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What is Chern-Simons theory expected to assign to a point?

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Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a $3$-dimensional topological quantum field theory (going down to surfaces), Chern-Simons theory.

Later other authors (Reshetikhin-Turaev, ???) described how to extend this theory down to $1$-manifolds. It's known that such a theory is determined by what it assigns to a circle $S^1$, which must be a modular tensor category; this category can be described either as a certain category of representations of the loop group $LG$ at level $k$ or as a certain category of representations of the quantum group $U_q(\mathfrak{g})$, where $q$ is a suitable function of $k$. The relationship between these two descriptions is unclear to me.

My impression is that it's expected that Chern-Simons theory extends all the way down to $0$-manifolds; that is, that it is a fully extended TQFT. By Lurie's classification, such a theory is completely determined by what it assigns to a point, which is a fully dualizable object in a suitable $3$-category.

What are some conjectural descriptions of this object?

The nLab is somewhat vague on this subject. Here's what I know:

The corresponding object for $3$-dimensional Dijkgraaf-Witten theory is known, although I'm not sure exactly who this is due to. Here $G$ is replaced by a finite group and $k$ is thought of as a class in $H^3(BG, \text{U}(1))$. $k$ is used to twist the associator on the monoidal category of $G$-graded vector spaces, giving a monoidal category (in fact a fusion category) of "twisted $G$-graded vector spaces," to be thought of as a fully dualizable object in the $3$-category of monoidal categories and bimodule bicategories over these (Douglas-Schommer-Pries-Snyder?), and I think this is what fully extended $3$-dimensional Dijkgraaf-Witten assigns to a point. Freed-Hopkins-Lurie-Teleman generalized this construction to the case that $G$ is a torus; here $G$-graded vector spaces are replaced by skyscraper sheaves on $G$. I don't know if this is expected to generalize.

This post imported from StackExchange at 2014-04-14 11:25 (UCT), posted by SE-user Qiaochu Yuan
asked Apr 14, 2014 in Theoretical Physics by Qiaochu Yuan (380 points) [ no revision ]
retagged Apr 14, 2014

1 Answer

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I have a proposal for what Chern-Simons should assign to a point:

The $\otimes$-category of (certain) representations of $\widetilde{\Omega G}$.

Here, $\Omega G$ is the based loop group, and the tilde indicates that one should take the central extension inherited from the level $k$ central extensions of $LG$.

Another way of phrasing the proposal, that also works when $G$ is not connected (Dijkgraaf-Witten theory being the special cases thereof when $G$ is finite) is to say that it's the category of (certain) vector bundles over the moduli space of $G$-bundles over $[0,1]$ trivialized at $\{0,1\}$.

The precise definition is spelled out at minute 50 of the following video: (and if you're gong to watch it, I recommend watching from the beginning)

This post imported from StackExchange at 2014-04-14 11:25 (UCT), posted by SE-user André Henriques
answered Apr 14, 2014 by André Henriques (210 points) [ no revision ]
Dear André, will you excuse me if I ask you what your mother tongue is? The only excuse for this indiscrete request is that I am pathologically interested in linguistics [and your English is too perfect to be native:-)]

This post imported from StackExchange at 2014-04-14 11:25 (UCT), posted by SE-user Georges Elencwajg

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