1307.4729 (Petr Dunin-Barkowski et al.)

Polynomiality of Hurwitz numbers, Bouchard-Mariño conjecture, and a
new proof of the ELSV formula    [PDF]

Petr Dunin-Barkowski, Maxim Kazarian, Nicolas Orantin, Sergey Shadrin, Loek Spitz

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new prove of the Bouchard-Mari\~no conjecture. After that, using the correspondence between the Givental group action and the topological recursion coming from matrix models, we prove the equivalence of the Bouchard-Mari\~no conjecture and the ELSV formula (it is a refinement of an argument by Eynard).

View original: http://arxiv.org/abs/1307.4729