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.

In this thesis the recently developed duality covariant approach to string and Mtheory is investigated. In this formalism the U-duality symmetry of M-theory or Tduality symmetry of Type II string theory becomes manifest upon extending coordinates that describe a background. The effective potential of Double Field Theory is formulated only up to a boundary term and thus does not capture possible topological effects that may come from a boundary. By introducing a generalised normal we derive a manifestly duality covariant boundary term that reproduces the known Gibbons-Hawking action of General Relativity, if the section condition is imposed. It is shown that the full potential can be represented as a sum of the scalar potential of gauged supergravity and a topological term that is a full derivative. The latter is written totally in terms of the geometric flux and the non-geometric Q-flux integrated over the doubled torus. Next we show that the Scherk-Schwarz reduction of M-theory extended geometry successfully reproduces known structures of maximal gauged supergravities. Local symmetries of the extended space defined by a generalised Lie derivatives reduce to gauge transformations and lead to the embedding tensor written in terms of twist matrices. The scalar potential of maximal gauged supergravity that follows from the effective potential is shown to be duality invariant with no need of section condition. Instead, this condition, that assures the closure of the algebra of generalised diffeomorphisms, takes the form of the quadratic constraints on the embedding tensor.

In the past few years, on-shell analytic methods have played a pivotal role in gauge theory calculations. Since the initial success of these methods in Standard Model physics, considerable activity has led to development and application in supersymmetric gauge theories. In particular, the maximally supersymmetric super Yang-Mills theory received much attention after the discovery of holographic dualities. Here, the spinor helicity formalism and on-shell superspace is described initially for four dimensions. The framework of general unitarity is shown to be a useful tool for calculating loop corrections of scattering amplitudes. Once the foundation is laid, application in three and six dimensions is explored. In six dimensions the case of interest is a theory with (1,1) supersymmetry which captures the dynamics of five-branes in string theory. In this setup the one-loop superamplitude with four and five external particles is calculated and checked for consistency. In three dimensions, the supersymmetric gauge theory that is supposed to describe the dynamics of M2-branes is considered. This particular theory is also related to M-theory via the holographic duality. The goal was to explore and determine the infra-red divergences of the theory. This was achieved by calculating the Sudakov form factor at two loops.

One Dollar Mandala is a string quartet in five movements. Each of the movements can function separately as a short piece and together they create a cyclical form that takes a little over half an hour in performance. The first movement, in its playful persistence pays homage to Gyorgi Kurtag's string quartet op.1. The second movement - under a working title "chorale" - alludes to rich harmonies of Bach and Scriabin that I can't stop admiring. The third movement, in rehearsals referred to as "mazurek" is a return to my roots of Polish folk dances and shows my admiration for Szymanowski's brilliant Mazurki op.50 for the piano. The fourth movement is a fragilite, a poem harking back to the sensual aspect of Scriabin's music. The final movement is a moto perpetuo exploring my fascination for the South Indian classical drumming tradition.

Double field theory and exceptional field theory are formulations of supergravity that make certain dualities manifest symmetries of the action. To achieve this, the geometry is extended by including dual coordinates corresponding to winding modes of the fundamental objects. This geometrically unifies the spacetime metric and the gauge fields (and their local symmetries) in a generalized geometry. Solutions to these extended field theories take the simple form of waves and monopoles in the extended space. From a supergravity point of view they appear as 1/2 BPS objects such as the string, the membrane and the fivebrane in ordinary spacetime. In this thesis double field theory and exceptional field theory are introduced, solutions to their equations of motion are constructed and their properties are analyzed. Further it is established how isometries in the extended space give rise to duality relations between the supergravity solutions. Extensions to these core ideas include studying Goldstone modes, probing singularities at the core of solutions and localizing them in winding space. The relation of exceptional field theory to F-theory is also covered providing an action for the latter and incorporating the duality between M-theory and F-theory.

We study the target space duality of classical two dimensional sigma models. The models with dilaton and tachyon field are analyzed. As a motivating example the historical electric-magnetic duality is presented. We review the construction of the duality transformation and the integrability conditions for the nonlinear sigma models with target spaces described by general metrics and antisymmetric two-forms. We generalize the formalism for the models whose actions contain the dilaton and tachyon field. For the dilaton field case it is required that the duality is a property solely of the target manifolds, independent of the world-sheet geometry. For both cases the duality transformation is established and the integrability conditions are calculated. The set of restrictions on geometrical data describing the models is obtained, the previously calculated condition on connections on target spaces is maintained in both cases.

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Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (D. Mus. Arts)--University of Washington, 2002. / Vita. Includes bibliographical references (leaves 122-126).