Ramifications of Hurwitz theory, KP integrability and quantum curves

Alexandrov, A.; Lewański, Danilo; Shadrin, Sergey

doi:10.1007/jhep05(2016)124

Cited by 49 publications

(105 citation statements)

References 49 publications

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“…In Section 3, we give evidence to support our main conjecture. First, we present the quantum curve for double Hurwitz numbers, which was first deduced by Alexandrov, Lewanski and Shadrin [1], and show that its semi-classical limit does indeed recover the spectral curve of equation (1). We then provide low genus evidence by calculating the free energies F 0,3 and F 1,1 and demonstrating that they are consistent with our main conjecture.…”

Section: Ramifications and Applicationssupporting

confidence: 76%

“…One can immediately observe that the semi-classical limit of the operator Q is simply y − P(x exp(sy)), for which we have the rational parametrisation of equation (1). The existence of the quantum curve lends strong support to Conjecture 3.…”

Section: The Quantum Curvementioning

confidence: 62%

“…x(z) = z exp(−sP(z)) and y(z) = P(z) (1) produces correlation differentials whose expansions at x i = 0 satisfy…”

Section: The Main Conjecturementioning

confidence: 99%

“…One can consider the semiclassical limit of the quantum curve, which is a plane curve that coincides with the original spectral curve in many cases. The quantum curve for double Hurwitz numbers was previously computed by Alexandrov, Lewanski and Shadrin [1] and its semi-classical limit is y = P(x exp(sy)), which recovers the spectral curve of equation (1). It has been proposed that the existence of a quantum curve can be used to predict the structure of topological recursion and the associated spectral curve [31].…”

Section: Evidence For the Conjecturementioning

confidence: 99%

“…More generally, one can consider P(z) = z a for a positive integer a and recover the fact that the topological recursion governs orbifold Hurwitz numbers [12,5]. Alexandrov, Lewanski and Shadrin propose a 1-parameter deformation of single Hurwitz numbers, via the enumeration of double Hurwitz numbers, weighted by c to the power of the colength of the ramification profile over 0 [1]. Our main conjecture recovers theirs by specialising the weights to q i = ic i−1 and sending d to infinity or, equivalently, by considering the case P(z) = z (1−cz) 2 .…”

Section: Ramifications and Applicationsmentioning

confidence: 99%

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Towards the topological recursion for double Hurwitz numbers

Do¹,

Karev

2018

Proceedings of Symposia in Pure Mathematics

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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 structure. In particular, they arise as intersection numbers on moduli spaces of curves and are governed by the topological recursion of Chekhov, Eynard and Orantin. Double Hurwitz numbers are defined analogously, but with prescribed ramification over both zero and infinity. Goulden, Jackson and Vakil have conjectured that double Hurwitz numbers also arise as intersection numbers on moduli spaces.In this paper, we repackage double Hurwitz numbers as enumerations of branched covers weighted by certain monomials and conjecture that they are governed by the topological recursion. Evidence is provided in the form of the associated quantum curve and low genus calculations. We furthermore reduce the conjecture to a weaker one, concerning a certain polynomial structure of double Hurwitz numbers. Via the topological recursion framework, our main conjecture should lead to a direct connection to enumerative geometry, thus shedding light on the aforementioned conjecture of Goulden, Jackson and Vakil.

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Section: Ramifications and Applicationssupporting

confidence: 76%

Section: The Quantum Curvementioning

confidence: 62%

“…x(z) = z exp(−sP(z)) and y(z) = P(z) (1) produces correlation differentials whose expansions at x i = 0 satisfy…”

Section: The Main Conjecturementioning

confidence: 99%

Section: Evidence For the Conjecturementioning

confidence: 99%

Section: Ramifications and Applicationsmentioning

confidence: 99%

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Towards the topological recursion for double Hurwitz numbers

Do¹,

Karev

2018

Proceedings of Symposia in Pure Mathematics

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Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

Gisonni

Грава

Ruzza

2020

Ann. Henri Poincaré

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We consider the Laguerre partition function and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was recently proven in Cunden et al. (Ann. Inst. Henri Poincaré D, to appear) that correlators have a topological expansion in terms of weakly or strictly monotone Hurwitz numbers that can be explicitly computed from our formulae. As a second result, we identify the Laguerre partition function with only positive couplings and a special value of the parameter $$\alpha =-1/2$$ α = - 1 / 2 with the modified GUE partition function, which has recently been introduced in Dubrovin et al. (Hodge-GUE correspondence and the discrete KdV equation. arXiv:1612.02333) as a generating function for Hodge integrals. This identification provides a direct and new link between monotone Hurwitz numbers and Hodge integrals.

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Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

Borot

Karev

et al. 2022

Math. Ann.

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Double Hurwitz numbers enumerate branched covers of $${{{\mathbb {C}}}}{{{\mathbb {P}}}}^1$$ C P 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 resolve these conjectures by a careful analysis of the semi-infinite wedge representation for double Hurwitz numbers. We prove an ELSV-like formula for double Hurwitz numbers, by deforming the Johnson–Pandharipande–Tseng formula for orbifold Hurwitz numbers and using properties of the topological recursion under variation of spectral curves. In the course of this analysis, we unveil certain vanishing properties of $$\Omega $$ Ω -classes.

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Ramifications of Hurwitz theory, KP integrability and quantum curves

Cited by 49 publications

References 49 publications

Towards the topological recursion for double Hurwitz numbers

Towards the topological recursion for double Hurwitz numbers

Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

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