Abstract
We study various non-perturbative approaches to the quantization of the Seiberg-Witten curve of mathcal{N} = 2, SU(2) super Yang-Mills theory, which is closely related to the modified Mathieu operator. The first approach is based on the quantum WKB periods and their resurgent properties. We show that these properties are encoded in the TBA equations of Gaiotto-Moore-Neitzke determined by the BPS spectrum of the theory, and we relate the Borel-resummed quantum periods to instanton calculus. In addition, we use the TS/ST correspondence to obtain a closed formula for the Fredholm determinant of the modified Mathieu operator. Finally, by using blowup equations, we explain the connection between this operator and the τ function of Painlevé III.
Highlights
The first approach is based on the quantum WKB periods and their resurgent properties. We show that these properties are encoded in the TBA equations of Gaiotto-Moore-Neitzke determined by the BPS spectrum of the theory, and we relate the Borel-resummed quantum periods to instanton calculus
Building on [6], Gaiotto considered in [7] the conformal limit of the TBA equations of [5] for an N = 2 supersymmetric gauge theory, and he conjectured that the resulting integral equations describe the quantum periods for the corresponding quantum SW curve
In this paper we have used non-perturbative techniques inspired by supersymmetric gauge theory and topological string theory to study the quantization of the Seiberg-Witten curve of N = 2, SU(2) super Yang-Mills theory, which gives the modified Mathieu operator
Summary
Our first approach to the quantum SW curve will be based on the so-called exact WKB method, see for example [20,21,22,23]. We can deform slightly the integration contour below or above the positive real axis, obtaining in this way the so-called lateral Borel resummations of the formal power series F ( ):. These lateral resummations are in general different, and their difference is defined as the Stokes discontinuity of F : disc(F )( ) = s+(F )( ) − s−(F )( ). Let us look at some examples of the Borel plane of the quantum periods for the modified Mathieu equation. The connection with SW theory gives very powerful information on this structure, which we will explore in detail
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