The Origin of Calabi-Yau Crystals in BPS States Counting

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 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.

arXiv comment53 pages; v3 minor corrections, references added

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Cited by in corpus (10)

• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• Wall-Crossing Effects on Quiver BPS Algebras
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues
• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• An Overview of Crystals and Double Quiver Yangians
• The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue
• Weyl Mutations in Quiver Yangians
• Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations