Wall-crossing effects on quiver BPS algebras

BPS states in supersymmetric theories can admit additional algebro-geometric structures in their spectra, described as quiver Yangian algebras. Equivariant fixed points on the quiver variety are interpreted as vectors populating a representation module, and matrix elements for the generators are then defined as Duistermaat-Heckman integrals in the vicinity of these points. The well-known wall-crossing phenomena are that the fixed point spectrum establishes a dependence on the stability (Fayet-Illiopolous) parameters ζ, jumping abruptly across the walls of marginal stability, which divide the ζ-space into a collection of stability chambers — “phases” of the theory. The standard construction of the quiver Yangian algebra relies heavily on the molten crystal model, valid in a sole cyclic chamber where all the ζ-parameters have the same sign. We propose to lift this restriction and investigate the effects of the wall-crossing phenomena on the quiver Yangian algebra and its representations — starting with the example of affine super-Yangian \({\text{Y}}\left({\widehat{\mathfrak{g}\mathfrak{l}}}_{1\left|1\right.}\right)\) . In addition to the molten crystal construction more general atomic structures appear, in other non-cyclic phases (chambers of the ζ-space). We call them glasses and also divide in a few different classes. For some of the new phases we manage to associate an algebraic structure again as a representation of the same affine Yangian \({\text{Y}}\left({\widehat{\mathfrak{g}\mathfrak{l}}}_{1\left|1\right.}\right)\) . This observation supports an earlier conjecture that the BPS algebraic structures can be considered as new wall-crossing invariants.