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In this thesis, we study the complex sine-Gordon model on a half line. The model in the bulk is an integrable (l+1) dimensional field theory which is U(1) gauge invariant and comprises a generalisation of the sine-Gordon theory. It accepts soliton and breather solutions. By introducing suitably selected boundary conditions we may consider the model on a half line. Through such conditions the model can be shown to remain integrable and various aspects of the boundary theory can be examined. The first chapter serves as a brief introduction to some basic concepts of integrability and soliton solutions. As an example of an integrable system with soliton solutions, the sine-Gordon model is presented both in the bulk and on a half line. These results will serve as a useful guide for the model at hand. The introduction finishes with a brief overview of the two methods that will be used on the fourth chapter in order to obtain the quantum spectrum of the boundary complex sine-Gordon model. In the second chapter the model is properly introduced along with a brief literature review. Different realisations of the model and their connexions are discussed. The vacuum of the theory is investigated. Soliton solutions are given and a discussion on the existence of breathers follows. Finally the collapse of breather solutions to single solitons is demonstrated and the chapter concludes with a different approach to the breather problem. In the third chapter, we construct the lowest conserved currents and through them we find suitable boundary conditions that allow for their conservation in the presence of a boundary. The boundary term is added to the Lagrangian and the vacuum is reexamined in the half line case. The reflection process of solitons from the boundary is studied and the time-delay is calculated. Finally we address the existence of boundary-bound states. In the fourth chapter we study the quantum complex sine-Gordon model. We begin with a brief overview of the theory in the bulk where the semi-classical spectrum and an exact S'-matrix are presented. Following that we use the stationary phase method to derive the semi-classical spectrum of boundary bound states. The bootstrap method is used as an alternative approach to obtain the same spectrum. The results are discussed and compared. The final chapter consists of a general discussion on open questions and problems of the model, and some proposals for further research.

Unitary models in two dimensions are classes of low dimensional theories which provide us with a convenient theoretical laboratory for studying various aspects of the theory of elementary particles. In this thesis, purely bosonic U(N) sigma models with the Wess-Zuraino-Witten (WZW) term in two-dimensional Euclidean space and the supersymmetric (Susy) U(N) σ models with and without this term are discussed. Particular attention is paid to the classical solutions of the equations of motion of these models. Due to the integrabihty of these models, we can associate with them a Lax-pair formalism. We observe that solutions of the Lax-pair equations of the U(N) a model provide us with solutions of the U(N) a model with the WZW-term. This is also the case for solutions of the Susy U(N) a model with the WZW-term which can be constructed from solutions of the Lax-pair equations of the Susy U(N) σ model. We present also some explicit solutions of the Susy U(N) a model without the WZW-term. Many properties of the constructed solutions for both the purely bosonic and Susy models are explored. In particular, we calculate the values of the action for some solutions and study the stability properties of these solutions and find that all the constructed solutions of these models correspond to the saddle points of the action. Finally we consider the hnearized fermion equations in the fixed background of a bosonic field. Special attention is paid to the case when the background field is given by a solution of the U(N) σ model with and/or without the WZW-term. Some classes of solutions of this problem are presented and their properties are discussed. We observe that a class of these solutions is related to the components of the energy-momentum tensor of the purely bosonic σ model and prove that some of these solutions are traceless.

Low dimensional models are generally regarded to be a convenient theoretical laboratory for studying various aspects of elementary particle theory. In this thesis, the extended solutions of one particular class of such models, namely the ₵p(^n-1) non-linear a-models in 2 dimensions, are discussed. Special attention is paid to the shape of these extended structures and their dependence on the parameters of the solutions. Time dependence is introduced into the models, and properties of the moving objects in these (2 + l)-dimensional theories are explored. In particular, the Hopf terms of the theories are investigated, and their relation to the spin of the extended solutions is discussed. Also the classical dynamics of these moving objects, and their explanation in terms of the geodesic motions on certain Hermitian and Kāhler manifolds is considered. Finally the embedding of the (₵p(^n-1)) solutions into the 2-dimensional U(n) chiral models is studied, paying particular attention to the stability of these embedded solutions in the larger group space, and to the number of independent negative modes of the fluctuation operator around these solutions.

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Low scale supersymmetry (SUSY) is a compelling solution to the electroweak hierarchy problem. However, increasingly strong limits on the masses of superpartners, first from LEP and now the LHC, mean that the simplest models require significant fine tuning. This thesis is dedicated to the study of a possible alternative low energy superpartner spectrum, natural SUSY, in which only superparticles directly involved in stabilising the electroweak scale are light, alleviating collider limits and potentially reducing tuning. After reviewing how low scale SUSY is motivated by the hierarchy problem, we build a model of SUSY breaking and mediation that successfully generates a natural SUSY spectrum. This also suppresses the first two generation fermion Yukawas, and leads to small parameters in the hidden sector, which are required for successful SUSY breaking. A challenge in models of natural SUSY is raising the physical Higgs mass to 125 GeV, and we study the possibility that this could occur through the addition of a singlet to the theory. If stops are very light, the coupling of the singlet to the Higgs needs to be so large that it becomes nonperturbative before the scale of grand unification, raising the concern that precision gauge coupling unification may be upset. However, we find that this is not necessarily the case. Rather it is possible this could correct for the present ∽ 3% discrepancy in the two-loop minimal supersymmetric model's unification prediction. We then turn to the fine tuning in models of natural SUSY, emphasising that this should be measured with respect to the theory's ultraviolet (UV) parameters. We show that the first two generation sfermions can be made relatively heavy, beyond LHC reach, without introducing tuning. However, the gluino generates a significant tuning through the stops during the renormalisation group flow. As a result, there is no fine tuning benefit in reducing the stop masses below (50 - 75)% of the weak scale gluino mass, and we obtain strong lower bounds on the tuning of theories compatible with collider limits. We also study theories with Dirac gauginos, which have relatively low fine tuning even if the scale of mediation is high. Finally, we consider the effect of relaxing a common assumption and allowing the hidden SUSY breaking sector to modify the running of the visible sector soft masses. This may plausibly occur in realistic models and could dramatically reduce the fine tuning of SUSY theories.

Determining cosmological parameters from current observational data requires knowledge of the primordial density perturbations generated during inflation. We begin by examining a model of inflation along a flat direction of the minimal supersymmetric Standard Model (MSSM) and the power spectrum of perturbations it can produce. We consider the fine-tuning issues associated with this model and discuss a modification of the potential to include a hybrid transition that reduces the fine-tuning, without affecting the viability of the model. However, supersymmetric flat directions might play a role in other models of inflation as well. In particular, they may cause a feature in the primordial power spectrum of perturbations, unlike the scale-free spectrum assumed in the standard Lambda Cold Dark Matter (LCDM) cosmological model. We then show that in the presence of such a feature, an alternative cosmological model with a large local void and no dark energy provides a good fit to both Type Ia supernovae and the cosmic microwave background (CMB) data from the WMAP satellite. Constraints from the locally measured Hubble parameter, baryon acoustic oscillations and primordial nucleosynthesis are also satisfied. This degeneracy motivates a search for other independent observational tests of LCDM. The integrated Sachs-Wolfe (ISW) imprint of large-scale structure on the CMB is one such test. The ISW imprint of superstructures of size ~100 Mpc/h at redshift z~0.5 has been detected with >4 sigma significance, however it has been noted that the signal is much larger than expected. We revisit the calculation using linear theory predictions in a LCDM cosmology and find the theoretical prediction is inconsistent by >3 sigma with the observation. If the observed signal is indeed due to the ISW effect then huge, extremely underdense voids are far more common in the observed universe than predicted by LCDM.

This thesis is dedicated to the study of a prominent class of dark matter (DM) models, in which the DM relic density is linked to the baryon asymmetry, often referred to as Asymmetric Dark Matter (ADM) theories. In ADM the relic density is set by a particle-antiparticle asymmetry, in direct analogue to the baryons. This is partly motivated by the observed proximity of the baryon and DM relic densities Ω_{DM} ≈ 5 Ω_{B}, as this can be explained if the DM and baryon asymmetries are linked. A general requisite of models of ADM is that the vast majority of the symmetric component of the DM number density, the DM-antiDM pairs, must be removed for the asymmetry to set the DM relic density and thus to explain the coincidence of Ω_{DM} and Ω_{B}. However we shall argue that demanding the efficient annihilation of the symmetric component leads to a tension with experimental constraints in a large class of models. In order to satisfy the limits coming from direct detection and colliders searches, it is almost certainly required that the DM be part of a richer hidden sector of interacting states. Subsequently, examples of such extended hidden sectors are constructed and studied, in particular we highlight that the presence of light pseudoscalars can greatly aid in alleviating the experimental bounds and are well motivated from a theoretical stance. Finally, we highlight that self-conjugate DM can be generated from hidden sector particle asymmetries, which can lead to distinct phenomenology. Further, this variant on the ADM scenario can circumvent some of the leading constraints.

This thesis contributes with a number of topics to the subject of string compactifications, especially in the instance of the E₈ × E₈ heterotic string theory compactified on smooth Calabi-Yau threefolds. In the first half of the work, I discuss the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties. The intricate structure of this plot is explained by the existence of certain webs of elliptic-K3 fibrations, whose mirror images are also elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fiber. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, give to the Hodge plot the appearance of a fractal. Moving on, I discuss a different type of web of manifolds, by looking at smooth Z₃-quotients of Calabi-Yau three-folds realised as complete intersections in products of projective spaces. Non-simply connected Calabi-Yau three-folds provide an essential ingredient in heterotic string compactifications. Such manifolds are rare in the classical constructions, but they can be obtained as quotients of homotopically trivial Calabi-Yau three-folds by free actions of finite groups. Many of these quotients are connected by conifold transitions. In the second half of the work, I explore an algorithmic approach to constructing E₈ × E₈ heterotic compactifications using holomorphic and poly-stable sums of line bundles over complete intersection Calabi-Yau three-folds that admit freely acting discrete symmetries. Such Abelian bundles lead to N = 1 supersymmetric GUT theories with gauge group SU(5) × U(4) and matter fields in the 10, ⁻10, ⁻5, 5 and 1 representations of SU(5). The extra U(1) symmetries are generically Green-Schwarz anomalous and, as such, they survive in the low energy theory only as global symmetries. These, in turn, constrain the low energy theory and in many cases forbid the existence of undesired operators, such as dimension four or five proton decay operators. The line bundle construction allows for a systematic computer search resulting in a plethora of models with the exact matter spectrum of the Minimally Supersymmetric Standard Model, one or more pairs of Higgs doublets and no exotic fields charged under the Standard Model group. In the last part of the thesis I focus on the case study of a Calabi-Yau hypersurface embedded in a product of four CP1 spaces, referred to as the tetraquadric manifold. I address the question of the finiteness of the class of consistent and physically viable line bundle models constructed on this manifold. Line bundle sums are part of a moduli space of non-Abelian bundles and they provide an accessible window into this moduli space. I explore the moduli space of heterotic compactifications on the tetraquadric hypersurface around a locus where the vector bundle splits as a direct sum of line bundles, using the monad construction. The monad construction provides a description of poly-stable S(U(4) × U(1))–bundles leading to GUT models with the correct field content in order to induce standard-like models. These deformations represent a class of consistent non-Abelian models that has co-dimension one in Kähler moduli space.

Identifying the relic particles that constitute the cold dark matter in our Universe is an outstanding problem in astro-particle physics. Direct detection experiments are among the most promising methods of detecting particle dark matter through non-gravitational interactions. In this thesis, the usual assumptions made when calculating the event rate at direct detection experiments are examined. Varying astrophysical parameters and the dark matter velocity distribution leads to significant changes in acceptance regions and exclusion curves for scenarios in which the tail of the velocity distribution is sampled; this includes 'light dark matter' (mass less than 10 GeV) and 'inelastic dark matter'. The DAMA and CoGeNT collaborations both report an annual modulation in their event rate that they attribute to dark matter. Two analyses of these experiments are performed. In the first, it is shown that these experiments can be compatible with each other and with the constraints from other direct detection experiments. This requires some isospin violation in the couplings of dark matter to protons and neutrons and a small inelastic splitting to boost the modulation fraction. The second analysis provides a comparison of the modulation signals free from all astrophysical parameters, under the assumption that dark matter scatters elastically. Again it is found that some isospin violation and a boosted modulation fraction is required in order that DAMA and CoGeNT are consistent with all experiments. A boosted modulation fraction may arise from a velocity distribution different from the Maxwell-Boltzmann distribution, which is usually assumed. Finally, a supersymmetric theory in which the dark matter candidate is a mixture of left- and right-handed sneutrino is considered. This theory has many novel signatures at colliders, indirect detection and direct detection experiments.