相关论文: On KP-integrable skew Hurwitz $\tau$-functions and…
We explain that the set of new integrable systems generalizing the Calogero family and implied by the study of WLZZ models, which was described in arXiv:2303.05273, is only the tip of the iceberg. We provide its wide generalization and…
There is now a renewed interest to the Hurwitz tau-function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a…
We suggest a two-matrix model depending on three (infinite) sets of parameters which interpolates between all the models proposed in arXiv:2206.13038, and defined there through $W$-representations. We also discuss further generalizations of…
We follow the general recipe for constructing commutative families of $W$-operators, which provides Hurwitz-like expansions in symmetric functions (Macdonald polynomials), in order to obtain a difference operator example that gives rise to…
We define cut-and-join operator in Hurwitz theory for merging of two branching points of arbitrary type. These operators have two alternative descriptions:(i) they have the GL characters as eigenfunctions and the symmetric-group characters…
In this paper we introduce a new family of the KP tau-functions. This family can be described by a deformation of the generalized Kontsevich matrix model. We prove that the simplest representative of this family describes a generating…
The basis elements spanning the Sato Grassmannian element corresponding to the KP $\tau$-function that serves as generating function for rationally weighted Hurwitz numbers are shown to be Meijer $G$-functions. Using their Mellin-Barnes…
$W$-representation realizes partition functions by an action of a cut-and-join-like operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for $\beta$- and $q,t$-deformations of the simplest…
The quantum spectral curve equation associated to KP $\tau$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning…
Schur functions are the common eigenfunctions of generalized cut-and-join operators which form a closed algebra. They can be expressed as differential operators in time-variables and also through the eigenvalues of auxiliary $N\times N$…
In this letter we continue the development of $W$-representations. We propose several generalizations of the known models, such as the hypergeometric Hurwitz $\tau$-functions. We construct $W$-representations for multi-character expansions,…
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…
Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the…
Since the ($\beta$-deformed) hermitian one-matrix models can be represented as the integrated conformal field theory (CFT) expectation values, we construct the operators in terms of the generators of the Heisenberg algebra such that the…
The KP $\tau$-function of hypergeometric type serving as generating function for quantum weighted Hurwitz numbers is used to compute the Baker function and the corresponding adapted basis elements, expressed as absolutely convergent Laurent…
Multiparametric families of hypergeometric $\tau$-functions of KP or Toda type serve as generating functions for weighted Hurwitz numbers, providing weighted enumerations of branched covers of the Riemann sphere. A graphical interpretation…
We study cut-and-join operators for spin Hurwitz partition functions. We provide explicit expressions for these operators in terms of derivatives in $p$-variables without straightforward matrix realization, which is yet to be found. With…
The KP and 2D Toda tau-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted…
In this paper we study matrix model realizations of Liouville conformal blocks with degenerate and irregular operators. The corresponding matrix model is Hermitian with a $\beta$-deformed measure and the degree of the potential corresponds…
Inspired by J. Novak's works on the asymptotic behavior of the BGW and the HCIZ matrix integrals \cite{[N0]} and by the algebraic and geometric properties of the Hurwitz numbers \cite{[IP]}, \cite{[LZZ]}, \cite{[LR]}, \cite{[OP]},…