ar X iv : h ep - t h / 00 11 07 2 v 2 2 3 Fe b 20 01 QCD 3 and the Replica Method

Using the replica method, we analyze the mass dependence of the QCD 3 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 QCD 3 partition function and the QCD 4 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 QCD 3 and QCD 4. Replica calculations, which are exact for QCD 4 at half-integer topological charge, thus result in exact expressions for the microscopic spectral density of the QCD 3 Dirac operator. Second, we derive Virasoro constraints for the QCD 3 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 QCD 3 partition function. Because of Duistermaat-Heckman localization exact results are obtained for the microscopic spectral density in this way.