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In this thesis we shall be concerned with the study of models which arise as a consequence of adopting discrete regularisations for various Euclidean space quantum field theories. Specifically, we employ a random triangulation of the continuum space, and define the fields only over nodes or links of the mesh. Lattice field theories, together with the Renormalisation Group, are introduced in the first chapter. Continuum physics is shown to depend on the positions and stabilities of zeroes of the β-function, which in turn requires a knowledge of the critical behaviour of the associated statistical model. In Chapter 2. we examine a theory of Dirac fermions in 2 + 1 dimensions on a random lattice. We investigate the behaviour of the 2-pt function and fermion condensate in the absence of any background gauge field. The results indicate certain doubling problems, generic to regular lattice formulations of fermion field theories, are evaded, at least at tree graph level. We then go on to examine the fermion vacuum currents in the presence of background fields with non-zero winding number. We are able to demonstrate the existence of a Chern-Simon's topological term in the gauge field effective action which yields parity violating vacuum currents. The magnitude of these are in agreement with certain continuum calculations. The final chapter concerns the properties of random surfaces. The particular class of models chosen originate as discretisations of Polyakov's string. The partition function is approximated by a sum over all possible random triangulations and an integral over vertex positions. The sum over random lattices is intended to mimick the functional integral over intrinsic metrics encountered in the continuum, and the model may also be pictured as 2D quantum gravity coupled to a scalar field. We consider the phase structure of the models when two forms of extrinsic curvature are added to the standard action. Monte-Carlo simulation indicates that with one type of curvature term a strong 2^nd order phase transition exists at finite coupling, leading to a new continuum limit for the model possessing long-range correlation properties. With the other type a much weaker higher order transition is observed. In this case the surface will be crumpled at long distance. We discuss the implications of these results for continuum surfaces.

This thesis is concerned with so called chiral gauge theories, also known as self dual gauge theories. In particular, the aim of this thesis to investigate the role that chiral gauge theories play in duality symmetries in lower dimensions through dimensional reduction. Chapter one serves as an introduction to the notions of duality in field and string theory. The problems of formulating well defined actions for self-dual gauge theories are introduced as well as a brief presentation of the different approaches used to over come these problems. Chapter two introduces dimensional reduction and demonstrates how duality symmetries arise from the dimensional reduction of self-dual theories in a variety of dimensions and on different compact spaces. Examples are presented where the couplings of the resulting theories are calculated explicitly in terms of the geometrical data of the compact space. The duality generators acting on these couplings are also calculated explicitly and related to the geometry/topology of the compact space. Chapter three deals with the idea of duality manifest actions and their relation to the self-dual theories in higher dimensions. Non-linear Born-Infeld type actions are introduced and again dimensional reduction is shown to play a role in the duality of the Born-Infeld action. This leads to a duality manifest version of the Born-Infeld action. Chapter four describes perhaps the main application of this thesis. The effective action of the M-theory five brane wrapped on a torus is identified with the effective action of the IIB D-3 brane dimensionally reduced on a circle (after some appropriate world volume dualizations). The IIB S-duality then arises as a result of the modular symmetry of the torus. The final chapter contains a brief summary and a hint of further directions for research that were outside the scope of this thesis.

Planar maximally supersymmetric Yang-Mills theory (N = 4 SYM) is a special quantum fi eld theory. A few of its remarkable features are conformal symmetry at the quantum level, evidence of integrability and, moreover, it is a prime example of the AdS/CFT duality. Triggered by Witten's twistor string theory [1], the past 15 years have witnessed enormous progress in reformulating this theory to make as many of these special features manifest, from the choice of convenient variables to recursion relations that allowed new mathematical structures to appear, like the Grassmannian [2]. These methods are collectively referred to as on-shell methods. The ultimate hope is that, by understanding N = 4 SYM in depth, one can learn about other, more realistic quantum fi eld theories. The overarching theme of this thesis is the investigation of how on-shell methods can aid the computation of quantities other than scattering amplitudes. In this spirit we study form factors and correlation functions, said to be partially and completely off-shell quantities, respectively. More explicitly, we compute form factors of half-BPS operators up to two loops, and study the dilatation operator in the SO(6) and SU(2j3) sectors using techniques originally designed for amplitudes. A second part of the work is dedicated to the study of scattering amplitudes beyond the planar limit, an area of research which is still in its infancy, and not much is known about which special features of the planar theory survive in the non-planar regime. In this context, we generalise some aspects of the on-shell diagram formulation of Arkani-Hamed et al. [3] to take into account non-planar corrections.

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Thesis (Ph. D.)--York University, 2000. Graduate Programme in Physics. / Typescript. Includes bibliographical references (leaves 100-105). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ59128

We study scalar-fermion models with Yukawa interaction on a space-time lat- tice. Such models can describe the Higgs sector of the Standard Model in the case when the Higgs particle is very heavy (few hundred GeV) and there are very heavy fermions whose masses are due to their Yukawa interactions with the Higgs field. We study a realistic model with four component scalar field as well as simplified models with one and two component scalar fields. We use a mean field approximation to calculate equations for critical lines in the large d (dimension of space-time) limit. These lines are in very good agreement with available Monte Carlo data for the models at d = 4. We calculate fermion correlation functions in the mean field and large d approximations to study properties of different phases in the lattice models. We find two distinct phases with vanishing expectation values of the scalar field. One (at small Yukawa coupling Y) contains massless fermions, while in the other (at large F) the fermions have masses larger than the scale given by the inverse lattice spacing. We find that in the latter phase fermions can form bosonic bound states. These states show up as poles in a four-fermion correlator. We discuss pos- sible continuum limits in the lattice scalar-fermion models. In particular, we show that a theory defined near the critical line separating the disordered phase from the phase with antiferromagnetic order is not unitary.

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Thesis (Ph.D.)--York University, 2006. Graduate Programme in Earth and Space Science. / Typescript. Includes bibliographical references (leaves 86-90). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR19789

Thesis (Ph.D.)--York University, 2007. Graduate Programme in Physics and Astronomy. / Typescript. Includes bibliographical references (leaves 211-217). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR29346