# Branes, Quivers, and BPS Algebras 3 of 4

Date: Wed, Aug 25, 2021

Location: PIMS, University of Saskatchewan, Online, Zoom

Conference: 2nd PIMS Summer School on Algebraic Geometry in High Energy Physics

Subject: Mathematics, Algebraic Geometry, Physics, Particle Physics and Quantum Field Theory

Class: Scientific

### Abstract:

This series of lectures covers a brief introduction into some algebro-geometric techniques used in the construction of BPS algebras. The starting point of our construction is a physical picture of D0-branes bound to D-branes of higher dimension. Using methods of the derived category of coherent sheaves, we are going to derive a framed quiver with potential describing supersymmetric quantum mechanics capturing the low-energy behavior of such D0-branes. For a large class of quivers, we are going to identify the space of BPS states with different melted-crystal configurations. Finally, by employing correspondences, we are going to construct an action of a BPS algebra known as the affine Yangian on the space of BPS states. The action of the affine Yangian factors through the action of various vertex operator algebras, Cherednik algebras, and more. This construction leads to an enormously rich interplay between physics, geometry and representation theory.