Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Preprint 2020-05

Gaetan Borot, Reinier Kramer, and Yannik Schüler


Higher Airy Structures and Topological Recursion for Singular Spectral Curves.


Abstract: We give elements towards theclassification of quantum Airy structures based on the 𝑊(𝔤𝔩𝑟)- algebras at self-dual level based on twisted modules of the Heisenberg VOA of 𝔤𝔩𝑟 for twists by arbitrary elements of the Weyl group 𝔖𝑟 . In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion à la Chekhov–Eynard–Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard–Eynard topological recursion (valid for smooth curves) to a large class of singular curves, and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves and give precise conjectures for application in open 𝑟 -spin intersection theory.


Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2020-05


88 pp.