The Quest for M-theory
In ten dimensions, there exist five consistent string theories and in eleven dimensions there is a unique supergravity theory. When trying to find the fundamental theory of nature this is clearly an "embarrassment of riches". In the mid 90s, however, it was discovered that these theories are all related via a kind of transformation called duality. The six theories are therefore only facets of a (largely unknown) underlying theory referred to as M-theory. This theory is intrinsically non-perturbative and therefore very hard to study. Witten has suggested that until we know more about M-theory, M can stand for `magic', `mystery' or `membrane', according to taste. In this thesis, comprising an introductory text and eight appended research papers, we are going to describe some of the methods used to study M-theory. Central to this analysis are non-perturbative, solitonic objects collectively referred to as p-branes, whose properties are studied in Papers I-IV and VI. In Paper II we generalize the Goldstone mechanism to the case of tensor fields of arbitrary rank, providing an understanding of the emergence of vector and tensor fields on branes in terms of broken symmetries. In Papers III and IV we find brane solutions with finite field strengths on the brane. Lately, noncommutative theories decoupled from closed strings have been discovered. These theories are defined on branes with critical field strengths and are studied and extended in Papers VI and VII.
In Paper V we generalize eleven dimensional supergravity to obtain the most general geometrical structure in eleven dimensional superspace. The motivation for this is to examine what constraints supersymmetry imposes on possible correction terms arising from M-theory. To facilitate this study the Mathematica package GAMMA, which is capable of performing gamma-matrix algebra and Fierz transformations, were developed and is presented in Paper VIII.