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The principal objects of study in this thesis are arbitrary spectral families, E, on a complex Banach space X. The central theme is the relationship between the geometry of X and the p-variation of E. We show that provided X is super- reflexive, then given any E, there exists a value 1 · p < 1, depending only on E and X, such that var p(E) < 1. If X is uniformly smooth we provide an explicit range of such values p, which depends only on E and the modulus of convexity of X*, delta X*(.).

We examine in detail the microscopic theories of Beliaev and Hugenholtz and Pines as applied to an interacting system of bosons in its ground state and demonstrate that these theories are strictly applicable to a zero density gas only. After providing a re-normalization of the time dependent perturbation theory of a many fermion system in its ground state, we reformulate the corresponding boson theory without ab initio approximations concerning the k=0 mode. The resulting boson perturbation theory has a diagrammatic structure which is topologically identical with the corresponding fermion theory and provided the implicit assumptions concerning convergence are valid, we conclude that a finite fraction of the particles in an arbitrary interacting Bose gas at T=ok will not condense into the k=0 mode.

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In this thesis some aspects of temperature corrections in string theory are analyzed: in particular, we study the thermal contributions to the 4 dimensional effective potential arising from string theory compactifications. String theory predicts that the spacetime has more than 4 dimensions; in particular, supersymmetric string theories are consistent only if the spacetime has 10 dimensions, 1 time plus 9 space directions. In order to describe the physics of our Universe with string theory we make 6 spatial directions very small, namely, we curl them into a 6-dimensional space. The resulting 4-dimensional theory depends on a large number of parameters which are massless scalar fields called moduli. The different values that the moduli can take represent both the possible deformations of the 6- dimensional compact space and the values of the coupling constants and masses in the 4-dimensional spacetime. Allowing them to have arbitrary values leads to a lack of predictability of various 4D physical quantities, to a huge vacuum degeneracy and to unobserved long range fifth forces. In the thesis we review some methods established in the literature in order to fix the moduli values and hence to fix a particular geometry and we investigate how the inclusion of temperature corrections alter their values and affect the geometry of the compact space. The analysis seems to suggest that at least in the specific compactification scenarios considered in this thesis, temperature corrections do not alter substantially the zero temperature results. In the final part of this work, we analyze instead an example in which the inclusion of temperature corrections alters dramatically the picture at zero temperature. In particular, we study an unstable system constituted by a pair of Dirichlet (D) and anti-D brane that, although being unstable at zero temperature, it can become stable once finite temperature corrections are switched on.

This thesis consists of two parts. In the rst part we investigate the worldvolume supersymmetry algebra of multiple membrane theories. We begin with a description of M-theory branes and their intersections from the perspective of spacetime and worldvolume supersymmetry algebras. We then provide an overview of the recent work on multiple M2-branes focusing on the Bagger-Lambert theory and its relation to the Nambu-Poisson M5-brane and the ABJM theory. The worldvolume supersymmetry algebras of these theories are explicitly calculated and the charges interpreted in terms of spacetime intersections of M-branes. The second part of the thesis looks at l3 p corrections to the supersymmetry transformations of the Bagger-Lambert theory. We begin with a review of the dNS duality transformation which allows a gauge eld to be dualised to a scalar eld in 2+1 dimensions. Applying this duality to 02 terms of the non-abelian D2-brane theory gives rise to the l3 p corrections of the Lorentzian Bagger-Lambert theory. We then apply this duality transformation to the 02 corrections of the D2-brane supersymmetry transformations. For the `abelian' Bagger-Lambert theory we are able to uniquely determine the l3 p corrections to the supersymmetry transformations of the scalar and fermion elds. Generalising to the `non-abelian' Bagger-Lambert theory we are able to determine the l3 p correction to the supersymmetry transformation of the fermion eld. Along the way make a number of observations relating to the implementation of the dNS duality transformation at the level of supersymmetry transformations.

Supersymmetry is an important concept in modern high energy physics. It has found many applications in theoretical considerations of supersymmetric gauge theories as well as in phenomenological approaches to physics beyond the Standard Model. In this report we discuss some recent progress in supersymmetric eld theories in four and six dimensions. After introducing basic ideas and properties of supersymmetry we review the concept of scattering amplitudes in maximally supersymmetric theories in four dimensions before constructing a related framework in six dimensions. Here, the spinor helicity formalism and on-shell superspace were recently developed for six-dimensional gauge theories with (1,1) supersymmetry. We combine these two techniques with (generalised) unitarity, which is a powerful technique to calculate scattering amplitudes in any massless theory. As an application we calculate one-loop superamplitudes with four and ve external particles in the (1,1) theory and perform several consistency checks on our results. Within the area of phenomenological applications of supersymmetric gauge theories, we brie y review basic properties of supersymmetry breaking and gauge mediation in four dimensions. An important recent development has been the concept of theories with broken supersymmetry and metastable vacua. By using the advances of Seiberg duality, we examine a metastable N = 1 Macroscopic SO(N) SQCD model of Intriligator, Seiberg and Shih (ISS). We introduce various baryon and meson deformations, including multitrace operators. In this setup, direct fundamental messengers and the symmetric pseudomodulus messenger mediate supersymmetry breaking to a minimal supersymmetric Standard Model. We compute gaugino and sfermion masses and compare them for each deformation type. We also explore reducing the rank of the magnetic

In this thesis we focus on the construction of models in which a supersymmetry breaking hidden sector is located on one fixed point of an extra dimensional interval and the effects are gauge mediated across this interval to the other fixed point where the supersymmetric standard model is located. We use the formalism of current correlators to encode supersymmetry breaking effects and explore flat, warped and deconstructed extra dimensions. We also apply these techniques to models of metastable supersymmetric breaking in N = 1 Supersymmetric Quantum Chromodynamics.