The ABCD of topological recursion

Abstract: Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of math-ph/0702045, seeing it as a quantization 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 in $V$ (the partition function) simultaneously annihilated by these operators. Finding and classifying quantum Airy structures modulo 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 classification for ${\rm dim}\,V = 2$. 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 $\mathbb{Z}{2}$-invariant version of the latter. This $\mathbb{Z}{2}$-invariant version in the case of a semi-simple Frobenius algebra corresponds to the topological recursion of math-ph/0702045.