# Higher Structures & Field Theory Seminar

### in brief

biweekly, **Thursdays 3-5 pm** central European time virtually

### What?

Originally, this seminar on quantum algebra, quantum topology and mathematical physics was supposed to strengthen regional connections by meeting once or twice per semester. However, due to well-known reasons, it will now take place virtually and more regularly instead. It will run biweekly, on Thursdays at 3-5 pm central European time.- We will start at 3pm with 10 minutes of “virtual tea” for informal converstions, so bring your tea and coffee!
- Then we will have a 50 minute talk, with a 5–10-minute break for discussions after the first half
- Finally, since we are all desperate for some more personal interactions, we will have some time for informal discussions and hopefully many questions after the talk, up to at the latest 5 pm.

### Where?

Of course we will meet virtually! We plan to use Big Blue Button, but reserve to switch to zoom in case of mishaps. More information on connecting can be found here: Technical information for speakersPlease sign up for our

**mailing list**as we will send out the connection information there. (Note that we have to accept your request to avoid spam, so if your email address is not an institutional address or self-explanatory about who you are, please leave your name and institution, or send us a clarifying email.)

### Who?

The speakers in the summer semester 2021 are:April 22 | Pavel Safronov (Edinburgh) | ## Rozansky--Witten TQFTIn this talk I will review a 3-dimensional TQFT defined by Rozansky and Witten. This theory is a 3-dimensional analog of the 2d B-model and it has recently seen a resurgence of interest due to its appearance in 3- dimensional mirror symmetry. I will recall what this is. I will also explain some old and new results on mathematical formalizations of TQFT invariants in dimensions ≤ 2. |

May 6 | Juliet Cooke (MPIM Bonn) | ## Skein Algebras, Askey-Wilson Algebras and the Five-Punctured SphereIn this talk I will discuss my work in progress with Abel Lacabanne on the relation between Kauffman bracket skein algebras of punctured spheres and Askey-Wilson algebras via Alekseev moduli algebras. Particular attention will be given to considering the five-punctured sphere which corresponds to the lowest higher-rank Askey-Wilson algebra. |

May 20 | Theo Johnson-Freyd (Dalhousie & PI) | ## Higher S-matricesEach fusion higher category has a "framed S-matrix" which encodes the commutator of operators of complementary dimension. I will explain how to construct and interpret this pairing, and I will emphasize that it may fail to exist if you drop semisimplicity requirements. I will then outline a proof that the framed S-matrix detects (non)degeneracy of the fusion higher category. This is joint work in progress with David Reutter. |

June 10 | Lóránt Szegedy (Vienna) | ## Fully extended 2d r-spin topological field theoriesI will discuss fully extended 2d topological field theories (TQFTs) with tangential structure in the 2-categorical setting. The tangential structures we consider are framing, orientation and r-spin structure, the latter is a generalisation of a spin structure with structure group being the r-fold cover of SO(2). I will list a number of natural examples of possible target 2-categories appearing in representation theory and algebraic geometry and identify algebraic structures that a TQFT provides. Finally I will sketch our proof of the cobordism hypothesis for r-spin TQFTs. This is joint work in progress with Nils Carqueville. |

June 24 | David Reutter (MPIM Bonn) | ## Minimal modular extensionsA braided fusion category is "slightly degenerate" if its Müger center is equivalent to the category of super vector spaces. In this talk, I will sketch a proof of the longstanding conjecture that any such braided fusion category admits a "minimal modular extension", i.e. an index-2 extension to a braided fusion category with trivial Müger center. Key players in this proof will be fusion 2-categories, their 2- categorical Drinfel’d centers, and various associated topological field theories. This is based on arXiv:2105.15167 and is joint work with Theo Johnson-Freyd. |

July 8 | Lukas Müller (MPIM Bonn) | ## Rigidity results for topological field theories and modular functorsTopological quantum field theories, as defined by Atiyah, are symmetric monoidal functors from a bordism category to vector spaces. The bordism category used by Atiyah is the homotopy category of a higher category of bordisms with diffeomorphisms and their isotopies as higher morphisms. Functors from the two dimensional higher bordism category to an appropriated 2-category of linear categories are one axiomatisation of modular functors appearing in conformal field theory. My talk will be concerned with the structure present on the category that a modular functor assigns to the circle. More precisely, we will show that it admits a balanced braided Grothendieck-Verdier structure, a generalisation of the concept of a ribbon category, introduced by Boyarchenko and Drinfeld. This turns out to be a consequence of a classification of cyclic algebras over the framed little disk operad in terms of balanced braided Grothendieck-Verdier categories. The talk is based on joint work with Lukas Woike. |

### Previously...

Nov. 25 | Konrad Waldorf (Greifswald) | ## Connes fusion of spinors on loop spaceI will talk about some progress with the problem to exhibit the 2d supersymmetric sigma model as a smooth and fully extended functorial field theory (FFT), which is part of the Stolz-Teichner programme. The spinor bundle on the loop space of a string manifold is the value of that FFT on circles. I describe a Connes fusion product on this spinor bundle, which produces the assignment of the FFT on a pair of pants, and at the same time gives an ansatz how to extend the FFT down to the point. This work combines operator algebras, infinite-dimensional representation theory and higher-categorical geometry, and is joint with Peter Kristel. |

Dec. 9 | Adrien Brochier (Paris 6) | ## Skein categories and higher genus associatorsThe theory of Drinfeld associators leads to universal representations of the categories of braids and tangles into some categories of Feynman diagrams. This provides powerful topological invariants, and is also deeply related to deformation-quantization. On the other hand, the formalism of skein categories, an ancestor of factorization homology, produces out of a ribbon category a certain TFT like construction of representations of braid groups and tangles in any oriented surface. Hence, plugging the category of diagrams into this machine one might hope to obtain higher genus analogs of Drinfeld associators. I'll explain why this doesn't quite works and how to make it work. We recover this way a combinatorial formula due to Calaque-Enriquez- Etingof for elliptic analogs of associators. Time permitting, I'll explain how this relates to quantizations of character varieties and to the Riemann-Hilbert correspondence. |

Jan. 13 | Danica Kosanovic (Paris 13) | ## Knot invariants from homotopy theoryEmbedding calculus of Goodwillie and Weiss is a certain homotopy theoretic technique for studying spaces of embeddings. When applied to the space of knots this method gives a sequence of knot invariants which are conjectured to be universal Vassiliev invariants. This is remarkable since such invariants have been constructed only rationally so far and many questions about possible torsion remain open. In this talk I will present some explicit computations and outline why these knot invariants are surjections. This confirms one half of the universality conjecture, and confirms it rationally, and p-adically in a range. We also prove some missing cases of the Goodwillie-Klein connectivity estimates. |

Jan. 27 | Matthias Ludewig (Regensburg) | ## Construction of the supersymmetric path integralThe task of rigorously constructing the path integral for the N=1/2 supersymmetric sigma-model has sparked a lot of research activity in the last 30 years, after it was used by Atiyah to give a short, but formal, proof of the Atiyah-Singer index theorem. In geometric terms, this path integral is just an integration functional for differential forms on the loop space of a spin manifold X, which is, however, ill-defined due to the infinite-dimensionality of the loop space. In this talk, we present a construction of this path integral using cyclic cohomology of the dg algebra of differential forms on X, which is connected to loop space forms via Chen’s iterated integral map. We then explain the connection to path integral formulae using Pfaffians and the Wiener measure. This is joint work with B. Güneysu and F. Hanisch. |

Feb. 10 | Tashi Walde (TU München) | ## Higher Segal spaces via higher excisionHigher Segal spaces form an interesting hierarchy of higher structures which generalize the classical Segal spaces used to encode homotopy coherent associative structures. In this talk I explain some basic aspects of their theory and show how one can understand higher Segal spaces conceptually in analogy to functor/manifold calculus. |