The ABCD of topological recursion

Abstract

Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of [43], seeing it as a quantisation of certain quadratic Lagrangians in T⁎V for some vector space V. KS topological recursion is a procedure which takes as initial data a quantum Airy structure — a family of at most quadratic differential operators on V satisfying some axioms — and gives as outcome a formal series of functions on V (the partition function) simultaneously annihilated by these operators. Finding and classifying quantum Airy structures modulo the gauge group action, is by itself an interesting problem which we study here. We provide some elementary, Lie-algebraic tools to address this problem, and give some elements of the classification for dimV=2. We also describe four more interesting classes of quantum Airy structures, coming from respectively Frobenius algebras (here we retrieve the 2d TQFT partition function as a special case), non-commutative Frobenius algebras, loop spaces of Frobenius algebras and a Z2-invariant version of the latter. This Z2-invariant version in the case of a semi-simple Frobenius algebra corresponds to the topological recursion of [43].