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.

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 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.

In this thesis we are concerned with the existence of infrared fixed points and the conformal window for gauge theories with fermions. We are particularly interested in those theories that are candidates for walking technicolor. We discuss the background of technicolor and the techniques relevant to a theoretical understanding of the conformal window. Following this we extend the ideas of metric confinement and causal analyticity to theories with fermions in non-fundamental representations. We use these techniques to, respectively, provide a lower bound on the lower end of the conformal window and to provide a measure of perturbativity. As well as analytic calculations we use lattice techniques to investigate two particular candidate theories for walking technicolor - SU(2) with two adjoint fermions and with six fundamental fermions. We use Schrodinger Functional techniques to investigate the running of the theory across a wide range of scales. We measure both the running of the coupling and an estimator for the fermion mass anomalous dimension, $gamma$. We find that both theories are consistent with an infrared fixed-point. However, paying particular attention to our error estimates, we are unable to absolutely confirm their existence. This is a not unexpected result for SU(2) with two adjoint fermions but is rather surprising for SU(2) with only six fundamental fermions. In the region where we are consistent with a fixed point we find $0.05<gamma<0.56$ for $SU(2)$ with two adjoint fermions and $0.135<gamma<1.03$ for $SU(2)$ with six fundamental fermions. The measurement of $gamma$ for $SU(2)$ with two adjoint fermions is the first determination of $gamma$ for any candidate theory of walking technicolor.

In this work we analyze various implications of the presence of large field vacum expectation values (VEVs) along supersymmetric flat directions during the early universe. First, we discuss supersymmetric leptogenesis and the gravitino bound. Supersym- metric thermal leptogenesis with a hierarchical right-handed neutrino mass spectrum normally requires the mass of the lightest right-handed neutrino to be heavier than about 109 GeV. This is in conflict with the upper bound on the reheating temperature which is found by imposing that the gravitinos generated during the reheating stage after inflation do not jeopardize successful nucleosynthesis. We show that a solution to this tension is actually already incorporated in the framework, because of the presence of flat directions in the supersymmetric scalar potential. Massive right- handed neutrinos are efficiently produced non-thermally and the observed baryon asymmetry can be explained even for a reheating temperature respecting the gravitino bound if two conditions are satisfied: the initial value of the flat direction must be close to Planckian values and the phase-dependent terms in the flat direction potential are either vanishing or sufficiently small. We then show that flat directions also contribute to the total curvature perturbation. Such perturbation is generated at the first oscillation of the flat direction condensate when the latter relaxes to the minimum of its potential after the end of inflation. If the contribution to the total curvature perturbation from supersymmetric flat direction is the dominant one, then a significant level of non-Gaussianity in the cosmological perturbation is also naturally expected. Finally, we argue that supersymmetric flat direction VEVs can decay non perturbatively via preheating even in the case where they undergo elliptic motion in the complex plane instead of radial motion through the origin. It has been generally argued that in this case adiabaticity is never violated and preheating is inefficient. Considering a toy U(1) gauge theory, we explicitly calculate the scalar potential, in the unitary gauge, for excitations around several flat directions. We show that the mass matrix for the excitations has non-diagonal entries which vary with the phase of the flat direction vacuum expectation value. Furthermore, this mass matrix has zero eigenvalues whose eigenstates change with time. We show that these light degrees of freedom are produced copiously in the non-perturbative decay of the flat direction VEV.

In the first part of this thesis we study baryonic U(1) symmetries dual to Betti multiplets in the AdS_4/CFT_3 correspondence for M2 branes at Calabi-Yau four-fold singularities. Such short multiplets originate from the Kaluza-Klein compactification of eleven-dimensional supergravity on the corresponding Sasaki-Einstein seven-manifolds. Analysis of the boundary conditions for vector fields in AdS_4 allows for a choice where wrapped M5 brane states carrying non-zero charge under such symmetries can be considered. We begin by focusing on isolated toric singularities without vanishing six-cycles, which we classify, and propose for them field theory duals. We then study in detail the cone over the well-known Sasaki-Einstein space Q^111, which is a U(1) fibration over CP^1 x CP^1 x CP^1. The boundary conditions considered are dual to a CFT where the gauge group is U(1)^2 x SU(N)^4. We find agreement between the spectrum of gauge-invariant baryonic-type operators in this theory and M5 branes wrapping five-cycles in the Q^111 space. Moreover, the physics of vacua in which these symmetries are spontaneously broken precisely matches a dual gravity analysis involving resolutions of the singularity, where we are able to match condensates of the baryonic operators, Goldstone bosons and global strings. We then study the implications of turning on a closed three-form with non-zero periods through torsion three cycles in the Sasaki-Einstein manifold. This three-form, otherwise known as torsion G-flux, non-trivially affects the supergravity dual of Higgsing, and we show that the supergravity and field theory analyses precisely match in an example based on the Sasaki-Einstein manifold Y^1,2(CP^2), which is a S^3 bundle over CP^2. We then explain how the choice of M-theory circle in the background can result in exotic renormalization group flows in the dual field theory, and study this in detail for the Sasaki-Einstein manifold Y^1,2(CP^2). We also argue more generally that theories where the resolutions have six-cycles are expected to receive non-perturbative corrections from M5 brane instantons. We give a general formula relating the instanton action to normalizable harmonic two-forms, and compute it explicitly for the Sasaki-Einstein Q^222 example, which is a Z_2 orbifold of Q^111 in which the free Z_2 quotient is along the R-symmetry U(1) fibre. The holographic interpretation of such instantons is currently unclear. In the second part of this thesis we study the breaking of baryonic symmetries in the AdS_5/CFT_4 correspondence for D3 branes at Calabi-Yau three-fold singularities. This leads, for particular vacuum expectation values, to the emergence of non-anomalous baryonic symmetries during the renormalization group flow. We identify these vacuum expectation values with critical values of the NS-NS B-field moduli in the dual supergravity backgrounds. We study in detail the C^3/Z_3 orbifold theory and the dual supergravity backgrounds that correspond to the breaking of the emerging baryonic symmetries, and identify the expected Goldstone bosons and global strings in the infra-red. In doing so we confirm the claim that the emerging symmetries are indeed non-anomalous baryonic symmetries.

We calculate the string tensions, mass spectrum, and deconfining temperatures of <i>SO(N</i>) gauge theories in 2+1 dimensions. After a review of lattice field theory, we describe how we simulate the corresponding lattice gauge theories, construct operators to project on to specific states, and extrapolate values to the continuum limit. We discuss how to avoid possible complications such as finite size corrections and the bulk transition. <i>SO(N</i>) gauge theories have become recently topical since they do not have a fermion sign problem, are orbifold equivalent to <i>SU(N</i>) gauge theories, and share a common large-<i>N</i> limit in their common sector of states with <i>SU(N</i>) gauge theories. This motivates us to compare the physical properties of <i>SO(N</i>) and <i>SU(N</i>) gauge theories between 'group equivalences', which includes Lie algebra equivalences such as <i>SO</i>(6) and <i>SU</i>(4), and particularly a large-<i>N</i> equivalence. We discuss the large-<i>N</i> orbifold equivalence between <i>SO(N</i>) and <i>SU(N</i>) gauge theories, which relates the large-<i>N</i> gauge theories perturbatively. Using large-<i>N</i> extrapolations at fixed 't Hooft coupling, we test to see if <i>SO(N</i>) gauge theories and <i>SU(N</i>) gauge theories share non-perturbative properties at the large-<i>N</i> limit. If these group equivalences lead to similar physics in the gauge theories, then we could imagine doing finite chemical potential calculations that are currently intractable in <i>SU(N</i>) gauge theories by calculating equivalent quantities in the corresponding <i>SO(N</i>) gauge theories. We show that the <i>SO(N</i>) and <i>SU(N</i>) values match between group equivalences and at the large-<i>N</i> limit.