# Higher Structures & Field Theory Seminar

### in brief

monthly, **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 once a month, 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 winter semester 2022/23 are:### Previously...

#### winter semester 2021

October 28 | Lukas Woike | ## Modular functors - multiplicative structures and generalizations beyond rigidityThe notion of a modular functor lies at the intersection of representation theory and low-dimensional topology. In short, it can be described as a consistent system of mapping class group representations, typically constructed from a given algebraic input datum, which for all available constructions is a modular tensor category (that one can obtain from certain Hopf algebras or vertex operator algebras). My talk will give an overview of certain meaningful generalizations of the traditional notion of a modular functor. In the first half of the talk, I will present the derived version of the modular functor associated to a non-semisimple modular tensor category and explain how it allows us to generalize the Verlinde formula to a statement about E_2-algebras. All these constructions rely on the rigidity of the input category (existence of duals), and in the second half of the talk, I will report on recent progress in relaxing this assumption to a Grothendieck-Verdier duality. This talk is based on joint work with Christoph Schweigert and Lukas Mü I will also briefly mention work in progress with Adrien Brochier. |

November 11 | Christoph Schweigert (Hamburg) | ## Bulk fields in conformal field theoryI will explain two recent developments concerning bulk fields in two-dimensional rational conformal field theories: the importance of the relative Serre functor to study bulk fields for logarithmic conformal field theories and the use of stringnet techniques to simplify the construction of correlators for semisimple modular tensor categories. |

November 25 | Marco de Renzi (Univ. of Zurich) | ## ETQFTs From Non-Semisimple Modular CategoriesWork of Bartlett, Douglas, Schommer-Pries, and Vicary shows that 3-dimensional ETQFTs (short for once-Extended Topological Quantum Field Theories) are classified by semisimple modular categories (appearing as the circle category of the corresponding theory). Although we will not aim for classification results, we will show how to use a (not necessarily semisimple) modular category C to build a 3-dimensional ETQFT defined on a 2-category of cobordisms that is not necessarily rigid. In this construction, it is the tensor ideal of projective objects Proj(C) that shows up as the circle category of the theory. From a topological point of view, concrete examples of non-semisimple modular categories associated with the small quantum group of sl(2) induce quantum representations of mapping class groups of surfaces whose properties are strikingly different from the ones of their semisimple counterparts. Based on joint work with A. Gainutdinov, N. Geer, B. Patureau, and I. Runkel. |

December 9 | Corina Keller (Montpellier) | ## Generalized Character Varieties and Quantization via Factorization HomologyFactorization homology is a local-to-global invariant which "integrates" disk algebras in symmetric monoidal higher categories over manifolds. In this talk I will focus on a particular instance of factorization homology on surfaces where the input algebraic data is a braided monoidal category. If one takes the representation category of a quantum group as an input, it was shown by Ben-Zvi, Brochier and Jordan (BZBJ) that categorical factorization homology quantizes the category of quasi-coherent sheaves on the moduli space of G-local systems. I will discuss two applications of the factorization homology approach for quantizing (generalized) character varieties. First, I will explain how to compute categorical factorization homology on surfaces with principal D-bundles decorations, for D a finite group. The main example comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We will see that in this case factorization homology gives rise to a quantization of Out(G)-twisted character varieties (This is based on joint work with Lukas Müller). In a second part we will consider surfaces that are decorated with marked points. It was shown by BZBJ that the algebraic data governing marked points are braided module categories and I will discuss an example related to the theory of dynamical quantum groups. |

January 20 | Michael Müger (Nijmegen) | ## On a conjecture of O. MathieuI will briefly talk about the Jacobian conjecture (JC) and the new approach to proving it suggested by Olivier Mathieu (Lyon, 1995/7). He stated a new conjecture concerning harmonic analysis on connected compact Lie groups and proved that it implies JC. So far, the Mathieu conjecture has been proven only in the abelian case. I will discuss attempts to prove the non-abelian version. |

February 3 | Elba Garcia-Failde (Paris Sorbonne IMJ-PRG) | ## Topological recursion and the r-spin cohomological field theoryNowadays there are deep connections among enumerative geometry, complex geometry, intersection theory of the moduli space of curves and integrable systems. In 1990, Witten conjectured that the generating series of intersection numbers of psi-classes is a tau function of the KdV hierarchy, which was first proved by Kontsevich using a cell decomposition of a combinatorial model of the moduli space of curves. In 2007, Chekhov, Eynard and Orantin introduced a procedure that associates a family of differentials to a Riemann surface with some extra data, which we call spectral curve. This method naturally fits in numerous algebro-geometric contexts, helping build relations among them. In the Witten—Kontsevich case, the Airy curve allows to build the connection with the 4 mentioned areas. After an introduction to topological recursion in general and the Witten—Kontsevich case in particular, I will introduce more general structures which help organising the intersection theory of the moduli space of curves: cohomological field theories. I will explain how they relate to topological recursion in a quite general framework. In joint work with R. Belliard, S. Charbonnier and B. Eynard and ongoing work with S. Charbonier, N. Chidambaram and A. Giacchetto, we extend both this relation and the Witten—Kontsevich result (r=2) to intersection numbers with Witten’s r-spin class, allowing us to complete the connections to the 4 mentioned areas in the context of Witten’s generalised conjecture for r>1. |

#### summer semester 2021

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. |

#### winter semester 2020/21

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. |