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.
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Acknowledgments
We would like to thank Ioana Coman, Dmitry Galakhov, Wei Li and Yehao Zhou for enjoyable discussions. JB and MY would like to thank Ulsan National Institute of Science and Technology for hospitality. RKS would like to thank Kavli IPMU for hospitality.
JB is supported by a JSPS fellowship. RKS is supported by a Basic Research Grant of the National Research Foundation of Korea (NRF2022R1F1A1073128). He is also supported by a Start-up Research Grant for new faculty at UNIST (1.210139.01), a UNIST AI Incubator Grant (1.230038.01) and UNIST UBSI Grants (1.230168.01, 1.230078.01), as well as an Industry Research Project (2.220916.01) funded by Samsung SDS in Korea. He is also partly supported by the BK21 Program (“Next Generation Education Program for Mathematical Sciences”, 4299990414089) funded by the Ministry of Education in Korea and the National Research Foundation of Korea (NRF). MY is supported in part by the JSPS Grant-in-Aid for Scientific Research (No. 19H00689, 19K03820, 20H05860, 23H01168), and by JST, Japan (PRESTO Grant No. JPMJPR225A, Moonshot R&D Grant No. JPMJMS2061).
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Bao, J., Seong, RK. & Yamazaki, M. The origin of Calabi-Yau crystals in BPS states counting. J. High Energ. Phys. 2024, 140 (2024). https://doi.org/10.1007/JHEP03(2024)140
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DOI: https://doi.org/10.1007/JHEP03(2024)140