We study the construction of non-supersymmetric Es® E8 heterotic string compactifications of symmetric orbifolds and tensor products of minimal superconformal theories with central charge c=9.The general formalism and features of both powerful techniques are presented and analyzed meticulously. Using the first method we classify all ~ (N=2,3,4 and 6) orbifolds which break space-time supersymmetry and provide us with a realistic chiral theory.Suprisinglyenough, we find two point groups of order 4 and one point group of order 6.The mechanism we propose to lift tachyons from the twisted sectors consists of a combination of the mass level matching principle with the requirement that the left-sector should be tachyon free. Modular invariance and equivalence relations associated with the shift;vectors of the Ea ® E8 lattice, help us to classify all possible shift;vectors which break the Ea ® E8 gauge group.For each viable shift vector we then detennine the massless spectrum of the symmetric Z6 orbifold since the Z4 case has been previously exhausted.The disentangle of representations from the "observable" and "hidden" sectors, the control of the number of chiral matter states both from untwisted and twisted sectors, as well as the gauge symmetry breaking are achieved by considering the presence of constant gauge-background fields (Wilson-lines).The problem of tachyons is resolved by taking advantage of the same method as the one suggested in the absence of Wilson-lines and a classification of all acceptable Wilson-lines and four-dimensional gauge groups is again carried out.Phenomenological implications of these models are discussed and some interesting features already known in string theory are explored. The second method although more complicated is simplified using orbifold techniques. Again space-time supersymmetry is broken, but now with the insertion of some discrete phases (torsions) in the partition function of the theory.The richness of this method leads to some computational difficulties which put restrictions on our ability to construct all of the models allowed by the theory. Therefore, we focus on a class of the so-called A-type invariants and examine how realistic the extracted models are by constructing their massless spectrum and the gauge group they correspond to.Three generation models do emerge in our analysis but further exploration excludes the possibility of identifying these with the standard model.

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In 1986 Polyakov published his theory of rigid string. I investigate the instantons associated with the consequent new fine structure of strings in four dimensional Euclidean space-time. I reduce the self-dual equation of rigid string instantons to a simple form and show that (p,q) torus knots satisfy the equation, thus forming an interesting new class of solutions. I calculate by computer the world-sheet self-intersection number of the first few such closed knotted strings and derive a very simple formula for the self-intersection number of a torus knot. I consider an interpretation in terms of the first Chem number and discover the empirical formula Q = q - p for the inslanton number, Q, of torus knots and links. In 1987 Biran, Floratos and Savvidy pioneered an approach for constructing self-dual equations for membranes. I present some new solutions for self-dual membranes in three dimensions. In 1989 Grabowski and Tze pointed out a new class of exceptional immersions for which self-dual equations can be constructed and for which there are no known non-trivial solutions. By analogy with (p,q) torus knots, I describe an algorithm for generating a class of potential solutions of self-dual lumps in eight dimensions. I show how these come to within a single sign change of solving all the required constraints and come very close to solving all the 32 self-dual (4;8)-brane equations.

This thesis explores the geometry of string theory backgrounds and the nongeometric features of string theory that arise due to T-duality. For this reason, it is divided into two complementary parts. Part I deals with the superalgebras of symmetries of string theory and M-theory backgrounds, the so-called Killing superalgebras. It is shown that one can define a Lie superalgebra consisting of the infinitesimal field-preserving isometries and the supersymmetries of the background. We also explore the extension of a Killing superalgebra with brane charges. Part II deals with non-geometric backgrounds. In particular, we adopt the framework of the doubled geometry, also known as the doubled torus. We analyze the hamiltonian dynamics of the system and quantize a model T-fold. Finally we extended the doubled torus system to include worldsheet supersymmetry. Throughout part II, we focus on the equivalence, classical and quantum, of the doubled formalism with the conventional formulation.

In an introductory chapter, a summary of the construction of string theories is given, for both the bosonic string and the RNS superstring. Relevant mathematical technology is introduced, including super-Riemann surfaces. Conformal field theory is discussed and BRST quantization of the string is explained. (Super) Schottky groups for the construction of higher-genus Riemann surfaces are introduced. As an example of the use of Schottky groups and super-Riemann surfaces, the one-loop gluon two point function is calculated from string theory. The incorporation of background gauge fields into string theory via nontrivial monodromies (twists) is discussed. The two loop Prym period matrix determinant is computed in the Schottky parametrization. The string theory model with N parallel separated D3-branes is introduced, and the formulae for the the vacuum amplitude are written down. A manifestly symmetric parametrization of two loop Schottky space is introduced. The relationship between worldsheet moduli and Feynman graph Schwinger times is given. The 0 ! 0 limit of the amplitude is written down explicitly. The lagrangian for the corresponding gauge theory is found, making use of a generalization of Gervais-Neveu gauge which accounts for scalar VEVs. Propagators in the given gauge field background are written down. All of the 1PI two-loop Feynman diagrams are written down, including diagrams with vertices with an odd number of scalars. Illustrative example Feynman graphs are computed explicitly in position space. These results are compared with the preceding string theory results and exact agreement is obtained for the 1PI diagrams. An example application is given: the computation of the function of scalar QED at two loops with the same methods, leading to the same result as found in the literature.

There has been great progress in recent years in the understanding of the mathematical structure of scattering amplitudes in Quantum Field Theory as well as the development of powerful methods for their calculation, particularly in the arena of N = 4 Super Yang-Mills where hidden and manifest symmetries lead to striking simplifications. In this thesis, we will discuss the extensions of such methods away from the case of on-shell amplitudes in conformal N = 4. After introducing the necessary mathematical background and physical setting, we consider in Chapter Three the form factors of BPS operators in N = 4 Super Yang- Mills. These objects have several physical applications, and share many properties with scattering amplitudes. However, they are off-shell, which makes them a natural starting point to set out in the direction of correlation functions. After demonstrating the computation of form factors by BCFW recursion and unitarity based methods, we go on to show how the scalar form factor can be supersymmetrised to encompass the full stress-tensor multiplet. In Chapter Four, we discuss the Sudakov form factor in ABJM Theory. This object, which first appears at two loops and controls the IR divergences of the theory, is computed by generalised unitarity. In particular, we note that the maximal transcendentality of three dimensional integrals is related to particular triple cuts. Finally, in Chapter Five we consider massive amplitudes on the Coulomb Branch of N = 4 at one loop. Here we find that vertex cut conditions inherited from the embedding of the theory in String Theory lead to a restricted class of massive integrals.