ar X iv : h ep - t h / 01 05 16 9 v 2 1 0 Ja n 20 02 QCD , Wick ’ s Theorem for KdV τ - functions and the String Equation

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 (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). 1. Recently G.Akemann and P.Damgaard [1] established two consistency relations for the partition functions of matrix models which connect finite-volume partition functions with different fermion numbers. In the course of their paper the similarity of these relations with different bilinear relations for the τ-functions of the KP hierarchy were noted and they conjectured they were in fact related. This letter provides a comment on the origin of these reduction formulae. We show they are direct corollaries of Wick's theorem and the string equations of the τ-function theory. Application of this theorem is possible because the partition functions of Cartanian matrix models are KP/Toda τ-functions (see [3, 4] and references therein, especially [5] and [6], for the generic theory and [7, 2] for applications to unitary-matrix models). The algebraic relation (" Consistency Condition I " of ref.[1]) is true for any 2-reduction of KP τ-function, which are essentially the KdV or mKdV reductions 1 , of which the Brezin-Gross-Witten (BGW) model [8] under actual examination in [1] is an example. The differential relation (" Consistency Condition II ") is the string equation, depending on the specifics of the particular model. (The BGW model is studied from the perspective of integrability theory in ref.[2].) These consistency conditions are imposed onto the QCD partition function as a function of fermionic masses µ i at very low energies, which is of the general form Z (N f) ν ({µ}) = det Φ({µ})/∆({µ 2 }). 1 The KdV and mKdV hierarchies are basically the same reduction of the (m)KP hierarchy. One deals with the mKdV case when one additionally considers the discrete (Toda) time " evolution ". While generically the evolution of the KP τ-function τn in this discrete time is governed by the Toda dynamics, in the case of the 2-reduction there exist only two different values of the zero …