Virasoro Constraints and Flavor-Topology Duality in QCD

TL;DR

This paper establishes Virasoro constraints for QCD-like partition functions, revealing a duality between flavor and topology that influences spectral sum-rules and is solved explicitly for a specific case using a Kontsevich model.

Contribution

It derives Virasoro constraints for QCD partition functions, uncovering a flavor-topology duality and providing explicit solutions for certain cases.

Findings

Virasoro constraints depend on topological charge through a specific combination.

The flavor-topology duality affects spectral sum-rules of the Dirac operator.

Explicit solution via a Generalized Kontsevich Model for a special case.

Abstract

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 $N_f +\beta\nu/2$ 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 ${\cal V}(X) = 1/X$.