Shivang Jindal (EPFL)

Title: The Loop-Nilpotent Cohomological Hall Algebra and Coulomb Branches

Abstract: Given a quiver with potential, Kontsevich and Soibelman defined its cohomological Hall algebra (CoHA) as a mathematical model for the algebra of BPS states. Since then, CoHAs have been found to have connections with several areas of mathematics, including cohomological Donaldson–Thomas theory, quantum groups, non-abelian Hodge theory, and cluster algebras. In certain cases, dimensional reduction identifies the CoHA with a preprojective CoHA, an algebra that appears naturally in the Schiffmann–Vasserot proof of the AGT conjecture and in the study of Nakajima quiver varieties.

The aim of this talk is to introduce a mix of these constructions, which we call the loop-nilpotent CoHA. We will describe this algebra using shuffle algebras. As a consequence, we characterize the preprojective BPS Lie algebra as a Lie subalgebra of a space of symmetric polynomials cut out by explicit degree and divisibility conditions. We then show that the loop-nilpotent CoHA is a universal algebra admitting a surjective homomorphism onto one half of a quantized Coulomb-branch algebra. This is joint work with Andrei Neguț.