Chern-Simons theory

Question

In Witten's paper on QFT and the Jones polynomial, he quantizes the Chern-Simons Lagrangian on $\Sigma\times \mathbb{R}^1$ for two case: (1) $\Sigma$ has no marked points (i.e., no Wilson loops) and (2) $\Sigma$ has marked points and each point has attached a representation of the gauge group. In case (1), Witten shows that the vector space should be the space of holomorphic sections of a determinant line bundle over the moduli space of flat connections. For the second case he states that the vector space should be the $G$-invariant subspace of the tensor product of all the representations associated to the marked points; by this I mean, if $\Sigma$ has $r$ marked points and each point has a rep $R_i$ then the quantum Hilbert space is $(\bigotimes_{i=1}^r R_i)^G$.

Does anyone know how to interpret this second case in terms of sections of some bundle? I mean, shouldn't the second case reduce to the first when you remove the marked points? Also, Witten states, immediately following case (2), that in the presence of no marked points the quantum Hilbert space is 1-dimensional. How can one see that from the formula for the quantum Hilbert space, $(\bigotimes_{i=1}^r R_i)^G$?

Answer 1

There are two ways to think of the Hilbert space as the space of sections of a line bundle.

First, the exponentiated Chern-Simons action on a manifold $\Sigma\times[0,1]$ is a section of the determinant line bundle $\mathcal{L}_\Sigma$ on the space of flat connections on $\Sigma$. Moreover, Wilson loops (which can be thought of as a 1d TFT) contribute $R_i$ each. So, the Hilbert space (before remembering gauge invariance) is $\Gamma(\mathcal{L}_\Sigma^k)\otimes\bigotimes_i R_i$. Now, if $\Sigma = S^2$, which is simply-connected, the space of flat connections is a point, so $\Gamma(\mathcal{L}_\Sigma^k)=\mathbf{C}$. Finally, gauge invariance picks out the $G$-invariants in $\bigotimes_i R_i$.

Note, that the Hilbert space for a non-simply-connected $\Sigma$ is nontrivial even without the punctures.

Another way to think of this Hilbert space is to recall the 2d CFT <-> 3d TFT correspondence. The idea here is the following. Correlation functions of a 2d CFT live in a certain bundle over the moduli space of complex curves $M_{g,n}$ called the bundle of conformal blocks. The Knizhnik-Zamolodchikov equations on correlation functions correspond to a (projectively) flat connection on this bundle. So, a 2d CFT associates global sections of this bundle to a topological surface $\Sigma$, this is the Hilbert space in a 3d TFT. In the case of the Chern-Simons theory, the associated 2d CFT is the Wess-Zumino-Witten model.

A down-to-earth description can be found in

S. Elitzur, G. Moore, A. Schwimmer, N. Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl Phys B326 (1989), 108.

Mathematically, this correspondence is an equivalence between modular functors (as defined by Segal in The definition of conformal field theory) and modular tensor categories which give rise to 3d TFTs (due to Reshetikhin and Turaev).

All of that is discussed in an excellent book Lectures on tensor categories and modular functors by Bakalov and Kirillov.