Extended Structures in Topological Quantum Field Theory

TL;DR

This paper explains how 3D topological quantum field theories relate to algebraic structures like quantum groups, including methods to incorporate central extensions and compute invariants of framed tangles, with examples and computational details.

Contribution

It provides an expository account connecting 3D topological field theories to quantum groups and details how to include central extensions and framed tangle invariants.

Findings

Establishes a correspondence between 3D TQFTs and quantum groups.

Demonstrates computation of invariants for specific finite examples.

Shows how to incorporate framing and central extensions into the theory.

Abstract

In this note, based on a conference talk, we show how a 3 dimensional topological field theory leads to an algebraic gadget roughly equivalent to a quantum group. This is an expository version of some material in hep-th/9212115 (where we also carry out computations for a specific finite example). We also explain how to incorporate the central extensions usually explained via ``framings'', and we show how to recover invariants of framed tangles. This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). 2 encapsulated postscript files were submitted separately in uuencoded tar-compressed format.