BPS states meet generalized cohomology

A bstract In this note we review a construction of BPS Hilbert space in an effective supersymmetric quiver theory with 4 supercharges. We argue abstractly that contains elements equivariant generalized cohomology $$ {E}_G^{\ast}\left(-\right) E G ∗ − the representation moduli giving concretely Dolbeault cohomology, K-theory or elliptic depending on spacial slice is compactified to point, circle torus respectively, and something more amorphous other cases. Furthermore instantons — basic contributors interface defects Berry connection induce algebra spaces representing Fourier-Mukai transforms descending over as its representation. cases when describes toric Calabi-Yau three-fold (CY 3 ) respective generalization Yangian discussed literature, general it given by abstract cohomological Hall algebra.