Perturbative Aspects of the Chern-Simons Topological Quantum Field Theory

We investigate the Feynman-diagram perturbative expansion of the Chem-Simons topological quantum field theory. After introducing the theory, we compute the one-loop expectation value for knots and links, recovering Gauss1 linking number formula for links and the self-linking number of a framed knot. The self-linking formula is shown to suffer from an anomaly proportional to the total torsion of the knot, whose definition requires ‘framing1 the knot. This explains the appearance of framings. In an appendix, we use these results to characterize the total torsion of a curve as the only parametrization independent quantity of vanishing scaling dimension having 'local1 variation, explaining why no further anomalies are expected. We then treat rigorously the two loop expectation value of a knot, finding it to be finite and invariant under isotopy. We identify the resulting knot invariant to essentially be the second coefficient of the Conway polynomial, in agreement with Witten's earlier non-perturbative computation. We give 'formal1 (namely, algebraic with missing analytical details) proofs that the perturbative expansion gives manifold and link invariants and suggest that a slight generalization of the Feynman rules of the Chern-Simons theory might still give knot invariants, possibly new. We discuss the relation between perturbation theory and the Vassiliev knot in­ variants, solving a related algebraic problem posed by Birman and Lin. We compute the stationary phase approximation to the Chem-Simons path in­ tegral with compact and non-compact gauge group, explaining the appearance of framings of 3-manifolds and the so called ‘shift in k \ and finding the result in the non-compact case not to be a simple analytic continuation of the result in the compact case. Finally we outline our expectation for the behavior of the theory beyond the oneand two-loop rigorous results.