Cohomological Hall Algebras in Mathematics and Physics

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

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.

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.

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 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.