Geometry and BPS spectra of 4d N = 2 theories

Abstract

In this thesis we present work done on N = 2 theories of class S with a focus on two projects. N = 2 theories have a large enough amount of supersymmetry to allow for exact results, while also being less restricted in terms of field contact as theories with more supercharges. The theories of class S can be obtained from string theory constructions, which assign a Riemann surface C to each theory. Physical phenomena and computations are then related to the geometry of C, in particular to the space of flat connections on C. One of the projects is about computing the effective twisted superpotential Weff, using a geometric method based on a proposal by Nekrasov, Rosly, and Shatashvili. We give a detailed explanation of the geometric recipe to compute Weff by first constructing Darboux coordinates for the moduli space of flat connections of the corresponding UV-surface C, and then evaluating these coordinates on oper connections by solving the differential equations on C, which can be found from quantization of the Seiberg-Witten curve Σ → C. We present results for the pure SU(2) theory in its weak coupling limit, the AD2 theory, and the Minahan-Nemeschansky E6 theory. The other project is about the BPS spectrum of the SU(2) N = 2∗ theory. We start with the theory on specific walls Ei in the Coulomb branch, on which the spectrum is known. We reproduce the results on these with quiver methods and then examine the perturbation from the wall, first by again using quiver methods and then by a python algorithm which implements the Kontsevich-Soibelmann wall crossing formula. We find very intricate behavior including highly unexpected wall crossings with negative pairing. We show a way to interpret these crossings as the reverse of the usual pairing 2 wall crossing formula, involving a collection of a priori unrelated states.