QCD(3) and the replica method

Using the replica method, we analyze the mass dependence of the QCD(3) part ition function in a parameter range where the leading contribution is from the zero momentum Goldstone fields. Three complementary approaches are cons idered in this article. First, we derive exact relations between the QCD(3) partition function and the QCD(4) partition function continued to half-int eger topological charge. The replica limit of these formulas results in exa ct relations between the corresponding microscopic spectral densities of QC D(3) and QCD(3). Replica calculations, which are exact for QCD(4) at half-i nteger topological charge, thus result in exact expressions for the microsc opic spectral density of the QCD(3) Dirac operator. Second, we derive Viras oro constraints for the QCD(3) partition function. They uniquely determine the small-mass expansion of the partition function and the corresponding su m rules for inverse Dirac eigenvalues. Due to de Wit-'t Hooft poles, the re plica 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) parti tion function. Because of Duistermaat-Heckman localization exact results ar e obtained for the microscopic spectral density in this way. (C) 2001 Elsev ier Science B.V. All rights reserved.