Algebraic aspects of conformal field theory

Abstract

This thesis presents an algebraic study of two areas of conformal field theory. The first part extends the theory of Galilean conformal algebras. These algebras are extended conformal symmetry algebras for two-dimensional quantum field theories, formed through a process known as Galilean contraction. Analogous to an Inonu-Wigner contraction, the Galilean contraction procedure takes two conformal symmetry algebras, equivalent up to central charge, as input and produces a new conformal symmetry algebra. We extend the theory of Galilean contractions in several directions. First, we develop the theory to allow input of any number of symmetry algebras. These generalised algebras have a truncated graded structure. We develop a theory for multi-graded Galilean algebras, whereby we can extend structures which are graded by sequences. Finally, we present a comprehensive analysis of the possible algebras which arise when the input algebras are no longer required to be equivalent. We refer to this as the asymmetric Galilean contraction. For each stage of generalisation, we present several pertinent examples, discuss the Sugawara construction of a Galilean Virasoro algebra given an affine Lie algebra, and apply our results to the W-algebra W3.The second part of this thesis presents an exploratory study of reducible but indecomposable modules of the N=2 superconformal algebras, known as staggered modules. These modules are characterised by a non-diagonalisable action of L0, the Virasoro zero-mode. Using recent results on the coset construction of N=2 minimal models, we are able to construct the first examples of staggered modules for the N=2 algebras. We determine the structure of a family of such modules for all admissible values of the central charge. Furthermore, we investigate the action of symmetries such as spectral flow on these modules. We present the results along with a range of examples, and discuss possible paths towards a classification of such modules for the Neveu-Schwarz and Ramond N=2 superconformal algebras.