Wall-Crossing Effects on Quiver BPS Algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910443823431680 |
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| author | Galakhov, Dmitry Morozov, Alexei Tselousov, Nikita |
| author_facet | Galakhov, Dmitry Morozov, Alexei Tselousov, Nikita |
| contents | 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 $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. 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 $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This observation supports an earlier conjecture that the BPS algebraic structures can be considered as new wall-crossing invariants. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_14600 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wall-Crossing Effects on Quiver BPS Algebras Galakhov, Dmitry Morozov, Alexei Tselousov, Nikita High Energy Physics - Theory Mathematical Physics Algebraic Geometry Quantum Algebra Representation Theory 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 $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. 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 $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This observation supports an earlier conjecture that the BPS algebraic structures can be considered as new wall-crossing invariants. |
| title | Wall-Crossing Effects on Quiver BPS Algebras |
| topic | High Energy Physics - Theory Mathematical Physics Algebraic Geometry Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2403.14600 |