Abstract

AbstractWe study monotone and strictly monotone Hurwitz numbers from a bosonic Fock space perspective. This yields to an interpretation in terms of tropical geometry involving local multiplicities given by Gromov-Witten invariants. Furthermore, this enables us to prove that a main result of Cavalieri-Johnson-Markwig-Ranganathan is actually equivalent to the Gromov-Witten/Hurwitz correspondence by Okounkov-Pandharipande for the equivariant Riemann sphere.

Highlights

  • Hurwitz numbers and Gromov-Witten invariants with target the Riemann sphere enumerate certain maps between Riemann surfaces

  • One successful approach to Hurwitz numbers has been achieved by means of tropical geometry, in which Hurwitz numbers are expressed as enumeration of maps between metric graphs with discrete data [5,8,9]

  • This interpretation has given rise to many interesting insights, such as a study of polynomial behaviour of double Hurwitz numbers [9] or the introduction of tropical mirror symmetry for elliptic curves involving quasi-modularity statements of certain tropical covers [4]. It was observed in [10] that tropical geometry gives rise to a graphical interface for the Gromov-Witten theory of curves

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Summary

Introduction

Hurwitz numbers and Gromov-Witten invariants with target the Riemann sphere enumerate certain maps between Riemann surfaces. This interpretation has given rise to many interesting insights, such as a study of polynomial behaviour of double Hurwitz numbers [9] or the introduction of tropical mirror symmetry for elliptic curves involving quasi-modularity statements of certain tropical covers [4] It was observed in [10] that tropical geometry gives rise to a graphical interface for the Gromov-Witten theory of curves. We derive a new tropical expression for monotone and strictly monotone Hurwitz numbers that shares several features with the results of [10], as the tropical covers are once again weighted by local multiplicities given by Gromov-Witten invariants This new interpretation has the advantage that the curves involved carry less non-geometric information than the ones in [12,13] and are closer to the tropical curves involved in [4,9]. This, in turn, establishes an equivalence between Theorem 5.3.4 in [10] and the original GW/H correspondence of [22]

Structure of the paper
Semi-infinite wedge formalism
Hurwitz numbers in the semi-infinite wedge formalism
Tropical curves
Bosonification
Newton’s identities: and h polynomials in terms of power sums p
Lascoux-Thibon operator: power sums p of the content in terms of F operators
Boson-Fermion correspondence: F operators in terms ofoperators
Putting the pieces together
Tropicalisation
Findings
Gromov-Witten theory and tropical curves
Full Text
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