Virasoro Constraints and Flavor-Topology Duality in QCD

We derive Virasoro constraints for the zero momentum part of the QCD-like partition functions in the sector of topological charge $\nu$. The constraints depend on the topological charge only through the combination $N_f+\beta \nu \mathrm{/}2$ where the value of the Dyson index $\beta$ is determined by the reality type of the fermions. This duality between flavor and topology is inherited by the small-mass expansion of the partition function and {\em all} spectral sum-rules of inverse powers of the eigenvalues of the Dirac operator. For the special case $\beta =2$ but arbitrary topological charge the Virasoro constraints are solved uniquely by a Generalized Kontsevich Model with potential ${\cal V}(X) = 1/X$.