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

Abstract

We study in detail the Schrödinger 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 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.

A new method for exact results on Quasinormal Modes of Black Holes

“…Thinking to gauge theory, generalizations of formula (12), already exist for the subcase of the N f = 1 gauge theory [32] (obtained in the limit y → +∞, Σ 4 → 0) and so we expect them to exist also for the whole N f = 2 theory and even more generally. In this way we expect that in general the integrable Bethe roots condition, which we have shown to follow straightforwardly from BHs physics, in gauge theory indeed corresponds to the quantization of the gauge A D period as stated in [4].…”

Wall-crossing of TBA equations and WKB periods for the third order ODE

“…See [70,71] for the case related to the second order ODE. It is also interesting to include more irregular/regular singular points in the potential, which will help us to study the four dimensional N = 2 super Yang-Mills theory [17,18,20,22,23].…”

“…One can determine the quantum periods directly by the Thermodynamic Bethe ansatz (TBA) equations [15] satisfied by the Y-functions. See also [16][17][18][19][20][21][22][23][24], for more general potential.…”

Anisotropic Kondo line defect and ODE/IM correspondence

“…Subsequent studies including exact WKB analysis, including at the self dual point b = 1 can also be found in e.g. [56][57][58][59]. Importantly, (3.7) takes an universal form as (3.5) since they both have 4d Chern Simons origin.…”