On finite-volume gauge theory partition functions

Abstract

We prove a Mahoux–Mehta-type theorem for finite-volume partition functions of SU(Nc≥3) gauge theories coupled to fermions in the fundamental representation. The large-volume limit is taken with the constraint V1/mπ4. 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 τ-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(Nc) gauge theories coupled to quenched fermions in the adjoint representation. The result coincides exactly with earlier results based on Random Matrix Theory.