id). Note that id ∈ Fun(C, C) is the unit element of the monoidal structure on Fun(C, C) given by the composition of functors, so by lemma 5.7 HH0 (C) is a commutative k-algebra. If C is a semisimple braided tensor category, we have another commutative algebra associated with C. Namely, the Grothendieck group K(C) inehrits a multiplication · from the tensor product on C which is commutative due to the existence of the braiding. Thus, (K(C), ·) is a commutative ring. Let C be a semisimple ribbon category. We are now going to define a ring morphism α : K(C) −→ HH0 (C). That is, for every object y ∈ C we have to define a natural endomorphism α(y) : id ⇒ id of the identity functor. We define its components α(y)x as follows: id⊗coev
σ⊗id
σ⊗id
id⊗δ⊗id
id⊗ev
α(y)x : x −−−−→ x ⊗ y ⊗ y ∨ −−−→ y ⊗ x ⊗ y ∨ −−−→ x ⊗ y ⊗ y ∨ −−−−→ x ⊗ (y ∨ )∨ ⊗ y ∨ −−−→ x. Since α(y) is a composition of natural transformations, it is itself natural. Pictorially, α(y)x is given by fig. 41.
TOPOLOGICAL QUANTUM FIELD THEORIES
85
y x
Figure 41. Definition of α. Example 5.31. α(1)x = idx and α(x)1 = dim(x) · id1 since we can untangle the braiding in this case. Lemma 5.32. The map α : K(C) → HH0 (C) is a morphism of rings. Proof. By rigidity and proposition 5.3 the tensor product preserves direct sums, so α sends sums in K(C) to sums in HH0 (C). The fact that it preserves multiplication follows from the hexagon axioms for the braiding and the balancing axioms and it shown pictorially in fig. 42. y1
y2 ◦
y1 y2 =
y1 ⊗ y2 =
Figure 42. α is multiplicative. Note that K(C) has rank n, the number of non-isomorphic simple objects of C. Similarly, HH0 (C) is a k-vector space of dimension n. Definition 5.33. A modular multitensor category is a ribbon multi-fusion category where α : K(C) ⊗Z k −→ HH0 (C) is an isomorphism. A modular tensor category is a modular multitensor category with a simple unit. Example 5.34. Suppose C is a symmetric fusion category with its canonical ribbon structure. −1 Since σx,y ∼ , by naturality we have α(y)x = dim(y) · idx . So, α : K(C) ⊗Z k −→ HH0 (C) = σy,x is only surjective in rank 1. In other words, the only modular tensor category which is symmetric is the category of finite-dimensional vector spaces Vec. Example 5.35. Suppose g is a simple Lie algebra and k ≥ 0 is an integer. One has the category Repk (b g) of integrable highest weight representations of the affine Lie algebra g at level k which is a modular tensor category (see [BK01, Theorem 7.0.1]). The construction of the braided monoidal structure is very nontrivial and we will not give it here. Let us just 0 c2 ) ∼ mention that Rep0 (b g) ∼ is a primitve 4th root of = Vec and Rep1 (sl = VecZ/2,ω where Bxx unity. Denote by di = dim(xi ) the dimensions of the simple objects of C.
86
PAVEL SAFRONOV
Definition 5.36. Let C be a modular tensor category. A global dimension D is a number such that X D2 = d2i . i
Definition 5.37. Let C be a modular tensor category. The Gaussian sums are defined to be X X p+ = θi d2i , p− = (θi )−1 d2i . i
i
We say C is anomaly-free if p = p . −
+
We refer to [BK01, Corollary 3.1.10] for the following statement. Proposition 5.38. One has D2 = p+ p− . Remark 5.39. Suppose C is an anomaly-free modular tensor category. Then p+ p− = (p+ )2 has a canonical square root given by p+ = p− . In this case we take the global dimension to be given by this choice. Example 5.40. Consider the ribbon fusion categories VecZ/2,ω from example 5.20. Recall that the nontrivial simple object x ∈ VecZ/2,ω is self-dual. The map ψx : x → x from 0 −1 0 0 −1 ) . In . Therefore, δx = θx · (Bxx ) = Bxx proposition 5.22 is given by ψx = ω(1, 1, 1) · (Bxx particular, θx dim(x) = 0 . Bxx 0 We have α(x)x = θx · Bxx idx and hence by example 5.31 α(1)1 α(1)x 1 1 α= = 0 −1 0 α(x)1 α(x)x θx · (Bxx ) θx · Bxx 0 2 0 This matrix is nonsingular iff (Bxx ) 6= 1, i.e. iff Bxx is a primitive 4th root of unity. We have X θ3 p+ = θi d2i = 1 + 0x 2 (Bxx ) i
while p− = θx2
X (θi )−1 d2i = 1 +
i 0 2 (Bxx )
θx . 0 )2 (Bxx
These are equal iff = 1, i.e. iff = 1. But in this case the category is not modular, so these modular tensor categories are anomalous. Example 5.41. Consider the ribbon fusion categories from example 5.21. The map ψx : x → x 0 −1 0 from proposition 5.22 is given by ψx = (Bxx ) . Therefore, δx = θx Bxx = 1. In particular, 1 dim(x) = = a + 1. a 0 x We have α(x)x = (aBxx + Bxx )idx . Therefore, 1 1 α= 0 x a + 1 aBxx + Bxx
TOPOLOGICAL QUANTUM FIELD THEORIES
87
x One can check that it is nonsingular for all choices of a and Bxx . In this way we get 4 modular tensor categories. We have X 1+a p+ = θi d2i = 1 + x 2 (Bxx ) i
and p− =
X
x 2 θi d2i = 1 + (Bxx ) (1 + a)
i
which are never equal. Thus, these 4 modular tensor categories are anomalous. Example 5.42. The smallest nontrivial modular tensor category which is anomaly-free has rank 4. For instance, one can consider Rep1 (so(16)). The reason modular tensor categories are called modular is that the Grothendieck ring K(C) ⊗Z k has a canonical action of SL2 (Z) which we will now define. Recall that the Grothendieck ring K(C) has a commutative multiplication · : K(C) ⊗Z K(C) −→ K(C) induced from the tensor product on C. Moreover, it has a trace φ : K(C) → Z given by P i Ni xi 7→ N0 , where N0 is the coefficient of the unit. Lemma 5.43. Let C be a modular tensor category. The multiplication · and the trace φ define the structure of a commutative Frobenius algebra on K(C). Proof. Let {xi } be the collection of simple objects. Suppose xi ⊗xj contains 1 as a summand. Then we have a nonzero map xi ⊗ xj → 1 which gives rise to a nonzero map xi → (xj )∨ . But since xi and xj are simple, this is an isomorphism. Therefore, φ(xi xj ) is nonzero only if xj = x∨i . From the proof of lemma 5.27 we know that xi ⊗ x∨i ∼ = 1 ⊕ y, where y does not have 1 as a summand. So, we have φ(xi xj ) = δij ∨ and hence θ is nondegenerate.
We will now introduce another commutative Frobenius structure on K(C). Recall that by lemma 5.27 we have di 6= 0. Define the new commutative product as xi ∗ xj = δij
D xi . di
Define the trace ρ : K(C) → k to be di . D In this way we get two nondegenerate pairings on K(C): φ(xi xj ) and ρ(xi ∗ xj ). In particular, they are related by an automorphism of K(C). The following is given by [BK01, Proposition 3.1.12]. ρ(xi ) =
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PAVEL SAFRONOV
Proposition 5.44. Define the S-matrix to be dj α(xi )xj Sij = D which gives an automorphism S : K(C) → K(C). Then S(x ∗ y) = S(x) · S(y),
φ(S(x)) = ρ(x).
Define the T -matrix to be the operator K(C) → K(C) given by T (xi ) = θi xi . The following is given by [BK01, Theorem 3.1.7]. Proposition 5.45. The S and T matrices satisfy the relations (ST )3 = S 2 ,
S 4 = 1.
Consider the group SL2 (Z) of integral 2×2-matrices of determinant 1. It has a presentation SL2 (Z) ∼ = {s, t|(st)3 = s2 , s4 = 1}. Under this isomorphism we have 0 −1 s= , 1 0
t=
1 1 0 1
.
Combining this with proposition 5.45 we obtain the following. Corollary 5.46. The S and T matrices give an action of SL2 (Z) on K(C) ⊗Z k. We will explain the geometric origin of the two commutative Frobenius structures and the SL2 (Z)-action in terms of the associated 3d TFTs. 5.5. Classification of 3-2-1 TFTs. We are now going to study extended oriented 3d TFTs, i.e. symmetric monoidal functors Bordor [1,3] → 2Vectk . The following result is proved in [Bar+15] (relying on a presentation of the oriented bordism bicategory from [Bar+]). Theorem 5.47 (Classification of oriented 3d TFTs). The bicategory Fun⊗ (Bordor [1,3] , 2Vectk ) is equivalent to the 2-groupoid of anomaly-free modular multitensor categories. We refer to [Bar+15] for the classification of oriented 3d TFTs with arbitrary target symmetric monoidal bicategor. The idea is that, first of all, the notion of a balanced braided monoidal category can be generalized to arbitrary target. The rigidity and semisimplicity can also be phrased in a bicategorical language by translating definition 5.9. This theorem has a long history which we will not review here. Let us just mention that the construction of a 3d TFT from a modular tensor category was first given in [Tur94] (following the earlier work [RT91]) and goes by the name “Reshetikhin–Turaev construction”. Let C be an anomaly-free modular tensor category and Z : Bordor [1,3] → 2Vectk the associated extended oriented 3d TFT. Then we have: • Z(S 1 ) = C. ` b → C. • Z sends the pair of pants S 1 S 1 → S 1 to the product ∆ : C⊗C ` tensor 1 1 1 • Z sends the reverse pair of pants S → S S to the right adjoint (equivalently, R b left adjoint) of the tensor product ∆ : C → C⊗C.
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89
• Z sends the disk ∅ → S 1 to the functor Vec → C given by including the unit. • Z sends the disk S 1 → ∅ to the right adjoint (equivalently, left adjoint) of the unit functor eR : C → Vec. For instance, splitting the two-sphere S 2 as the composition of two disks ∅ → S 1 → ∅ we obtain that Z(S 2 ) ∼ = EndC (1). 3 −1 One can also show that Z(S ) = D . We have a functor or Bordor 2 −→ Bord[1,3] given by M 7→ S 1 × M . Therefore, every extended oriented 3d TFT gives rise to an extended oriented 2d TFT. We will call this procedure “compactification on the circle”. Let Z2 : Bordor 2 → 2Vectk be the corresponding extended oriented 2d TFT. By construction we have Z2 (pt) = Z(S 1 ) = C. By theorem 3.14 we know that Z2 (S 1 ) ∼ = dim(Z2 (pt)) is a commutative Frobenius algebra. By proposition 4.46 we get that Z(S 1 × S 1 ) = Z2 (S 1 ) ∼ = K(C) ⊗Z k. In particular, we obtain a commutative Frobenius structure on K(C) ⊗Z k. One can show that the multiplication is given by x ∗ y and the trace is ρ as defined in the previous section. However, since Z(S 1 × S 1 ) ∼ = K(C) ⊗Z k, we get an action of π0 (Diff or (T 2 )) ∼ = SL2 (Z) on K(C) ⊗Z k. This explains the SL2 (Z)-action on the Grothendieck group of a modular tensor category constructed in corollary 5.46. Moreover, the two commutative Frobenius structures come from the two circles on the torus while the S matrix exchanges the two circles. One can explicitly compute invariants of oriented 3-manifolds in the following way. Definition 5.48. A 3-dimensional handlebody is a compact oriented 3-manifold with boundary which admits an excellent function of indices ≤ 1. For instance, the solid ball B 3 is a 3-dimensional handlebody with boundary S 2 . Definition 5.49. Let M be a closed 3-manifold. A Heegaard splitting of M is given by the following data: • Two 3-dimensional handlebodies V, W with a diffeomorphism f : ∂V ∼ = ∂W of their boundaries. • A diffeomorphism a M∼ W. =V f
The following is given by [Sav12, Theorem 1.1]. Theorem 5.50. Any closed oriented 3-manifold admits a Heegaard splitting. Given a Heegaard splitting of M , we can write Z(M ) as the composite Z(V ) Z(W ) k −−−→ Z(∂V ) ∼ = Z(∂W ) −−−→ k.
Here is a basic statement about the associated 3-manifold invariants we can prove.
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PAVEL SAFRONOV
Proposition 5.51. Let C be a modular tensor category and Z the corresponding oriented 3d TFT. Let M1 and M2 be closed oriented 3-manifolds. Then Z(M1 ]M2 ) =
Z(M1 )Z(M2 ) . Z(S 3 )
Proof. Let M10 = M1 − B 3 considered as a bordism from ∅ to S 2 . Let M20 = M2 − B 3 considered as a bordism from S 2 to ∅. ` Then M1 ]M2 = M20 ◦ M10 . The invariant Z(M1 ]M2 S 3 ) = Z(M1 ]M2 )Z(S 3 ) is given by the composite Z(M 0 )
Z(M 0 )
Z(η)
Z()
1 2 k −−−→ Z(S 2 ) −−−→ k −−→ Z(S 2 ) −−→ k,
where η and are 3-balls considered as bordisms from ∅ to S 2 and from S 2 to ∅. Since C is modular, Z(S 2 ) ∼ = EndC (1) is one-dimensional, so the above composite is equal to the composite Z(M 0 )
Z()
Z(η)
Z(M 0 )
1 2 k −−−→ Z(S 2 ) −−→ k −−→ Z(S 2 ) −−−→ k ` which computes the invariant Z(M1 M2 ) = Z(M1 )Z(M2 ).
5.6. Knot invariants. Definition 5.52. An (oriented) knot is a smooth (oriented) embedding of S 1 into S 3 . An ` ` 1 (oriented) link is a smooth (oriented) embedding of S · · · S 1 into S 3 . The unknot is given by an embedding S 1 ,→ R2 ,→ R3 ,→ S 3 . We are interested in knots up to isotopy. In other words, we want to understand π0 (Emb(S 1 , S 3 )). Definition 5.53. A knot invariant is a locally-constant function on Emb(S 1 , S 3 ). If two knots are isotopic, they have the same value of the knot invariant. However, note that two non-isotopic knots can have the same value of the knot invariant. Definition 5.54. A framed knot is a knot which extends to an embedding of S 1 × D into S 3. Definition 5.55. Let K ⊂ R3 be an (oriented) link. Choose a projection R3 → R2 such that K ⊂ R3 → R2 is an immersion whose only singularities are double points. The link diagram is the immersion K → R2 where we record overcrossings or undercrossings at the double points. See figs. 43 to 46 for examples of knot diagrams. Given two isotopic links, they might have very different links diagrams. Examples of local moves which preserve isotopy classes of links are shown in figs. 47 to 50 and are known as the Reidemeister moves. Suppose a link diagram is oriented. Then it induces a canonical framing on the link as follows. At every point we take one of the basis vectors to be orthogonal to the plane and the other to be orthogonal to the tangent vector at that point. Then the move (R1) changes the framing on the knot while (R1’), (R2) and (R3) preserve the framings.
TOPOLOGICAL QUANTUM FIELD THEORIES
Figure 43. Unknot.
Figure 44. Hopf link.
Figure 45. Left-handed trefoil.
Figure 46. Right-handed trefoil.
=
=
Figure 47. Reidemeister move (R1).
Figure 48. Reidemeister move (R1’).
=
Figure 49. Reidemeister move (R2).
91
=
Figure 50. Reidemeister move (R3).
Theorem 5.56 (Reidemeister, Alexander–Briggs). Given two link diagrams, the corresponding links are isotopic iff the diagrams are related by a sequence of Reidemeister moves (R1), (R2) and (R3). The framed links are isotopic iff the diagrams are related by a sequence of Reidemeister moves (R1’), (R2) and (R3). Choose an orientation of the link diagram. Then double points can be of two kinds which we call overcrossings and undercrossings (see figs. 51 and 52).
Figure 51. Overcrossing.
Figure 52. Undercrossing.
Definition 5.57. Let K ⊂ R3 be a knot. Its writhe w(K) is the number of overcrossings minus the number of undercrossings. Proposition 5.58. The writhe is a framed knot invariant.
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PAVEL SAFRONOV
Proof. It is easy to see that the writhe is preserved under the Reidemeister moves (R1’)-(R3), so it is a framed knot invariant by theorem 5.56. Remark 5.59. The writhe is not an invariant of unframed knots since it changes by 1 under (R1). Let us introduce another framed knot invariant known as the Kauffman bracket. Definition 5.60. Let L ⊂ R3 be a link. Its Kauffman bracket is the unique Laurent polynomial hLi ∈ Z[A, A−1 ] satisfying the axioms (1) hempty diagrami = 1. (2) For any diagram D we have D E D = (−A2 − A−2 )hDi. (3) It satisfies the local relation D E
D =A
E
+ A−1
D
E
.
To show uniqueness, observe that the last relation (known as the skein relation) allows us to reduce the number of crossings in a given component by 1. Proposition 5.61. The Kauffman bracket is invariant under the Reidemeister moves (R2) and (R3). Let us see what happens under (R1): D E D =A
E −2
= (A(−A = −A−3
+ A−1
D E
D E −A )+A ) 2
−1
D E
Therefore, the Kauffman bracket is merely a framed link invariant. We can normalize it by writhe to get a knot invariant. Definition 5.62. Let K ⊂ R3 be a knot. Its Jones polynomial is the Laurent polynomial VK (t) ∈ Z[t1/2 , t−1/2 ] defined by VK (t) = (−A3 )−w(K) hKi A2 =t−1/2 . By construction the Jones polynomial is invariant under all three Reidemeister moves, so it is a knot invariant. Example 5.63. The Kauffman bracket of the left-handed trefoil knot (see fig. 45) is A7 −A3 −A−5 . It has writhe −3, so the Jones polynomial is V (t) = (−A16 + A12 + A4 ) 2 −1/2 = −t−4 + t−3 + t−1 . A =t
TOPOLOGICAL QUANTUM FIELD THEORIES
93
Similarly, the Jones polynomial of the right-handed trefoil (see fig. 46) is V (t) = −t4 + t3 + t. In particular, the left- and right-handed trefoils are not isotopic. Let us give another way to present knots. Denote by xk ∈ R2 the points with coordinates (k, 0). `
Definition 5.64. A braid on n strands is an embedding (i1 , . . . , in ) : [0, 1] with the following properties for any k: • ik (0) ⊂ {0} × {x1 , . . . , xn }. • ik (1) ⊂ {1} × {x1 , . . . , xn }. • ik : [0, 1] ,→ [0, 1] × R2 → [0, 1] has no critical points.
n
,→ [0, 1]×R2
Given a braid on n strands, we obtain a link by connecting its boundary components as shown in fig. 53.
Figure 53. Closure of a braid. Theorem 5.65 (Alexander). Any link can be obtained as a closure of a braid. Let Bn be the set of isotopy classes of braids on n strands. It has a group structure given by stacking the braids. Here is a topological interpretation of Bn . Let Conf n (R2 ) be the configuration space of n unordered points in R2 so that Conf n (R2 ) ∼ = ((R2 )×n − ∆)/Sn , where ∆ ⊂ (R2 )×n is the diagonal R2 → (R2 )×n . We can then identify Bn ∼ = π1 (Conf n (R2 )), where the basepoint is given by the configuration of points (x1 , . . . , xn ). Let σi ∈ Bn (where 1 ≤ i ≤ n − 1) be the braid given by the identity on all strands except for i and i + 1 where it is given by the permutation with a single overcrossing. Proposition 5.66 (Artin). The group Bn is generated by σ1 , . . . σn−1 subject to the relations (1) σi σj = σj σi for |i − j| ≥ 2. (2) σi σi+1 σi = σi+1 σi σi+1 for 1 ≤ i ≤ n − 2. We have a morphism Bn → Sn which remembers permutations of the points (x1 , . . . , xn ). On the level of presentations this corresponds to imposing the extra relation σi2 = 1 for every 1 ≤ i ≤ n − 1. We also have an inclusion Bn ⊂ Bn+1 given by adding a trivial strand on the right.
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PAVEL SAFRONOV
It is easy to see that an isotopy of braids gives rise to an isotopy of their closures. However, if two braid closures are isotopic, the braids might not be isotopic. Theorem 5.67 (Markov). Consider two braids in Bn . Their braid closures are isotopic as oriented links iff they are related by a sequence of the following Markov moves: • (M1) ab ↔ ba for a, b ∈ Bn . • (M2) bσn ↔ b ↔ bσn−1 for b ∈ Bn where the relation takes place in Bn+1 . Using this theorem we are going to show how to construct invariants of links from a ribbon category C and an object x ∈ C. For simplicity we assume that the ribbon category is strict as a monoidal category, but this is not essential. We define a homomorphism from the free group F hσ1 , . . . , σn−1 i F hσ1 , . . . , σn−1 i −→ HomC (x⊗n , x⊗n ) by sending σi to the braiding applied to factors i and i + 1. Proposition 5.68. The map F hσ1 , . . . , σn−1 i → HomC (x⊗n , x⊗n ) descends to a map Bn → HomC (x⊗n , x⊗n ). Proof. We have to check that this assignment preserves the relations in the braid group. The relation σi σj = σj σi for |i − j| ≥ 2 follows from commutativity of the diagram (x ⊗ x) ⊗ y ⊗ (x ⊗ x)
σx,x ⊗idy⊗x⊗x
/
(x ⊗ x) ⊗ y ⊗ (x ⊗ x)
idx⊗x⊗y ⊗σx,x
(x ⊗ x) ⊗ y ⊗ (x ⊗ x)
σx,x ⊗idy⊗x⊗x
/
idx⊗x⊗y ⊗σx,x
(x ⊗ x) ⊗ y ⊗ (x ⊗ x)
which expresses the fact that the tensor product is a functor C × C → C. The hexagon relation gives σx,x⊗x = (σx,x ⊗ id) ◦ (id ⊗ σx,x ) : x⊗3 → x⊗3 . Naturality of the braiding gives a commutative diagram x⊗x⊗x
σx,x⊗x
/
x⊗x⊗x
id⊗σx,x
x⊗x⊗x
σx,x⊗x
/
σx,x ⊗id
x⊗x⊗x
These two relations imply σi σi+1 σi = σi+1 σi σi+1 .
We have a morphism HomC (x⊗n , x⊗n ) −→ HomC (x⊗(n−1) , x⊗(n−1) ) given by taking the trace (see definition 5.28) over the last component. Iterating it, we obtain a morphism HomC (x⊗n , x⊗n ) −→ HomC (1, 1). Consider the composite n IC,x : Bn −→ HomC (x⊗n , x⊗n ) −→ HomC (1, 1).
TOPOLOGICAL QUANTUM FIELD THEORIES
95
n Proposition 5.69. We have the following invariance properties of IC,x : n n (ba) for every a, b ∈ Bn . (ab) = IC,x • IC,x n+1 n • IC,x (b) = dim(x) · IC,x (b) for every b ∈ Bn , where we view it as an element of Bn+1 on the left-hand side. • Suppose x ∈ C is simple. Then n+1 n+1 IC,x (bσn ) = θx · IC,x (b).
for every b ∈ Bn . Proof. (1) This property follows from proposition 5.29. n (2) Adding a trivial strand adds a factor of tr(1) = dim(x) to IC,x (b). (3) This follows from proposition 5.23. Therefore, the invariant IC,x gives rise to a framed link invariant IC,x (K). If x ∈ C is a simple object, we can normalize IC,x by the writhe to get an oriented link invariant. Example 5.70. Suppose C = Repq sl2 . We have an equivalence of categories Repq sl2 ∼ = Rep sl2 (not of ribbon categories!), so we may consider the two-dimensional representation x ∈ C of sl2 which is a simple object. Then by [Oht02, Theorem 4.19] we have IC,x (K) = (−1)]K+w(K) hKi|A=q1/4 , where ]K is the number of components of K, w(K) is the writhe of K and h−i is the Kauffman bracket. 5.7. Knot invariants from TFTs. Suppose K ⊂ S 3 is a framed knot. Then by definition the boundary of the tubular neighborhood of K becomes diffeomorphic to T 2 . Changing framing of a given knot will then correspond to a Dehn twist of T 2 . Let C be an anomaly-free modular tensor category and Z the associated extended oriented 3d TFT. Given a framed knot K ⊂ S 3 , we obtain a functional Z(S 3 − K) : Z(T 2 ) ∼ = K(C) ⊗Z k −→ k where S 3 − K is considered as a cobordism from T 2 to ∅. An object x ∈ C has a class [x] ∈ K(C) in the Grothendieck group, so IC,x (K) =
Z(S 3 − K)([x]) Z(S 3 )
is a framed knot invariant. Lemma 5.71. Suppose x ∈ C is a simple object and K a framed knot. Let K 0 be the framed knot obtained from K by adding a single twist in the framing. Then IC,x (K 0 ) = θx IC,x (K). Proof. A twist in the framing corresponds to applying a Dehn twist to the boundary torus T 2 . Such a diffeomorphism induces the map T : K(C) → K(C) which sends [x] 7→ θx [x].
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PAVEL SAFRONOV
Let us now explain where the skein relation for the invariant IC,x (K) comes from following [Wit89, Section 4.1]. Consider a sphere S 2 and recall that Z(S 2 ) ∼ (1) = k. Now = EndC` 2 1 consider a sphere Sn with n disks removed considered as a cobordism (S ) n → ∅. One can show that under Z it goes to the functor Z(Sn2 ) : C⊗n −→ Vec given by x1 , . . . , xn 7→ HomC (x1 ⊗ · · · ⊗ xn , 1). Let B 3 ⊂ S 3 be a ball whose boundary S 2 intersects the knot at 4 points, two of which are positively-oriented and two are negatively-oriented. Then IC,x (K − B 3 ) : Z(S42 )(x, x, x∨ , x∨ ) → k IC,x (K ∩ B 3 ) ∈ Z(S42 )(x, x, x∨ , x∨ ) IC,x (K) = IC,x (K − B 3 ) ◦ IC,x (K ∩ B 3 ). b 2 has simple objects x0 , . . . , xk , Example 5.72. The modular tensor category C = Repk sl where x0 is the unit object. The fusion rules are min(s+t,2k−s−t)
M
xs ⊗ xt =
xu .
u=|s−t|, u−s−t∈2Z
For instance, we see that any simple object is self-dual since xs ⊗ xs contains the unit. Moreover, x1 ⊗ x1 = x0 if k = 1 and x1 ⊗ x1 = x0 ⊕ x2 if k ≥ 2. In particular, the vector space Z(S 2 )(x1 , x1 , x∨ , x∨ ) ∼ = HomC (x1 ⊗ x1 ⊗ x∨ ⊗ x∨ , 1) ∼ = HomC (x1 ⊗ x1 , x1 ⊗ x1 ) 4
1
1
1
1
is one-dimensional if k = 1 and two-dimensional if k ≥ 2. Note, however, that C is not anomaly-free, but we will ignore this fact. Let us now assume that we have chosen C and a simple object x ∈ C so that Z(S42 )(x, x, x∨ , x∨ ) is two-dimensional. Then the value of IC,x on the three pictures ,
,
gives three vectors ψ, ψ1 , ψ2 ∈ Z(S42 )(x, x, x∨ , x∨ ). Since the latter vector space is twodimensional, we have a relation αψ + βψ1 + γψ2 = 0. Therefore, we obtain a skein relation αIC,x + βIC,x
+ γIC,x
= 0.
We can obtain S 3 by gluing two solid tori Tsolid along a diffeomorphism T 2 → T 2 realized by the S-matrix. The vector Z(Tsolid ) ∈ Z(T 2 ) ∼ = K(C) ⊗Z k corresponds to the class of the unit element 1 ∈ C. Then Z(S 3 ) = hZ(Tsolid ), SZ(Tsolid )i,
TOPOLOGICAL QUANTUM FIELD THEORIES
97
where the pairing is hx, yi = ρ(x ∗ y), is equal to Z(S 3 ) = h1, S1i = S00 =
1 . D
Similarly, if Kunk ⊂ S 3 is the unknot, we have h1, Sxi IC,x (Kunk ) = = DS0x = dim(x). h1, S1i b 2 for k ≥ 2 and let Example 5.73. Consider the modular tensor category C = Repk sl x = x1 ∈ C. Introduce the variable 2πi q = exp . k+2 Then it is shown in [Wit89, Section 4.1] that IC,x (K) = −VK (q −1 )q 3w(K)/4 = (−1)w(K) hKi|A=q1/4 . This result should be compared with example 5.70. The relationship between these two is given by a certain equivalence between the ribbon categories of representation of the quantum group and that of affine Lie algebras (see [KL91] and [Fin96]). 6. Higher categories Recall that we began with the bordism category Cobn defined as follows: • Its objects are closed (n − 1)-manifolds. • Its morphisms are diffeomorphism classes of bordisms. One can then consider extending the bordism category “down” or “up”. Extending the bordism category “down” means considering the extended bordism bicategories Bord[n−2,n] where objects are closed (n − 2)-manifolds. We might also try to extend it further down by considering a tricategory Bord[n−3,n] where objects are closed (n − 3)manifolds, 1-morphisms are bordisms between them and so on. As one can see from the classification of topological field theories (compare theorem 3.29 and theorem 4.67), the structure of the fully extended bordism category is much simpler than that of the partially extended ones. So, in dimension n we might want to consider the n-category Bordn . One of the problems we run into is even defining what a completely weak n-category is. That is, we have to provide associators for composition of k-morphisms which satisfy certain coherences up to higher cells. We have given such a definition for a bicategory in definition 4.1. One can also give a reasonably explicit definition of a tricategory (see e.g. [Gur06]). However, already the definition of a tetracategory is too complicated to work with. One way around is to use an inductive definition, such as the theory of iterated Segal spaces or Θn -spaces. We can also extend the bordism category “up” as follows. Let Bord2[n−1,n] ⊂ Bord[n−1,n+1] be the full sub bicategory which has the same objects and 1-morphisms, but the only 2morphisms are given by mapping cylinders. Then Bord2[n−1,n] becomes what is known as a (2, 1)-category, i.e. a bicategory where all 2-morphisms are 2-isomorphisms. Moreover, instead of modding out by diffeomorphisms of bordisms, we remember a part of the homotopy type of the diffeomorphism group. We might imagine continuing this process to build an
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(∞, 1)-category Bord∞ [n−1,n] where we remember the full homotopy type of the diffeomorphism groups. Finally, one may imagine combining the two procedures to build an (∞, n)-category Bord∞ n of fully extended bordisms. 6.1. ∞-categories. Definition 6.1. Let f : X → Y be a continuous morphism of topological spaces. It is a weak equivalence if the induced morphism πn (X, x) −→ πn (Y, f (x)) is an isomorphism for all n ≥ 0. Recall that to a topological space X we can associate the following objects: • The set π0 (X) of connected components. • The fundamental groupoid π≤1 (X). • The fundamental 2-groupoid π≤2 (X). It is clear that a weak equivalence of spaces induces an equivalence of the fundamental (2-)groupoids. One may imagine continuing this process to arrive at the notion of a fundamental ∞-groupoid π≤∞ (X) of a topological space which will again only depend on the weak equivalence class. The following definition is known as the homotopy hypothesis. Definition 6.2. An ∞-groupoid is a topological space considered up to weak equivalence. Remark 6.3. One can make this definition precise by defining the model category of ∞groupoids as the model category of topological spaces, but we will not discuss it here. We can regard any groupoid G as an ∞-groupoid, i.e. a topological space, by considering its classifying space BG. This functor will be implicit from now on. Let C be a category. We can consider the set of objects Ob C ∈ Set. Note, however, that equivalent categories may have non-isomorphic sets of objects. Instead, we can consider the groupoid of objects C∼ which is simply the subcategory with the same objects and whose only morphisms are isomorphisms in C. This defines a functor (−)∼ : Cat → Gpd which sends equivalences of categories to equivalences of groupoids. In this section we will define the notion of an ∞-category so that the objects in such form an ∞-groupoid. Moreover, compositions in an ∞-category will be defined and associative only up to weak equivalence. For n ≥ 0 we denote by [n] ∈ Cat the category given by the poset (0 → 1 → · · · → n). Definition 6.4. The simplex category ∆ is the category whose objects are categories [n] and morphisms given by functors. Definition 6.5. Let E be a category. A simplicial object of E is a functor ∆op → E. Thus, a simplicial object x• : ∆op → E of E is given by specifying a collection of objects xn = x• ([n]) of E together with maps between them corresponding to functors between the categories [n]. For instance, a simplicial groupoid is a simplicial object in Gpd and a simplicial space is a simplicial object in the category of topological spaces Top.
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Definition 6.6. Let C be a category. The classifying diagram NR C is the simplicial groupoid given by ∼ NR n C = Fun([n], C) . We are now going to describe some important properties of the simplicial groupoid NR C. The groupoid NR 2 C has objects given by pairs of morphisms x0 → x1 → x2 in C. The set of morphisms x0 → x1 → x2 is the set of pairs of morphisms, so we have a fiber square /
Ob Fun([2], C)
Ob Fun([1], C)
/
Ob Fun([1], C)
Ob Fun([0], C)
where • The top horizontal map is given by sending x0 → x1 → x2 to x1 → x2 . It is induced by the functor [1] → [2] which sends i 7→ i + 1. • The left vertical map is given by sending x0 → x1 → x2 to x0 → x1 . It is induced by the functor [1] → [2] which sends i 7→ i. • The right vertical map is given by sending x1 → x2 to x1 . It is induced by the functor [0] → [1] given by 0 7→ 0. • The bottom horizontal map is given by sending x0 → x1 to x1 . It is induced by the functor [0] → [1] given by 0 7→ 1. One can observe that the same statement holds for groupoids. Namely, the diagram of groupoids /
Fun([2], C)∼
Fun([1], C)∼
/
Fun([1], C)∼
Fun([0], C)∼
is homotopy Cartesian in the sense of definition 3.34 where the maps are as before. We can generalize the notion of a homotopy Cartesian diagram to topological spaces. Ψ
Φ
Definition 6.7. Let X2 − →X ← − X1 be morphisms of topological spaces. The homotopy pullback is the topological space X1 ×hX X2 = X1 ×X Map([0, 1], X) ×X X2 , where the two maps Map([0, 1], X) → X are given by evaluating at 0 and 1. Definition 6.8. Let X0
˜ Ψ
/
˜ Φ
X1
X2
Ψ
/
Φ
X
be a diagram of topological spaces. It is homotopy Cartesian if the natural morphism X0 → X1 ×hX X2 is a weak equivalence.
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We can now consider more generally simplicial spaces satisfying analogs of the previous conditions. The following notion was introduced by Rezk in [Rez01]. We also refer to [CS15] for an expository account. Definition 6.9. A Segal space is a simplicial space X• such that for every n, m the diagram Xn+m
Xn
/
/
Xm
X0
is homotopy Cartesian. Here the top morphism is induced by the functor [m] → [n + m] given by i 7→ i, the left morphism is induced by the functor [n] → [n + m] given by i 7→ i + m, the right morphism is induced by the functor [0] → [m] given by 0 7→ m and the bottom morphism is induced by the functor [0] → [n] given by 0 7→ 0. We will think about a Segal space as follows: • X0 is the space of objects. • X1 is the space of morphisms. • X2 is the space of composable pairs of morphisms x → y → z (this follows from the Segal condition). • ... Note that we have three functors [1] → [2]: • 0 7→ 0, 1 7→ 1 which induces a map X2 → X1 given informally by sending x → y → z to x → y. • 0 7→ 1, 1 7→ 2 which induces a map X2 → X1 given informally by sending x → y → z to y → z. • 0 7→ 0, 1 7→ 2 which induces a map X2 → X1 given informally by sending x → y → z to x → z. Definition 6.10. Let X• be a Segal space. We define the underlying ∞-groupoid X ∼ to be X0 . Given two objects x, y ∈ X0 their mapping space is the ∞-groupoid HomX• (x, y) = {x} ×hX0 X1 ×hX0 {y}. We can extract an ordinary category from a Segal space as follows. Definition 6.11. Let X• be a Segal space. Its homotopy category is the category Ho X• whose objects are points of X0 and morphisms are given by HomHo X• (x, y) = π0 (HomX• (x, y)). A point in X1 gives rise to a morphism in Ho X• . Let X1inv ⊂ X1 be the subspace of morphisms which become invertible in Ho X• . Observe that the natural map X0 → X1 induced by the unique functor [1] → [0] factors as X0 → X1inv → X1 . Definition 6.12. Let X• be a Segal space. It is complete if the map X0 → X1inv is a weak equivalence.
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The notion of a complete Segal space is a model of (∞, 1)-categories (also known as ∞categories), see e.g. [Toë05]. Note that given any Segal space there is a universal way to define its completion which will be a complete Segal space. Example 6.13. Let C be a category. Then its classifying diagram NR C is a complete Segal space. This gives a way to embed categories into (∞, 1)-categories. Example 6.14. If C is any category with a wide subcategory W ⊂ C we can build a complete Segal space C[W −1 ]. For instance, for C = Top, the category of topological spaces, and W ⊂ Top the subcategory with the same objects whose morphisms are weak equivalences, we obtain a complete Segal space S = Top[W −1 ] of ∞-groupoids. Similarly, for C = Chk , the category of chain complexes of k-vector spaces, and W ⊂ Chk the subcategory of quasiisomorphisms, i.e. maps of chain complexes inducing an isomorphism on cohomology, we obtain a complete Segal space Chk = Chk [W −1 ]. Let CSS be the category of complete Segal spaces. Just like for ordinary categories we can talk about equivalences of complete Segal spaces. Our next goal is to define symmetric monoidal (∞, 1)-categories. Consider the category FinSet∗ of pointed finite sets. We denote by hmi ∈ FinSet∗ the finite set {0, . . . , m} pointed by 0. We have m morphisms γk : hmi → h1i for 1 ≤ k ≤ m given by sending k to 1 and the rest to 0. Definition 6.15. A symmetric monoidal (∞, 1)-category is a functor F : FinSet∗ → CSS such that the map γ1 × · · · × γm : F (hmi) → F (h1i) × · · · × F (h1i) is an equivalence of complete Segal spaces for every m ≥ 0. We call F (h1i) the underlying (∞, 1)-category of a symmetric monoidal (∞, 1)-category. Let C be the underlying (∞, 1)-category of a symmetric monoidal (∞, 1)-category F : FinSet∗ → CSS. Then F (h2i) ∼ = C × C. The map h2i → h1i which sends 1 7→ 1, 2 7→ 2 defines the tensor product functor C × C → C. Note that F (h0i) is the trivial (∞, 1)-category with a single object. Then the unique map h0i → h1i defines the unit object in C. Remark 6.16. One can give a similar definition for ordinary categories in which case we recover the notion of an unbiased symmetric monoidal category (see [Lei04, Chapter 3.1] for the non-symmetric version). Using the coherence theorem for symmetric monoidal categories one can show that the notion of an unbiased symmetric monoidal category is equivalent to the ordinary one. In particular, the classifying diagram of a symmetric monoidal category is a symmetric monoidal (∞, 1)-category. Conversely, if C is a symmetric monoidal (∞, 1)category, the homotopy category Ho C is a symmetric monoidal category. Definition 6.17. Let C be a symmetric monoidal (∞, 1)-category and x ∈ C an object. It is dualizable if x ∈ Ho C is dualizable. Let Cf d ⊂ C be the full subcategory where we restrict to dualizable objects.
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6.2. Higher bordism categories. We are now ready to define the bordism (∞, 1)-category Bord∞ 1 . We will only give a sketch of definitions referring to [Lur09] and [CS15] for more complete accounts. Let V be a finite-dimensional real vector space. We let PBordV1,k be the set parametrizing (M, {t0 ≤ · · · ≤ tk }), where ti are real numbers and M ⊂ V ×R is an n-dimensional manifold satisfying the following properties: • The map M ⊂ V × R → R is proper. • The map M → R does not have ti as critical values. We can endow PBordV1,k with a topology and define PBord1,k to be the inductive limit over all finite-dimensional vector spaces V . See [Lur09] Proposition 6.18. PBord1,• is a Segal space. We denote by Bord∞ 1 the corresponding complete Segal space which we think of as an (∞, 1)-category. Moreover, one can construct a symmetric monoidal structure on Bord∞ 1 by sending a pointed finite set I ∈ FinSet∗ to the space of embedded manifolds M together with a surjective map π0 (M ) → I − ∗. Finally, one can also construct a symmetric monoidal (∞, 1)-category of oriented (equivr alently, framed) bordisms Bord∞,f . 1 We have equivalences of symmetric monoidal categories fr Ho Bord∞ ∼ Ho Bord∞,f r ∼ = Bord1 , = Bord , where Bord1 = Cob1 and defined in section 1.3.
1 fr Bord1
1
=
Cobf1 r
1
are the symmetric monoidal categories of bordisms
6.3. Cobordism hypothesis. Let C be a symmetric monoidal (∞, 1)-category and consider r r a symmetric monoidal functor Z : Bord∞,f → C. In particular, if pt+ ∈ Bord∞,f is the 1 1 Z(pt+ ) ∈ C. We refer to [Lur09] and [Har12] for the following statement. Theorem 6.19 (1d cobordism hypothesis). One has an equivalence of (∞, 1)-categories Fun⊗ (Bord∞,f r , C) ∼ = (Cf d )∼ . 1
If C is an ordinary symmetric monoidal category, the (∞, 1)-category of symmetric monoidal r → C is equivalent to the ordinary category of symmetric monoidal functors functors Bord∞,f 1 ∞,f r ∼ fr Ho Bord1 = Bord1 → C. In particular, theorem 6.19 reduces in this case to theorem 2.30. Let us consider the following example. Let C = Mor∞ k be the symmetric monoidal (∞, 1)category whose objects are dg algebras over k and 1-morphisms are given by bimodules. The symmetric monoidal structure is given by the tensor product of algebras over k. Such a symmetric monoidal (∞, 1)-category has been constructed in [Hau17]. We will not need all the details of this construction, but we will use the following two important properties of the construction: (1) There is a functor of symmetric monoidal categories Ho Mor∞ ∼ = Ho Mork , k
where Mork is the Morita bicategory defined in section 4.4.
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(2) The ∞-groupoid HomMor∞ (k, k) is equivalent to Ch∼ , the ∞-groupoid of chain comk plexes. We have shown in section 4.4 that every object of Mork is dualizable and similarly every object of Mor∞ k is dualizable as well. Therefore, by theorem 6.19 for any object A ∈ Mor∞ we can canonically assign a 1-dimensional oriented topological field theory k ∞,f r ∞ ZA : Bord1 → Mork . Definition 6.20. Let A ∈ Mor∞ k be a dg algebra. Its Hochschild homology is defined to be HH• (A) = dim(A) = ZA (S 1 ) ∈ HomMor∞ (k, k) ∼ = Ch∼ . k Remark 6.21. If A is an ordinary associative algebra, then the zeroth homology of HH• (A) coincides with the zeroth Hochschild homology (see definition 4.32). By construction HomBord∞,f r (∅, ∅) ∼ =
a
BDiff or (S 1 )n ,
1
n≥0 or
1
where BDiff (S ) is the classifying space of oriented diffeomorphisms of S 1 . The n-th connected component corresponds to a closed 1-manifold with n connected compo`of this ` space 1 1 nents, i.e. to S · · · S . One can show that the inclusion of rotations S 1 ,→ Diff or (S 1 ) is a weak equivalence, so we get a map BS 1 −→ HomBord∞,f r (∅, ∅). 1
The TFT ZA then gives a map of ∞-groupoids BS 1 ∼ (k, k) ∼ = HomBord∞,f r (∅, ∅) −→ HomMor∞ = Vect∼ k 1
1
which sends the basepoint of BS to ZA (S 1 ) = HH• (A). Such a map is equivalent to the data of the action of group algebra of chains C• (S 1 , k) on HH• (A). If k has characteristic zero, one can identify the algebra C• (S 1 , k) with k[B] with deg(B) = −1. Thus, we obtain a degree −1 operation on the Hochschild homology complex. This operation is known as the Connes operator, see e.g. [Wei94, Section 9]. 6.4. Even higher categories. So far we have discussed the theory of (∞, 1)-categories which allowed us to avoid modding out by diffeomorphisms of bordisms in the definition of the bordism category. We may also be interested in its extended version, i.e. we are looking for an (∞, n)-category Bord∞ n which has the following informal description: • Its objects are 0-manifolds, • Its morphisms are 1-dimensional cobordisms, • ... • Its n-morphisms are n-dimensional cobordisms, • Its (n + 1)-morphisms are diffeomorphisms between n-dimensional cobordisms, • ... One of the models of (∞, n)-categories convenient for describing bordisms is the theory of n-fold complete Segal spaces. Namely, these are functors (∆op )×n → Top satisfying analogues of the Segal and completeness conditions. We refer to [Lur09] and [CS15] for details. We will use the following properties of (∞, n)-categories:
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• Given an (∞, n)-category C, we have its ∞-groupoid of objects C∼ . • Given a pair of objects x, y in a symmetric monoidal (∞, n)-category C, the mapping space HomC (x, y) is an (∞, n − 1)-category. • Given a (symmetric monoidal) (∞, n)-category C, we can extract a (symmetric monoidal) bicategory Ho2 C called the homotopy bicategory . Moreover, (∞, n)-categories can be organized into an ∞-category Cat(∞,n) . Definition 6.22. Let C be an (∞, n)-category. We say C admits adjoints for 1-morphisms if Ho2 C admits adjoints. We say C admits adjoints for k-morphisms if the (∞, n − 1)category HomC (x, y) admits adjoints for (k − 1)-morphisms. We say C admits adjoints if it admits adjoints for k-morphisms for 0 < k < n. Definition 6.23. Let C be a symmetric monoidal (∞, n)-category. We say C has duals if every object of the symmetric monoidal category Ho C is dualizable. For any symmetric monoidal (∞, n)-category C there is a universal (∞, n)-category Cf d of fully dualizable objects which admits adjoints and has duals and which has a functor Cf d → C. In fact, there is a filtration Cf d = Cndual ⊂ C(n−1)dual ⊂ ... ⊂ C0dual = C, where Ckdual has duals and adjoints for m-morphisms for m < k. We call objects of Ckdual the k-dualizable objects of C. Just as in section 1.3, one can consider G-structures (for G → GL(n, R) a group homomorphism) on bordisms giving rise to a symmetric monoidal (∞, n)-category Bord∞,G . n ∞,f r For instance, the (∞, n)-category of framed bordisms Bordn corresponds to the choice of the trivial subgroup ∗ ⊂ GL(n, R). The following theorem is proved by Lurie, see [Lur09, Theorem 2.4.6]. Theorem 6.24 (n-dimensional cobordism hypothesis). Let C be a symmetric monoidal (∞, n)-category. There is an equivalence of (∞, n)-categories r Fun⊗ (Bord∞,f , C) ∼ = (Cf d )∼ n
which on the level of objects is given by Z 7→ Z(pt+ ). Definition 6.25. Let G be a topological group and C an (∞, n)-category. A G-action on C is a functor BG −→ Cat(∞,n) which takes value C at the basepoint. Given such an action, one can define an (∞, n)-category ChG of homotopy fixed points of G. If C is an ordinary category, it reduces to definition 2.34. Theorem 6.26 (n-dimensional cobordism hypothesis for bordisms with a G-structure). Let C be a symmetric monoidal (∞, n)-category. There is a G-action on the ∞-groupoid (Cf d )∼ of fully dualizable objects, so that there is an equivalence of (∞, n)-categories Fun⊗ (Bord∞,G , C) ∼ = ((Cf d )∼ )hG . n
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Example 6.27. There is a tricategory (and hence an (∞, 3)-category) TC of finite tensor categories (see [DSS13, Section 3.2] for the precise assumptions). One can show the following: • Every object of TC is 2-dualizable (in particular, dualizable). • 3-dualizable objects of TC are separable finite tensor categories. For instance, fusion categories are 3-dualizable. By theorem 6.26 we have a GL(3, R)-action on the 3-groupoid (TCf d )∼ . One can now study the data necessary to describe homotopy fixed points for some subgroups of GL(3, R). For instance, it is conjectured in [DSS13] that an SO(2)-fixed point is described by a pivotal structure on the tensor category and an SO(3)-fixed point is described by a spherical structure on the tensor category. We refer to [Eti+15, Section 4.7] for these notions. References [Abr96]
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[Wit88] [Wit89]
107
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