**Quantum Topology Biennial: Focus on Representation Theory**

**Titles and Abstracts**

**Christian Blanchet** (Université Paris Cité)-Heisenberg homologies of configurations in ribbon graphs

Abstract. The Heisenberg homologies program aims to study homologies of surface configurations with local coefficients given by representations of the Heisenberg quotient of the surface braid group.

We will overview the basic constructions, discuss the connections with TQFTs and present computational results in the case of ribbon graphs.

**Pablo Boixeda Alvarez**(Yale University)-The center of the small quantum group and affine springer fibers

**Tudor Dimofte** (University of Edinburgh)-Spark algebras and quantum groups

Abstract: Tannakian duality represents a category C as modules for the endomorphism algebra of a "fiber functor" C -> Vect. I'll describe a setup that produces fiber functors for braided tensor categories of line operators (a.k.a. Z(S^1)) in 3d TQFT, and explicitly allows one to construct quasi-triangular Hopf algebras (or loosely: "quantum groups") that represent these categories. I'll apply this to a situation where a 3d TQFT was expected to exist on physical grounds, but the "quantum groups" were not known: the topological twists of 3d supersymmetric gauge theories. I also hope to touch on connections between this construction and Kazhdan-Lusztig correspondences for logarithmic VOA's. (Based on work with T. Creutzig and W. Niu.)

**Azat Gainutdinov** (Université de Tours)-Non-semisimple link and manifold invariants

Abstract: I will talk about link and three-manifold invariants defined in terms of a non-semisimple finite ribbon category C together with a choice of tensor ideal and modified trace. If the ideal is all of C, these invariants agree with those defined by Lyubashenko in the 90’s, and as we show, they only depend on the Grothendieck class of the objects labelling the link. These invariants are therefore not able to determine non-split extensions. However, we observed an interesting phenomenon: if one chooses an intermediate proper ideal between C and the minimal ideal of projective objects, the invariants do distinguish non-trivial extensions. This is demonstrated in the case of C being the super-modular category of an exterior algebra. That is why these invariants deserve to be called “non-semisimple”. This is a joint work with J. Berger and I. Runkel.

**Eugene Gorsky** (UC Davis)-Splitting maps in link Floer homology and integer points in permutahedra

Abstract: We study maps in Heegaard Floer homology corresponding to generalized crossing changes. In particular, we associate the splitting maps for the torus link T(n,n) to integer points in the (n−1)-dimensional permutahedron, and relate Heegaard Floer homology of T(n,n) to certain determinantal ideals defined by Haiman. These results can be used to define colored Heegaard Floer homology. This is a joint work with Akram Alishahi and Beibei Liu.

**Ivan Losev** (Yale University)-Quantum category O vs affine Hecke category.

Abstract: I will establish an equivalence between a block of the quantum category O at an odd root of unity and the heart of the "new" t-structure on a suitably singular affine Hecke category. This is based on arXiv:2310.03153.

**Alexei Oblomkov**(University of Massachusetts Amherst)- Soergel bimodules, matrix factorizations and Hilbert schemes.

$SBim_n$ to $MF_n$. We compose this functor with the Chern functor $CH: MF_n \to Coh^{per}(Hilb_n(C^2)$ to obtain a two-periodic complex of sheaves $S_b$ for a braid $b\in Br_n$ such that $H^*(S_b)$ is equal to the triply graded homology of $b$. Some explicit examples of $S_b$ will be shown.

**Andrei Negut** (Massachusetts Institute of Technology)-AHA and EHA

Abstract: In a joint project with Eugene Gorsky, we seek to describe the trace of the affine Hecke category (in type A) in relation to the elliptic Hall algebra, using shuffle algebras, integral forms, convex paths, and other gadgets.

**Jacob Rasmussen** (Cambridge University)-Bordered-sutured Floer homology as a sutured TQFT

Abstract: The hat flavor of Heegaard Floer homology should fit into an extended "sutured" TQFT whose 2-3 part is Zarev's bordered-sutured Floer homology. Such sutured TQFTs have not been much studied, but since Heegaard Floer homology is very interesting, it seems natural to ask if there are more of them. I'll explain what I mean by a sutured TQFT and give a categorification-friendly interpretation of bordered-sutured theory.

**Simon Riche**(Université Clermont Auvergne)- Koszul duality for general Coxeter groups

Abstract: Koszul duality for derived categories of constructible complexes on flag varieties of reductive groups was initially constructed by Beilinson-Ginzburg-Soergel to explain some combinatorial identities in category O of the associated complex semisimple Lie algebra. This construction was later generalized to Kac-Moody groups by Bezrukavnikov-Yun. Here we will explain how one can make sense of this duality for general Coxeter groups, and essentially arbitrary realizations. The replacement for semisimple complexes (or parity complexes) is provided by the Elias-Williamson diagrammatic category. This talk will be based on joint work with Cristian Vay, and also earlier work with Pramod Achar.

**Wolfgang Soergel** (Universität Freiburg)-Langlands philosophy and Koszul duality revisited.

Abstract: New progress in the theory of motives allows for a much more concrete version of my old conjecture in the form of a nice equivalence of triangulated categories. This builds on work of V.Gajda and J.Eberhardt.

**Joshua Sussan **(CUNY)-Non-semisimple Hermitian TQFTs

Abstract: We endow a category of representations of the unrolled quantum group for sl(2) with a Hermitian structure. This leads to a non-degenerate Hermitian pairing on the state spaces for surfaces coming from a non-semisimple TQFT. We study the resulting unitary representations of the braid group and show that the representations sometimes have dense image. This is joint with Nathan Geer, Aaron Lauda, and Bertrand Patureau-Mirand.

**Kostiantyn Tolmachov** (University of Edinburgh)-Equivariant derived category of a reductive group as a categorical center

Abstract: There is a classical relationship between representations of the Iwahori-Hecke algebra associated with a Weyl group of a split reductive group G, defined over a finite field, and the (principal series) representations of the corresponding finite group of Lie type. I will discuss a categorification of this relationship in the context of various triangulated categories of constructible sheaves on the group G. In particular, I will present a new approach to connecting the categories of character sheaves to a version of a categorical center of the constructible Hecke category. Based on a joint work with R. Bezrukavnikov, A. Ionov, and Y. Varshavsky. Time permitting, I will also talk about an approach to the Koszul duality for the y-ified Hecke category and symmetries of Khovanov-Rozansky homology, based on the work in progress joint with Q. Ho, A. Ionov and P. Li.

**Emmanuel Wagner** (Université Paris Cité): From representation theory to topology: there and back again

Abstract: In this talk, I will explain how foam evlauation allows to see an sl(2) action on the equivariant gl(n) Khovanov-Rozansky link homologies. We will also see how to extend the previous action functorially. Joint work with You Qi, Louis-Hadrien Robert and Joshua Sussan.

**Weiqiang Wang** (University of Virginia)-Nil-Brauer category and i-quantum groups

Abstract: i-Quantum groups are generalizations of quantum groups. In this talk I will introduce the nil-Brauer category and explain how the monoidal category of finitely generated graded projective modules for the nil-Brauer categorifies the i-quantum group of split rank one, with the indecomposables categorifying the i-canonical basis. We will introduce standard modules for nil-Brauer which categorifies the PBW basis for the i-quantum groups. Similarly defined standard modules for the 2-category of sl_2 categorify the PBW basis of modified quantum sl_2. Joint work with Jon Brundan and Ben Webster.

**Paul Wedrich **(University of Hamburg)-From link homology to TQFTs

Abstract: Skein theory offers several plausible strategies for extending link homology theories, such as Khovanov homology, to topological quantum field theories in 4 or 5 dimensions. In this talk, I will focus on a categorified analog of a TQFT of Turaev-Viro type. Joint work with Matthew Hogancamp and David Rose.

**Stephan Wehrli** (Syracuse University)-Categorified traces and colored sl_2 knot homology

Abstract: In this talk, I will compare three different models for colored sl_2 knot homology, including Khovanov’s nonreduced model and Cooper-Krushkal’s categorified projector model. In the case of the unknot, I will prove that all three models become equivalent when formulated in the quantum annular setting. I will preface the proof of this result with a general discussion of categorified trace functions on bicategories. By using observations from our proof, I will further describe the classes of the Cooper-Hogancamp projectors in the quantum horizontal trace. As an application, I will compute the full quantum Hochschild homology of Khovanov’s arc ring H^n.

This is joint work with Anna Beliakova, Matthew Hogancamp, and Krzysztof Putyra.