Report
This paper is a follow-up of a previous paper co-authored by one of the authors, in which the concept of higher quantum Airy structures was introduced. This quadratic quantum Airy structures introduced by Kontsevich and Soibelman generalize the Virasoro constraints on the tau-functions related to matrix models and to the theory of intersections, while the higher quantum Airy structures generalize the W-constraints. In the previous paper, the quantum Airy structure was associated with a W-algebra for the Lie-algebra gl(n) and an automorphism induced by the Coxeter element of the Weyl group. Such quantum Airy structure describes the critical points of the two-matrix model. In the present paper, a more general automorphism with several cycles of the same length is considered. A classification of the higher quantum Airy structures obtained in this way is presented and some examples are given. The relation to the topological recursion is discussed, and a conjecture is advanced that the generalisation considered corresponds to topological recursion associated with a reducible algebraic curve. An interesting point not discussed in the paper is which type large-N matrix integrals are described by reducible algebraic curves.
I think that this paper contains original results and contributes to a larger program of finding a complete classification of the higher quantum Airy structures and their relation to the topological recursion. I recommend publication.
