Exact Results on Moduli Spaces of Supersymmetric Gauge Theories

Abstract

In this thesis, certain exact results in supersymmetric gauge theories are discussed. In these theories, holomorphic gauge invariant operators play a central role in understanding the structure of the space of solutions to vacuum equations, known as the moduli space. We focus on a technique to count such operators with various quantum numbers. The counting can be done by computing a partition function, known as the Hilbert series, which counts all holomorphic gauge invariant operators carrying a speci ed set of global U(1) charges. The Hilbert series can be computed exactly for various gauge theories. In Part I of this thesis, we compute the Hilbert series of four dimensional N = 1 supersymmetric QCD with classical gauge groups. In part II, we count chiral operators on the one instanton moduli space on R4 and study the hypermultiplet moduli spaces of a large class of N = 2 supersymmetric gauge theories in four dimensions. We demonstrate that the Hilbert series not only contains information about the spectrum of operators in the theory, but it also carries geometrical properties of the moduli space, e.g. the dimension. It is also an indicator of whether the moduli space is Calabi-Yau. Moreover, Hilbert series can be used as a primary tool to test various dualities in gauge theories and in string theory.