Branes, Quivers and BPS Algebras

TL;DR

This paper introduces algebraic geometric methods for constructing BPS algebras using derived categories, quiver representations, and crystal configurations, linking brane models to rich algebraic structures.

Contribution

It presents a novel framework connecting derived categories, quiver quantum mechanics, and algebraic structures like affine-Yangians in the context of BPS algebras.

Findings

Derived category models branes in toric Calabi-Yau three-folds

Vacua correspond to critical equivariant cohomology of quiver moduli

Construction of affine-Yangian representations from algebraic geometry correspondences

Abstract

These lecture notes cover a brief introduction into some of the algebro-geometric techniques used in the construction of BPS algebras. The first section introduces the derived category of coherent sheaves as a useful model of branes in toric Calabi-Yau three-folds. This model allows a rather simple derivation of quiver quantum mechanics describing low-energy dynamics of various brane systems. Vacua of such quantum mechanics can be identified with the critical equivariant cohomology of the moduli space of quiver representations. These are often counted by various crystal configurations. Using correspondences in algebraic geometry, one can construct rich families of affine-Yangian representations. We conclude with an exploration of different algebraic structures naturally appearing in our story. The material was covered in a 4-lecture mini-course within the Second PIMS Summer School on Algebraic Geometry in High-Energy Physics. The text contains some new ideas, examples and remarks that are going to be covered in detail in a joint work with Dylan Butson.