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.