Chern-Simons Theory in Lens Spaces from 2d Yang-Mills on the Cylinder

TL;DR

This paper establishes a connection between 2d Yang-Mills theory on the cylinder and Chern-Simons theory in lens spaces, revealing new insights into their mathematical structure and relations to conformal field theory.

Contribution

It demonstrates that 2d Yang-Mills on the cylinder computes Chern-Simons invariants in lens spaces via a specific operator, generalizing known relations with WZW models and topological theories.

Findings

Partition function reduces to U=ST^pS operator in Chern-Simons theory.

U operator yields partition functions and Wilson loop expectations in lens spaces.

Modular properties of 2dYM are linked to affine character transformations.

Abstract

We use the relation between 2d Yang-Mills and Brownian motion to show that 2d Yang-Mills on the cylinder is related to Chern-Simons theory in a class of lens spaces. Alternatively, this can be regarded as 2dYM computing certain correlators in conformal field theory. We find that the partition function of 2dYM reduces to an operator of the type U=ST^pS in Chern-Simons theory for specific values of the YM coupling but finite k and N. U is the operator from which one obtains the partition function of Chern-Simons on S^3/Z_p, as well as expectation values of Wilson loops. The correspondence involves the imaginary part of the Yang-Mills coupling being a rational number and can be seen as a generalization of the relation between Chern-Simons/WZW theories and topological 2dYM of Witten, and Blau ant Thompson. The present reformulation makes a number of properties of 2dYM on the cylinder explicit. In particular, we show that the modular transformation properties of the partition function are intimately connected with those of affine characters.