Exact WKB methods in $SU(2)$ $N_f=1$

Abstract

We study in detail the Schrödinger equation corresponding to the four dimensional $SU(2)$ $\mathcal{N}=2$ SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schrödinger operator using the TS/ST correspondence and Zamolodchikov's TBA, again finding good agreement. In addition, we explore two aspects of the relationship between singularities of the Borel transformed WKB series and BPS states: BPS states of the 4d theory are related to singularities in the Borel transformed WKB series for the quantum periods, and BPS states of a coupled 2d+4d system are related to singularities in the Borel transformed WKB series for local solutions of the Schrödinger equation.

arXiv comment63 pages, 18 figures

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• Holographic thermal correlators from supersymmetric instantons
• On the resurgent structure of quantum periods
• Exponential Networks, WKB and Topological String
• Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization
• Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs
• The Exact WKB analysis and the Stokes phenomena of the Unruh effect and Hawking radiation
• Anisotropic Kondo line defect and ODE/IM correspondence
• Les Houches lectures on non-perturbative Seiberg-Witten geometry