Lecture 15: Duality We Ended the Last Lecture by Introducing One of The

Lecture 15: Duality

We ended the last lecture by introducing one of the main characters in the remainder of the course, a topological quantum field theory (TQFT). At this point we should, of course, elaborate on the definition and give examples, background, motivation, etc. I will not do so in these notes. Instead I refer you to the expository paper [F1] as well as to the beginning sections of [L1]. There are many other references with great expository material. In this lecture we explore the finiteness property satisfied by a TQFT, which is encoded via duality in symmetric monoidal categories.

Some categorical preliminaries We begin with a standard notion which you’ll ﬁnd in any book which contains a chapter on categories, including books on category theory.

Definition 15.1. Let C, D be categories. A functor F : C → D is an equivalence if there exist a functor G: D → C, and natural isomorphisms G ◦ F → idC and F ◦ G → idD. Proposition 15.2. A functor F : C → D is an equivalence if and only if it satisfies: (i) For each d ∈ D there exist c ∈ C and an isomorphism f(c) → d ∈ D; and (ii) For each c1, c2 ∈ C the map of hom-sets F : C(c1, c2) → DF (c1), F (c2) is a bijection. If F satisfies (i) it is said to be essentially surjective and if it satisfies (ii) it is fully faithful.

Exercise 15.3. Prove Proposition 15.2.

Next we spell out the answer to Exercise 13.45. It is part of the deﬁnition of a TQFT.

Deﬁnition 15.4. Let C, D be symmetric monoidal categories and F, G: C → D symmetric monoidal functors. Then a symmetric monoidal natural transformation η : F → G is a natural transformation such that the diagrams

(15.5) F (1C )

1D η(1C )

G(1C ) and

ψ (15.6) F (y1) ⊗ F (y2) F (y1 ⊗ y2)

η⊗η η ψ G(y1) ⊗ G(y2) G(y1 ⊗ y2) commute for all y1,y2 ∈ C.

Bordism: Old and New (M392C, Fall ’12), Dan Freed, October 27, 2012 1 2 D. S. Freed

TQFT’s as a symmetric monoidal category X(n) Fix a bordism category B = Bordhn−1,ni and a symmetric monoidal category C. We now explain that topological quantum field theories F : B → C are objects in a symmetric monoidal category. A morphism F → G is as defined in Definition 15.4. The monoidal product of theories F1, F2 is defined by

(F1 ⊗ F2)(Y )= F1(Y ) ⊗ F2(Y ) (15.7) (F1 ⊗ F2)(X)= F1(X) ⊗ F2(X) for all objects Y ∈ B and morphisms (X : Y0 → Y1) ∈ B. The tensor unit 1 is the trivial theory

1(Y ) = 1C (15.8) 1(X)=id1C for all Y ∈ B and (X : Y0 → Y1) ∈ B. We denote the symmetric monoidal category of TQFT’s as

X(n) ⊗ X(n) (15.9) TQFTn = TQFTn [C] = Hom (Bordhn−1,ni,C).

The short form of the notation is used if the tangential structure X(n) and codomain category C are clear.

(15.10) Endomorphisms of the trivial theory. Suppose η : 1 → 1 in TQFTn. Then for all Y ∈ X(n) Bordhn−1,ni we have η(Y ) ∈ C(1C , 1C ) = End(1C ). Note that if C = Ab is the category of abelian groups, then End(1C )= Z. So η is a numerical invariant of closed (n − 1)-manifolds. Furthermore, if X : Y0 → Y1 then by the naturality condition (13.18) we ﬁnd that η(Y0) = η(Y1). This shows that η factors down to a homomorphism of monoids

X(n) (15.11) η :Ωn−1 −→ End(1C ).

Now by Lemma 1.30, and its generalization to manifolds with tangential structure, we know that X(n) every element of Ωn−1 is invertible. It follows that the image of η consists of invertible elements. We say, simply, that η is invertible. In other words, an endomorphism of 1 is a bordism invariant of the type studied in the first half of the course. A topological quantum field theory, then, is a “categorified” bordism invariant.

Exercise 15.12. What is a topological quantum ﬁeld theory whose codomain category has as objects the set of integers and only identity arrows?

The invertibility observed in (15.10) is quite general.

Theorem 15.13. A morphism (η : F → G) ∈ TQFTn is invertible. TQFTn is a groupoid. The two statements are equivalent. We prove Theorem 15.13 at the end of this lecture. Bordism: Old and New (Lecture 15) 3

(15.14) Central problem. Given a dimension n, a tangential structure X(n), and a codomain X(n) category C we can ask to “compute” the groupoid TQFTn [C]. This is a vague problem whose solution is an equivalent groupoid which is “simpler” than the groupoid of topological quantum ﬁeld theories. It has a nice answer when n = 1. In the oriented case it is a generalization of the SO theorem that Ω0 is the free abelian group with a single generator pt+. There is also a nice answer in the oriented case for n = 2.

Finiteness in TQFT To motivate the abstract formulation of ﬁniteness in symmetric monoidal categories, we prove the following simple proposition. For simplicity we omit any tangential structure.

Proposition 15.15. Let F : Bordhn−1,ni → VectC be a TQFT. Then for all Y ∈ Bordhn−1,ni the vector space F (Y ) is ﬁnite dimensional. !

Figure 26. Some elementary bordisms

Figure 27. The S-diagram

n−1 n−1 Proof. Fix Y ∈ Bordhn−1,ni and let V = F (Y ). Let c: ∅ → Y ∐ Y and e: Y ∐ Y → ∅ be the bordisms pictured in Figure 26. The manifold Y is depicted as a point, and each bordism has underlying manifold with boundary [0, 1] × Y . The composition depicted in Figure 27 is diﬀeomorphic to the identity bordism idY : Y → Y . Under F it maps to idV : V → V (see (13.40)). On the other hand, the composition maps to

id ⊗F (c) F (e)⊗id (15.16) V −−−−−−→V V ⊗ V ⊗ V −−−−−−→V V

C C ′ ′′ ′ ′′ Let the value of F (c): → V ⊗ V on 1 ∈ be P vi ⊗ vi for some ﬁnite set of vectors vi, vi ∈ V . i Then equating (15.16) with the identity map we ﬁnd that for all ξ ∈ V we have

′ ′′ (15.17) ξ = X e(ξ, vi)vi , i ′′ and so the ﬁnite set of vectors {vi } spans V . This proves that V is ﬁnite dimensional. Exercise 15.18. Prove that F (c) and F (e) are inverse bilinear forms. 4 D. S. Freed

Duality data and dual morphisms We abstract the previous argument by singling out those objects in a symmetric monoidal category which obey a ﬁniteness condition analogous to that of a ﬁnite dimensional vector space.

Deﬁnition 15.19. Let C be a symmetric monoidal category and y ∈ C. (i) Duality data for y is a triple of data (y∨, c, e) in which y∨ is an object of C and c, e are ∨ ∨ morphisms c: 1C → y ⊗ y , e: y ⊗ y → 1C . We require that the compositions

c⊗id id ⊗e (15.20) y −−−→y y ⊗ y∨ ⊗ y −−−−→y y

and

id ∨ ⊗c e⊗id ∨ (15.21) y∨ −−−−−→y y∨ ⊗ y ⊗ y∨ −−−−→y y∨

be identity maps. If duality data exists for y, we say that y is dualizable. f (ii) A morphism of duality data (y∨, c, e) → (˜y∨, c,˜ e˜) is a morphism y∨ −→ y∨ such that the diagrams f

∨ (15.22) c y ⊗ y

1C idy ⊗f c˜ y ⊗ y∨ f and

∨ (15.23) y ⊗ y e

f⊗idy 1C y∨ ⊗ y e˜ f commute. c is called coevaluation and e is called evaluation. We now express the uniqueness of duality data. As duality data is an object in a category, as deﬁned in Deﬁnition 15.19, we cannot say there is a unique object. Rather, here we have the strongest form of uniqueness possible in a category: duality data is unique up to unique isomorphism.

Deﬁnition 15.24. Let C be a category.

(i) If for each pair y0,y1 ∈ C the hom-set C(y0,y1) is either empty or contains a unique element, we say that C is a discrete groupoid.

(ii) If for each pair y0,y1 ∈ C the hom-set C(y0,y1) has a unique element, we say that C is contractible. Bordism: Old and New (Lecture 15) 5

A discrete groupoid is equivalent to a set (Example 13.10). A contractible groupoid is equivalent to a category with one object and one morphism, the categorical analog of a point.

Proposition 15.25. Let C be a symmetric monoidal category and y ∈ C. Then the category of duality data for y is either empty or is contractible.

The proof is a homework problem (Problem Set #2). A morphism between dualizable objects has a dual.

Deﬁnition 15.26. Let y0,y1 ∈ C be dualizable objects in a symmetric monoidal category and ∨ ∨ ∨ f : y0 → y1 a morphism. The dual morphism f : y1 → y0 is the composition

id ∨ ⊗c0 id ∨ ⊗f⊗id ∨ e1⊗id ∨ ∨ y1 ∨ ∨ y1 y0 ∨ ∨ y0 ∨ (15.27) y1 −−−−−→ y1 ⊗ y0 ⊗ y0 −−−−−−−−−→ y1 ⊗ y1 ⊗ y0 −−−−−→ y0

∨ ∨ In the deﬁnition we choose duality data (y0 , c0, e0), (y1 , c1, e1) for y0,y1. Exercise 15.28. Check that this deﬁnition agrees with that of a dual linear map for C = Vect. Also, spell out the consequence of Proposition 15.25 for the dual morphism.

Duality in bordism categories We already encountered dual manifolds and dual bordisms in Deﬁnition 1.22, Remark 1.24, and (2.20). In this subsection we prove the following.

X(n) Theorem 15.29. Every object in a bordism category Bordhn−1,ni is dualizable. X X(n) Proof. If (n) is the trivial tangential structure BO(n) → BO(n), so Bordhn−1,ni = Bordhn−1,ni is the bordism category of (unoriented) manifolds, then for any closed (n − 1)-manifold Y we have Y ∨ = Y with coevaluation and evaluation as in Figure 26. In the general case, an object (Y,θ) ∈ X(n) Bordhn−1,ni is a closed (n − 1)-manifold Y equipped with a classifying map

(15.30) R ⊕ TY θ S(n)

Y X(n) to the universal bundle (9.59); cf. (9.60). Its dual (Y,θ)∨ = (Y,θ∨) has the same underlying manifold and classifying map θ∨ the composition

−1⊕id (15.31) R ⊕ TY T Y R ⊕ TY θ S(n)

id Y Y Y X(n) 6 D. S. Freed

The coevaluation and evaluation (X,p,θ0,θ1) are as depicted in Figure 26. In both cases X = [0, 1] × Y . For the coevaluation p: ∂X → {0, 1} is the constant function 1, and for the evaluation it is the constant function 0. For the evaluation

θ0 : [0, 1) × Y ∐ [0, 1) × Y −→ [0, 1] × Y (15.32) (t,y) 7−→ (t/4,y) (t,y) 7−→ (1 − t/4,y)

with natural lifts to the X(n)-structures. The formula for θ1 for the coevaluation is similar.

Exercise 15.33. Write the map on X(n)-structures explicitly. Note the − sign in the diﬀerential of the last formula in (15.32) matches the −1 in the ﬁrst map of (15.31).

Proof of Theorem 15.13 We ﬁrst prove the following.

Proposition 15.34. Let B,C be symmetric monoidal categories, F, G: B → C symmetric monoidal functors, and y ∈ B dualizable. Then (i) F (y) ∈ C is dualizable. (ii) If η : F → G is a symmetric monoidal natural transformation, then η(y): F (y) → G(y) is invertible.

∨ ∨ Proof. If (y , c, e) is duality data for y, then F (y ), F (c), F (e) is duality data for F (y). This proves (i). For (ii) we claim that η(y∨)∨ is inverse to η(y). Note that by Deﬁnition 15.26, η(y∨)∨ is a map G(y∨)∨ → F (y∨)∨, and since G(y∨) = G(y)∨ it may be interpreted as a map G(y) → F (y). Let ∨ ∨ c: 1B → y ⊗ y and e: y ⊗ y → 1B be coevaluation and evaluation. Consider the diagram

(15.35) id ⊗F (c) id ⊗η(y∨)⊗id G(e)⊗id G(y) G(y) ⊗ F (y∨) ⊗ F (y) G(y) ⊗ G(y∨) ⊗ F (y) F (y)

∨ id ⊗η(y )⊗η(y) id ⊗ id ⊗η(y) η(y) id ⊗G(c) G(e)⊗id G(y) ⊗ G(y∨) ⊗ G(y) G(y)

We claim it commutes. The left triangle commutes due to the naturality of η applied to the ∨ coevaluation c: 1B → y ⊗ y . The next triangle and the right square commute trivially. Now starting on the left, the composition along the top and then down the right is the composition η(y) ◦ η(y∨)∨. The composition diagonally down followed by the horizontal map is the identity, by G applied to the S-diagram relation (15.20) (and using (13.40)). A similar diagram proves that η(y∨)∨ ◦ η(y) = id.

Theorem 15.13 is an immediate consequence of Theorem 15.29 and part (ii) of (11.71). Part (i) implies the following. Bordism: Old and New (Lecture 15) 7

X(n) Theorem 15.36. Let C be a symmetric monoidal category and F : Bordhn−1,ni → C be a topological X(n) quantum ﬁeld theory. Then for all Y ∈ Bordhn−1,ni, the object F (Y ) ∈ C is dualizable.

References [F1] D. S. Freed, The cobordism hypothesis. http://www.ma.utexas.edu/users/dafr/cobordism.pdf. [L1] Jacob Lurie, On the classiﬁcation of topological ﬁeld theories, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 129–280. arXiv:0905.0465.