Search

The triad refers to embedding the Macdonald polynomials into the Noumi-Shiraishi functions and their reduction to solutions of simple linear equations at particular values of $t$. It provides an alternative definition of Macdonald theory. We discuss lifting the triad to an elliptic generalization of the Noumi-Shiraishi functions. The central unknown ingredient is linear equations, for which we discuss various possible approaches, including immediate elliptic deformation of periodicity conditions, (elliptic) Ding-Iohara-Miki algebra operators, and elliptic Kostka coefficients., Comment: 13 pages

A series of two-particle examples of the Ruijsenaars pq-duality is considered in detail, the dual Hamiltonians are constructed. Of special interest is the case of the sinh-Gordon model., Comment: 5 pages

Some eigenvalue matrix models possess an interesting property: one can manifestly define the basis where all averages can be explicitly calculated. For example, in the Gaussian Hermitian and rectangular complex models, averages of the Schur functions are again expressed through the Schur functions. However, so far this property remains restricted to very particular (e.g. Gaussian) measures. In this paper, we extend this observation to unitary matrix integrals, where one could expect that this restriction is easier to lift. We demonstrate that this is indeed the case, only this time the Schur averages are linear combinations of the Schur functions. Full factorization to a single item in the sum appears only on the Miwa locus, where at least one half of the time-variables is expressed through matrices of the same size. For unitary integrals, this is a manifestation of the de Wit-t'Hooft anomaly, which prevents the answer to be fully analytic in the matrix size $N$. Once achieved, this understanding can be extended back to the Hermitian model, where the phenomenon looks very similar: beyond Gaussian measures superintegrability requires an additional summation., Comment: 10 pages

The modern version of conformal matrix model (CMM) describes conformal blocks in the Dijkgraaf-Vafa phase. Therefore it possesses a determinant representation and becomes a Toda chain $\tau$-function only after a peculiar Fourier transform in internal dimensions. Moreover, in CMM Hirota equations arise in a peculiar discrete form (when the couplings of CMM are actually Miwa time-variables). Instead, this integrability property is actually independent on the measure in the original hypergeometric integral. To get hypergeometric functions, one needs to pick up a very special $\tau$-function satisfying an additional "string equation". Usually, its role is played by the lowest $L_{-1}$ Virasoro constraint, but, in the Miwa variables, it turns into a finite-difference equation with respect to the Miwa variables. One can get rid of these differences by rewriting the string equation in terms of some double ratios of the shifted $\tau$-functions, and then these ratios satisfy more sophisticated equations equivalent to the discrete Painlev\'e equations by M. Jimbo and H. Sakai ($q$-PVI equation). They look much simpler in the $q$-deformed ($"5d"$) matrix model, while in the "continuous" limit $q\longrightarrow 1$ to $4d$ one should consider the Miwa variables with non-unit multiplicities, what finally converts the simple discrete Painlev\'e $q$-PVI into sophisticated differential Painlev\'e VI equations, which will be considered elsewhere., Comment: 13 pages