Mapping class group and U(1) Chern-Simons theory on closed orientable surfaces

TL;DR

This paper explores the quantization of U(1) Chern-Simons theory on closed surfaces, revealing how the mapping class group is represented in the quantum theory and uncovering a duality in the representations related to the Chern-Simons level.

Contribution

It demonstrates the consistent representation of the mapping class group in quantum U(1) Chern-Simons theory and identifies a duality in the representations based on the level parameter.

Findings

Mapping class group representations are consistent under quantum deformation.

A quantization condition on the Chern-Simons level k is established.

A duality between levels k and 1/k in the representations is found.

Abstract

U(1) Chern-Simons theory is quantized canonically on manifolds of the form $M=\mathbb{R}\times\Sigma$, where $\Sigma$ is a closed orientable surface. In particular, we investigate the role of mapping class group of $\Sigma$ in the process of quantization. We show that, by requiring the quantum states to form representation of the holonomy group and the large gauge transformation group, both of which are deformed by quantum effect, the mapping class group can be consistently represented, provided the Chern-Simons parameter $k$ satisfies an interesting quantization condition. The representations of all the discrete groups are unique, up to an arbitrary sub-representation of the mapping class group. Also, we find a $k\leftrightarrow1/k$ duality of the representations.