W-algebras and topological recursion

I will describe the notion of quantum Airy structures (proposed by Kontsevich and Soibelman). These are collections of differential operators acting on formal functions on a vector space V which generate an ideal in the Weyl algebra of V, and are of the form L_i = d/dx_i + higher order. To any Airy structure one can associate a partition function Z, which is the formal function annihilated simultaneously by these operators. This algebraic setting immediately lead to the computation of Z by a "topological recursion".

Then I will show how to obtain examples of quantum Airy structures from modules of W-algebras of simply-laced Lie algebras for the self-dual level. In particular for W(gl_r), we prove an equivalence between the W-constraints and the Bouchard-Eynard topological recursion associated with spectral curves with arbitrary ramifications (i.e. a period computation).

Very often, the partition function has an enumerative geometry interpretation on the moduli space of curves: we will revisit in this context some results and new questions put forward by this approach.

This is based on a joint work with Vincent Bouchard, Thomas Creutzig, Nitin Chidambaram and Dmitry Noshchenko.