Topological quantum field theory and invariants of graphs for quantum groups

TL;DR

This paper develops a topological quantum field theory framework for 3-manifolds using generalized 6j-symbols, connecting quantum group invariants with graph-based models and providing computational methods for state space dimensions.

Contribution

It introduces a new formulation of TQFTs incorporating colored graphs and demonstrates their relation to existing ribbon graph models for quantum groups at roots of unity.

Findings

Provides calculational methods for state space dimensions

Establishes a relation between new models and Reshetikhin-Turaev constructions

Examples based on deformations of classical groups at roots of unity

Abstract

On basis of generalized 6j-symbols we give a formulation of topological quantum field theories for 3-manifolds including observables in the form of coloured graphs. It is shown that the 6j-symbols associated with deformations of the classical groups at simple even roots of unity provide examples of this construction. Calculational methods are developed which, in particular, yield the dimensions of the state spaces as well as a proof of the relation, previously announced for the case of $SU_q(2)$ by V.Turaev, between these models and corresponding ones based on the ribbon graph construction of Reshetikhin and Turaev.