Roughly, 2-Segal sets are simplicial sets such that higher-dimensional simplices can be uniquely described by triangulated polygons formed out of 2-simplices. In a sense that I will make precise, 2-Segal sets can be viewed as categorified associative algebras. As a TQFT Club member, you might ask, “Are there 2-Segal sets that correspond to (commutative) Frobenius algebras?” The answer is yes, commutativity and Frobenius structures come from asking the simplicial set to possess additional compatible structure maps. I’ll give an overview of these correspondences as well as some background as to how I arrived at this topic from the world of Poisson geometry. This is based on joint works with Ivan Contreras, Walker Stern, and Sophia Marx.

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.