# Analyticity of the free energy for quantum Airy structures

## TL;DR

This paper proves that the free energy of finite-dimensional quantum Airy structures is analytic at each order of the $ar{ ext{h}}$ expansion, providing a formalism that clarifies the semiclassical geometry and addresses divergence issues.

## Contribution

It introduces a new formalism for topological recursion equations that reveals semiclassical geometry and proves analyticity of free energy in quantum Airy structures.

## Key findings

- Free energy is analytic at each finite order of the $ar{ ext{h}}$ expansion.
- A differential operator formalism satisfying a Lie algebra cocycle condition is developed.
- Resummation techniques are applied to divergent $ar{ ext{h}}$ series.

## Abstract

It is shown that the free energy associated to a finite dimensional Airy structure is an analytic function at each finite order of the $\hbar$ expansion. Semiclassical series itself is in general divergent. Calculations are facilitated by putting the topological recursion equations into a form exhibiting more explicitly the semiclassical geometry. This formulation involves certain differential operators on the characteristic variety, which are found to satisfy a Lie algebra cocycle condition. It is proven that this cocycle is a coboundary. Developed formalism is applied in specific examples. In the case of a divergent $\hbar$ series, a simple resummation is performed. Analytic properties of the obtained partition functions are investigated.