# Cohomological Hall Algebras in Mathematics and Physics

Cohomological Hall Algebras in Mathematics and Physics

## On one example of a chiral Lie group

Friday Mar 01, 2019
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We quantize the Khesin-Zakharevich Poisson-Lie group of pseudo-differential symbols. This is a joint work with A.Linshaw

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## Fusion Hall algebra and shuffle conjectures

Friday Mar 01, 2019
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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.

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## TBA

Friday Mar 01, 2019
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## Connecting affine Yangians with W-algebras

Thursday Feb 28, 2019
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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

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## COHA and AGT for Spiked Instantons

Thursday Feb 28, 2019
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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.

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## Glueing W-algebras

Thursday Feb 28, 2019
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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.

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## Partition functions and the McKay correspondence

Thursday Feb 28, 2019
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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

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## An old-fashioned view of BPS-algebras

Wednesday Feb 27, 2019
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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.

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

Wednesday Feb 27, 2019
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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:
>

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## Comultiplications on cohomological Hall algebras and vertex algebras

Wednesday Feb 27, 2019
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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.

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## LECTURES ON-DEMAND

### Gabriela González, Louisiana State University

Speaker: Gabriela Gonzalez