QCD3 and the replica method

Abstract

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.

Random Matrices with Merging Singularities and the Painlevé V Equation

“…The formulas (5.25), (5.26) which we obtained in the proof of Theorem 1.1 for t < 0, hold also in the case t > 0, if we replace Ψ (1) by Ψ (2) . We obtain from (3.11) and condition d) in the RH problem for Ψ + that as z → √ t,…”

Factorization of correlation functions and the replica limit of the Toda lattice equation

“…It has long been known [2][3][4][5][6][7][8][9][10][11][12] that this is the case for perturbative calculations such as for the expansion of the resolvent in an asymptotic series in 1/x. Problems are known to occur when one attempts to derive exact nonperturbative results for the quenched average resolvent [4,7].…”

“…For instance, it was shown that bosonic and fermionic replica limits lead to different results for the two point function of the Gaussian unitary ensemble and that neither limits lead to the correct result [4]. The original critique [4] motivated an insightful and challenging debate [5][6][7][8][9][10][23][24][25]] the details of which have been accounted for in the introduction of [26]. The debate lead to the generally accepted belief that, although the replica trick correctly reproduces the asymptotic expansion of the microscopic spectral correlation functions, it can not be trusted for nonperturbative calculations except in a few cases, when the series can be resummed [24] or simply terminates [7,9].…”

Character expansions for the orthogonal and symplectic groups

“…(For the application of Virasoro constraints on the effective finite volume partition function see, for example, Refs. [33][34][35]). …”