Related papers: Brezin-Gross-Witten tau function and isomonodromic…

Conformal blocks and their AGT relations to LMNS integrals and Nekrasov functions are best described by "conformal" (or Dotsenko-Fateev) matrix models, but in non-Gaussian Dijkgraaf-Vafa phases, where different eigenvalues are integrated…

In this work we find the isomonodromic (Jimbo-Miwa) tau-function corresponding to Frobenius manifold structures on Hurwitz spaces. We discuss several applications of this result. First, we get an explicit expression for the G-function…

Correlation function of twist operators is a natural quantity of interest in two-dimensional conformal field theory (2d CFT) and finds relevance in various physical contexts. For computing twist operator correlators associated with generic…

We exhibit the Kontsevich matrix model with arbitrary potential as a BKP tau-function with respect to polynomial deformations of the potential. The result can be equivalently formulated in terms of Cartan-Pl\"ucker relations of certain…

We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlev\'e VI and the Garnier system.…

Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a $\textit{group element}$, i.e. satisfying the property $\Delta(X) = X\otimes X$, then their generating functions obey…

The $\tau$-function theory of Painlev\'e systems is used to derive recurrences in the rank $n$ of certain random matrix averages over U(n). These recurrences involve auxilary quantities which satisfy discrete Painlev\'e equations. The…

Connections between classical and quantum integrable systems are analyzed from the viewpoint of Slavnov products of Bethe states. It is well known that, modulo model dependent aspects, the functional structure of Slavnov products generally…

We establish an explicit relationship between the partition function of certain special cubic Hodge integrals and the generalized Brezin--Gross--Witten (BGW) partition function, which we refer to as the Hodge-BGW correspondence. As an…

We study the generalized matrix model which corresponds to the n-point toric Virasoro conformal block. This describes four-dimensional N=2 SU(2)^n gauge theory with circular quiver diagram by the AGT relation. We first verify that it is…

This paper is a brief review of recent results on the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements in a given highest-weight representation of a universal enveloping algebra ${\cal…

The Baxter-Bazhanov-Stroganov model (also known as the \tau^(2) model) has attracted much interest because it provides a tool for solving the integrable chiral Z_N-Potts model. It can be formulated as a face spin model or via cyclic…

We argued in [Proc. Sympos. Pure Math., Vol. 103, American Mathematical Society, Providence, RI, 2021, 1-66, arXiv:1912.06504] that, when a certain sub-exponential growth property holds, the Donaldson-Thomas invariants of a 3-Calabi-Yau…

We say that a function F(tau) obeys WDVV equations, if for a given invertible symmetric matrix eta^{alpha beta} and all tau \in T \subset R^n, the expressions c^{alpha}_{beta gamma}(tau) = eta^{alpha lambda} c_{lambda beta gamma}(tau) =…

For a class of generalized integrable hierarchies associated with affine (twisted or untwisted) Kac-Moody algebras, an explicit representation of their local conserved densities by means of a single scalar tau-function is deduced. This…

A $(q,t)$-deformation of the 2d Toda integrable hierarchy is introduced by enhancing the underlying symmetry algebra $\mathfrak{gl}(\infty)\simeq \text{q-W}_{1+\infty}$ to the quantum toroidal $\mathfrak{gl}(1)$ algebra. The…

We introduce and investigate a family of tau-functions of the 2D Toda hierarchy, which is a natural generalization of the hypergeometric family associated with Hurwitz numbers. For this family we prove a skew Schur function expansion…

A direct relation between the conformal formalism for 2d-quantum gravity and the W-constrained KP hierarchy is found, without the need to invoke intermediate matrix model technology. The Kontsevich-Miwa transform of the KP hierarchy is used…

The Wirtinger integral is one of the integral representations of the Gauss hypergeometric function. Its integrand is given by a product of complex powers of theta functions. We study the structure of the twisted homology and cohomology…

In this paper, as a step towards a unified mathematical treatment of the gauge functionals from quantum field theory that have found profound applications in mathematics, we generalize the Seiberg-Witten functional that in particular…