Modular Categories and TQFTs Beyond Semisimplicity

TL;DR

This paper reviews recent advances in extending 3D Topological Quantum Field Theories using generalized, non-semisimple modular categories, broadening the scope of link invariants in quantum topology.

Contribution

It summarizes recent developments that relax semisimplicity in modular categories, enabling new constructions of TQFTs beyond traditional frameworks.

Findings

Extended TQFTs to non-semisimple categories

Enhanced link invariants in quantum topology

Showcased the richness of Vladimir Turaev's framework

Abstract

Vladimir Turaev discovered in the early years of quantum topology that the notion of modular category was an appropriate structure for building 3-dimensional Topological Quantum Field Theories (TQFTs for short) containing invariants of links in 3-manifolds such as Witten-Reshetikhin-Turaev ones. In recent years, generalized notions of modular categories, which relax the semisimplicity requirement, have been successfully used to extend Turaev's construction to various non-semisimple settings. We report on these recent developments in the domain, showing the richness of Vladimir's lineage.