Abstract

Moduli spaces of complex surfaces are hard and computing all their GW invariants is for the moment out of reach. The GW of certain surfaces however can be reduced to the GW theory of curves together with an extra spin structure. For us this is particularly interesting as it eventually boils down to spin Hurwitz numbers and topological recursion of Eynard and Orantin. Hence we decided to attempt the extension of the Okounkov-Pandharipande trilogy on the GW of curves to the spin setting (we are now 2 papers in, looking for some extra ideas to unlock the last one!). Current results involve arguably the hardest part already, i.e. the spin Riemann sphere, describing its integrability, ELSV formulae, and providing an algorithm to compute the GW invariants. This reproves a conjecture by Maulik-Pandharipande in degrees 1 and 2 and generalises it to higher degrees.