Quantum Seiberg-Witten periods for N=2 SU(N) SQCD around the superconformal point

Abstract

We study the quantum Seiberg-Witten periods of N = 2 superconformal field theories which are obtained by taking the scaling limit of N = 2 SU (N c ) SQCD around the superconformal fixed point. The quantum Seiberg-Witten curves of these superconformal field theories are shown to be classified into the Schrödinger type and the SQCD type, which depend on flavor symmetry at the fixed point. We study the quantum periods and compute the differential operators which relate the quantum periods to the classical ones up to the fourth-order in the deformation parameter. IntroductionThe low-energy description of N = 2 supersymmetric gauge theories has singularities in the strong coupling region [1,2]. In particular, there appears an IR fixed point at a locus of the Coulomb branch, where mutually non-local BPS particles become massless [3,4,5].The dynamics around the fixed point is described by an interacting N = 2 superconformal field theory called the Argyres-Douglas (AD) theory.One can study the BPS spectrum of the AD theory from the Seiberg-Witten (SW) curve, which is obtained by degeneration of the SW curve of the UV gauge theory. It is also realized from the compactification of the six-dimensional N = (2, 0) superconformal field theory on a punctured Riemann surface [6,7,8] and classified in [9,10,11,12].One can also study the SW theories in the Nekrasov-Shatashvili (NS) limit of the Omega-background [13]. The Omega background deforms four dimensional spacetime by the two-dimensional torus action [14]. The NS limit is defined by the limit where one of the deformation parameters goes to zero. In this limit, the deformed theory is reformulated by the quantum SW curve [15], which is a differential equation obtained by the quantization of the symplectic structure based on the SW differential. The deformation parameter plays a role of the Planck constant.The WKB analysis of the quantum SW curve determines the deformed SW periods [16,17,18]. In the weak coupling region, the quantum corrections to the SW periods agree with those obtained from the NS limit of the Nekrasov partition function [15,19,20,21].In the strong coupling region such as the massless monopole/dyon point, the quantum SW periods for the SU(2) SQCD with N f ≤ 4 have been studied in [22,23,24]. The relation to the topological strings have been studied in [25,26]. (See also [27,28,29,30] for its relation to CFT.)In the previous papers [31,32], we have studied the quantum SW curves for the AD theories associated with N = 2 SU(2) SQCD, the (A 1 , A r )-type and (A 1 , D r )-type AD theories. We have calculated the quantum corrections to the SW periods up to the fourth order in the deformation parameter and confirmed that they are consistent with the scaling limit of the quantum periods of the UV theories. Recently the exact WKB periods for the quantum SW curve of (A 1 , A r )-type AD theory are shown to be determined by the

Argyres-Douglas theories, S-duality and AGT correspondence

“…• It would be interesting to study the Nekrasov-Shatashvili limit [37] of the Ω-deformed (A 3 , A 3 ) theory. In this limit, combining the results of [38][39][40][41] with our formula, one can evaluate the deformed prepotential of the (A 3 , A 3 ) theory including both the perturbative and instanton parts.…”

Argyres-Douglas Theories, S-duality and AGT Correspondence

“…• It would be interesting to study the Nekrasov-Shatashvili limit [34] of the Ω-deformed (A 3 , A 3 ) theory. In this limit, combining the results of [35][36][37][38] with our formula, one can evaluate the deformed prepotential of the (A 3 , A 3 ) theory including both the perturbative and instanton parts.…”

WKB periods for higher order ODE and TBA equations

“…The higher order ODEs also appear as the quantum Seiberg-Witten (SW) curves which describe the low-energy effective action of N = 2 supersymmetric gauge theories in the Nekrasov-Shatashvili limit of the Omega background [39,40]. In particular, for the Argyres-Douglas (AD) theory which is obtained by the scaling limit of the gauge theories [41][42][43], the quantum SW curve becomes the higher order ODE [44][45][46][47][48]. The quantum SW curves for a class of AD theories are labeled by a pair of Lie algebras (G, G ) [49], where the quantum SW curve for the (A 1 , A r )-type AD theory corresponds to the second order ODE with A r -type superpotential which is a polynomial potential of order r + 1.…”