# Cohomological Hall Algebras in Mathematics and Physics

COVID-19 information for researchers and visitors

Conference Date:
Monday, February 25, 2019 (All day) to Friday, March 1, 2019 (All day)
Scientific Areas:
Mathematical Physics

This workshop will bring together leading mathematicians and physicists interested in the Cohomological Hall algebra, as it appears in the study of moduli spaces and in gauge and string theory.

• Alexander Braverman, Perimeter Institute & University of Toronto
• Kevin Costello, Perimeter Institute
• Thomas Creutzig, University of Alberta
• Ben Davison, University of Edinburgh
• Pavel Etingof, Massachusetts Institute of Technology
• Davide Gaiotto, Perimeter Institute
• Sergei Gukov, California Institute of Technology
• Mikhail Kapranov, Kavli IPMU, Tokyo
• Fyodor Malikov, University of Southern California
• Anton Mellit, University of Vienna
• Andrei Negut, Massachusetts Institute of Technology
• Natalie Paquette, California Institute of Technology
• Nicolo' Piazzalunga, Stony Brook University
• Miroslav Rapcak, Perimeter Institute
• Francesco Sala, Kavli IPMU, Tokyo
• Yan Soibelman, Kansas State University
• Balazs Szendroi, University of Oxford
• Yegor Zenkevich, Institute for Theoretical and Experimental Physics & Milano Bicocca University
• Gufang Zhao, University of Melbourne

• Ian Le, Perimeter Institute
• Thomas Przezdziecki, Max Planck Institute
• Alexander Weekes, Perimeter Institute
• Junya Yagi, Perimeter Institute
• Philsang Yoo, Yale University

Monday, February 25, 2019

 Time Event Location 9:00 – 9:25am Registration Reception 9:25 – 9:30am Yan Soibelman, Kansas State UniversityWelcome and Opening Remarks Sky Room 9:30 – 10:30am Yan Soibelman, Kansas State UniversityAn introduction to Cohomological Hall algebras and their representations Sky Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00 – 12:00pm Davide Gaiotto, Perimeter InstituteGauge theory, vertex algebras and COHA Sky Room 12:00 – 2:00pm Lunch Bistro – 2nd Floor 2:00 – 3:00pm Nicolo' Piazzalunga, Stony Brook UniversityMagnificent Four with color Sky Room 3:00 – 3:15pm Break Bistro – 1st Floor 3:15 – 4:15pm Yegor Zenkevich,Institute for Theoretical and Experimental Physics& Milano Bicocca UniversityNetworks of intertwiners, 3d theories and superalgebras Sky Room 4:15 – 4:45pm Coffee Break Bistro – 1st Floor 4:45 – 5:45pm Mikhail Kapranov, Kavli IPMU, TokyoCOHA of surfaces and factorization algebras Sky Room

Tuesday, February 26, 2019

 Time Event Location 9:30 – 10:30am Ben Davison, University of EdinburghYangians from cohomological Hall algebras Sky Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00 – 12:00pm Sergei Gukov, California Institute of TechnologyAlgebraic structures of T[M3] and T[M4] Sky Room 12:00 – 12:10pm Conference Photo TBA 12:10 – 2:30pm Lunch Bistro – 2nd Floor 2:30 – 3:30pm Francesco Sala, Kavli IPMU, TokyoCategorification of 2d cohomological Hall algebras Sky Room 3:30 – 4:00pm Coffee Break Bistro – 1st Floor 4:00 – 5:00pm Pavel Etingof, Massachusetts Institute of TechnologyShort star-products for filtered quantizations Sky Room 5:30 – 8:00pm Banquet Bistro – 2nd Floor

Wednesday, February 27, 2019

 Time Event Location 9:30 – 10:30am Kevin Costello, Perimeter InstituteCohomological hall algebras from string and M theory Sky Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00 – 12:00pm Gufang Zhao, University of MelbourneComultiplications on cohomological Hall algebras and vertex algebras Sky Room 12:00 – 2:00pm Lunch Bistro – 2nd Floor 2:00 – 3:30pm Mikhail Kapranov, Kavli IPMU, TokyoColloquiumPerverse sheaves, perverse schobers and  physical "theories". Time Room 3:30 – 4:00pm Coffee Break Bistro – 1st Floor 4:00 – 5:00pm Natalie Paquette, California Institute of TechnologyAn old-fashioned view of BPS-algebras Sky Room

Thursday, February 28, 2019

 Time Event Location 9:30 – 10:30am Balazs Szendroi, University of OxfordPartition functions and the McKay correspondence Sky Room 10:30 – 11:00am Coffee Break Sky Room 11:00 – 12:00pm AThomas Creutzig, University of AlbertaGlueing W-algebras Sky Room 12:00 – 2:30pm Lunch Bistro – 2nd Floor 2:30 – 3:30pm Miroslav Rapcak, Perimeter InstituteCOHA and AGT for Spiked Instantons Sky Room 3:30 – 4:00pm Coffee Break Bistro – 1st Floor 4:00 – 5:00pm Andrei Negut, Massachusetts Institute of TechnologyConnecting affine Yangians with W-algebras Sky Room

Friday, March 1, 2019

 Time Event Location 9:30 – 10:30am Alexander Braverman, Perimeter Institute & University of Toronto Sky Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00 – 12:00pm Anton Mellit, University of ViennaFusion Hall algebra and shuffle conjectures Sky Room 12:00 – 2:30pm Lunch Bistro – 2nd Floor 2:30 – 3:30pm Fyodor Malikov, University of Southern CaliforniaOn one example of a chiral Lie group Sky Room 3:30 – 4:00pm Coffee Break Bistro – 1st Floor

Kevin Costello, Perimeter Institute

Cohomological hall algebras from string and M theory

I will outline a framework to understand certain COHAS from a mathematical incarnation of string theory and M theory. As an application, I will give a conjectural description of the COHA of the resolved conifold.

Thomas Creutzig, University of Alberta

Glueing W-algebras

S-duality predicts rather surprising isomorphisms of extensions of W-algebras. The aim of this talk is to present some explanation. Firstly, I will explain the concept of glueing W-algebras along certain categories of modules and then I will introduce what we call a W-algebra translation functor.

Ben Davison, University of Edinburgh

Yangians from cohomological Hall algebras

I will explain various features of the preprojective CoHA, a kind of universal algebra of correspondences generalising the algebras of endomorphisms of cohomology of quiver varieties considered by Nakajima. In particular I will focus on features of this algebra that become visible after viewing it as a dimensional reduction of the Kontsevich-Soibelman) critical CoHA associated to a related 3-dimensional Calabi-Yau category. Many nice features emerge from this view, e.g. an embedding in a related shuffle algebra, a formula for the graded dimension of the algebra, a flat non cocommutative deformation, a cocommutative coproduct, a geometric doubling procedure, the PBW theorem, an isomorphism with the Yangian considered by Maulik and Okounkov... I will focus on the perverse filtration, which is the key feature of my joint work with Sven Meinhardt, and gives rise to most of the above

Pavel Etingof, Massachusetts Institute of Technology

Short star-products for filtered quantizations

Let $A$ be a filtered Poisson algebra with Poisson bracket ${ , }$ of degree -2. A star product on $A$ is an associative product $*: A\otimes A\to A$ given by $a*b=ab+\sum_{i\ge 1}C_i(a,b)$, where $C_i$ has degree $-2i$ and $C_1(a,b)-C_1(b,a)={a, b}$. We call the product * "even", if $C_i(a,b)=(-1)^iC_i(b,a)$ for all $i$, and call it "short",  if $C_i(a,b)=0$ whenever $i> min(deg(a),deg(b))$.

Motivated by three-dimensional $N=4$ superconformal field theory, In 2016 Beem, Peelaers and Rastelli considered short even star-products for homogeneous symplectic singularities (more precisely, hyperK\"ahler cones) and conjectured that that they exist and depend on finitely many parameters. We prove the dependence on finitely many parameters in general and existence for a large class of examples, using the connection of this problem with zeroth Hochschild homology of quantizations suggested by Kontsevich.

Beem, Peelaers and Rastelli also computed the first few terms of short quantizations for Kleinian singularities of type A, which were later computed to all orders by Dedushenko, Pufu and Yacoby. We will discuss some generalizations of these results.

This is joint work with Eric Rains and Douglas Stryker.

Davide Gaiotto, Perimeter Institute

Gauge theory, vertex algebras and COHA

I will discuss some of the relations between supersymmetric  gauge theories, vertex algebras and COHA.

Sergei Gukov, California Institute of Technology

Algebraic structures of T[M3] and T[M4]

The talk will focus on tensor categories associated with 3d N=2 theories and chiral algebras associated with 2d N=(0,2) theories, as well as their combinations that involve 3d N=2 theories "sandwiched" by half-BPS boundary conditions and interfaces. Such situations, originally studied in a joint work with A.Gadde and P.Putrov, have a variety of applications, including applications to topology of 3-manifolds and 4-manifolds where Kirby moves translate into novel dualities of 3d N=2 and 2d N=(0,2) theories and where the corresponding algebraic structures can be related to COHAs. After reviewing some elements of that story going back to 2013, I will focus on the latest developments in the area of "3d Modularity" where mock Jacobi forms, SL(2,Z) Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs make a surprising appearance (based on recent and ongoing work with M.Cheng, S.Chun, F.Ferrari, S.Harrison and B.Feigin).

Mikhail Kapranov, Kavli IPMU, Tokyo

COHA of surfaces and factorization algebras

This is a report on joint work with E. Vasserot. For a smooth quasiprojective surface S we consider the stack Coh(S) of coherent sheaves on S with compact support and make the Borel-Moore homology of Coh(S) into an associative algebra using a derived version of the Hall multiplication diagram. Subcategories in Coh(S) given by various conditions on dimension of support give rise to various types of Hecke operators. In particular, we consider the COHA R(S) of sheaves with 0-dimensional support and show that its chain level lift is a factorization coalgebra on S with values in homotopy associative algebras. This allows us to find the size (graded dimension) of R(S).

Fyodor Malikov, University of Southern California

On one example of a chiral Lie group

We quantize the Khesin-Zakharevich Poisson-Lie group of pseudo-differential symbols. This is a joint work with A.Linshaw

Anton Mellit, University of Vienna

Fusion Hall algebra and shuffle conjectures

The classical Hall algebra of the category of representations of one-loop quiver is isomorphic to the ring of symmetric functions, and Hall-Littlewood polynomials arise naturally as the images  of objects. I will talk about a second "fusion" product on this algebra, whose structure constants are given by counting of bundles with nilpotent endomorphisms on P^1 with restrictions at 0, 1 and infinity. The two products together make up a structure closely related to the elliptic Hall algebra. In the situations when bundles can be explicitly enumerated, I will explain how this leads to q,t-identities conjectured by combinatorists, such as the shuffle conjecture and its generalizations. This is a joint project with Erik Carlsson.

Andrei Negut, Massachusetts Institute of Technology

Connecting affine Yangians with W-algebras

Alday-Gaiotto-Tachikawa connect instanton counts in gauge theory with conformal blocks for W-algebras. We realize this mathematically by relating q-deformed W-algebras with the affine q-Yangians that control gauge theory, thus offering an affine, q-deformed generalization of the well-known Brundan-Kleshchev construction in type A

Natalie Paquette, California Institute of Technology

An old-fashioned view of BPS-algebras

The notion of the algebra of BPS states goes back to work of Harvey and Moore in the late 90's. Explicit computations in perturbative heterotic string theory point to an algebraic structure isomorphic to a Generalized Kac-Moody (GKM) algebra in that context; at the same time, rather mysteriously, denominators of GKMs furnish signed counts of BPS states in certain supersymmetric string vacua. The most famous example of this is the reciprocal of the Igusa cusp form of weight 10, which governs some BPS state counts related to string theory on the product of a nonsingular K3 surface and an elliptic curve E. Other denominator formulas, more generally, produce interesting examples of Siegel automorphic forms, and have enumerative geometric interpretations. I will review recent progress understanding the role of GKMs in some special string vacua related to Monstrous and Conway moonshine modules (based on work with Persson & Volpato, and Harrison & Volpato), as well as a new class of putative GKM denominators obtained from a large class of quotients of K3xE (based on work with Volpato & Zimet).

Nicolo' Piazzalunga, Stony Brook University

Magnificent Four with color

I will present the rank N magnificent four theory, which is the supersymmetric localization of U(N) super-Yang-Mills theory with matter on a Calabi-Yau fourfold, and conjecture an explicit formula for the partition function Z: it has a free-field representation, and surprisingly it depends on Coulomb and mass parameters in a simple way. Based on joint work with N.Nekrasov.

Miroslav Rapcak, Perimeter Institute

COHA and AGT for Spiked Instantons

The well-known AGT correspondence relates $\mathcal{W}_N$-algebras and supersymmetric gauge theories on $\mathbb{C}^2$. Embedding $\mathbb{C}^2$ as a coordinate plane inside $\mathbb{C}^3$, one can associate the COHA to $\mathbb{C}^3$ and derive the corresponding $\mathcal{W}_N$ as a truncation of its Drinfeld double. Building up on Zhao's talk, I will discuss a generalization of this story, where $\mathbb{C}^2$ is replaced by a more general divisor inside $\mathbb{C}^3$ with three smooth components supported on the three coordinate planes. Truncations of the Drinfeld double lead to a three-parameter family of algebras $\mathcal{W}_{L,M,N}$ determining the vertex algebras associated to Nekrasov's spiked instantons. Many interesting questions emerge when considering a general Calabi-Yau three-fold instead of $\mathbb{C}^3$. I will discuss a class of vertex algebras conjecturally arising from divisors inside more general toric Calabi-Yau three-folds.

Francesco Sala, Kavli IPMU, Tokyo

Categorification of 2d cohomological Hall algebras

Let $\mathcal{M}$ denote the moduli stack of either coherent sheaves on a smooth projective surface or Higgs sheaves on a smooth projective curve $X$. The convolution algebra structure on the Borel-Moore homology of $\mathcal{M}$ is an instance of two-dimensional cohomological Hall algebras.

In the present talk, I will describe a full categorication of the cohomological Hall algebra of $\mathcal{M}$. This is achieved by exhibiting a derived enhancement of $\mathcal{M}$. Furthermore, this method applies also to several other moduli stacks, such as the moduli stack of vector bundles with flat connections on $X$ and the moduli stack of finite-dimensional representations of the fundamental group of $X$. In the second part of the talk, I will focus on the case of curves and discuss some relations between the Betti, de Rham, and Dolbeaut categorified cohomological Hall algebras. This is based on a work in progress with Mauro Porta.

Yan Soibelman, Kansas State University

An introduction to Cohomological Hall algebras and their representations

I am going to review for non-experts the notion of Cohomological Hall algebra (COHA) introduced in my joint paper with Maxim Kontsevich in 2010 (see  arXiv:1006.2706).Restricting considerations to the case of COHA of quivers with potential I will recallsome structural results, like e.g. dimensional reduction of COHA from 3d to 2d.

Then I plan to discuss a class of representations of COHA in the cohomology of the modulispaces of framed stable objects of a 3d Calabi-Yau category endowed with a stability structure, following my paper arXiv:1404.1606.

Finally, if time permits, I will discuss some examples of representations of COHA and its double, including my recent joint work with Miroslav Rapcak, Yaping Yang and Gufang Zhao on the relation of COHA with affine Yangians and moduli spaces of Nekrasov spiked instantons (see arXiv:1810.10402). As will be explained in other talks at this conference this relation gives rise to a class of representations of the vertex algebra at the corner" (see Gaiotto and Rapcak,  arXiv:1703.00982). Other classes of representations of the VOA at the corner are conjecturally related to  the action of the double of spherical COHA on the cohomology of Hilbert schemes of non-reduced divisors in toric Calabi-Yau 3-folds.

Balazs Szendroi, University of Oxford

Partition functions and the McKay correspondence

I will explain some results on certain sheaf-theoretic partition functions defined on Calabi-Yau orbifolds, and their connection to the McKay correspondence, the representation theory of affine Lie algebras, and cohomological Hall algebras. Based on joint work with Gyenge and Nemethi, respectively Davison and Ongaro

Yegor Zenkevich, Institute for Theoretical and Experimental Physics & Milano Bicocca University

Networks of intertwiners, 3d theories and superalgebras

Refined topological vertex formalism allows one to conveniently compute partition functions of topological strings on toric CY backgrounds. These partition functions reproduce instanton partition functions of 5d N=1 gauge theories, obtained from the CY by the geometric engineering procedure. In the algebraic language the vertices can be described as intertwiners of Fock representations of a quantum toroidal algebra. I will present a «Higgsed» version of refined topological vertex formalism which computes vortex partition functions of certain N=2* 3d theories, and show how it naturally arises in the algebraic approach. The new formalism gives a streamlined way to write down the screening charges of a general class of q-deformed W-algebras, including those associated with superalgebras. The obtained partition functions are automatically eigenfunctions of Ruijsenaars-Schneider Hamiltonians or their supersymmetric generalizations.

Gufang Zhao, University of Melbourne

Comultiplications on cohomological Hall algebras and vertex algebras

In this talk we will explore a factorization structure of the cohmological Hall algebra (COHA) of a quiver, and the occurrence of the same structure from Beilinson-Drinfeld Grassmannians. In particular, in collaboration with Mirkovic and Yang, we identified a Drinfeld-type comultiplication on the COHA with the factorizable line bundle on the zastava space. I will discuss one aspect of a recent joint work with Rapcak, Soibelman, and Yang, which can be reviewed as a construction of a vertex algebra from the standard comultiplication on the double COHA of the Jordan quiver. The standard comultiplication differs from the Drinfeld comultiplication by an S-operator. The relevant vertex algebra is the VOA at the corner of Rapcak-Gaiotto.

## On one example of a chiral Lie group

Friday Mar 01, 2019
Speaker(s):

We quantize the Khesin-Zakharevich Poisson-Lie group of pseudo-differential symbols. This is a joint work with A.Linshaw

Scientific Areas:

## Fusion Hall algebra and shuffle conjectures

Friday Mar 01, 2019
Speaker(s):

The classical Hall algebra of the category of representations of one-loop quiver is isomorphic to the ring of symmetric functions, and Hall-Littlewood polynomials arise naturally as the images of objects. I will talk about a second "fusion" product on this algebra, whose structure constants are given by counting of bundles with nilpotent endomorphisms on P^1 with restrictions at 0, 1 and infinity. The two products together make up a structure closely related to the elliptic Hall algebra.

Scientific Areas:

## TBA

Friday Mar 01, 2019
Speaker(s):
Scientific Areas:

## Connecting affine Yangians with W-algebras

Thursday Feb 28, 2019
Speaker(s):

Alday-Gaiotto-Tachikawa connect instanton counts in gauge theory with conformal blocks for W-algebras. We realize this mathematically by relating q-deformed W-algebras with the affine q-Yangians that control gauge theory, thus offering an affine, q-deformed generalization of the well-known Brundan-Kleshchev construction in type A

Scientific Areas:

## COHA and AGT for Spiked Instantons

Thursday Feb 28, 2019
Speaker(s):

The well-known AGT correspondence relates $\mathcal{W}_N$-algebras and supersymmetric gauge theories on $\mathbb{C}^2$. Embedding $\mathbb{C}^2$ as a coordinate plane inside $\mathbb{C}^3$, one can associate the COHA to $\mathbb{C}^3$ and derive the corresponding $\mathcal{W}_N$ as a truncation of its Drinfeld double. Building up on Zhao's talk, I will discuss a generalization of this story, where $\mathbb{C}^2$ is replaced by a more general divisor inside $\mathbb{C}^3$ with three smooth components supported on the three coordinate planes.

Scientific Areas:

## Glueing W-algebras

Thursday Feb 28, 2019
Speaker(s):

S-duality predicts rather surprising isomorphisms of extensions of W-algebras. The aim of this talk is to present some explanation. Firstly, I will explain the concept of glueing W-algebras along certain categories of modules and then I will introduce what we call a W-algebra translation functor.

Scientific Areas:

## Partition functions and the McKay correspondence

Thursday Feb 28, 2019
Speaker(s):

I will explain some results on certain sheaf-theoretic partition functions defined on Calabi-Yau orbifolds, and their connection to the McKay correspondence, the representation theory of affine Lie algebras, and cohomological Hall algebras. Based on joint work with Gyenge and Nemethi, respectively Davison and Ongaro

Scientific Areas:

## An old-fashioned view of BPS-algebras

Wednesday Feb 27, 2019
Speaker(s):

The notion of the algebra of BPS states goes back to work of Harvey and Moore in the late 90's. Explicit computations in perturbative heterotic string theory point to an algebraic structure isomorphic to a Generalized Kac-Moody (GKM) algebra in that context; at the same time, rather mysteriously, denominators of GKMs furnish signed counts of BPS states in certain supersymmetric string vacua.

Scientific Areas:

## Perverse sheaves, perverse schobers and physical "theories"

Wednesday Feb 27, 2019
Speaker(s):

The mathematical concept of sheaves is a tool for
> describing global structures via local data. Its generalization, the
> concept of perverse sheaves, which appeared originally in the study of
> linear PDE, turned out to be remarkably useful in many diverse areas
> of mathematics. I will review these concepts as well as a more recent conjectural categorical generalization, called perverse schobers.
> One reason for the interest in such structures is the remarkable
> parallelism between:
>

Scientific Areas:

## Comultiplications on cohomological Hall algebras and vertex algebras

Wednesday Feb 27, 2019
Speaker(s):

In this talk we will explore a factorization structure of the cohmological Hall algebra (COHA) of a quiver, and the occurrence of the same structure from Beilinson-Drinfeld Grassmannians. In particular, in collaboration with Mirkovic and Yang, we identified a Drinfeld-type comultiplication on the COHA with the factorizable line bundle on the zastava space. I will discuss one aspect of a recent joint work with Rapcak, Soibelman, and Yang, which can be reviewed as a construction of a vertex algebra from the standard comultiplication on the double COHA of the Jordan quiver.

Scientific Areas:

## Pages

Scientific Organizers:

• Kevin Costello, Perimeter Institute
• Yan Soibelman, Kansas State University
• Yaping Yang, University of Melbourne