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
Summary
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]
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