Integrability of zero-dimensional replica field theories at beta=1

TL;DR

This paper demonstrates that certain replica field theories for real symmetric random matrices are integrable, forming a supersymmetric Pfaff-KP hierarchy, which reproduces key eigenvalue correlation functions nonperturbatively.

Contribution

It proves the integrability of zero-dimensional replica sigma models at beta=1 and connects them to a supersymmetric Pfaff-KP hierarchy, enabling exact calculations.

Findings

Replica partition functions form a supersymmetric Pfaff-KP hierarchy.

The formalism reproduces the nonperturbative eigenvalue correlation formula.

Implications for integrable structures in random matrix theory are discussed.

Abstract

Building on insights from the theory of integrable lattices, the integrability is claimed for nonlinear replica sigma models derived in the context of real symmetric random matrices. Specifically, the fermionic and the bosonic replica partition functions are proven to form a single (supersymmetric) Pfaff-KP hierarchy whose replica limit is shown to reproduce the celebrated nonperturbative formula for the density-density eigenvalue correlation function in the infinite-dimensional Gaussian Orthogonal Ensemble. Implications of the formalism outlined are briefly discussed.