# The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures

@article{Mulase2013TheSC, title={The spectral curve and the Schr{\"o}dinger equation of double Hurwitz numbers and higher spin structures}, author={M. Mulase and S. Shadrin and L. Spitz}, journal={Communications in Number Theory and Physics}, year={2013}, volume={7}, pages={125-143} }

We derive the spectral curves for q-part double Hurwitz numbers, r-spin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0, 1)-geometry. We quantize this family of spectral curves and obtain the Schrodinger equations for the partition function of the corresponding Hurwitz problems. We thus confirm the conjecture for the existence of quantum curves in these generalized Hurwitz number cases.

#### 50 Citations

Quantum curves for simple Hurwitz numbers of an arbitrary base curve

- Mathematics, Physics
- 2013

Various generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, the Eynard-Orantin topological recursion, an… Expand

Chiodo formulas for the r-th roots and topological recursion

- Physics, Mathematics
- 2015

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… Expand

Topological recursion and a quantum curve for monotone Hurwitz numbers

- Mathematics, Physics
- 2014

Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data.… Expand

Quantizing Weierstrass

- Mathematics, Physics
- 2016

We study the connection between the Eynard-Orantin topological recursion and quantum curves for the family of genus one spectral curves given by the Weierstrass equation. We construct quantizations… Expand

Weighted Hurwitz Numbers and Topological Recursion

- Mathematics
- 2018

The KP and 2D Toda $$\tau $$ τ -functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme.… Expand

Ramifications of Hurwitz theory, KP integrability and quantum curves

- Physics, Mathematics
- 2015

A bstractIn 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… Expand

Monotone Orbifold Hurwitz Numbers

- Mathematics
- 2015

In general, the Hurwitz numbers count the branched covers of the Riemann sphere with prescribed ramification data or, equivalently, the factorizations of a permutation with prescribed cycle structure… Expand

Enumerative Geometry, Tau-Functions and Heisenberg–Virasoro Algebra

- Mathematics, Physics
- 2014

In this paper we establish relations between three enumerative geometry tau-functions, namely the Kontsevich–Witten, Hurwitz and Hodge tau-functions. The relations allow us to describe the… Expand

Quantum curves for the enumeration of ribbon graphs and hypermaps

- Mathematics, Physics
- 2013

The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite… Expand

QUANTUM CURVE AND BILINEAR FERMIONIC FORM FOR THE ORBIFOLD GROMOV-WITTEN THEORY OF P[r]

- 2019

We construct the quantum curve for the Baker-Akhiezer function of the orbifold Gromov-Witten theory of the weighted projective line P[r]. Furthermore, we deduce the explicit bilinear Fermionic… Expand

#### References

SHOWING 1-10 OF 49 REFERENCES

The spectral curve of the Eynard-Orantin recursion via the Laplace transform

- Mathematics
- 2012

The Eynard-Orantin recursion formula provides an effective tool for certain enumeration problems in geometry. The formula requires a spectral curve and the recursion kernel. We present a uniform… Expand

Hurwitz numbers, matrix models and enumerative geometry

- Mathematics, Physics
- 2007

We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric… Expand

Double Hurwitz numbers via the infinite wedge

- Mathematics
- 2010

We derive an algorithm to produce explicit formulas for certain generating functions of double Hurwitz numbers. These formulas generalize a formula in the work of Goulden, Jackson, and Vakil for one… Expand

On double Hurwitz numbers with completed cycles

- Mathematics, Computer Science
- J. Lond. Math. Soc.
- 2012

A geometric interpretation of these generalized Hurwitz numbers is given and a cut-and-join operator for completed (r + 1)-cycles is derived, which proves a strong piecewise polynomiality property in the sense of Goulden-Jackson-Vakil. Expand

Gromov-Witten theory, Hurwitz theory, and completed cycles

- Mathematics
- 2002

We establish an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles. The… Expand

Orbifold Hurwitz numbers and Eynard-Orantin invariants

- Mathematics, Physics
- 2012

We prove that a generalisation of simple Hurwitz numbers due to Johnson, Pandharipande and Tseng satisfy the topological recursion of Eynard and Orantin. This generalises the Bouchard-Marino… Expand

Mirror symmetry for orbifold Hurwitz numbers

- Mathematics, Physics
- 2013

We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a dierential recursion, which is then proved to be equivalent to the… Expand

Spectral curves and the Schroedinger equations for the Eynard-Orantin recursion

- Mathematics, Physics
- 2012

It is predicted that the principal specialization of the partition function of a B-model topological string theory, that is mirror dual to an A-model enumerative geometry problem, satisfies a… Expand

Abelian Hurwitz-Hodge integrals

- Mathematics
- 2008

Hodge classes on the moduli space of admissible covers with monodromy group G are
associated to irreducible representations of G. We evaluate all linear Hodge integrals over moduli
spaces of… Expand

Intersection numbers of spectral curves

- Mathematics, Physics
- 2011

We compute the symplectic invariants of an arbitrary spectral curve with only 1 branchpoint in terms of integrals of characteristic classes in the moduli space of curves. Our formula associates to… Expand