QCD, Wick's Theorem for KdV $\tau$-functions and the String Equation

TL;DR

This paper interprets consistency conditions of QCD partition functions using integrable hierarchies, showing their algebraic relations stem from Wick's theorem and the string equation in the context of KdV tau-functions.

Contribution

It reveals the algebraic structure behind QCD partition functions through integrable hierarchies, connecting Wick's theorem and the string equation to matrix models and KdV tau-functions.

Findings

Consistency conditions are derived from integrable hierarchy principles.

Wick's theorem explains the algebraic relations in fermionic correlators.

The string equation encodes the derivative-based consistency condition.

Abstract

Two consistency conditions for partition functions established by Akemann and Dam-gaard in their studies of the fermionic mass dependence of the QCD partition function at low energy ({\it a la} Leutwiller-Smilga-Verbaarschot) are interpreted in terms of integrable hierarchies. Their algebraic relation is shown to be a consequence of Wick's theorem for 2d fermionic correlators (Hirota identities) in the special case of the 2-reductions of the KP hierarchy (that is KdV/mKdV). The consistency condition involving derivatives is an incarnation of the string equation associated with the particular matrix model (the particular kind of the Kac-Schwarz operator).