Cohomological Hall Algebras in Mathematics and Physics

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.

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.

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).

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

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).

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.

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.

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.

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.

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: > > (1) The purely mathematical classification theory of perverse sheaves > on a complex plane with several singular points > (Gelfand-MacPherson-Vilonen). > > (2) The "infrared" analysis of > 2d supersymmetric theories (Gaiotto-Moore-Witten). > > I will explain this parallelism which suggests that the infrared > analysis should be formulated in terms of a perverse schober. This is > based on work in progress with Y. Soibelman and L. Soukhanov.

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).