Abstract
We introduce a class of 4-dimensional crystal melting models that count the BPS bound state of branes on toric Calabi-Yau 4-folds. The crystalline structure is determined by the brane brick model associated to the Calabi-Yau 4-fold under consideration or, equivalently, its dual periodic quiver. The crystals provide a discretized version of the underlying toric geometries. We introduce various techniques to visualize crystals and their melting configurations, including 3-dimensional slicing and Hasse diagrams. We illustrate the construction with the D0-D8 system on \({\mathbb{C}}\)4. Finally, we outline how our proposal generalizes to arbitrary toric CY 4-folds and general brane configurations.
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Acknowledgments
We would like to thank Yang-Hui He, Eduardo García-Valdecasas, Nikita Nekrasov, Nicolo Piazzalunga and, specially, Xingyang Yu for enjoyable and useful discussions. We are also grateful to Dongwook Ghim, Azeem Hasan, Sangmin Lee, Rak-Kyeong Seong and Cumrun Vafa for earlier collaborations on related topics. This work is supported by the U.S. National Science Foundation grants PHY-2112729 and DMS-1854179.
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Franco, S. 4d crystal melting, toric Calabi-Yau 4-folds and brane brick models. J. High Energ. Phys. 2024, 91 (2024). https://doi.org/10.1007/JHEP03(2024)091
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DOI: https://doi.org/10.1007/JHEP03(2024)091