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.
Highlights
This paper concerns a detailed exploration of the relation between Schrödinger equations and N = 2 supersymmetric gauge theories, from several different points of view
We study in detail the Schrödinger equation corresponding to the four dimensional SU(2) N = 2 SQCD theory with one flavour
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
Summary
The classical periods Zγ control the central charges and masses of BPS states, while the quantum periods Πγ can be computed in terms of the Nekrasov-Shatashvili instanton partition function.. The classical periods Zγ control the central charges and masses of BPS states, while the quantum periods Πγ can be computed in terms of the Nekrasov-Shatashvili instanton partition function.3 This connection has been derived from various different points of view, including direct gauge theory computations and class S constructions using the AGT correspondence; see e.g. 1.2 The SU(2) Nf = 1 equation In this paper we consider a specific example of (1.1): This equation corresponds to N = 2 supersymmetric Yang-Mills theory with gauge group SU(2) and Nf = 1;4 Λ ∈ C× is the dynamical scale, m ∈ C is the hypermultiplet mass, and E ∈ C parameterizes the Coulomb branch
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