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

Abstract

We show that the each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials.

Voros Coefficients and the Topological Recursion for a Class of the Hypergeometric Differential Equations associated with the Degeneration of the 2-dimensional Garnier System

“…For more details about the topological recursion, see, e.g., the review paper [EO2]. The main results of this paper as well as those of [IKoT1,IKoT2] strengthen the interplay between the WKB theory and the topological recursion and these interplays are expected to produce more profound insights in these theories.…”

“…In [IKoT1,IKoT2] we showed that there is an interesting relation between the exact WKB theory and the topological recursion for the confluent family of the Gauss hypergeometric differential equations, that is, we verified that the Voros coefficients of (confluent) hypergeometric equations are expressed as the difference values of the free energy of the spectral curve obtained as the classical limit of the equations. In this paper we discuss the extension of this result to a family of hypergeometric differential equations associated with 2-dimensional degenerate Garnier systems.…”

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In my joint papers with Iwaki and Koike ([IKoT1,IKoT2]) we found an intriguing relation between the Voros coefficients in the exact WKB analysis and the free energy in the topological recursion introduced by Eynard and Orantin in the case of the confluent family of the Gauss hypergeometric differential equations. In this paper we discuss its generalization to the case of the hypergeometric differential equations associated with 2-dimensional degenerate Garnier systems.

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Trinion conformal blocks from topological strings

“…Another technique for the reconstruction of the topological string partition function from the curve Σ is the topological recursion [61,62]. This method has recently been applied to the case at hand in [63,64]. The result can be represented in the form…”

Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter

“…Recently, new insights into the relationship between cluster algebra and Voros coefficients have been obtained in [19]. Furthermore, it is known that Voros coefficient of the Gauss hypergeometric differential equation can be described using the free energy in the topological recursion [18]. Besides, the Voros coefficients have also appeared in the physics literature as the spectral coordinates [15].…”