Brownian Motion, Chern-Simons Theory, and 2d Yang-Mills

TL;DR

This paper reveals deep connections between Brownian motion, Chern-Simons theory, and 2d Yang-Mills, showing how probabilistic models can encode gauge theory partition functions and link invariants.

Contribution

It establishes a precise correspondence between Brownian motion probabilities and gauge theory quantities, including partition functions and knot invariants, unifying these concepts within a probabilistic framework.

Findings

Brownian motion probabilities match Chern-Simons partition functions.

Expectations of certain Brownian motion configurations correspond to knot invariants.

Matrix model calculations relate to probability additivity laws.

Abstract

We point out a precise connection between Brownian motion, Chern-Simons theory on S^3, and 2d Yang-Mills theory on the cylinder. The probability of reunion for N vicious walkers on a line gives the partition function of Chern-Simons theory on S^3 with gauge group U(N). The probability of starting with an equal-spacing condition and ending up with a generic configuration of movers gives the expectation value of the unknot. The probability with arbitrary initial and final states corresponds to the expectation value of the Hopf link. We find that the matrix model calculation of the partition function is nothing but the additivity law of probabilities. We establish a correspondence between quantities in Brownian motion and the modular S- and T-matrices of the WZW model at finite k and N. Brownian motion probabilitites in the affine chamber of a Lie group are shown to be related to the partition function of 2d Yang-Mills on the cylinder. Finally, the random-turns model of discrete random walks is related to Wilson's plaquette model of 2d QCD, and the latter provides an exact two-dimensional analog of the melting crystal corner. Brownian motion provides a useful unifying framework for understanding various low-dimensional gauge theories.