No description available.

No description available.

No description available.

No description available.

No description available.

By formulating the six dimensional (0,2) superconformal field theory X[j] on a Riemann surface decorated with certain codimension two defects, a multitude of four dimensional N=2 supersymmetric field theories can be constructed. In this dissertation, various aspects of this construction are investigated in detail for j=A,D,E. This includes, in particular, an exposition of the various partial descriptions of the codimension two defects that become available under dimensional reductions and the relationships between them. Also investigated is a particular observable of this class of four dimensional theories, namely the partition function on the four sphere and its relationship to correlation functions in a class of two dimensional non-rational conformal field theories called Toda theories. It is pointed out that the scale factor that captures the Euler anomaly of the four dimensional theory has an interpretation in the two dimensional language, thereby adding one of the basic observables of the 4d theory to the 4d/2d dictionary. / text

No description available.

No description available.

This thesis investigates the algebraic structure of certain quantum field theories in one space and one time dimension. These theories are integrable - essentially, highly constrained and therefore soluble. Thus, instead of having to use perturbative techniques, it is possible to conjecture their exact 5-matrices, which have the property that they are factorized into two-particle 5-matrices. In particular, there are two types of such theory: in one, scattering is purely elastic, whilst in the other, there is additional structure dictated by the Yang-Baxter equation. This thesis explores the algebraic structure of the latter and its links with the former. We begin, in chapter one, with an informal summary of the development of the subject, followed by a more mathematical exposition in chapter two. Chapter three constructs explicitly some exact factorized 5-matrices with Yang-Baxter structure, and comments on their features, both intrinsic and in relation to purely elastic 5-matrices. In particular, there is an unexplained close correspondence between the mass spectra and particle fusings in the two types of theory. The next three chapters attempt to shed some light on these features. Chapter four constructs similar 5-matrices, but based on quantum-deformed algebras rather than classical algebras. In chapter five we describe the structure of the 5-matrices when the particles they describe transform in irreducible representations of classical algebras. This leads us to consider the Yangian algebra, the representation theory of which underlies Yang-Baxter dependent 5-matrices, and which we therefore review briefly. We begin chapter six by reviewing the work which shows that the Yangian is also the charge algebra of the integrable quantum field theory, and subsequently show that the Yangian is also to a great extent present in the corresponding classical theory. We conclude with a brief seventh chapter describing the outlook for further research, followed by appendices containing respectively details of the Lagrangians of some integrable quantum field theories, a continuum formulation of the quantum inverse problem, explicit expressions for some of the R-matrices computed in the text, and a summary of known solutions of the Yang-Baxter equation.

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.