Gaussian measure on the dual of $\mathrm{U}(N)$, random partitions, and topological expansion of the partition function

TL;DR

This paper investigates a Gaussian measure on the dual of U(N), revealing its connection to random partitions and topological expansions, and provides a rigorous proof of gauge/string duality in 2D Yang-Mills theory.

Contribution

It introduces a Gaussian measure on the dual of U(N), analyzes its properties, and establishes a topological expansion of the partition function linked to ramified coverings, confirming gauge/string duality.

Findings

Coupling of random partitions vanishes as N approaches infinity.

Partition function admits a 1/N asymptotic expansion with topological coefficients.

Provides a rigorous proof of gauge/string duality for 2D U(N) Yang-Mills theory.

Abstract

We study a Gaussian measure with parameter $q\in(0,1)$ on the dual of the unitary group of size $N$: we prove that a random highest weight under this measure is the coupling of two independent $q$-uniform random partitions $\alpha,\beta$ and a random highest weight of $\mathrm{U}(1)$. We prove deviation inequalities for the $q$-uniform measure, and use them to show that the coupling of random partitions under the Gaussian measure vanishes in the limit $N\to\infty$. We also prove that the partition function of this measure admits an asymptotic expansion in powers of $1/N$, and that this expansion is topological, in the sense that its coefficients are related to the enumeration of ramified coverings of elliptic curves. It provides a rigorous proof of the gauge/string duality for the Yang-Mills theory on a 2D torus with gauge group $\mathrm{U}(N),$ advocated by Gross and Taylor \cite{GT,GT2}.