Abstract

The exact WKB analysis is a method to analyze differential equations with a small parameter. The main ingredient of the exact WKB analysis is a formal solution, called a WKB solution. When we study differential equations by using the exact WKB analysis, the so-called Voros coefficients play an important role. The Voros coefficient is defined as a contour integral of the logarithmic derivative of WKB solutions.  On the other hand, the topological recursion introduced by B. Eynard and N. Orantin ([EO]) is a recursive algorithm to construct a formal solution to the loop equations that the correlation functions of the matrix model satisfy.  The quantization scheme connects WKB solutions with the topological recursion. It is found that WKB solutions can be constructed via the topological recursion ([BE] etc.).  In this talk, we prove that the Voros coefficients for hypergeometric differential equations are described by the generating functions of free energies defined in terms of the topological recursion. Furthermore, as its applications we show the following objects can be explicitly computed for hypergeometric equations: (i) three-term difference equations that the generating function of free energies satisfies, (ii) explicit form of the free energies, and (iii) explicit form of Voros coefficients ([IKT],[T]).  References and B.Eyanard, Reconstructing WKB from topological recursion, Journal de l’Ecole polytechnique—Mathematiques, 4 (2017), 845—908.  [EO] B.Eynard and  N.Orantin, Invariants of algebraic curves and topological expansion, Communications in Number Theory and Physics, 1 (2007), 347—452. [IKT] K.Iwaki, T.Koike and Y.-M.Takei, Voros coefficients for the hypergeometric differential equations and Eynard-Orantin’s topological recursion,  Part I, arXiv:1805.10945, & Part II, Journal of Integrable Systems, 3 (2019), 1—46.  [T] Y.-M.Takei, Voros Coefficients and the Topological Recursion for a Class of the Hypergeometric Differential Equations associated with the Degeneration of the 2-dimensional Garnier System, arXiv: 2005.08957.