The q-deformed conformal matrix model partition function, after a Fourier transform, is a Toda tau-function whose shifted ratios satisfy the discrete Painleve q-PVI equation, with the string equation supplied by Virasoro constraints.
On a c-number quantum $\tau$-function
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abstract
We first review the properties of the conventional $\tau$-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 $\tau$-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.
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Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models
The q-deformed conformal matrix model partition function, after a Fourier transform, is a Toda tau-function whose shifted ratios satisfy the discrete Painleve q-PVI equation, with the string equation supplied by Virasoro constraints.