Fermionic approach to weighted Hurwitz numbers and topological recursion

Abstract

A fermionic representation is given for all the quantities entering in the generating function approach to weighted Hurwitz numbers and topological recursion. This includes: KP and 2D Toda $\tau$-functions of hypergeometric type, which serve as generating functions for weighted single and double Hurwitz numbers; the Baker function, which is expanded in an adapted basis obtained by applying the same dressing transformation to all vacuum basis elements; the multipair correlators and the multicurrent correlators. Multiplicative and differential recursion relations are deduced for the adapted bases and their duals, and a Christoffel-Darboux type formula is derived for the pair correlator. The quantum and classical spectral curves linking this theory with the topological recursion programme are derived, as well as the generalized cut and join equations. The results are detailed for four special cases: the simple single and double Hurwitz numbers, the weakly monotone case, corresponding to signed enumeration of coverings, the strongly monotone case, corresponding to Belyi curves and the simplest version of quantum weighted Hurwitz numbers.

57 pages. Details added to examples; sign errors corrected; consistency of notation corrected

References in corpus (4)

• On KP-integrable Hurwitz functions
• Matrix Models for Random Partitions
• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra
• Fermionic approach to weighted Hurwitz numbers and topological recursion

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• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
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• Simple maps, Hurwitz numbers, and Topological Recursion
• KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model
• Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integrals
• Simple formulas for constellations and bipartite maps with prescribed degrees
• Generating weighted Hurwitz numbers
• Matrix model generating function for quantum weighted Hurwitz numbers
• Generalized $\tilde{W}$ algebras
• KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions
• 2D Toda $\tau$ Functions, Weighted Hurwitz Numbers and the Cayley Graph: Determinant Representation and Recursion Formula