Categories of Line Defects and Cohomological Hall Algebras

TL;DR

This paper explores the relationship between line defect categories and Cohomological Hall Algebras in 4D supersymmetric quantum field theories, proposing a conjectural functor linking these structures and analyzing their properties.

Contribution

It introduces a conjectural monoidal functor from line defect categories to bimodule categories over BPS algebras, and studies its properties and examples.

Findings

Proposes a conjectural functor linking line defects to BPS algebra bimodules.

Describes images of simple objects under the functor and their monoidal structure.

Tests the conjecture using equivariant Witten indices.

Abstract

Any four-dimensional Supersymmetric Quantum Field Theory with eight supercharges can be associated to a monoidal category of BPS line defects. Any Coulomb vacuum of such a theory can be conjecturally associated to an ``algebra of BPS particles'', exemplified by certain Cohomological Hall Algebras. We conjecture the existence of a monoidal functor from the category of line defects to a certain category of bimodules for the BPS Algebra in any Coulomb vacuum. We describe images of simple objects under the conjectural functor and study their monoidal structure in examples. We conjecture that the functor may be an equivalence of dg-categories and test the conjecture at the level of the equivariant Witten indices of the spaces of morphisms.