The origin of Calabi-Yau crystals in BPS states counting

Bao, Jiakang; Seong, Rak-Kyeong; Yamazaki, Masahito

doi:10.1007/jhep03(2024)140

J. High Energ. Phys.

2024

DOI: 10.1007/jhep03(2024)140

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The origin of Calabi-Yau crystals in BPS states counting

Jiakang Bao,

Rak-Kyeong Seong,

Masahito Yamazaki

Abstract: We study the counting problem of BPS D-branes wrapping holomorphic cycles of a general toric Calabi-Yau manifold. We evaluate the Jeffrey-Kirwan residues for the flavoured Witten index for the supersymmetric quiver quantum mechanics on the worldvolume of the D-branes, and find that BPS degeneracies are described by a statistical mechanical model of crystal melting. For Calabi-Yau threefolds, we reproduce the crystal melting models long known in the literature. For Calabi-Yau fourfolds, however, we find that th… Show more

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Wall-crossing effects on quiver BPS algebras

Galakhov,

Morozov,

Tselousov

2024

J. High Energ. Phys.

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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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Wall-crossing effects on quiver BPS algebras

Galakhov,

Morozov,

Tselousov

2024

J. High Energ. Phys.

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Simple representations of BPS algebras: the case of $$Y(\widehat{\mathfrak {gl}}_2)$$

Galakhov,

Morozov,

Tselousov

2024

Eur. Phys. J. C

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BPS algebras are the symmetries of a wide class of brane-inspired models. They are closely related to Yangians – the peculiar and somewhat sophisticated limit of DIM algebras. Still they possess some simple and explicit representations. We explain here that for $$Y(\widehat{\mathfrak {gl}}_r)$$ Y ( gl ^ r ) these representations are related to Uglov polynomials, whose families are also labeled by natural r. They arise in the limit $$\hbar {\longrightarrow } 0$$ ħ ⟶ 0 from Macdonald polynomials, and generalize the well-known Jack polynomials ($$\beta $$ β -deformation of Schur functions), associated with $$r=1.$$ r = 1 . For $$r=2$$ r = 2 they approximate Macdonald polynomials with the accuracy $$O(\hbar ^2),$$ O ( ħ 2 ) , so that they are eigenfunctions of two immediately available commuting operators, arising from the $$\hbar $$ ħ -expansion of the first Macdonald Hamiltonian. These operators have a clear structure, which is easily generalizable, – what provides a technically simple way to build an explicit representation of Yangian $$Y(\widehat{\mathfrak {gl}}_2),$$ Y ( gl ^ 2 ) , where $$U^{(2)}$$ U ( 2 ) are associated with the states $$|\lambda {\rangle },$$ | λ ⟩ , parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables $$p_{2n+1}$$ p 2 n + 1 can be expressed through mutually commuting operators from Yangian, however even time-variables $$p_{2n}$$ p 2 n are inexpressible. Implications to higher r become now straightforward, yet we describe them only in a sketchy way.

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The origin of Calabi-Yau crystals in BPS states counting

Cited by 2 publications

References 68 publications

Wall-crossing effects on quiver BPS algebras

Wall-crossing effects on quiver BPS algebras

Simple representations of BPS algebras: the case of $$Y(\widehat{\mathfrak {gl}}_2)$$

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