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.

53 pages; v3 minor corrections, references added

References in corpus (21)

• Witten Index and Wall Crossing
• Brane Tilings
• Non-commutative Donaldson-Thomas theory and the conifold
• Brane Tilings and Their Applications
• Crystal Melting and Toric Calabi-Yau Manifolds
• Massive type IIA string theory cannot be strongly coupled
• Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions
• Brane Brick Models in the Mirror
• Quiver Yangian and Supersymmetric Quantum Mechanics
• Shifted Quiver Yangians and Representations from BPS Crystals
• Emergent Calabi-Yau Geometry
• Crystal Melting and Wall Crossing Phenomena
• A Note on Quiver Quantum Toroidal Algebra
• ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds
• Fano 3-Folds, Reflexive Polytopes and Brane Brick Models
• Quiver Yangians and Crystal Melting: A Concise Summary
• Multiple D4-D2-D0 on the Conifold and Wall-crossing with the Flop
• Brane Brick Models for the Sasaki-Einstein 7-Manifolds Y^{p,k}(CP^1 x CP^1) and Y^{p,k}(CP^2)
• Mass Deformations of Brane Brick Models
• Charging solid partitions
• On the Master Space for Brane Brick Models

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