Analyticity of the free energy for quantum Airy structures

It is shown that the free energy associated to a finite-dimensional Airy structure is an analytic function at each finite order of the [inline-formula]-expansion. Its terms are interpreted as objects living on the zero locus of the classical hamiltonians. The geometry of this variety is studied. The structure of singularities of the free energy is described. To this end topological recursion equations are expressed in a form particularly suitable for semiclassical analysis. It involves a differential operator which is a deformation of the exterior derivative. Its local properties are derived. The developed formalism is applied in several examples. Global properties of the obtained partition functions are investigated. In the case of a divergent [inline-formula]-series, a simple resummation is performed.