Aspects of Holomorphic Sectors In Supersymmetric Theories

Abstract

In this dissertation, we discuss some aspects of theories with extended supersymmetry that have interesting, exactly calculable holomorphic sectors. The two classes of theories we consider are d = 4, N = 2 effective supergravities that describe Calabi-Yau compactifications of a Type IIA superstring, and two-dimensional theories with N = (0, 2) supersymmetry. In the first case, we study higher-derivative couplings in the 4d N = 2 superpotential (as well as 2d N = (2, 2) superpotential in the presence

of D4 branes), which is a holomorphic function of chiral superfields. It is described by the Gopakumar-Vafa formula in terms of BPS spectra of M-theory compactifications (and Ooguri-Vafa formula for the 2d N = (2, 2) case). In the second class of theories, we study another holomorphic object known as a chiral algebra, which emerges in the cohomology of one supercharge of a two-dimensional theory with N = (0, 2) supersymmetry.

In chapter two, we describe a detailed derivation of the Gopakumar-Vafa formula, as well as explain the Ooguri-Vafa formula at the end. The main idea of the derivation is to compute the effective superspace action on a properly chosen background due to BPS states winding the M-theory circle. A lot of technical and conceptual details, such as how supersymmetry of the background determines the action for BPS particles,

why and in which limit the computation makes sense, are explained along the way.

In chapter three, we explore chiral algebras of N = (0, 2) theories. We explain why these objects are invariant along the RG flows and study some of their general properties. We give more details for theories known as N = (0, 2) Landau-Ginzburg (LG) models, and later we specialize to N = (2; 2) supersymmetry, for which we consider some concrete examples, such as LG models which flow to N = 2 minimal models in the infrared.