Towards an orbifold generalization of Zvonkine's $r$-ELSV formula

Abstract

We perform a key step towards the proof of Zvonkine's conjectural $r$-ELSV formula that relates Hurwitz numbers with completed $(r+1)$-cycles to the geometry of the moduli spaces of the $r$-spin structures on curves: we prove the quasi-polynomiality property prescribed by Zvonkine's conjecture. Moreover, we propose an orbifold generalization of Zvonkine's conjecture and prove the quasi-polynomiality property in this case as well. In addition to that, we study the $(0,1)$- and $(0,2)$-functions in this generalized case and we show that these unstable cases are correctly reproduced by the spectral curve initial data.

20 pages

References in corpus (6)

• Double ramification cycles on the moduli spaces of curves
• Equivariant Gromov-Witten theory of one dimensional stacks
• Ramifications of Hurwitz theory, KP integrability and quantum curves
• Chiodo formulas for the r-th roots and topological recursion
• Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson-Pandharipande-Tseng formula
• Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck's dessins d'enfants

Cited by in corpus (10)

• Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type
• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
• Double Hurwitz numbers: polynomiality, topological recursion and intersection theory
• Spectral curves and $W$-representations of matrix models
• Combinatorics of Bousquet-Mélou-Schaeffer numbers in the light of topological recursion
• Special cases of the orbifold version of Zvonkine's $r$-ELSV formula
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Generalized $\tilde{W}$ algebras
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
• Completed Cycles Leaky Hurwitz Numbers