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The Origin of Calabi-Yau Crystals in BPS States Counting
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The Origin of Calabi-Yau Crystals in BPS States Counting
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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 the crystal does not contain the full information for the BPS degeneracy and we need to explicitly evaluate non-trivial weights assigned to the crystal configurations. Our discussions treat Calabi-Yau threefolds and fourfolds on equal footing, and include discussions on elliptic and rational generalizations of the BPS states counting, connections to the mathematical definition of generalized Donaldson-Thomas invariants, examples of wall crossings, and of trialities in quiver gauge theories.
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Cited by 1 Pith paper
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Quiver BPS Indices from Crystal Profiles
Elliptic genera and lower-dimensional BPS indices equal discrete sums over molecule boundaries in crystals defined by Jeffrey-Kirwan residues, generalizing Nekrasov Young-diagram formulas.
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