Since the mid 1980s, there have been hints of a deep connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) in which cocycles are 2-dimensional supersymmetric field theories. Basic properties of these field theories lead to expected integrality and modularity properties, but the abundant torsion in TMF has always been mysterious. In this talk, I will describe deformation invariants of 2-dimensional field theories that realize some of the torsion in TMF.

I will discuss a construction of Rozansky-Witten models with target T^*C^n as extended TQFTs. The starting point is a 3-category, whose objects are affine RW models, and whose morphisms are defects of any codimension. By truncation, we obtain a (non-semisimple) 2-category CC of bulk theories, surface defects, and isomorphism classes of line defects. Through a systematic application of the cobordism hypothesis we construct a unique extended oriented 2-dimensional TQFT valued in CC for every affine Rozansky–Witten model.

We develop a general theory of 3-dimensional "orbifold completion", to describe (generalised) orbifolds of topological quantum field theories as well as all their defects. Given a semistrict 3-category T with adjoints for all 1- and 2-morphisms, we construct the 3-category T_orb as a Morita category of certain E_1-algebras in T which encode triangulation invariance. We prove that in T_orb again all 1- and 2-morphisms have adjoints and that it contains T as a full subcategory. Applications include defect TQFTs for state sum models and Reshethikin-Turaev theory. This is joint work with Lukas Müller.

Liouville CFT has a number of interesting features, such as the non-existence of a vacuum vector, and the non-surjectiveness of the state-field correspondence. I will explain which properties of Liouville CFT one might expect to find in general unitary non-compact functorial field theories.

A theorem of Deligne suggests that the complex numbers are not algebraically closed in a 1-categorical sense but that their 1-categorical algebraic closure is the category sVec of complex super vector spaces. In fact, this property uniquely (up to non-unique isomorphism) characterizes sVec amongst complex-linear symmetric monoidal categories.  In these talks, we will outline work in progress on constructing complex-linear symmetric n-categories which are higher categorical analogues of sVec in that they are uniquely characterized by being the n-categorical separable closure of the complex numbers. We will explore the resulting higher-categorical absolute Galois group of the complex numbers, and outline a construction of that group very much akin to the surgery-theoretic description of the stable piecewise linear group PL.

We define a Turaev-Viro-Barrett-Westbury state sum model of triangulated 3-manifolds with surface defects (oriented 2d surfaces), line defects and point defects (graphs on the defect surfaces). Surface defects are labeled with bimodule categories over spherical fusion categories with bimodule traces, line and point defects with bimodule functors and bimodule natural transformations.
The state sum uses generalised 6j-symbols that encode the coherence isomorphisms of the defect data. We prove the triangulation independence of the state sum and show that it can be computed with polygon diagrams that satisfy the cutting and gluing identities for polygon presentations of oriented surfaces. State sums detect the genus of a defect surface and are sensitive to its embedding. Defect lines on defect surfaces with trivial defect data define ribbon invariants for the centre of the underlying spherical fusion category.
Reference: C. Meusburger, State sum models with defects based on spherical fusion categories, Adv. Math. 429 (2023), DOI: 10.1016/j.aim.2023.109177

I will argue that the Morita 4-category of braided fusion 1-categories is equivalent to the Morita 4-category of fusion 2-categories. Thanks to the work of Brochier, Jordan, and Snyder, this implies that every fusion 2-category is fully dualizable, and therefore yields a fully extended framed 4D TFT. Further, this equivalence of 4-categories can be leveraged to study the equivalence classes of these TFTs.

Associated to a certain subquotient of the category of representations of the small quantum group at a root of unity is an invertible 4d TQFT known as Crane–Yetter. In fact, the non-semisimplified representation category is invertible in the Morita theory of braided tensor categories: under the cobordism hypothesis this defines a non-semisimple invertible TQFT. Such an invertible theory assigns to a closed 3-manifold a 1-dimensional vector space. In this talk, we define a relative TQFT which can be seen as varying the non-semisimple Crane-Yetter theory over the character stack: it assigns to a closed 3-manifold a line bundle on its G-character stack. We construct this theory by analysing invertibility of a 1-morphism in the Morita theory of symmetric tensor categories, coming from representations of Lusztig's quantum group at a root of unity regarded as a bimodule for Rep(G) using the quantum Frobenius map.

Chern-Simons theory for the double of a Lie bialgebra has two natural boundary conditions, given by the Lie algebra and its dual. Using these boundary conditions for punctured disks and bordisms between them as a motivation, we are led to a definition of a nerve of Hopf algebra. We prove an equivalence between Hopf algebras and such nerves and give an application to quantization, using Drinfeld associators. Based on joint work with Pavol Ševera arXiv:1906.10616.