I will describe some requirements on a non semi-simple ribbon category that ensure its admissible skein modules form the state spaces of a 3+1-dimensional TQFT.

The study of topological defects in quantum field theory has seen a wealth of activity recently leading to many interesting insights, for example the explicit realisation of non-invertible topological defects in higher dimensional QFTs via the gauging of higher form symmetries, or the description of the higher algebraic structures inherent in these topological defects. In this talk, I would like to focus on low-dimensional examples, where such defects and their properties have been investigated for some time already. I would like to exhibit some of the properties of topological defects in two-dimensional conformal field theory and in three-dimensional topological field theory, and show some of the structural insights into 2d CFT and 3d TFT one can gain with the help of defects. In this way, the well-understood low-dimensional case might serve as a source of ideas and as a test case for higher dimensional constructions.

Defining internal symmetry in a quantum theory through the lens of topological defects opens the door to generalised notions of symmetry, including some arising from non-invertible transformations, and enables a calculus that leverages well-established methods from topological quantum field theory. In d spatial dimensions, the framework of fusion d-category theory is believed to offer an axiomatisation for finite non-invertible symmetries. Though seemingly exotic, such non-invertible symmetries can be shown to naturally arise as dual symmetries upon gauging invertible symmetries. In this talk, I will present a framework to systematically investigate these aspects in quantum lattice models.

We explain the construction of the stringor bundle on a string manifold recently given in joint work with Peter Kristel and Konrad Waldorf. We start by discussing the spinor bundle on the loop space of a string manifold, together with its fusion product. Then we explain how the stringor bundle on the manifold itself can be obtained using a regression procedure.

Perhaps the most (mathematically) well understood 3d TQFT is that of Reshetikhin–Turaev. Famously, the input data for their construction is that of a semisimple modular tensor category (MTC). Attempts at generalizing this construction to the non-semisimple case date back to the 90's with work of Hennings, Lyubashenko and Kerler–Lyubashenko. However, only partial results were achieved. This was until De Renzi et al. defined a 3d TQFT from such non-semisimple modular categories. Importantly, they had to impose an admissibility condition on the cobordism categories they use. My work has been in the direction of defining a once-extended 3d TQFT from this data. However, Bartlett et al. proved that such TQFTs are classified by semisimple modular categories. We will investigate the most natural method of circumventing this. This will lead to the notion of noncompact TQFT. I will then proceed to talk about my work on constructing such a TQFT from the data of a (potentially) non-semisimple MTC, with an emphasis on the key ingredients of this construction. Time permitting, I will also discuss how to extract 3-manifold invariants and a modified trace from such a noncompact TQFT.