# Categorical braid group actions and categorical representation theory.

**Conference details**

Department of Mathematics and Statistics, University of Massachusets, Amherst, USA.

June 14 -- June 18, 2021

Organizing committee: Eugene Gorsky, Ben Elias, Andrei Negut, Alexei Oblomkov, Lev Rozansky, Anne Schilling.

NSF FRG Conference: **Categorical braid group actions and categorical representation theory****.**

**Organizing committee:**** **

Eugene Gorsky, Ben Elias, Andrei Negut, Alexei Oblomkov, Lev Rozansky, Anne Schilling.

**Aim**

**Aim**

The goal of this conference is to bring together researchers working in different aspects of Geometric Representation Theory, Algebraic geometry, Topology and to facilitate communication and interaction between them.

Links to the videos of talks are posted on titles and abstracts page. If you spoke at this conference and would like to provide notes to your talk, please email the organizers.

**Dates**

**Dates**

The conference is held virtually in the period **June**** 1****4****-****18****, 20****21**.

**Contact**

**Contact**

If you have questions, please email **@****math.umass****.edu** the user **oblomkov**

**Other Information**

**Other Information**

Support for this conference was provided by the National Science Foundation by NSF grant DMS-0968646. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

**List of speakers**

**List of speakers**

Nick Addington (U of Oregon) Roman Bezrukavnikov (MIT) Nils Carqueville (U of Vienna) Mikhail Gorsky ( UPJV, Amiens)

Iva Halacheva (NorthEastern U) Tina Kanstrup (U of Massachusetts). Mikhail Khovanov (Columbia U) Oscar Kivinen (U of Toronto)

Yakov Kononov (Columbia U) Ivan Losev (Yale U) Anton Mellit (U of Vienna) Simon Riche (U Clermont Auvergne)

Peng Shan (Tsinghua U) Catharina Stroppel (U of Bonn) Yukinobu Toda (IPMU) Kostya Tolmachov (U of Toronto)

Minh-Tam Trinh (MIT) Yu Zhao (MIT) Ben Webster (U of Waterloo, PI)

**Registration **

**Registration**

**Schedule of Talks**

**Schedule of Talks**

We are planning to have four one hour talks each day. The breaks between the talk could provide time for further discussions. The schedule below is tentative.

The schedule is in the Eastern Time . Talks will be recorded and available for the participants.

**Zoom Links:**

**Zoom Links:**

9:00 AM-10:25 AM. 10:45 AM-12:10 PM 12:30 PM-1:55 PM. 6:00 PM-7:25 PM 7:25 PM-9:10 PM

*Passcode.* Dimension of the group algebra of the rank six exceptional Weyl group.

**Titles and abstracts**

**Titles and abstracts**

* Nicolas Addington: *A categorical sl_2 action on some moduli spaces of sheaves.

We study certain sequences of moduli spaces of sheaves on K3 surfaces, building on work of Markman, Yoshioka, and Nakajima. We show that these sequences can be given the structure of a geometric categorical sl_2 action in the sense of Cautis, Kamnitzer, and Licata. As a corollary, we get an equivalence between derived categories of some moduli spaces that are birational via stratified Mukai flops. I'll spend most of my time on a nice example. This is joint with my student Ryan Takahashi.

* Roman Bezrukavnikov: *Affine braid group actions and Hodge structures.

Affine braid groups act on derived category of coherent sheaves and representations. I will discuss the natural variation of mixed Hodge structure underlying the induced action on the Grothendieck group and its conjectural relation to properties of the above action on the categories

**Nils ****Carqueville*** : *Extended spin TQFTs from matrix factorisations.

For every tangential structure, there is a corresponding 2-category of bordisms. An extended TQFT is a symmetric monoidal 2-functor whose domain is such a bordism 2-category. We will explain how, in the case of spin structures, these TQFTs are determined by duality and fixed point data. Then, by choosing appropriate codomain 2-categories, we can show that every Landau-Ginzburg model is naturally a spin extended TQFT, while truncated affine Rozansky-Witten models can be described as oriented extended TQFTs (joined work with Ilka Brunner, Daniel Roggenkamp and Lóránt Szegedy).

* Mikhail Gorsky: *Weave calculus and braid varieties.

* Iva Halacheva:* A cactus group action on crystals through categorical braid group actions.

Suppose g is a semisimple, simply-laced Lie algebra, V is a representation of its quantum group, and C is an abelian categorification of V. Then there is a categorical braid group action on the bounded derived category D^b(C). It is realized through work of Chuang-Rouquier, who introduce autoequivalences on D^b(C) called Rickard complexes. Cautis-Kamnitzer show that the Rickard complexes satisfy the braid relations. I will discuss the collection of Rickard complexes corresponding to the longest elements of certain parabolic Weyl groups. In joint work with Licata, Losev and Yacobi, we show that they recover an action of the cactus group on the crystal for V, originally defined via generalized Schützenberger involutions.

* Tina Kanstrup: *Link homologies and Hilbert schemes via representation theory

The aim of this joint work in progress with Roman Bezrukavnikov is to unite different approaches to Khovanov-Rozansky triply graded link homology. The original definition is completely algebraic in terms of Soergel bimodules. It has been conjectured by Gorsky, Negut and Rasmussen that it can also be calculated geometrically in terms of cohomolgy of sheaves on Hilbert schemes. Motivated by string theory Oblomkov and Rozansky constructed a link invariant in terms of matrix factorizations on related spaces and later proved that it coincides with Khovanov-Rozansky homology. In this talk I’ll discuss a direct relation between the different constructions and how one might invent these spaces.

* Mikhail Khovanov:* Foam evaluation and annular homology.

We'll explain recent joint work with Rostislav Akhmechet on constructing Asaeda-Przytycki-Sikora annular homology, its U(1)*U(1) equivariant version and their sl(3) counterparts via foam evaluation in the 3-space with a distinguished line.

* Oscar Kivinen: *A coherent-constructible correspondence for Hilbert schemes of points

I will describe a Springer-theoretic construction relating homologies of Hilbert schemes of points on singular curves and (quasi-)coherent sheaves on the Hilbert scheme of points on the plane. Time permitting I will discuss the relationship to braids. This is joint work with Gorsky and Oblomkov.

* Yakov Kononov: *Pursuing quantum difference equations.

The talk is based on joint work with A.Smirnov. We obtain a factorization theorem about the limit of elliptic stable envelopes to a point on a wall in H^2(X,R),

which generalizes the result of M.Aganagic and A.Okounkov. This approach allows us to extend the action of quantum groups, quantum Weyl groups,

R-matrices, etc., to actions on the K-theory of the symplectic dual variety. In the case of the Hilbert scheme of points in the plane, our results imply

the conjectures of E.Gorsky and A.Negut. As another application of this technique, we gain a better geometric understanding of the wall crossing operators

and the quantum difference equations.

* Ivan Losev: * Springer, Procesi and Cherednik.

The talk is based on a joint work with Pablo Boixeda Alvarez, arXiv:2104.09543. We study equivariant Borel-Moore homology of certain affine Springer fibers and relate them to global sections of suitable vector bundles arising from Procesi bundles on Q-factorial terminalizations of symplectic quotient singularities. This relation should give some information on the center of the principal block of the small quantum group. Our main technique is based on studying bimodules over Cherednik algebras.

* Anton Mellit*: Extra structures on the cohomology of character and braid varieties.

**Simon Riche:*** ** *Modular perverse sheaves on affine flag varieties and geometry of the dual group.

I will discuss a work in progress with Bezrukavnikov aiming at the construction of an equivalence relating the two natural categorifications of the affine Hecke algebra, in the case of positive-characteristic coefficients. I will present the results obtained so far, and explain in particular how they mimic the traditional Soergel theory in the study of perverse sheaves on flag varieties.

* Peng Shan: *Coherent categorification of quantum loop algebras.

We explain an equivalence of categories between a module category of quiver Hecke algebras associated with the Kronecker quiver and a category of equivariant perverse coherent sheaves on the nilpotent cone of type A. This provides a link between two different monoidal categorifications of the open quantum unipotent cell of affine type A_1, one given by Kang-Kashiwara-Kim-Oh-Park in terms of quiver Hecke algebras, the other given by Cautis-Williams in terms of equivariant perverse coherent sheaves on affine Grassmannians. This is a joint work with Michela Varagnolo and Eric Vasserot.

* Catharina Stroppel:* Ringel duality via braid group actions and coends.

* Yukinobu Toda: * Categorified Hall products in Donaldson-Thomas theory and wall-crossing

I will define categorified Donaldson-Thomas invariants for semistable locus on (-1)-shifted cotangent derived stacks as certain singular support quotients, which are regarded as gluing of dg-categories of matrix factorizations via Koszul duality. Then I will explain that they admit the categorified Hall product structure, induced by Halpern-Leistner's theory of theta stacks, which generalizes Porta-Sala's two dimensional categorified Hall products. The categorified Hall products are used to construct semi-orthogonal decompositions of categorical DT invariants, which give window theorem for them. As an application of window theorem via categorified Hall products, I will show the existence of fully-faithful functors of categorical Pandharipande-Thomas invariants on local surfaces under wall-crossing.

* Kostya Tolmachov:* Monodromic Hecke categories and Khovanov-Rozansky homology.

Khovanov-Rozansky homology is a knot invariant which, by the result of Khovanov, can be computed as the Hochschild cohomology functor applied to Rouquier complexes of Soergel bimodules. I will describe a new geometric model for the Hochschild cohomology of Soergel bimodules, living in the monodromic Hecke category. I will also explain how it allows to identify objects representing individual Hochsсhild cohomology groups as images of explicit character sheaves. Based on the joint work with Roman Bezrukavnikov.

* Minh-Tam Trinh: * From the Hecke Category to the Unipotent Locus.

If W is the Weyl group of a split semisimple group G, then the Hecke category of W can be built from pure perverse sheaves on the double flag variety of G. By developing a formalism of generalized realization functors, we construct a monoidal trace from the Hecke category to a category of bigraded modules over a certain graded ring: namely, the self-Ext's of the G-equivariant Springer sheaf. Conjecturally, it recovers the underived horizontal trace studied by Queffelec-Rose-Sartori and Gorsky-Hogancamp-Wedrich. We prove that: (1) The trace of the Rouquier complex of a positive braid \beta is the equivariant Borel-Moore homology of a generalized Steinberg scheme Z(\beta), equipped with its weight grading. (2) Our trace contains, as a summand, the one used by Webster-Williamson to construct Khovanov-Rozansky homology. Thus, the Khovanov-Rozansky homology of \beta is completely encoded in the Springer theory of Z(\beta).

As applications: (3) We are led to conjecture a new homeomorphism of stacks that should "explain" the Serre duality of KR homology under the full twist. (4) We discover a parabolic-induction formula that is not so obvious from KR homology by itself. (5) We obtain a new proof of a result of Varagnolo-Vasserot about when simple spherical modules of rational DAHA's are finite-dimensional. (6) We are led to conjecture a "Betti" analogue of Oblomkov-Yun's geometric construction of DAHA modules.

**Ben Webster: **** **Knot homology from coherent sheaves on Coulomb branches

Recent work of Aganagic details the construction of a homological knot invariant categorifying the Reshetikhin-Turaev invariants of miniscule representations of type ADE Lie algebras, using the geometry and physics of coherent sheaves on a space which one can alternately describe as a resolved slice in the affine Grassmannian, a space of G-monopoles with specified singularities, or as the Coulomb branch of the corresponding 3d quiver gauge theories. We give a construction of this invariant using an algebraic perspective on BFN's construction of the Coulomb branch, and in fact extend it to an invariant of annular knots. This depends on the theory of line operators in the corresponding quiver gauge theory and their relationship to non-commutative resolutions of these varieties (generalizing Bezrukavnikov's non-commutative Springer resolution).

**Yu Zhao: **** **** **A Weak Categorical Quantum Toroidal Action on the Derived Categories of Hilbert Schemes

The quantum toroidal algebra is the affinization of the quantum Heisenberg algebra. Schiffmann-Vasserot, Feigin-Tsymbaliuk and Negut studied the quantum toroidal algebra action on the Grothendieck group of Hilbert schemes of points on surfaces, which generalized the action by Nakajima and Grojnowski in cohomology. In this talk, we will categorify the above quantum toroidal algebra action in the weak sense. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semi-divisorial log terminal singularities.

**Participants**

**Participants**

Nick Addington, Roman Bezrukavnikov, Nils Carqueville, Ben Elias, Eugene Gorsky, Mikhail Gorsky, Iva Halacheva, Tina Kanstrup, Mikhail Kohvanov, Oscar Kivinen, Yakov Kononov, Ivan Losev, Anton Mellit, Andrei Negut, Alexei Oblomkov, Anne Schilling, Simon Riche, Lev Rozansky, Peng Shan, Catharina Stroppel, Yukinobu Toda, Kostya Tolmachov, Minh-Tam Trinh, Yu Zhao, Yasuyoshi Yonezawa, Ben Webster; Nitu Kitchloo, Andrew Adair, Pedro Vaz, Quoc Ho, José Simental Rodríguez, Léo Schelstraete, Wenjun Niu, Jacob Caudell, Haihan Wu, Dori Bejleri, Arthur Wang, Christine Lee, Jeeuhn Kim, Alexander Shapiro, Changjian Su, Xinchun Ma, Noah Arbesfeld, Aaron Mazel-Gee, Olivier Dudas, Zhaoting Wei, Oded Yacobi, Chris Fraser, Cailan Li, Francesco Sala, Ilya Dumanski, Samuel Lopes, Tom Sutherland, Eric Vasserot, Kevin McGerty, Michela Varagnolo, Robert Laugwitz, Xiaobin Li, Elijah Bodish, Ben Cooper, Victor Ginzburg, Toshiyuki Tanisaki, Max Vargas, Ryosuke Kodera, Emmanuel WAGNER, Jack Wagner, Chris Bowman, Ryszard Rubinsztein, Mee Seong Im.