On finite-volume gauge theory partition functions

Abstract

We prove a Mahoux–Mehta-type theorem for finite-volume partition functions of SU( N c ≥3) gauge theories coupled to fermions in the fundamental representation. The large-volume limit is taken with the constraint V≪1/ m π 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 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( N c ) gauge theories coupled to quenched fermions in the adjoint representation. The result coincides exactly with earlier results based on Random Matrix Theory.