Chern-Simons Theory, 2d Yang-Mills, and Lie Algebra Wanderers

TL;DR

This paper reveals deep connections between Chern-Simons theory, 2d Yang-Mills, and Brownian motion, showing how observables relate to counting paths in Lie algebra chambers and linking to matrix models and non-commutative geometry.

Contribution

It establishes a novel correspondence between Chern-Simons theory and Brownian motion in Lie algebra chambers, introducing fermionic formulations and matrix models for computation.

Findings

Exact enumeration of Chern-Simons observables via Brownian paths

Fermionic formulation linking Chern-Simons to non-commutative geometry

Development of matrix models for Brownian motion averages

Abstract

We work out the relation between Chern-Simons, 2d Yang-Mills on the cylinder, and Brownian motion. We show that for the unitary, orthogonal and symplectic groups, various observables in Chern-Simons theory on S^3 and lens spaces are exactly given by counting the number of paths of a Brownian particle wandering in the fundamental Weyl chamber of the corresponding Lie algebra. We construct a fermionic formulation of Chern-Simons on $S^3$ which allows us to identify the Brownian particles as B-model branes moving on a non-commutative two-sphere, and construct 1- and 2-matrix models to compute Brownian motion ensemble averages.