Finite Volume Chiral Partition Functions and the Replica Method

In the framework of chiral perturbation theory we demonstrate the equivalence of the supersymmetric and the replica methods in the symmetry breaking classes of Dyson indices β = 1 and β = 4. Schwinger-Dyson equations are used to derive a universal differential equation for the finite volume partition function in sectors of fixed topological charge, ν. All dependence on the symmetry breaking class enters through the Dyson index β. We utilize this differential equation to obtain Virasoro constraints in the small mass expansion for all β and in the large mass expansion for β = 2 with arbitrary ν. Using quenched chiral perturbation theory we calculate the first finite volume correction to the chiral condensate demonstrating how, for all βthere exists a region in which the two expansion schemes of quenched finite volume chiral perturbation theory overlap.