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 $\hbar$ 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.