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.

Article Outline

1. Introduction

2. The QCD₃ partition function

3. The QCD₃–QCD₄ connection

4. Applying the replica method: general results

4.1. The microscopic spectral density

4.2. Higher order correlation functions

5. Virasoro constraints and spectral sum rules

6. The replica limit of the QCD₃ finite-volume partition function

6.1. Small-mass expansion

6.2. Large-mass expansion

7. Conclusions

Acknowledgements

Appendix A. Relations between QCD₃ and QCD₄ determinants

Appendix B. Relations between QCD₃ and QCD₄ from Random Matrix Theory

Appendix C. Partition function and spectral sum rules to sixth order

References