Non-Perturbative Aspects of Supersymmetric Gauge Theories and String Theory

by

John Ma Pierre

Introduction

For the past few decades physicists have used the tools of quantum field theory to study the high energy, small-scale structure of the universe.  Two promising candidates, string theory and supersymmetry, have also emerged as possible essential ingredients of a unified theory of all fundamental interactions.  Most of our understanding of these areas has been obtained using perturbation theory which by construction is a largely incomplete description of a quantum theory.  Many of the outstanding questions in theoretical physics  such as the dynamics of quarks inside a nucleus, the full  quantum nature of black holes, or the origin of our 4-dimensional spacetime will probably be answered only when we have complete non-perturbative formulations of the theories that we use to study them.

What is remarkable is that much of the recent progress in understanding non-perturbative aspects of string theory and supersymmetric gauge theories has been made in parallel, using each to gain knowledge and insights about the other.  There are various reasons for this intimate connection between supersymmetric gauge theories and string theory.  One is that supersymmetric gauge theories arise as low energy effective descriptions of compactified string theories in limits where gravity decouples.  Another reason is that superstring theories can be formulated in backgrounds that contain D-branes,  and supersymmetric gauge theories serve as effective world volume theories for these D-branes.  In addition to these direct examples, it is sometimes the case that intuition about non-perturbative physics that is gained in one area can be directly applied to the other.  An example of this is the guiding principle that singularities in the quantum moduli space of a low energy effective theory signal the appearance of new massless states.  This was seen to be a generic phenomena in supersymmetric gauge theories and was subsequently applied to the resolution of conifold singularities by massless black holes in string theory.

What has become clear is that by studying the seemingly disparate subjects of string theory, supergravity, and supersymmetric gauge theories we are uncovering aspects of a single underlying theory.  Each description has a separate regime of validity corresponding to different limits in the moduli space of the underlying theory.
The main challenge for the future is to precisely define this fundamental theory, and to show that it naturally leads to the observed physics of our universe.

In this dissertation we study some non-perturbative aspects of  N = 1 and N = 2 supersymmetric gauge theories and the physics of branes in the context of  string theory, M-theory, and black holes in supergravity.  In Chapter II, we discuss exact non-perturbative results in N = 1 supersymmetric gauge theories with exceptional gauge groups.  Theories with $N=1$ supersymmetry (such as the Minimal Supersymmetric Standard Model) are widely believed to be relevant to describing physics at energies above the TeV scale. Exceptional gauge groups are of interest because they naturally appear in string theory and E6, for example, may hold promise as a grand unified gauge group.  Beyond that, it is hoped that by studying many examples we may learn something new about the strong coupling dynamics of gauge theories such as new phases, new mechanisms for dynamical supersymmetry breaking, identifying generic properties, and uncovering the fundamental concepts behind duality. We treat the case with gauge group G2 in detail, presenting the various phenomena that appear as we add flavors of fundamental quarks.  We will encounter examples with dynamically generated superpotentials, modified and unmodified quantum moduli spaces, non-abelian coulomb phases, and strong-weak coupling duality.

In Chapter III, we study the N = 2 supersymmetric gauge theory with gauge group G2.  N = 2 theories have just the ``right'' amount of supersymmetry to keep quantum effects under control so that they may be solved exactly, and yet they are not overconstrained, which allows interesting strong coupling dynamics to occur.  We begin by reviewing Seiberg-Witten theory and go on to deduce the exact quantum moduli space of vacua in the Coulomb branch by applying the method of confining phase superpotentials.  We identify singularities in the moduli space which have dual descriptions in terms of massless monopoles and dyons. Some connections between N = 2 and N = 1 theories are also discussed.

In Chapter IV, we move on to string theory and M-theory.  String theories are of great interest because they are consistent theories of quantum gravity and they have the potential to reproduce the gauge group and matter content of the Standard Model at low energies. The five known superstring theories in ten dimensions are linked together by various dualities, which include a phase in which an eleventh dimension opens up.  This phase, known as M-theory, arises as the strong coupling limit of the IIA and E8 x E8 heterotic string theories.  It has eleven dimensional supergravity as its low energy effective theory, and in some sense can be thought of as the most symmetric phase.  We discuss the matrix formulation of M-theory also known as M(atrix) theory, which is conjectured to be a full quantum description of  M-theory in the infinite momentum frame.  We review the relationship between the various BPS p-branes in IIA string theory and its eleven dimensional M-theory limit and study the scattering of these branes in string theory and M(atrix) theory, concentrating mainly on the interactions of eight-branes.

In Chapter V, we use D-branes to probe a black hole which carries 0- and 6-brane charge and examine its relationship to a possible bound state configuration of D0-branes and D6-branes.  An appropriate configuration of D-branes can provide a weakly coupled microscopic description of a black hole which is intrinsically a strongly-coupled macroscopic object. Black holes serve as useful theoretical laboratories in which to test the effects of a quantum theory of gravity such as string theory. We show that the long distance interactions between the supergravity black hole and p-brane probes are exactly reproduced using the D-brane configuration and string theory.