Abstract
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.
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Acknowledgments
We would like to thank Alexey Litvinov for useful comments on the draft. Our work is partly supported by grants RFBR 21-51-46010 ST_a (D.G., A.M., N.T.), by the grants of the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” (A.M., N.T.). This research was also partly supported by the Ministry of Science and Higher Education of the Russian Federation, agreement 075-15-2022-289 date 06/04/2022 (D.G., N.T.).
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Galakhov, D., Morozov, A. & Tselousov, N. Wall-crossing effects on quiver BPS algebras. J. High Energ. Phys. 2024, 118 (2024). https://doi.org/10.1007/JHEP05(2024)118
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DOI: https://doi.org/10.1007/JHEP05(2024)118