BPS states meet generalized cohomology

Abstract

Abstract In this note we review a construction of a BPS Hilbert space in an effective supersymmetric quiver theory with 4 supercharges. We argue abstractly that this space contains elements of an equivariant generalized cohomology theory $$ {E}_G^{\ast}\left(-\right) $$ E G ∗ − of the quiver representation moduli space giving concretely Dolbeault cohomology, K-theory or elliptic cohomology depending on the spacial slice is compactified to a point, a circle or a torus respectively, and something more amorphous in other cases. Furthermore BPS instantons — basic contributors to interface defects or a Berry connection — induce a BPS algebra on the BPS Hilbert spaces representing Fourier-Mukai transforms on the quiver representation moduli spaces descending to an algebra over $$ {E}_G^{\ast}\left(-\right) $$ E G ∗ − as its representation. In the cases when the quiver describes a toric Calabi-Yau three-fold (CY3) the algebra is a respective generalization of the quiver BPS Yangian algebra discussed in the literature, in more general cases it is given by an abstract generalized cohomological Hall algebra.