### Chern-Simons States from L-infinity Bundles, III: States over the Circle

#### Posted by Urs Schreiber

I am continuing to look at the structure of $(4-d)$-states of Chern-Simons theory over $d$-dimensional manifolds by using the $L_\infty$-algebraic model of the Chern-Simons 3-bundle ($\sim$ 2-gerbe) with connection over $B G$ discussed in section 7.3 of *$L_\infty$-Connections and Applications* (pdf, blog, arXiv).

The program is: transgress this 3-bundle as in section 9.2. to spaces of maps from a $d$-dimensional parameter space, and then compute the collection of $(4-d)$-sections following the $L_\infty$-algebraic description for computing $\infty$-sections of the general concept of $\infty$-sections.

Last time I looked at the sections over 2-dimensional surfaces, that computation remaining somewhat inconclusive, as the holomorphic structure on the transgressed 1-bundle does not appear yet.

Today I instead looked at the states over 1-dimensional manifolds: over circles.

I find in section 4.1 of

*Sections and covariant derivatives of $L_\infty$-algebra connections*

(pdf, blog)

that the sections of the 2-bundle obtained from transgression of the Chern-Simons 3-bundle to the configuration space over the circle come from bundles of representations of the Kac-Moody central extension of the Lie algebra of loops over the space of $g$-holonomies over the circle.

And I think this is what the result should be, though a couple of details deserve a closer look.

One nice byproduct is this:

the computation explicitly gives a derivation in the present context of the fact that the 2-cocycle on the loop group

$tg_{S^1} \mu(f,g) = \int_{S^1} \langle f \wedge d g \rangle$

comes from the very transgression we are talking about of the 3-cocycle

$\mu(x,y,z) = \langle x, [y,z]\rangle$

on the semisiple Lie algebra $g$ that the Chern-Simons 3-bundle is governed by.

This crucially depends on some gymnastics with that “almost-internal-hom”

$(A, B) \mapsto \Omega^\bullet(\mathrm{maps}(A,B))$ of dg-algebras which we talked about in Transgression of $n$-Transport (section 5.1).

Posted at February 4, 2008 5:26 PM UTC
## Re: Chern-Simons States from L-infinity Bundles, III: States over the Circle

Actually, there is this useful reformulation of the very fact addressed above:

according to section 8 we can regard the Chern-Simons 3-bundle on $B G$ as the obstruction to lifting the universal $G$ bundle $E G \to B G$ to a $String(G)$-2-bundle.

Now as we hit everything in sight with

$\Omega^\bullet(maps(--, \Omega^\bullet(S^1)))$

we transgress everything to loop space:

now we have an $\Omega G$-bundle over $\Omega B G$ and the CS 3-bundle becomes a 2-bundle aka 1-gerbe which is the lifting gerbe obstructing the lift of the loop group bundle to the corresponding centrally extended loop group bundle.

So the fact that Chern-Simons theory over the circle involves bundles whose fibers are reps of the centrally extended loop group is just another incarnation of the fact that a string-structure on a Spin manifold $X$ is equivalently

- the obstruction to lifting the Spin-bundle on $X$ to a String-2-bundle on $X$

- the obstruction to lifting the $\Omega Spin$-bundle on $\Omega X$ to a $\hat \Omega Spin$-bundle.

That’s probably clear to anyone who has ever thought about it, but it’s kind of nice to see how it arises here by just hitting $L_\infty$-algebra connection descent objects with the transgression functor

$\Omega^\bullet(-- ,\Omega^\bullet(S^1)) \,.$