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We develop the theory of quantization of spectral curves via the topological recursion. We formulate a quantization scheme to spectral curves which is not necessarily admissible in the sense of Bouchard-Eynard 2015. Our main result establishes a relation between the Voros coefficient for quantum curves and the free energies are formulated for spectral curves associated with a confluent family of Gauss hypergeometric differential equations. As an application, we find explicit formulas of free energies for those spectral curves in terms of the Bernoulli numbers.

journal of integrable systems

Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations