# CHERN–SIMONS THEORY AND KNOT INVARIANTS

# CHERN–SIMONS THEORY AND KNOT INVARIANTS

This chapter presents various aspects of Chern-Simons theory. In a groundbreaking paper, Witten (1989) showed that Chern-Simons gauge theory, which is a quantum field theory in three dimensions, provides a physical description of a wide class of invariants of three-manifolds and of knots and links in three-manifolds. The partition function and correlation functions of Wilson loops in Chern-Simons theory can be computed in a variety of ways. In order to define the partition function of Chern-Simons theory at the quantum level, one has to specify a framing of the three-manifold. It turns out that the evaluation of correlation functions also involves a choice of framing of the knots. A good starting point for understanding framing is to take Chern-Simons theory with gauge group *U*. The relation between Chern-Simons theory and string theory involves the vacuum expectation values of Wilson loop operators for arbitrary irreducible representations of *U(N)*. This means that *N* has to be bigger than the number of boxes of any representation under consideration.

*Keywords:*
Chern-Simons theory, knot invariants, group factors, partition function, Wilson loops, matrix models, canonical quantization, three-manifold

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