On a c-number quantum $τ$-function

TL;DR

This work develops a $c$-number quantum $\tau$-function as a finite-difference ($q$-deformed) generalization of the conventional $\tau$-functions of the KP/Toda hierarchies. It shows that a $\tau^{(q)}$ can be obtained from the classical Toda-lattice $\tau$ by a Miwa-type transformation relating time variables, with $t_k = \frac{1}{k} \frac{((1-q)T_1)^k}{1-q^k}$, and that higher times $T_k$ correspond to insertions of $k$-fermion non-local operators, preserving the free-fermion framework through a universal time reparameterization. The paper also discusses the interpretation of these $q$-deformed objects within the standard determinant/Hirota structure and explores implications for $q$-matrix models and $q$-free fields, suggesting how Ward identities and $q$-Virasoro constraints might arise. Overall, it provides a bridge between classical Toda/$\tau$-functions and their $q$-deformed, discrete counterparts while keeping the familiar Grassmannian formalism intact.

Abstract

We first review the properties of the conventional $τ$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve however the concept of operator-valued $τ$-function nor the one associated with non-Cartanian (level $k\ne1$) algebras. The present study could be useful to understand better $q$-free fields and their relation to ordinary free fields.