Categories of Line Defects and Cohomological Hall Algebras

TL;DR

The work develops a comprehensive framework linking 4D ${ m N}=2$ theories to a monoidal category of BPS line defects via the Cohomological Hall Algebra (CoHA) as the model BPS algebra. It introduces a conjectural monoidal functor RG$_u$ that maps line defects to bimodules over the BPS algebra in a given Coulomb vacuum, realized concretely through framed quivers and spherical CoHA/bimodule constructions, and tested in explicit quivers ($A_1$, $A_2$, $A_3$) with PBW factorizations, spectrum generators, and tensor products. The paper provides detailed statements about stability, mutations, factorization, and IR-equivalence via Schur indices, arguing for a quasi-isomorphism between Line and the bimodule category and outlining refinements to incorporate PBW structures and $oldsymbol{ m \,Gamma}^ullet$-equivariance. These results offer a concrete algebraic route to understand line defects in non-Lagrangian theories and suggest deep links between IR data, wall-crossing, and categorical equivalences in 4D ${ m N}=2$ physics. The framework paves the way for systematic computations of framed BPS degeneracies, tensor products, and morphisms across multiple quiver variants through the spectrum generator and its mutations, with potential connections to Koszul duality and higher categorical dualities in twisted theories.

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.