QM Research Seminar: Introduction to the Witten-Reshetikhin-Turaev TQFT

Speaker: William Elbæk Mistegård (University of Southern Denmark)

Abstract:

It was envisioned by Witten that quantum Chern-Simons field theory is a 2+1 dimensional topological quantum field theory (TQFT) in the sense of Atiyah. Further, Witten argued that the link invariant known as the Jones polynomial, can be computed in this TQFT as the expectation value of Wilson loop operators. This gave an intrinsic definition of the Jones polynomial, as opposed to the known combinatorial definition, and thus solved an important problem in topology. Subsequently, a full TQFT was constructed mathematically rigorously by Reshetikhin and Turaev, using surgery presentations of three-manifolds and modular tensor categories. This is now known as the Witten-Reshetikhin-Turaev TQFT. To a closed oriented three-manifold M, the WRT-TQFT associates a topological invariant in the form of a complex number. This is called the quantum invariant of M. These invariants are essentially the building blocks of the WRT-TQFT. The goals of this introductory talk are to introduce the quantum invariants and to present the axioms of TQFT (or more precisely, the axioms of a modular functor).