Affine BPS algebras and W algebras

Abstract: One may associate to a quiver Q with potential W a certain Lie algebra, called the BPS Lie algebra. On the one hand this Lie algebra generates the Kontsevich-Soibelman cohomological Hall algebra (CoHA) associated to Q and W under a Yangian-type PBW theorem, and on the other hand it partially categorifies the BPS invariants of the Jacobi algebra associated to (Q,W). For special choices of (Q,W), the resulting cohomological Hall algebra is isomorphic to the cohomological Hall algebra studied by Schiffamnn and Vasserot in their solution of the AGT conjecture. I will explain how a special case of a joint result with Kinjo enables us to affinize the BPS Lie algebra for these CoHAs, and express the CoHA as a universal enveloping algebra. I will explain how for the three-loop quiver with its canonical cubic potential, a one-parameter deformation of the affine BPS Lie algebra recovers one half of the Lie algebra of differential operators on the complex torus. In particular, this implies that the associated CoHA is spherically generated, so that we can use a result of Rapčák, Soibelman, Yang and Zhou to completely describe the fully deformed version of this algebra in terms of half of the affine Yangian of gl(1).

algebraic geometrysymplectic geometry

Audience: researchers in the topic

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