Cut-and-join structure and integrability for spin Hurwitz numbers

Abstract

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 odd partitions (OP) and the strict partitions (SP). The Q Schur functions Q_R with R\in SP are common eigenfunctions of cut-and-join operators W_Δwith Δ\in OP. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a τ-function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.

22 pages

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• Connection between cut-and-join and Casimir operators
• BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic $N$-Point Functions
• Spin Hurwitz theory and Miwa transform for the Schur Q-functions
• Natanzon-Orlov model and refined superintegrability
• Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewanski conjecture
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Kac-Schwarz Operators of Type $B$, Quantum Spectral Curves, and Spin Hurwitz Numbers
• Connected $(n,m)$-Point Functions of Diagonal $2$-BKP Tau-Functions, and Spin Double Hurwitz Numbers
• Notes about KP/BKP correspondence
• Averaging method in combinatorics of symmetric polynomials
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy