### States of Chern-Simons Theory

#### Posted by Urs Schreiber

In the thread $L_\infty$ associated Bundles and Sections I started talking about how to compute the space of states of Chern-Simons theory using the $L_\infty$-algebraic model for the Chern-Simons 2-gerbe with connection on $B G$ that we describe in *$L_\infty$-connections and applications* (pdf, blog, arXiv).

Prompted by a request for more references on this question which I received, I shall try to collect some (incomplete) list of literature here, with some comments.

The general setup is as follows, and the various approaches to it may differ in terms of which concrete models are used to **make sense** of the various objects mentioned now:

For $G$ any suitably well behaved Lie group, there is supposed to be a canonical (family of) line 3-bundles (= 2-gerbes) $CS$ with connection (“and curving”) $\nabla$ over the space $B G$.

$\array{ CS_\nabla \\ \downarrow \\ B G } \,.$

For any 2-dimensional manifold $\Sigma$ we should be able to transgress this to a line bundle with connection $tg_\Sigma CS_\nabla$ on a suitable space $Maps(\Sigma, B G)$ of maps from $\Sigma$ to $B G$

$\array{ tg_\Sigma CS_\nabla &&& \Sigma \times Maps(\Sigma, B G) &&& CS_\nabla \\ & \searrow & {}^{p_2} \swarrow && \searrow^{ev} & \swarrow \\ && Maps(\Sigma, B G) && B G } \,.$

There happens to be a complex structure appearing which makes this a holomorphic line bundle. The **space of states** of Chern-Simons theory over $\Sigma$ is supposed to be the space of *holomorphic sections*
$Z(\Sigma) := \Gamma_{hol}(tg_\Sigma CS)$
of $tg_\Sigma CS_\nabla$.

This space, in turn, has a “holographically” related interpretation in terms of the space of “pre-correlators” of another theory which comes from a line 2-bundle (= gerbe) on $G$: Wess-Zumino-Witten theory.

While much of the literature addresses both the Chern-Simons as well as the Wess-Zumino-Witten aspect, here I will concentrate mostly on the Chern-Simons aspect.

The observation which got all this started is the one in

Edward Wittem
*Quantum Field Theory and the Jones Polynomial*

(1989)

(pdf)

Here the quantum space of states of Chern-Simons theory was first analyzed and the relation to the conformal blocks of WZW theory observed.

More details on the computations appearing there appeared shortly afterwards in

Shmuel Elitzur, Gregory Moore, Adam Schwimmer, Nathan Seiberg
*Remarks on the canonical quantization of the Chern-Simons-Witten theory*

(1989)

(pdf)

A very detailed analysis is given in

Krzystof Gawedzki and Antti Kupiainen
*$SU(2)$ Chern-Simons states at genus zero*

(1991)

(…)

and

Fernando Falceto and Krzystof Gawedzki
*Chern-Simons states at genus one*

(1992)

(arXiv)

and

Krzysztof Gawedzki
*$SU(2)$ WZW Theory at Higher Genera*

(1994)

(arXiv).

A good general overview about what’s going on, including lots of further references, can be obtained from the first five pages of the second one.

Chern-Simons theory is governed by an element in the fourth integral cohomology of $B G$ – the *level*. These elements classify abelian 2-gerbes (= line 3-bundle) on $B G$.

As far as I am aware, the first article which observes that Chern-Simons theory is therefore a theory involving 2-gerbes is

Jean-Luc Brylinski and Dennis McLaughlin
*A geometric construction of the first Pontryagin class*

(1993)

(review, pdf)

which closes with the remark

We conclude from this discussion that 2-gerbes are the fundamental geometric objects in Chern-Simons theory. A similar observation has been made by D. Kazhdan.

A detailed discussion of the relevant transgression of this 2-gerbe is on page 133 of

Jean-Luc Brylinski and Dennis McLaughlin
*The geometry of degree-4 characteristic classes and of line bundles on loop spaces II*

(1996)

(review, pdf).

More recent approaches tend to put a stronger emphasis on this higher structure.

In the last section of

Stephan Stolz and Peter Teichner
*What is an elliptic object*

(2002)

(pdf)

there are some indications about how Chern-Simons theory is related to String-(2-)bundles.

A bundle gerbe-theoretic discussion of the Chern-Simons 2-gerbe is given in

Alan L. Carey, Stuart Johnson, Michael K. Murray, Danny Stevenson, Bai-Ling Wang
*Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories*

(2004)

arXiv.

I had reported a while ago on a visionary talk by Michael Hopkins, where he indicated how he conceives Chern-Simons theory in the context of $\infty$-functors

Michael Hopkins

Lecture: Topological Aspects of topological field theory

(2006)

Introduction and Outlook

Infinity-Catgeory Description

Chern-Simons

There is of course much more literature. But I’ll leave it at that for the moment.

## Re: States of Chern-Simons Theory

Wow thanks for all these references Urs! This is great.