Cut-and-join equation for monotone Hurwitz numbers revisited

Abstract

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. Our proof in particular uses a combinatorial technique developed by Han. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of the topological recursion for monotone Hurwitz numbers, obtained first by Do, Dyer, and Mathews.

arXiv comment7 pages. v2: Added a second proof of lemma 2.3, using Jucys-Murphy elements, and expanded motivation

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• On the Complex Asymptotics of the HCIZ and BGW Integrals
• Quantization of Harer-Zagier formulas
• $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
• Log topological recursion through the prism of $x-y$ swap
• Topological Recursion for the extended Ooguri-Vafa partition function of colored HOMFLY-PT polynomials of torus knots
• Topological recursion for monotone orbifold Hurwitz numbers: a proof of the Do-Karev conjecture
• $\mathfrak{b}$-Hurwitz numbers from refined topological recursion