On Finite-Volume Gauge Theory Partition Functions

TL;DR

This paper proves a theorem relating finite-volume gauge theory partition functions to spectral correlations, connecting to integrable hierarchies and confirming results via Random Matrix Theory.

Contribution

It establishes a Mahoux-Mehta--type theorem for finite-volume SU(N_c) gauge theories, linking correlation functions to integrable systems and deriving spectral densities from partition functions.

Findings

Expressed k-point correlation functions in terms of 2-point functions.

Connected finite-volume partition functions to KP hierarchy and $ au$-functions.

Derived spectral density matching Random Matrix Theory results.

Abstract

We prove a Mahoux-Mehta--type theorem for finite-volume partition functions of SU(N_c\geq 3) gauge theories coupled to fermions in the fundamental representation. The large-volume limit is taken with the constraint V << 1/m_{\pi}^4. The theorem allows one to express any k-point correlation function of the microscopic Dirac operator spectrum entirely in terms of the 2-point function. The sum over topological charges of the gauge fields can be explicitly performed for these k-point correlation functions. A connection to an integrable KP hierarchy, for which the finite-volume partition function is a $\tau$-function, is pointed out. Relations between the effective partition functions for these theories in 3 and 4 dimensions are derived. We also compute analytically, and entirely from finite-volume partition functions, the microscopic spectral density of the Dirac operator in SU(N_c) gauge theories coupled to quenched fermions in the adjoint representation. The result coincides exactly with earlier results based on Random Matrix Theory.