In this paper we revisit several recent results on monotone and strictly monotone Hurwitz numbers, providing new proofs. In particular, we use various versions of these numbers to discuss methods of derivation of quantum spectral curves from the point of view of KP integrability and derive new examples of quantum curves for the families of double Hurwitz numbers.
We analyze Chiodo's formulas for the Chern classes related to the r -th roots of the suitably twisted integer powers of the canonical class on the moduli space of curves. The intersection numbers of these classes with ψ-classes are reproduced via the Chekhov-Eynard-Orantin topological recursion. As an application, we prove that the Johnson-Pandharipande-Tseng formula for the orbifold Hurwitz numbers is equivalent to the topological recursion for the orbifold Hurwitz numbers. In particular, this gives a new proof of the topological recursion for the orbifold Hurwitz numbers.
Abstract. In this paper we present an example of a derivation of an ELSVtype formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of Chekhov, Eynard, and Orantin, where the main new step compared to the existing proofs is a direct combinatorial proof of their quasipolynomiality. Spectral curve topological recursion leads to a formula for the orbifold Hurwitz numbers in terms of the intersection theory of the moduli space of curves, which, in this case, appears to coincide with a special case of the Johnson-Pandharipande-Tseng formula.
We study the Masur-Veech volumes MV g,n of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus g with n punctures. We show that the volumes MV g,n are the constant terms of a family of polynomials in n variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of [11] proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in [3]. We also obtain an expression of the area Siegel-Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur-Veech volumes, and thus of area Siegel-Veech constants, for low g and n, which leads us to propose conjectural formulas for low g but all n.
We study spin Hurwitz numbers, which count ramified covers of the Riemann sphere with a sign coming from a theta characteristic. These numbers are known to be related to Gromov-Witten theory of Kähler surfaces and to representation theory of the Sergeev group, and are generated by BKP tau-functions. We use the latter interpretation to give polynomiality properties of these numbers and we derive a spectral curve which we conjecture computes spin Hurwitz numbers via a new type of topological recursion. We prove that this conjectural topological recursion is equivalent to an ELSV-type formula, expressing spin Hurwitz numbers in terms of the Chiodo class twisted by the 2-spin Witten class.
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