Turaev-Viro theory as an extended TQFT

Abstract

In recent years, the application of quantum groups to the study of low-dimensional topology has become an active topic of research. In three-dimensions, these yield the well-known Reshetihkin-Turaev (RT) invariants, which are a mathematical formulation of Chern- Simons theory and Turaev-Viro (TV) theory, which is a convergent form of the Ponzano-Regge state sum formula from Quantum Gravity. Both RT and TV are more than invariants;they have a far richer structure known as a Topological Quantum Field Theory (TQFT). We describe Turaev-Viro theory as an extended (3-2-1)-TQFT and use this description to prove a theorem relating it to RT theory. Turaev-Viro theory has several equivalent descriptions. We examine Kitaev's toric code from quantum computation and the Levin- Wen model from condensed matter physics and show that these theories are extended TQFTs coinciding with TV theory.