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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

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.

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.

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

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).

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.

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.

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.

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.

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.

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).