Quantum Invariants of 3-Manifolds Arising from Non-Semisimple Categories

TL;DR

This survey discusses new quantum invariants of 3-manifolds derived from non-semisimple categories, extending classical invariants and providing finer distinctions among manifolds.

Contribution

It introduces non-semisimple Reshetikhin-Turaev-type invariants and extends them to TQFTs, revealing richer topological information than traditional invariants.

Findings

Non-semisimple invariants extend classical quantum invariants.

These invariants can distinguish lens spaces where previous invariants could not.

A machinery for constructing invariants from general ribbon categories is provided.

Abstract

This survey covers some of the results contained in the papers by Costantino, Geer and Patureau (https://arxiv.org/abs/1202.3553) and by Blanchet, Costantino, Geer and Patureau (https://arxiv.org/abs/1404.7289). In the first one the authors construct two families of Reshetikhin-Turaev-type invariants of 3-manifolds, $\mathrm{N}_r$ and $\mathrm{N}^0_r$, using non-semisimple categories of representations of a quantum version of $\mathfrak{sl}_2$ at a $2r$-th root of unity with $r \geqslant 2$. The secondary invariants $\mathrm{N}^0_r$ conjecturally extend the original Reshetikhin-Turaev quantum $\mathfrak{sl}_2$ invariants. The authors also provide a machinery to produce invariants out of more general ribbon categories which can lack the semisimplicity condition. In the second paper a renormalized version of $\mathrm{N}_r$ for $r \not\equiv 0 \; (\mathrm{mod} \; 4)$ is extended to a TQFT, and connections with classical invariants such as the Alexander polynomial and the Reidemeister torsion are found. In particular, it is shown that the use of richer categories pays off, as these non-semisimple invariants are strictly finer than the original semisimple ones: indeed they can be used to recover the classification of lens spaces, which Reshetikhin-Turaev invariants could not always distinguish.