Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory

TL;DR

The paper constructs a complete family of cut-and-join operators $\hat{\cal W}(\Delta)$ for Hurwitz-Kontsevich theory and proves two equivalent realizations: (i) eigenfunctions given by GL$(\infty)$ characters $\chi_R(t)$ with eigenvalues $\varphi_R(\Delta)$, and (ii) differential $W$-type operators $:\widetilde{D(\Delta)}:$ acting on time-variables, most transparent in matrix Miwa variables $p_k=kt_k$. These operators generate a commutative associative algebra—the Universal Hurwitz Algebra—and Hurwitz numbers are realized as the image under a distinguished linear form on the Young-diagram space. The authors define Hurwitz partition functions $\mathcal{Z}(t,t',...|\beta)$ and show that exponentiating the cut-and-join operators yields Kontsevich-Hurwitz tau-functions, connecting Hurwitz counting to KP/Toda hierarchies and to matrix-model frameworks. They also establish a detailed dictionary between permutation compositions, Feynman diagram techniques, and operator algebra, and discuss integrability properties, illustrating a robust algebraic-and-analytic structure for universal Hurwitz numbers with potential extensions to open strings and matrix-model representations.

Abstract

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 as eigenvalues; (ii) they can be represented as differential operators of the $W$-type (in particular, acting on the time-variables in the Hurwitz-Kontsevich tau-function). The operators have the simplest form if expressed in terms of the matrix Miwa-variables. They form an important commutative associative algebra, a Universal Hurwitz Algebra, generalizing all group algebra centers of particular symmetric groups which are used in description of the Universal Hurwitz numbers of particular orders. This algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear form on the linear space of Young diagrams, evaluated at the product of all diagrams, which characterize particular ramification points of the covering.

Figures (1)

• Figure 1: Composition of two permutations: of the cycle (123) with the order 6 cycle. At the same time, these Feynman diagrams contribute to multiplication of the normal ordered matrix differential operators $\hat{\cal W}([3])$ and $\hat{\cal W}([6])$.