通过 theta 特征重新审视 Hurwitz 理论与 ELSV

Polynomiality of Hurwitz numbers, Bouchard-Mariño conjecture, and a new proof of the ELSV formula

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new proof of the Bouchard-Mariño conjecture. After that, using the correspondence between the Givental group action and the topological recursion coming from matrix models, we prove the equivalence of the Bouchard-Mariño conjecture and the ELSV formula (it is a refinement of an argument by Eynard).

一种新的加权双Hurwitz数族及一个包含Ω类的新的ELSV型公式

A new family of weighted double Hurwitz numbers and a new ELSV-type formula with $Ω$-classes

本文研究了一种新的加权双Hurwitz数族，并在此基础上发展了与超几何（Orlov--Scherbin）KP tau函数和曲线模空间的交理论相关的技术，推导出一个新的显式ELSV型公式，该公式涉及所谓的Ω类。

环路方程与Zvonkine的$qr$-ELSV公式的一个证明

Loop equations and a proof of Zvonkine's $qr$-ELSV formula

本文证明了Zvonkine的$r$-ELSV公式及其orbifold推广$qr$-ELSV公式，将Hurwitz数与Chiodo类相交数联系起来。

Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson-Pandharipande-Tseng formula

In this paper we present an example of a derivation of an ELSV-type 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 quasi-polynomiality. 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 ca…

On the Goulden-Jackson-Vakil conjecture for double Hurwitz numbers

Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers. In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first …

双重赫尔维茨数：多项式性、拓扑递归与相交理论

Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

本文通过半无限楔表示分析双重赫尔维茨数，证明其多项式性结构并确认其受拓扑递归支配，进而推导出 ELSV 型公式的初步版本，同时揭示了 Chiodo 类的消失性质。