Expectation values of polynomials and moments on general compact Lie groups

TL;DR

This paper introduces a unified framework for calculating expectation values of polynomials and moments on compact Lie groups, extending classical results and providing explicit formulas for various groups and measures.

Contribution

It develops a general representation-theoretic approach to compute expectation values on compact Lie groups, including new formulas for Wilson loops and Weingarten functions.

Findings

Derived expectation formulas for classical and exceptional Lie groups.

Constructed Weingarten functions for general compact Lie groups.

Extended classical results to new settings involving Haar measure, Brownian motion, and Wilson action.

Abstract

We develop a powerful framework to calculate expectation values of polynomials and moments on compact Lie groups based on elementary representation-theoretic arguments and an integration by parts formula. In the setting of lattice gauge theory, we generalize expectation value formulas for products of Wilson loops by Chatterjee and Jafarov to arbitrary compact Lie groups, and study explicit examples for many classical compact Lie groups and the exceptional Lie group $G_2$. Extending classical results by Collins and L\'evy, we use our framework to derive expectation value formulas of polynomials of matrix coefficients under the Haar measure, Brownian motion, and the Wilson action. In particular, we construct Weingarten functions for general compact Lie groups by studying the underlying tensor invariants, and apply this to $\mathrm{SU}(N)$ and $G_2$.