Lecture 3

Braided tensor categories

3.1 Representations of loop groups

To make this lecture as concrete as possible I shall begin by describing some remarkable properties of the representations of loop groups. A few preliminary explanations are needed, though they are irrelevant to the main idea.

We consider unitary representations of the group LG of smooth loops in a compact connected Lie group G. We are only interested in so-called positiveenergy representations: I shall explain the meaning of that presently. Positive energy representations are necessarily projective, i.e. they are really representations of a central extension of LG by the circle group 𝕋. The isomorphism class of the extension which acts is called the level of the representation. If G is simply connected it is completely determined by the topological type of the central extension as a circle-bundle on LG, i.e. by its Chern class in H^{2}(LG; ℤ) ≅ H^{3}(G; ℤ). There are only finitely many different irreducible representations of LG at each level.

Let us fix a level k, and let R be the category of all level k representations of LG which are finite sums of irreducibles. Then the surprising fact is that a smooth cobordism Σ : S^{1} ⇝ S^{1} induces an additive functor U_{Σ} : R → R, and, more generally, a cobordism

\begin{aligned}Σ : & S^{1}⊥⊥ ⋯ ⊥⊥S^{1} & ⇝ & S^{1}⊥⊥ ⋯ ⊥⊥S^{1} \\ & ⟵ p ⟶ & & ⟵ q ⟶ \end{aligned}

induces a functor U_{Σ} : R_{p} → R_{q}, where R_{p} is the category of representations of (LG)^{p} = LG × ⋯ × LG which are of level k on each factor. Composition of cobordisms Σ corresponds to composition of the operators U_{Σ}, and a “trivial” cobordism Σ = S×I induces the identity functor. Furthermore, a diffeomorphism f : Σ → Σ′ between two cobordisms Σ, Σ′ : S_{0} ⇝ S_{1} which is the identity on S_{0} and S_{1} induces a transformation of functors T_{f} : U_{Σ} → U_{Σ′}.

In the last sentence I have oversimplified slightly. I should have required the cobordisms Σ to be rigged surfaces. A rigging of a surface Σ is something analogous to a choice of a simply connected covering space of Σ — I shall give a precise definition below. The choice of rigging does not affect the functor U_{Σ} up to isomorphism, but the diffeomorphisms f can be lifted to the rigging in more than one way (in fact, a sequence of ways forming a ℤ-torsor, if Σ is connected), and changing the lift changes T_{f} by multiplication by a root of unity.

Postponing discussion of technicalities, let us consider the general implications of the existence of the functors U_{Σ}.

The first observation is that there is a bi-additive functor R×R → R induced by a pair-of-pants cobordism Σ : S^{1}⊥⊥S^{1} → S^{1}. It is called fusion, and is unique up to an isomorphism of functors, as Σ is unique up to diffeomorphism. It is a kind of tensor product on the category R, and I shall write it (H_{1}, H_{2}) ↦ H_{1}∗H_{2}. (The usual tensor product of Hilbert spaces does not give us a tensor product on R, for the level of H_{1} ⊗ H_{2} is the sum of the levels of H_{1} and of H_{2}). The same arguments that we used in Lecture 1 tell us that fusion is associative and commutative up to isomorphism. The commutativity isomorphism

T_{θ} : H_{1} ∗ H_{2} → H_{2} ∗ H_{1},

however, is induced by a diffeomorphism θ of the pair of pants which interchanges the two incoming circles while being the identity on the outgoing one. Thus T_{θ}^{2} = T_{θ^{2}} need not be the identity. We can, however, say something about it. For this we need a general observation.

Proposition 3.1.1 A finite dimensional representation of the group Diff (Σ rel ∂Σ), for any manifold Σ, is necessarily trivial on the identity component, i.e. it factorizes through the mapping-class group Γ_{Σ rel ∂Σ} = π_{0} Diff (Σ rel ∂Σ).

I shall give the proof at the end of this section. We apply the result as follows. As any object H of our category R is a finite sum of irreducibles, its group of automorphisms is a product of finite dimensional general linear groups, and so 3.1.1 tells us that for any action of Diff(Σ rel ∂Σ) on H the operator T_{f} depends only on the isotopy class of f. In particular every H has a canonical automorphism ρ_{H} which is obtained by choosing an isomorphism between H and U_{A}(H), where A = S^{1}×I, and transferring to H the action of the standard “Dehn twist” f : A → A which generates the mapping-class group Γ_{A rel ∂A} ≅ ℤ. The diffeomorphism θ^{2} of Σ is isotopic to the Dehn twist in an annulus around the outgoing boundary circle of Σ, and so

T_{θ}^{2} = ρ_{H_{1}∗H_{2}}. (3.1.2)

Of course ρ_{h} must simply be multiplication by some scalar λ on each irreducible summand P of H.

More generally, for any H the object

H^{∗p} = H ∗ ⋯ ∗_{⟵p⟶} H

depends on a choice of a surface Σ_{p} which is a disc with p holes. The relevant mapping-class group is the group diffeomorphisms of Σ_{p} which are allowed to permute the p incoming circles. This is the braid group Br_{p} on p strings, and it acts on H^{∗p}. The fact that it is the braid group Br_{p} rather than the symmetric group S_{p} which permutes the factors of H^{∗p} expresses the less-than-perfect commutativity of the operation of fusion, and explains the name “braided tensor category”.

If Σ is a surface with p boundary circles, all outgoing, then U_{Σ} is a functor R^{⊗0} → R^{⊗p}. As R^{⊗0} is the category of finite dimensional vector spaces it is natural to identify U_{Σ} with the object U_{Σ}(ℂ) of R^{⊗p}: it is a unitary representation of (LG)^{p} with an interwining action of Γ_{Σ rel ∂Σ}. If Σ is closed then U_{Σ} is simply a finite dimensional unitary representation of Γ_{Σ} (actually a projective representation because of the question of rigging).

The representation theory of loop groups thus provides us unexpectedly with a panoply of representations of braid groups and mapping-class groups. But in fact much more is true. If Σ and Σ′ are cobordisms from S_{0} to S_{1} then not only does a diffeomorphism f : Σ → Σ′ give us a transformation T_{f} from U_{Σ} to U_{Σ′}, but a cobordism M : Σ ⇝ Σ′ does so too. (A cobordism between Σ, Σ′ : S_{0} ⇝ S_{1} means a 3-manifold M with an isomorphism

∂M ≅ Σ̄ ∪ ((S̄_{0}⊥⊥S_{1}) × I) ∪ Σ′

where Σ̄ and Σ′ are joined by a cylinder along their common boundary

∂Σ = ∂Σ′ = ((S̄_{0}⊥⊥S_{1}) × I.)

In particular, if Σ is a closed surface, so that U_{Σ} is a vector space, then a 3manifold M with ∂M = Σ gives us a vector T_{M} ∈ U_{Σ}, and a closed 3−manifold M gives us a number T_{M} ∈ ℂ. This is the next surprise: we now have a 3dimensional topological field theory in the sense of Lecture 1, and it turns out to be Chern-Simons theory, i.e.

T_{M} = ∫ e^{2πiCS(A)}DA, (3.1.3)

where, for a connection A on a G-bundle on the 3-manifold M,

CS(A) = ∫_{M}(⟨A, dA⟩ + ⟨A, [A, A]⟩). (3.1.4)

The integral in (3.1.4) is over all G-bundles-with-connection A on M, while in (3.1.4) the connection A is written as a Lie-algebro-valued 1-form, which is only legitimate if the bundle is trivial. Like the category R of representations, the Chern-Simons action (3.1.4) depends on a choice of level, which enters into the formula implicitly as the choice of the inner product on the Lie algebra.

3.2 Category-valued field theories

Let us now try to give a definition of a category-valued topological field theory modelled on the properties of the representations of loop groups. The values will be assumed to be ℂ-linear categories. I shall take this to mean not only that the morphisms between any two objects form a complex vector space, and composition is bilinear, but also that

(i) any two objects have a direct sum, and

(ii) idempotents split, i.e. if f : A → A is a morphism such that f^{2} = f then A = ker(f) ⊕ im(f)

Two such categories C_{1} and C_{2} have a tensor product C_{1} ⊗ C_{2}, which is unique up to canonical equivalence. Any object of C_{1} ⊗ C_{2} is a summand in a finite sum

⨁ P_{i} ⊗ Q_{i} ,

where the P_{i} are objects of C_{1} and the Q_{i} are objects of C_{2}. As an example, if C_{1} and C_{2} are the categories of finite dimensional representations of compact groups G_{1} and G_{2} then C_{1} ⊗ C_{2} is the corresponding category for G_{1} × G_{2}.

Let Vect denote the category of finite dimensional complex vector spaces. Then there is an equivalence

Vect ⊗ C ⟶ C

for any linear category C (for it makes sense to tensor an object of C by an object of Vect). I shall denote the category of additive functors C → Vect by C^{∗}.

I shall define a category-valued n-dimensional topological field theory as the following data.

(i) A functor E from closed oriented (n − 1)-manifolds (and diffeomorphisms) to linear categories. It is required to take disjoint unions to tensor products, which implies that Ψ(∅) = Vect.

(ii) An additive functor Ψ_{Y} : E(X_{0}) → E(X_{1}) for each cobordism Y : X_{0} ⇝ X_{1}, such that

Ψ_{Y_{2}} ∘ Ψ_{Y_{1}} = Ψ_{Y_{2}∘Y_{1}}. (3.2.1)

If S_{0} is empty then Ψ_{Y} will be identified with the object Ψ_{Y} (ℂ) of E(X_{1}) : thus Ψ_{Y} is a finite dimensional vector space if Y is closed.

I shall assume the same kind of compatibility between the data as in Lecture 1. In any discussion of categories it is quite difficult to decide how much to make explicit, and how much is better left to the reader’s imagination. In the present situation it does seem important to say that the cobordisms Y : X_{0} ⇝ X_{1} between given manifolds X_{0}, X_{1} themselves form a category Cob(X_{0}; X_{1}), in which the morphisms are diffeomorphisms Y → Y′, and that Y ↦ Ψ_{Y} is a functor from Cob(X_{0}; X_{1}) to the category of functors E(X_{0}) → E(X_{1}). Composition of cobordisms is really only defined up to isomorphism: to say that Y_{3} is Y_{2} ∘ Y_{1} really means that there is a map f : Y_{1}⊥⊥Y_{2} → Y_{3} with certain properties, and (3.2.1) really means that f induces an isomorphism of functors

f_{∗} : Ψ_{Y_{2}} ∘ Ψ_{Y_{1}} → Ψ_{Y_{3}}.

I do not think there are any interesting subtleties hidden here. The point to keep in mind is that the group of diffeomorphisms of a closed n-manifold Y acts on the vector space Ψ_{Y}, and if Y : X_{0} ⇝ X_{1} then Diff(Y rel ∂Y) — the group of diffeomorphisms which fix the boundary — acts on Ψ_{Y} (E) for each object E of E(X_{0}).

The analogue of the non-degeneracy assumption (1.1.7) is

Assumption 3.2.2 The cobordism X × I : X ⇝ X induces a functor E(X) → E(X) which is equivalent to the identity.

Notice that we do not want to assume that the equivalence is given: in the loop group example we get an equivalence for each choice of a complex structure on X × I.

As in each of the situations we considered in Lecture 1 there is an analogue of the finiteness - and - duality result . I shall say that two linear categories C_{1} and C_{2} are in duality if there is a bi-additive functor C_{1} × C_{2} → Vect and on object Δ of C_{1} ⊗ C_{2} which induce inverse equivalences

C_{1} ⇄ C_{2}^{∗} and C_{2} ⇄ C_{1}^{∗}.

The existence of a duality if very restrictive (cf. (1.1.8)):

Proposition 3.2.3 If C_{1} and C_{2} are in duality then each is semisimple — i.e. each object is a finite sum of irreducible objects — and has, up to isomorphism, only finitely many irreducible objects.

Corollary 3.2.4 If E is a category-valued topological field theory then each category E(X) is semisimple with finitely many irreducible objects, and the cobordism X × I : X̄⊥⊥X → ∅ induces a duality

E(X̄) × E(X) → Vect,

which will be denoted (A, B) ↦ ⟨A, B⟩.

Proof of (3.2.3) The object Δ of C_{2}⊗C_{1} is a summand in a finite sum ⨁ P_{i}⊗Q_{i}, and it follows at once that each object A of C_{1} is a summand in

⨁⟨A, P_{i}⟩ ⊗ Q_{i}.

Thus each A is a summand in a finite sum of copies of Q, where Q = ⊕Q_{i}. Replacing Q by P = ⊕P_{i}, the same holds for the objects of C_{2}.

Because C_{1} → C_{2}^{∗} is an equivalence to functor A ↦ ⟨P, A⟩ identifies C_{1} with a subcategory of the category of finite dimensional vector spaces. Let R denote the ring End_{C_{1}}(Q), which is a finite dimensional algebra over ℂ. The functor

A ↦ Hom_{C_{1}}(Q; A)

from C_{1} to finitely generated projective right R-modules is faithful, and it is easy to see that it is an equivalence of categories. On the other hand C_{1} is equivalent to C_{2}^{∗}, which is an abelian category, and an abelian category in which every object is projective is clearly semisimple.

For semisimple categories C with finitely many irreducibles the contravariant functor C^{op} → C^{∗} given by

A ↦ {B ↦ Hom_{C}(A; B)}

is an equivalence. Putting C(X)^{op} ≃ C(X)^{∗} together with C(X̄) ≃ C(X)^{∗} we see that one can define a functor A ↦ A^{∗} which is an equivalence C(X)^{op} → C(X̄). It is characterized by

⟨A^{∗}, B⟩ ≅ Hom_{C(X)}(A; B).

Because the functor A ↦ A^{∗} must induce an isomorphism

Hom_{C(X)}(A; B) → Hom_{C(X̄)}(B^{∗}; A^{∗}),

while the symmetry of ⟨A^{∗}, C^{∗}⟩ gives us

Hom_{C(X)}(A; C^{∗}) ≅ Hom_{C(X̄)}(C; A^{∗}),

we have a natural isomorphism A → A^{∗∗}.