States of Chern-Simons Theory

February 1, 2008

Posted by Urs Schreiber

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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.

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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.

Posted at February 1, 2008 4:10 PM UTC

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7 Comments & 8 Trackbacks

Re: States of Chern-Simons Theory

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

Posted by: Bruce Bartlett on February 4, 2008 5:47 PM | Permalink | Reply to this

Read the post Chern-Simons States from L-infinity Bundles, III: States over the Circle
Weblog: The n-Category Café
Excerpt: On computing the states of Chern-Simons theory over the circle from the L-infinity algebraic model of the Chern-Simons 3-bundle over BG.
Tracked: February 4, 2008 6:10 PM

Re: States of Chern-Simons Theory

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Some other random references : Atiyah’s beautiful Bourbaki summary of Witten’s viewpoint on Chern-Simons theory, Freed’s Classical Chern-Simons Theory I and II and Axelrod’s thesis.

Here’s something I’ve never understood about Chern-Simons theory. It’s a TQFT, so we want to calculate the vector space assigned to a surface $\Sigma$ . As good geometers, we will never be caught without a complex structure, so we hastily add one. Then we make our moduli space, which has a line bundle sitting on it, and we take the space of sections. So to every pair $(\Sigma, \tau)$ where $\Sigma$ is a surface and $\tau$ is a complex structure for it, we have a nice vector space:

(1)

(\Sigma, \tau) \mapsto Z(\Sigma, \tau).

These vector spaces fit together into a vector bundle over Teichmuller space (the space of complex structures on $\Sigma$ ), and hooray there is a canonical flat connection on this vector bundle. At this point everyone seems happy, because this roughly means “it didn’t matter which complex structure we chose”.

But I still want to know… what is the vector space we assign to $\Sigma$ ? We can’t take the space of flat sections under this connection, because the topology of Teichmuller space is nontrivial (isn’t it?), and so there will be holonomy; i.e. there won’t be any globally flat sections, only locally flat sections, right? It seems like the vector space still depends on the complex structure we chose… it’s just that it doesn’t depend on it locally.

Posted by: Bruce Bartlett on February 4, 2008 8:20 PM | Permalink | Reply to this

Re: States of Chern-Simons Theory

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As good geometers, we will never be caught without a complex structure, so we hastily add one.

That’s the very point I need to better understand.

From the point of view of canonical quantization, the complex structure arises from the fact that we recognize that the space of $g$ -valued 1-forms on the surface is not to be regarded as config space, but actually as phase space. So there is a symplectic form around which can be seen to relate to a complex structure on the surface.

I follow this from the point of view of canonical quantization of the CS Lagrangian.

But I am looking for something “better”. Whatever that will mean.

I am guessing that we want to consider an auxiliary $ISO(3)$ -connection on the 3-manifold which will take care of that.

But right now I am somewhat confused about how best to think of this complex structure from the point of view of Chern-Simons theory.

Posted by: Urs Schreiber on February 4, 2008 8:31 PM | Permalink | Reply to this

Re: States of Chern-Simons Theory

As good geometers, we will never be caught without a complex structure, so we hastily add one.

sasy rather either

As physicists
OR
As *algebraic* geometers

Posted by: jim stasheff on February 5, 2008 1:16 AM | Permalink | Reply to this

Re: States of Chern-Simons Theory

because the topology of Teichmuller space is nontrivial (isn’t it?)

In fact it isn’t, rather it’s the universal covering space of the moduli space. Sorry, no good reference at hand, I’m sure someone else has though.

Posted by: Jens on February 5, 2008 1:08 PM | Permalink | Reply to this

Re: States of Chern-Simons Theory

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A nice talk given last month : Remarks on Chern-Simons theory, Dan Freed, talk at the 25th anniversary celebrations of the MSRI.

Abstract:

The Chern-Simons invariant was introduced into differential geometry in the early 1970s. Its quantum embodiment at the end of the 1980s quickly became a poster child quantum field theory for mathematicians: not only does it place knot polynomials in a manifestly three-dimensional context, but it also reveals many algebraic and topological aspects of quantum field theory in general. The mathematics involved strays far from its differential-geometric origins. I will review some developments in this area and use Chern-Simons theory as a window into the broader math-physics interaction.

Unfortunately my internet is currently a bit slow to view this talk properly.

Posted by: Bruce Bartlett on February 5, 2008 8:12 AM | Permalink | Reply to this

Re: States of Chern-Simons Theory

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Hi Bruce,

thanks for the link to this very nice talk!

Unfortunately my internet is currently a bit slow to view this talk properly.

The slides take an unusual long time to load. It’s maybe the fancy background.

The slides are a little similar to those of Freed’s Andrejewski lecture.

Posted by: Urs Schreiber on February 5, 2008 11:32 AM | Permalink | Reply to this

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February 1, 2008