QCD3 and the Replica Method

TL;DR

This paper employs the replica method to analyze the QCD3 partition function, deriving exact relations with QCD4, and obtaining spectral density results through multiple approaches including Virasoro constraints and localization techniques.

Contribution

It introduces novel exact relations between QCD3 and QCD4 partition functions and spectral densities, and develops methods to compute spectral properties using the replica approach.

Findings

Exact relations between QCD3 and QCD4 partition functions.

Explicit expressions for the microscopic spectral density of QCD3.

Small and large mass expansions of the resolvent obtained from replica limits.

Abstract

Using the replica method, we analyze the mass dependence of the QCD3 partition function in a parameter range where the leading contribution is from the zero momentum Goldstone fields. Three complementary approaches are considered in this article. First, we derive exact relations between the QCD3 partition function and the QCD4 partition function continued to half-integer topological charge. The replica limit of these formulas results in exact relations between the corresponding microscopic spectral densities of QCD3 and QCD4. Replica calculations, which are exact for QCD4 at half-integer topological charge, thus result in exact expressions for the microscopic spectral density of the QCD3 Dirac operator. Second, we derive Virasoro constraints for the QCD3 partition function. They uniquely determine the small-mass expansion of the partition function and the corresponding sum rules for inverse Dirac eigenvalues. Due to de Wit-'t Hooft poles, the replica limit only reproduces the small mass expansion of the resolvent up to a finite number of terms. Third, the large mass expansion of the resolvent is obtained from the replica limit of a loop expansion of the QCD3 partition function. Because of Duistermaat-Heckman localization exact results are obtained for the microscopic spectral density in this way.