Topological Quantum Field Theories -- A Meeting Ground for Physicists and Mathematicians

TL;DR

This paper explores how topological quantum field theories, especially Chern-Simons theories, serve as a bridge between physics and mathematics, enabling exact solutions and new invariants for knots, links, and three-manifolds, with applications in quantum gravity.

Contribution

It provides a comprehensive overview of Chern-Simons theories as tools for understanding low-dimensional topology and their exact solutions, linking quantum field theory with knot and three-manifold invariants.

Findings

Exact solutions for Chern-Simons theories enable computation of knot invariants.

Wilson loop expectation values produce polynomial invariants like the Jones polynomial.

Perturbative analysis relates to Vassiliev invariants and applications in quantum gravity.

Abstract

Topological quantum field theories can be used as a powerful tool to probe geometry and topology in low dimensions. Chern-Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of quantum field theories which can be exactly (non-perturbatively) and explicitly solved. Abelian Chern-Simons theory provides a field theoretic interpretation of the linking and self-linking numbers of a link. In non-Abelian theories, vacuum expectation values of Wilson link operators yield a class of polynomial link invariants; the simplest of them is the famous Jones polynomial. Other invariants obtained are more powerful than that of Jones. Powerful methods for completely analytical and non-perturbative computation of these knot and link invariants have been developed. In the process answers to some of the open problems in knot theory are obtained. From these invariants for unoriented and framed links in $S^3$, an invariant for any three-manifold can be easily constructed by exploiting the Lickorish-Wallace surgery presentation of three-manifolds. This invariant up to a normalization is the partition function of the Chern-Simons field theory. Even perturbative analysis of the Chern-Simons theories are rich in their mathematical structure; these provide a field theoretic interpretation of Vassiliev knot invariants. Not only in mathematics, Chern-Simons theories find important applications in three and four dimensional quantum gravity also.