QCD, Wickʹs theorem for KdV τ-functions and the string equation

Two consistency conditions for partition functions established by Akemann and Damgaard 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 Wickʹs 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).