Atlantes Hurwitz numbers and transalgebraic topological recursion


Hurwitz numbers count covers of Riemann surfaces with given ramification data. The case of covers of $\mathbb{P}^1$ with one variable ramifications and a uniform ramification elsewhere is particularly interesting, as these numbers are related to the KP integrable hierarchy. A large family of such Hurwitz problems can be computed using topological recursion: a universal procedure, recursive on the Euler characteristic of the surfaces involved, which requires a spectral curve as input. The case of Atlantes Hurwitz numbers evaded this approach up to now, as it seemed to have the same spectral curve as completed cycles Hurwitz numbers. I will introduce all these notions, and explain how Atlantes Hurwitz numbers do satisfy a variant of topological recursion, with a transalgebraic spectral curve, and this distinguishes it from the completed cycles case. This is joint work with Vincent Bouchard and Quinten Weller.