Geometrical Actions on Tubelike Submanifolds and Topological Defects in Anti de Sitter Space

In physical theories where the energy (action) is localized near a submanifold of a constant curvature space, there is a universal expression for the energy (or the action). One can derive a multipole expansion for the energy that has a finite number of terms, and depends on intrinsic geometric invariants of the submanifold and extrinsic invariants of the embedding of the submanifold. Chapter 2 expands upon prior work in an attempt try to develop a theory of emergent gravity arising from the embedding of a submanifold into an ambient space equipped with a quantum field theory. The theoretical method presented here requires a generalization of a formula by Hermann Weyl. While the prior work discussed the framework in Euclidean (Minkowski) space, here it is discussed how this framework generalizes to spaces of constant sectional curvature. The focus is primarily on anti de Sitter space. It is then discussed how such a theory can give rise to a cosmological constant and Planck mass that are within reasonable bounds of the experimental values. One configuration that supports the application of the adapted Weyl formula is that of a topological defect. These are also interesting to study in their own right. In chapter 3 exact analytical solutions for a class of SO(l) Higgs field theories are found in a non-dynamic background n-dimensional anti de Sitter space. These finite transverse energy solutions are maximally symmetric p-dimensional topological defects wheren = (p + 1) + l. The radius of curvature of anti de Sitter space provides an extra length scale that allows one to study the equations of motion in a limit where the masses of the Higgs field and the massive vector bosons are both vanishing. This is affectionately dubbed the double BPS limit. In anti de Sitter space, the equations of motion depend on both p and l. The exact analytical solutions are expressed in terms of standard special functions. The known exact analytical solutions are for kink-like defects (p = 0; 1; 2; : : : ; l = 1), vortex-like defects (p = 1; 2; 3; l = 2), and the ’tHooftPolyakov monopole (p = 0; l = 3). In certain cases where the author did not find an analytic solution, numerical solutions to the equations of motion are presented instead. The asymptotically exponentially increasing volume of anti de Sitter space imposes different constraints than those found in the study of defects in Minkowski space. These solutions are seen as the first step in a perturbative analysis. In chapter 4 the author examines codimension{1 topological defects whose associated world line is geodesically embedded in AdS2. This discussion extends the study of exact analytical solutions in the preceding chapter. Here the idea is to study the linear perturbations about the zeroth order kink-like solution and verify that they are stable. There is also a discussion on general features of the perturbation expansion to all orders. Finally, in chapter 5 the discussion is extended to kink-like (codimension 1) topological defects whose world brane is AdSq embedded into AdSq+1. In this chapter the main result is that all perturbations to the mass of the field are stable to first order and the author finds an explicit expression for the form of the first-order correction