Communications in Number Theory and Physics

Volume 7 (2013)

Number 1

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

Pages: 125 – 143

DOI: http://dx.doi.org/10.4310/CNTP.2013.v7.n1.a4

Authors

M. Mulase (Department of Mathematics, University of California at Davis)

S. Shadrin (Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands)

L. Spitz (Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands)

Abstract

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 Schrödinger 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.

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