Airy structures for semisimple Lie algebras

TL;DR

This paper classifies Airy structures for finite-dimensional simple Lie algebras, identifying only a few that admit such structures and exploring their properties using representation theory and semiclassical analysis.

Contribution

It provides a complete classification of Airy structures for simple Lie algebras and extends the analysis to semisimple cases, revealing their abundance and properties.

Findings

Only rak{sl}_2, rak{sp}_4, and rak{sp}_{10} admit Airy structures.

Each of these admits exactly two non-equivalent Airy structures.

Semisimple Lie algebras have countably infinite non-equivalent Airy structures.

Abstract

We give a complete classification of Airy structures for finite-dimensional simple Lie algebras over $\mathbb C$, and to some extent also over $\mathbb R$, up to isomorphisms and gauge transformations. The result is that the only algebras of this type which admit any Airy structures are $\mathfrak{sl}_2$, $\mathfrak{sp}_4$ and $\mathfrak{sp}_{10}$. Among these, each admits exactly two non-equivalent Airy structures. Our methods apply directly also to semisimple Lie algebras. In this case it turns out that the number of non-equivalent Airy structures is countably infinite. We have derived a number of interesting properties of these Airy structures and constructed many examples. Techniques used to derive our results may be described, broadly speaking, as an application of representation theory in semiclassical analysis.