## June 17, 2008

### Teleman on Topological Construction of Chern-Simons Theory

#### Posted by Urs Schreiber

Imagine me drowned in chocolate. Or similar. Here: drowned in interesting stuff (at the Hausdorff institute). I didn’t know that I could get too much of it. But this is getting close. :-)

Today Constantin Teleman gave a talk on ongoing joint work with Dan Freed, Michael Hopkins and Jacob Lurie on describing Chern-Simons theory as an extended QFT – or as an $n$-tiered QFT as they sometimes say.

We have talked about that a lot here already, and most of the things in his talk, except for a new construction at the very end, we have seen here in one form or other before. In particular, with each talk like this I hear I am being reminded of Bruce Bartlett’s mysteriously unpublished PhD thesis which contains various of the central ideas appearing here.

Here is an attempt at a quick transcript of the notes that I took in the talk. The main point is towards the end, where a candidate construction for the 2-category assigned by Chern-Simons theory to the point is given. I don’t think I’ll make it that far. But it is supposedly a generalization of the situation for the Dijkgraaf-Witten case with a finite gauge group.

After the talk I asked Constantin Teleman about his opinion about the observation which I made in the entry with the curious title 2-Monoid of observables on String-G, where I pointed out that Simon Willerton’s rephrasing of the Freed-Hopkins-Teleman result has a nice generalization from finite to Lie groups as follows:

for $\mathrm{String}\left(G\right)$ the strict String Lie 2-group and $B\mathrm{String}\left(G\right)$ its incarnation as a one-object 2-groupoid, we have, up to dealing with technicalities, that Chern-simons theory assigns to the point

$\mathrm{Rep}\left(\mathrm{String}\left(G\right)\right):=2{\mathrm{Funct}}^{\infty }\left(B\mathrm{String}\left(G\right),2\mathrm{Vect}\right)$

and to the circle the transgression obtained from homming into

${\mathrm{TwistedVectBund}}^{G}\left(G\right)={\mathrm{Funct}}^{\infty }\left(\mathrm{Funct}\left(BZ,B\mathrm{String}\left(G\right)\right),\mathrm{Hilb}\right)$

following some general pattern.

I am not sure if I expressed myself well in the attempt to propose this as a useful reformulation which may point in interesting directions. I think Constantin Teleman replied that this is what they are doing anyway.

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 $h\in {H}^{4}\left(BG,ℤ\right)$ 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

${\int }_{\Sigma }{\mathrm{ev}}^{*}h\in {H}^{2}\left(G{\mathrm{Bund}}_{\mathrm{flat}}\left(\Sigma \right),ℤ\right)$

on the space of flat $G$-bundles on the surface, which arises by transgressing the given 4-class $h\in {H}^{4}\left(BG,ℤ\right)$ through the correspondence $G{\mathrm{Bund}}_{\mathrm{flat}}\left(\Sigma \right)←G{\mathrm{Bund}}_{\mathrm{flat}}\left(\Sigma \right)\to BG\phantom{\rule{thinmathspace}{0ex}}.$

To a circle it assigns a linear category, namely the modular tensor category $\mathrm{Rep}\left(LG\right)$ 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 $\mathrm{Vect}\left[G\right]-\mathrm{Mod}$ 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 ${\mathrm{Vect}}^{G}\left[G\right]$ 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 $U\left(1\right)$ obtained from the twist by using ${H}^{4}\left(BG,ℤ\right)\simeq \mathrm{Hom}\left({H}^{3}\left(BG,ℤ\right),U\left(1\right)\right)$ and then sending the fundamental class of $X$ along $\left[X\right]\stackrel{{P}_{*}}{↦}{H}^{3}\left(BG\right)\stackrel{h}{↦}U\left(1\right)\phantom{\rule{thinmathspace}{0ex}},$ 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 ${\mathrm{HH}}^{*}\left(A\right)\simeq {\mathrm{HH}}_{*}\left(A\right)\phantom{\rule{thinmathspace}{0ex}}.$

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 $𝒞\left[G\right]$ which one can think of as $G$-equivariant ${L}^{2}$-function on $G$, ${L}^{2}\left(G{\right)}^{G}$.

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

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

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 ${A}^{×}$ the group of invertible elements, we get the strict 2-group $\mathrm{AUT}\left({A}^{×}\right)$ coming from the crossed module $\mathrm{AUT}\left({A}^{×}\right)=\left({A}^{×}\stackrel{\mathrm{Ad}}{\to }\mathrm{Aut}\left(A\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

(I think the point here was to secretly talk about weak 2-functors $BG\to B\mathrm{AUT}\left({A}^{×}\right)$).

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 ${\mathrm{Vect}}^{n}$, 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 ${\mathrm{Vect}}^{G}\left[G\right]$

$\xi :2\mathrm{Rep}\left(G\right)\to {\mathrm{Vect}}^{G}\left[G\right]\phantom{\rule{thinmathspace}{0ex}}.$

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 $\tau \in {H}^{4}\left(BG,ℤ\right)$ 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 $\tau$

${}^{\tau }{K}_{G}\left(G\right)\simeq \left[{}^{\tau }{\mathrm{Rep}}_{\mathrm{pos}\mathrm{energy}}\left(LG\right)\right]\phantom{\rule{thinmathspace}{0ex}}.$

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.

Posted at June 17, 2008 6:33 PM UTC

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### Re: Teleman on Topological Construction of Chern-Simons Theory

Now done. Sorry for the delay.

Posted by: Urs Schreiber on June 18, 2008 10:22 AM | Permalink | Reply to this

### Re: Teleman on Topological Construction of Chern-Simons Theory

how weak is weak?

Posted by: jim stasheff on June 27, 2008 5:30 PM | Permalink | Reply to this

### Re: Teleman on Topological Construction of Chern-Simons Theory

how weak is weak?

You mean as in “weak 2-functor” and “weak 2-representation”?

Here weak is meant as in the ordinary context of bicategories. This means that these functors respect, composition of 1-morphisms only up to a coherent 2-isomorphism – the associator –, respect identity 1-morphisms only up to a coherent 2-morphisms – the unitor (or something) and so on.

The important point for weak 2-representations of ordinary 1-groups is that associator: its presence amounts to twisting an ordinary representation by a cocycle.

Posted by: Urs Schreiber on June 27, 2008 7:24 PM | Permalink | Reply to this

### Re: Teleman on Topological Construction of Chern-Simons Theory

Urs wrote:

The important point for weak 2-representations of ordinary 1-groups is that associator: its presence amounts to twisting an ordinary representation by a cocycle.

associator? as opposed to a representator?
cf. an H-map??

Posted by: jim stasheff on June 28, 2008 2:53 AM | Permalink | Reply to this

### Re: Teleman on Topological Construction of Chern-Simons Theory

associator? as opposed to a representator?

Actually I made a mistake: I should have said “compositor” instead of “associator”: the compositor being the “homotopy” which related the composition of the representation of two group elements with the representation of their composition.

So this would justly also be addressed as the “representator”, I suppose.

Posted by: Urs Schreiber on June 29, 2008 5:57 PM | Permalink | Reply to this

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