Matrix models and the (q-)Painlevé VI equation

The well-known GIL formula (O. Gamayun, N. Iorgov, O. Lisovyy) states that the general tau function of the sixth Painlevé equation is given by the Fourier transform of a conformal block. There is a similar statement about the AGT dual Nekrasov functions. On the other hand, under a certain integrality condition on conformal dimensions, the conformal block has a representation in the form of the Dotsenko-Fateev integral – the partition function of the logarithmic matrix model. It turns out that from this point of view, the Painlevé equation is a reduction of the Hirota equations of the Toda hierarchy using the Virasoro conditions, which are satisfied by the partition functions of the matrix model. The statement is formulated in the most controlled way after the q-deformation, so I will start with it and show that all structures are conserved in the continuous limit and in the pure gauge theory limit.