  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.
