# Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

@article{Borot2020DoubleHN, title={Double Hurwitz numbers: polynomiality, topological recursion and intersection theory}, author={Gaetan Borot and Norman Do and Maksim Karev and Danilo Lewa'nski and Ellena Moskovsky}, journal={arXiv: Algebraic Geometry}, year={2020} }

Double Hurwitz numbers enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification over two points and simple ramification elsewhere. In contrast to the single case, their underlying geometry is not well understood. In previous work by the second- and third-named authors, the double Hurwitz numbers were conjectured to satisfy a polynomiality structure and to be governed by the topological recursion, analogous to existing results concerning single Hurwitz numbers. In this paper, we…

## 12 Citations

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

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Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type

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We study the n-point differentials corresponding to Kadomtsev–Petviashvili tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their…

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—We derive a new explicit formula in terms of sums over graphs for the n-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev–Petviashvili tau…

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We derive a new explicit formula in terms of sums over graphs for the $n$-point correlation functions of general formal weighted double Hurwitz numbers coming from the Orlov-Scherbin partition…

On the Complex Asymptotics of the HCIZ and BGW Integrals

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In this paper, we prove a longstanding conjecture on the asymptotic behavior of a pair of oscillatory matrix integrals: the Harish-Chandra/Itzykson-Zuber (HCIZ) integral, and the Brezin-Gross-Witten…

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We derive an analogue of the famous Harer-Zagier formula in the q-deformed Hermitian Gaussian matrix model. This fully describes single-trace correlators and opens a road to q-deformations of…

Pluricanonical cycles and tropical covers

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A BSTRACT . We extract a system of numerical invariants from logarithmic intersection theory on pluricanonical double ramiﬁcation cycles, and show that these invariants exhibit a number of properties…

On some hyperelliptic Hurwitz-Hodge integrals

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This short note addresses Hodge integrals over the hyperelliptic locus. Recently Afandi computed, via localisation techniques, such one-descendant integrals and showed that they are Stirling numbers.…

Ju n 20 21 T OPOLOGICAL RECURSION FOR FULLY SIMPLE MAPS FROM CILIATED MAPS

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Ordinary maps satisfy topological recursion for a certain spectral curve (x,y). We solve a conjecture from [5] that claims that fully simple maps, which are maps with non selfintersecting disjoint…

## References

SHOWING 1-10 OF 72 REFERENCES

Towards the topological recursion for double
Hurwitz numbers

- MathematicsProceedings of Symposia in Pure Mathematics
- 2018

Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich…

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

- MathematicsAdvances in Mathematics
- 2022

Chamber Structure of Double Hurwitz numbers

- Mathematics
- 2010

Double Hurwitz numbers count covers of the projective line by genus g curves with assigned ramification profiles over 0 and infinity, and simple ramification over a fixed branch divisor. Goulden,…

Simple Maps, Hurwitz Numbers, and Topological Recursion

- MathematicsCommunications in Mathematical Physics
- 2020

We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study…

Hurwitz numbers

- Mathematics
- 2011

From basic settings the Hurwitz numbers counting unique ramified coverings over PC, the Riemann sphere, with arbitrary ramification over 0 and∞, plus finite other simple ramifications. After a broad…

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

- MathematicsJ. Lond. Math. Soc.
- 2015

For orbifold Hurwitz numbers, a new proof of the spectral curve topological recursion is given 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.

Monotone Hurwitz Numbers in Genus Zero

- MathematicsCanadian Journal of Mathematics
- 2013

Abstract Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified…