Non-compact 3D TQFT and non-semisimplicity

TL;DR

This paper constructs a non-compact 3D TQFT from non-semisimple modular tensor categories, linking it to Lyubashenko's invariants and modified traces, and develops methods for decomposing 3-manifolds within this framework.

Contribution

It introduces a new non-compact 3D TQFT framework based on non-semisimple categories, connecting it to existing invariants and developing manifold decomposition techniques.

Findings

The TQFT recovers Lyubashenko's 3-manifold invariants.

Decomposition method for closed 3-manifolds using 2-morphism generators.

Solid torus value relates to modified trace data.

Abstract

We define a once extended non-compact 3-dimensional TQFT $\mathcal{Z}$ from the data of a (potentially) non-semisimple modular tensor category. This is in the framework of generators and relations of [Bartlett et al., arxiv:1509.06811 (2015)], having disallowed generating 2-morphisms whose source is the empty. Moreover, we show that the projective mapping class group representations this TQFT gives rise to, are dual to those of [Lyubashenko, arXiv:hep-th/9405167 (1994)] and [De Renzi et al., arXiv:2010.14852 (2020)]. We develop a method to decompose a closed 3-manifold in terms of 2-morphism generators. We use this to compute the value of $\mathcal{Z}$ on 3-manifolds, explaining why it should recover Lyubashenko's 3-manifold invariants [Lyubashenko, arXiv:hep-th/9405167 (1994)]. Finally, we explain that the value of the non-compact TQFT on the solid torus recovers the data of a modified trace [Geer et al., arXiv:0711.4229 (2007)].