Spin Hurwitz theory and Miwa transform for the Schur Q-functions

Abstract

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 $N\times N$ matrices $X$, known as Miwa variables. Relevant for the cubic Kontsevich model and also for spin Hurwitz theory is an alternative set of Schur Q-functions. They appear in representation theory of the Sergeev group, which is a substitute of the symmetric group, related to the queer Lie superalgebras $\mathfrak{q}(N)$.. The corresponding spin $\hat{\mathcal{W}}$-operators were recently found in terms of time-derivatives, but a substitute of the Miwa parametrization remained unknown, which is an essential complication for the matrix model technique and further developments. We demonstrate that the Miwa representation, in this case, involves a fermionic matrix $\Psi$ in addition to $X$, but its realization using supermatrices is {\it not} quite naive.

7 pages

References in corpus (7)

• On the complete perturbative solution of one-matrix models
• Superintegrability of Kontsevich matrix model
• The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations
• Connection between cut-and-join and Casimir operators
• Hypergeometric functions related to Schur Q-polynomials and BKP equation
• Recursion Formulas for Spin Hurwitz Numbers
• Notes about KP/BKP correspondence

Cited by in corpus (4)

• Hunt for 3-Schur polynomials
• Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}({\widehat{\mathfrak{gl}}}_{1|1})$
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
• Itzykson-Zuber correlators from character expansion