A positive formula for volumes of moduli spaces of flat unitary connections on compact surfaces

TL;DR

This paper presents a positive, explicit formula for the volume of moduli spaces of flat unitary connections on compact surfaces, connecting geometric, combinatorial, and probabilistic perspectives.

Contribution

It introduces a novel positive formula for these volumes using polytopes and honeycombs, extending the work of Knutson and Tao to moduli space volumes.

Findings

Explicit positive volume formula for moduli spaces

Connection between volumes and honeycomb polytopes

Positive formula for U(n) Yang-Mills marginals

Abstract

We provide a manifestly positive expression for the volume of the moduli spaces of flat $\mathrm{U}(n)$-valued connections on punctured compact oriented surfaces. This volume is obtained by summing volumes of explicit polytopes describing coloured honeycombs on a polygon, in the spirit of the work of Knutson and Tao describing the spectrum of the sum of two hermitian matrices. As a corollary, we also provide a positive formula for marginals of the $\mathrm{U}(n)$-valued Yang-Mills measure on a compact oriented surface in terms of the probability distribution of an explicit path process.