The n-Category Café

Here is the beginning of my transcript. My personal comments set in italics

Goal: Give a topological construction of Chern-Simons theory of a compact Lie group as a 3-2-1-0 TFT.

(The latter is the Freed-school language of saying that we have something like an $\infty$-functor on $n$-dimensional manifolds which assigns data in the given dimensions.)

Status: since 1992, due to Reshitikhin-Turaev, Chern-Simons theory exists as a 3-2-1 TFT, built from an element $h \in H^4(B G, \mathbb{Z})$ for $G$ a connected compact and simply connected Lie group. (For finite groups the analoguous construction is due to Dijkgraaf-Witten and Freed-Quinn.)

This construction uses quantum groups. For the pusposes of this talk, quantum groups will not count as “topological objects” and a different construction is being sought.

Recall Chern-Simons theory.

To a 3-fold it assigns a complex number: the Chern-Simons invariant.

To a surface it assigns a vector space, namely the space of holomorphic sections of the line bundle with class

$$\int_\Sigma ev^* h \in H^2(G Bund_{flat}(\Sigma), \mathbb{Z})$$

on the space of flat $G$-bundles on the surface, which arises by transgressing the given 4-class $h \in H^4(B G, \mathbb{Z})$ through the correspondence $$G Bund_{flat}(\Sigma) \leftarrow G Bund_{flat}(\Sigma) \to B G \,.$$

To a circle it assigns a linear category, namely the modular tensor category $Rep(L G)$ of positive energy projective reps of the loop group of $G$.

So far this is well known since Reshitikhin-Turaev who proved a theorem to the extent that every semisimple modular tensor category gives rise to a 3-2-1 TFT in the above sense.

The goal is to understand what it assigns to the point.

Situation for finite groups. For $G$ replaced by a finite group the situation is a bit better understood.

Here we have a 3-2-1-0 TFT realization of Dijkgraaf-Witten theory, where to the point one associates the 2-category $$Vect[G]-Mod$$ of module categories over the monoidal category of $G$-graded vector spaces (the “group 2-ring” over $G$).

(This is the same as weak 2-reps of $G$.)

To the circle one assigns the category $Vect^G[G]$ of vector bundles over $G$ which are equiuvariant with respect to the adjoint action of $G$ on itself. This is naturally a braided monoidal category

(under the natural fusion operation).

To a closed 3-fold $X$, the finite group theory assigns the number of (necessarily flat) $G$-bundles over it, wheighted by one over the number of automorphisms of each such $G$-bundle and times the element in $U(1)$ obtained from the twist by using $$H^4(B G, \mathbb{Z}) \simeq Hom(H^3(B G, \mathbb{Z}), U(1))$$ and then sending the fundamental class of $X$ along $$[X] \stackrel{P_*}{\mapsto} H^3(B G) \stackrel{h}{\mapsto} U(1) \,,$$ where the first map denotes push-forward along the classifiying map of the given $G$-bundle.

(The curious weighting here is the natural measure on the configuration space, coming from the fact that we can understand the path-integral here as a categorical push-forward as recalled in section 1.4 of $\Sigma$-models and nonabelian differential cohomology, compare also the push-forward operation in Groupoidification, definition 5 in HDA VII).

Aspects of the 2-dimensional theory

Next Constantin Teleman recalled some aspects of extended 2-dimensional TFT, alluding mainly to Kevin Costello’s work on 2-dimensional TCFT.

In such a 2-1-0 theory we assign

a complex number to a surface,

a vector space to a circle namely, in this case, the Hochschild cohomology of some algebra, which in nice cases happens to be isomorphic to the Hoshschild homology $$HH^*(A) \simeq HH_*(A) \,.$$

To the point we assign the linear category of $A$-modules.

topological Yang-Mills

Next example: topological Yang-Mills theory. (No details were given, just the following:)

To a surface assign the number of flat $G$-bundles for $G$ a finite group, or else the relative volume of the moduli space of flat connections for $G$ a compact Lie group.

To a circle assign the center of the group algebra $\mathcal{C}[G]$ which one can think of as $G$-equivariant $L^2$-function on $G$, $L^2(G)^G$.

To a point assign, for $G$ finite dimensional, the category of $G$-representations, thought of here best as the category of $\mathcal{C}[G]$-modules.

This setup can be twisted with a class in $H^3(B G, \mathbb{Z})$, corresponding t a choice of central extension of $G$ by $U(1)$.

example for 2-groups and 2-reps

Next Constantin Teleman talked about an example for a 2-group. I think I can summarize this simply by saying:

For any algebra $A$ with $A^\times$ the group of invertible elements, we get the strict 2-group $$AUT(A^\times)$$ coming from the crossed module $$AUT(A^\times) = (A^\times \stackrel{\mathrm{Ad}}{\to} Aut(A)) \,.$$

A projective 2-representation of an ordinary group $G$ on the algebra $A$ os a group homomorphism $$G \to \mathrm{Out}(A)$$

(I think the point here was to secretly talk about weak 2-functors $\mathbf{B} G \to \mathbf{B} AUT(A^\times)$).

Due to a theorem by Ostrik, all 2-representations of a group $G$ on a semisimple 2-category, such as Kapranov-Voevodsky 2-vector spaces $Vect^n$, are induced from 1-reps of subgroups of $G$.

(typing this I realize that I may be missing some details of what Teleman discussed here)

Now Ganter and Kapranov discussed characters of 2-representations

(and Bruce Bartlett has expanded greatly on that work, maybe he’ll chime in and provides us with more details)

Such a character is a map from 2-representations of $G$ to that category $Vect^G[G]$

$$\xi : 2Rep(G) \to Vect^G[G] \,.$$

One problem to be dealt with here is: such characters do not span all class 2-functions.

(I think the problem alluded to here is that 2-representations on KV-2-vector spaces or other semisimple categories are too restricted. My saying.)

Back to Chern-Simons

statement: “it is credible that $G$-linear categories are the right notion for the assignment by Chern-Simons theory to a point”.

Recall the Freed-Hopkins-Teleman theorem, which says that for $G$ a compact Lie group and $\tau \in H^4(B G, \mathbb{Z})$ we have that the K-class of $\tau$-twisted $G$-equivariant vector bundles on $G$ are isomorphic to the positive energy reps of the loop group at level $\tau$

$${}^\tau K_G(G) \simeq [{}^\tau Rep_{pos energy}(L G)] \,.$$

Notice that the right hand side is the “K-group” (the isomorphism classes) in the modular tensor category of such reps which we mentioned before.

slogan: “reps of the loop group of $G$ are 2-reps of $G$”

(one way to make this slogan precise is the realization of the strict String 2-group in terms of loop groups)

Now finally came the construction of the 2-category supposed to be the right assignment to the point by Chern-Simons. Somehow it involved looking at skyscraper sheaves on the goup (with attention restricted to the case where the group is a torus) and somehow twist the group action on the fibers of these skyscraper sheaves. Apparently this is supposed to be directly analogous to the construction in the finite group case. But I need to check this, maybe with Chris Schommer-Pries, before I write more about this.