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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
- Tadashi Okazaki, 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 University |
Sky Room |
9:30 – 10:30am |
Yan Soibelman, Kansas State University |
Sky Room |
10:30 – 11:00am |
Coffee Break |
Bistro – 1^{st} Floor |
11:00 – 12:00pm |
Davide Gaiotto, Perimeter Institute |
Sky Room |
12:00 – 2:00pm |
Lunch |
Bistro – 2^{nd} Floor |
2:00 – 3:00pm |
Nicolo' Piazzalunga, Stony Brook University |
Sky Room |
3:00 – 3:15pm |
Break |
Bistro – 1^{st} Floor |
3:15 – 4:15pm |
Yegor Zenkevich, |
Sky Room |
4:15 – 4:45pm |
Coffee Break |
Bistro – 1^{st} Floor |
4:45 – 5:45pm |
Mikhail Kapranov, Kavli IPMU, Tokyo |
Sky Room |
Tuesday, February 26, 2019
Time |
Event |
Location |
9:30 – 10:30am |
Ben Davison, University of Edinburgh |
Sky Room |
10:30 – 11:00am |
Coffee Break |
Bistro – 1^{st} Floor |
11:00 – 12:00pm |
Sergei Gukov, California Institute of Technology |
Sky Room |
12:00 – 12:10pm |
Conference Photo |
TBA |
12:10 – 2:30pm |
Lunch |
Bistro – 2^{nd} Floor |
2:30 – 3:30pm |
Francesco Sala, Kavli IPMU, Tokyo |
Sky Room |
3:30 – 4:00pm |
Coffee Break |
Bistro – 1^{st} Floor |
4:00 – 5:00pm |
Pavel Etingof, Massachusetts Institute of Technology |
Sky Room |
5:30 – 8:00pm |
Banquet |
Bistro – 2^{nd} Floor |
Wednesday, February 27, 2019
Time |
Event |
Location |
9:30 – 10:30am |
Kevin Costello, Perimeter Institute |
Sky Room |
10:30 – 11:00am |
Coffee Break |
Bistro – 1^{st} Floor |
11:00 – 12:00pm |
Gufang Zhao, University of Melbourne |
Sky Room |
12:00 – 2:00pm |
Lunch |
Bistro – 2^{nd} Floor |
2:00 – 3:30pm |
Mikhail Kapranov, Kavli IPMU, Tokyo |
Time Room |
3:30 – 4:00pm |
Coffee Break |
Bistro – 1^{st} Floor |
4:00 – 5:00pm |
Natalie Paquette, California Institute of Technology |
Sky Room |
Thursday, February 28, 2019
Time |
Event |
Location |
9:30 – 10:30am |
Balazs Szendroi, University of Oxford |
Sky Room |
10:30 – 11:00am |
Coffee Break |
Sky Room |
11:00 – 12:00pm |
AThomas Creutzig, University of Alberta |
Sky Room |
12:00 – 2:30pm |
Lunch |
Bistro – 2^{nd} Floor |
2:30 – 3:30pm |
Miroslav Rapcak, Perimeter Institute |
Sky Room |
3:30 – 4:00pm |
Coffee Break |
Bistro – 1^{st} Floor |
4:00 – 5:00pm |
Andrei Negut, Massachusetts Institute of Technology |
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 – 1^{st} Floor |
11:00 – 12:00pm |
Anton Mellit, University of Vienna |
Sky Room |
12:00 – 2:30pm |
Lunch |
Bistro – 2^{nd} Floor |
2:30 – 3:30pm |
Fyodor Malikov, University of Southern California |
Sky Room |
3:30 – 4:00pm |
Coffee Break |
Bistro – 1^{st} 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.
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.
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 $i> min(deg(a),deg(b))$.
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.
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.
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.
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.
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.
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.
Gauge theory, vertex algebras and COHA
I will discuss some of the relations between supersymmetric gauge theories, vertex algebras and COHA.
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.
Pages
Scientific Organizers:
- Kevin Costello, Perimeter Institute
- Yan Soibelman, Kansas State University
- Yaping Yang, University of Melbourne