Exact WKB methods in SU(2) Nf=1

We study in detail the Schrodinger equation corresponding to the four dimensional SU(2) 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 Schrodinger 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 Schrodinger equation.

Automated summary

We study in detail the Schrodinger equation corresponding to the four dimensional SU(2) N = 2 SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways and find good agreement. We also study the exact quantization condition for the spectrum and explore the relationship between singularities of the Borel transformed WKB series and BPS states.