TROPICAL OPEN HURWITZ NUMBERS
Grigory Mikhalkin · 2016
Abstract. We give a tropical interpretation of Hurwitz numbers extending the one discovered in [CJM]. In addition we treat a generalization of Hurwitz numbers for surfaces with boundary which we call open Hurwitz numbers. Hurwitz numbers are defined as the (weighted) number of ramified coverings of a compact closed oriented surface S of a given genus having a given set of critical values with given ramification profiles. These numbers have a long history, and have connections to many areas of mathematics, among which we can mention algebraic geometry, topology, combinatorics, and representation theory (see [LZ04] for example). Here we define a slight generalization of these numbers that we call open Hurwitz numbers. To do so, we fix not only points on S and ramification profiles, but also a collection of disjoint circles on S and the behavior of the coverings above each of these circles. Note that the total space of the ramified coverings we consider now is allowed to have boundary components. We also define tropical open Hurwitz numbers, and establish a correspondence with their complex counterpart. This can simply be seen as a translation in the tropical language of the computation of open Hurwitz numbers by cutting S along a collection of circles. A decomposition of S into pairs of pants reduces the problem to the enumeration of ramified coverings of the sphere S2 with 3 critical values. In the particular case where all ramification points are simple, except maybe two of them, we recover the tropical computation of double Hurwitz numbers in [CJM]. This note is motivated by the forthcoming paper [BBM] where the computation of genus 0 char-acteristic numbers of CP 2 is reduced to enumeration of floor diagrams and computation of genus 0 open Hurwitz numbers. We would like to thank Arne Buchholz and Hannah Markwig who pointed out an inaccuracy in the first version of the discussion at the end of the paper. 1. Open Hurwitz numbers The data we need to define open Hurwitz numbers are • S an oriented connected closed compact surface; • L a finite collection of disjoint smoothly embedded circles in S; we denote by S the surface