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The Origin of Calabi-Yau Crystals in BPS States Counting

Jiakang Bao, Rak-Kyeong Seong, Masahito Yamazaki

TL;DR

The paper develops a unified localization-based approach to counting BPS D-brane bound states on general toric Calabi–Yau manifolds by evaluating Jeffrey–Kirwan residues of the flavored Witten index. It shows that BPS degeneracies are captured by crystal-melting models, with CY3 crystals reproducing known results and CY4 crystals requiring additional rational weights to encode full degeneracies. By extending to elliptic and rational generalizations, the work links BPS counting to generalized Donaldson–Thomas invariants and wall crossing, and reveals a rich chamber structure including quiver trialities. The analysis integrates brane brick models, 2d N=(0,2) quivers, and non-commutative DT4 frameworks, offering a comprehensive toolkit for understanding BPS spectra in fourfold geometries and their mathematical counterparts.

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.

The Origin of Calabi-Yau Crystals in BPS States Counting

TL;DR

The paper develops a unified localization-based approach to counting BPS D-brane bound states on general toric Calabi–Yau manifolds by evaluating Jeffrey–Kirwan residues of the flavored Witten index. It shows that BPS degeneracies are captured by crystal-melting models, with CY3 crystals reproducing known results and CY4 crystals requiring additional rational weights to encode full degeneracies. By extending to elliptic and rational generalizations, the work links BPS counting to generalized Donaldson–Thomas invariants and wall crossing, and reveals a rich chamber structure including quiver trialities. The analysis integrates brane brick models, 2d N=(0,2) quivers, and non-commutative DT4 frameworks, offering a comprehensive toolkit for understanding BPS spectra in fourfold geometries and their mathematical counterparts.

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.
Paper Structure (41 sections, 94 equations, 4 figures)

This paper contains 41 sections, 94 equations, 4 figures.

Table of Contents

  1. Introduction and Summary
  2. Toric CY4 and 2d N=(0,2) Theories
  3. Quivers.
  4. The $J$-and $E$-terms.
  5. Periodic Quivers.
  6. Brane Brick Models.
  7. Dimensional Reduction.
  8. Triality.
  9. The 4d Crystals from BPS States Countings
  10. JK Residues for Supersymmetric Indices
  11. The integrand: $Z_{\textrm{1-loop}}$
  12. The space $\mathfrak{M}$
  13. The Jeffrey-Kirwan Residue
  14. Remark
  15. Flags and JK Residues
  16. ...and 26 more sections

Figures (4)

  • Figure 1: Quiver diagram for the $\mathbb{C}^4$ brane brick model.
  • Figure 2: The periodic quiver for the $\mathbb{C}^4$ brane brick model.
  • Figure 3: Illustration of $J$- and $E$-term plaquettes that correspond to a single Fermi field in the brane brick model.
  • Figure 7: The wall crossing of the conifold$\times\mathbb{C}$ quiver. Here, we are only showing the chambers that descend from the cyclic chambers of the 4d $\mathcal{N}=1$ quiver. The labels indicate the multiplicities of the edges. These were also recently studied in Cao:2020huo.