Chern-Simons theory as a theory of knots and links

Abstract

Topological quantum field theories provide a powerful tool in physics, for the study of geometry and topology of low dimensions. Cohomological or Witten-type field theories in four dimensions gives Donaldson invariants associated with four manifolds. Three-dimensional Chern-Simons theory is a schwarz-type topological field theory which provides a natural framework for the study of knots and links. In this thesis the researcher has obtained generalised invariants and studied some exciting problems to determine their powerfulness in comparison with already well-known polynomial invariants viz., Jones, Homfly and Kauffman polynomials. The Salient features of knot theory, braid groups and Chern-Simon's theory are reviewed. The essential details of obtaining Jones polynomial recursion relation from SU(2) (SU(N)) Chern-Simons theory are presented. The non-perturbative results have been obtained by exploiting the connection between its Hilbert Space of states with the space of correlator conformal blocks of the corresponding Wess-Zumino conformal field theory. With this method the generalised recursion relation from Chern-Simons theory based on compact semi-simple group is derived. A direct method of computing new generalised knot and link invariantshas been developed. Some building blocks for directly determining invariants of complicated knots / links has been presented. The generalised invariants for a special class of SU(N) representations has been tabulated for some knots upto eight crossings. The minimal models superconformal models and WN models which admit coset representation of the Wess-Zumino model are studied as examples. It is deduced that the invariants from these rational conformal theories are just the products of the corresponding Wess-Zumino invariants contained in the coset. These invariants hence have an equivalent Chern-Simons description. Some important issues pertinent to the question, ' how powerful are these generalised Chern-Simons field theoretic invariants in classifying knots and links?', are addressed in this thesis.