Cohomological Hall algebras and categories of line operators of 4d, N= 2 gauge theories

We study standard and costandard modules of the cohomological Hall algebra associated with a quiver with potential. We will discuss their properties, such as equivariant formality and highest weight generation. As an application, we prove a Kirwan surjectivity statement for the nilpotent Grothendieck-Springer fiber in the Grassmannian.

We then use these representation-theoretic results to construct a triangulated category equipped with a monoidal structure, which we call the category of lines. For a quiver of ADE Dynkin type, we establish a monoidal equivalence between the category of lines and the category of 1/2 BPS line operators in 4d, N=2 pure gauge theory, as constructed by Braverman, Finkelberg, and Nakajima and further developed by Cautis and Williams.

This talk is based on ongoing joint work with Ryo Fujita, Yan Soibelman, and Gufang Zhao.

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