Abstract
Solid partitions are the 4D generalization of the plane partitions in 3D and Young diagrams in 2D, and they can be visualized as stacking of 4D unit-size boxes in the positive corner of a 4D room. Physically, solid partitions arise naturally as 4D molten crystals that count equivariant D-brane BPS states on the simplest toric Calabi-Yau fourfold, ℂ4, generalizing the 3D statement that plane partitions count equivariant D-brane BPS states on ℂ3. In the construction of BPS algebras for toric Calabi-Yau threefolds, the so-called charge function on the 3D molten crystal is an important ingredient — it is the generating function for the eigenvalues of an infinite tower of Cartan elements of the algebra. In this paper, we derive the charge function for solid partitions. Compared to the 3D case, the new feature is the appearance of contributions from certain 4-box and 5-box clusters, which will make the construction of the corresponding BPS algebra much more complicated than in the 3D.
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J.A. Harvey and G.W. Moore, Algebras, BPS states, and strings, Nucl. Phys. B 463 (1996) 315 [hep-th/9510182] [INSPIRE].
J.A. Harvey and G.W. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998) 489 [hep-th/9609017] [INSPIRE].
M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Num. Theor. Phys. 5 (2011) 231 [arXiv:1006.2706] [INSPIRE].
H. Ooguri and M. Yamazaki, Crystal Melting and Toric Calabi-Yau Manifolds, Commun. Math. Phys. 292 (2009) 179 [arXiv:0811.2801] [INSPIRE].
H. Ooguri, P. Sulkowski and M. Yamazaki, Wall Crossing As Seen By Matrix Models, Commun. Math. Phys. 307 (2011) 429 [arXiv:1005.1293] [INSPIRE].
W. Li and M. Yamazaki, Quiver Yangian from Crystal Melting, JHEP 11 (2020) 035 [arXiv:2003.08909] [INSPIRE].
M. Rapcak, Y. Soibelman, Y. Yang and G. Zhao, Cohomological Hall algebras, vertex algebras and instantons, Commun. Math. Phys. 376 (2019) 1803 [arXiv:1810.10402] [INSPIRE].
M. Rapcak, Branes, Quivers and BPS Algebras, arXiv:2112.13878 [INSPIRE].
N. Nekrasov, Magnificent four, Adv. Theor. Math. Phys. 24 (2020) 1171 [arXiv:1712.08128] [INSPIRE].
M.R. Douglas, Branes within branes, NATO Sci. Ser. C 520 (1999) 267 [hep-th/9512077] [INSPIRE].
M.R. Douglas and G.W. Moore, D-branes, quivers, and ALE instantons, hep-th/9603167 [INSPIRE].
H. Awata and H. Kanno, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [INSPIRE].
H. Kanno, Quiver matrix model of ADHM type and BPS state counting in diverse dimensions, PTEP 2020 (2020) 11B104 [arXiv:2004.05760] [INSPIRE].
N. Nekrasov and N. Piazzalunga, Magnificent Four with Colors, Commun. Math. Phys. 372 (2019) 573 [arXiv:1808.05206] [INSPIRE].
G. Bonelli, N. Fasola, A. Tanzini and Y. Zenkevich, ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds, J. Geom. Phys. 191 (2023) 104910 [arXiv:2011.02366] [INSPIRE].
R.J. Szabo and M. Tirelli, Instanton Counting and Donaldson-Thomas Theory on Toric Calabi-Yau Four-Orbifolds, arXiv:2301.13069 [INSPIRE].
T. Kimura, Double Quiver Gauge Theory and BPS/CFT Correspondence, SIGMA 19 (2023) 039 [arXiv:2212.03870] [INSPIRE].
N. Piazzalunga, The one-legged K-theoretic vertex of fourfolds from 3d gauge theory, arXiv:2306.12405 [INSPIRE].
N. Nekrasov and N. Piazzalunga, Global magni4icence, or: 4G Networks, arXiv:2306.12995 [INSPIRE].
T. Kimura and G. Noshita, Gauge origami and quiver W-algebras, arXiv:2310.08545 [INSPIRE].
Y. Cao and M. Kool, Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds, Adv. Math. 338 (2018) 601 [arXiv:1712.07347] [INSPIRE].
Y. Cao and M. Kool, Counting zero-dimensional subschemes in higher dimensions, J. Geom. Phys. 136 (2019) 119 [arXiv:1805.04746] [INSPIRE].
Y. Cao and M. Kool, Curve counting and DT/PT correspondence for Calabi-Yau 4-folds, Adv. Math. 375 (2020) 107371 [arXiv:1903.12171] [INSPIRE].
Y. Cao and G. Zhao, Quasimaps to quivers with potentials, arXiv:2306.01302 [INSPIRE].
S. Franco, D. Ghim, S. Lee, R.-K. Seong and D. Yokoyama, 2d (0, 2) Quiver Gauge Theories and D-Branes, JHEP 09 (2015) 072 [arXiv:1506.03818] [INSPIRE].
S. Franco, S. Lee and R.-K. Seong, Brane Brick Models, Toric Calabi-Yau 4-Folds and 2d (0, 2) Quivers, JHEP 02 (2016) 047 [arXiv:1510.01744] [INSPIRE].
S. Franco and A. Hasan, Graded Quivers, Generalized Dimer Models and Toric Geometry, JHEP 11 (2019) 104 [arXiv:1904.07954] [INSPIRE].
S. Franco and X. Yu, BFT2: a general class of 2d \( \mathcal{N} \) = (0, 2) theories, 3-manifolds and toric geometry, JHEP 08 (2022) 277 [arXiv:2107.00667] [INSPIRE].
R.J. Szabo and M. Tirelli, Noncommutative Instantons in Diverse Dimensions, arXiv:2207.12862 [INSPIRE].
B. Szendroi, Non-commutative Donaldson-Thomas invariants and the conifold, Geom. Topol. 12 (2008) 1171 [arXiv:0705.3419] [INSPIRE].
A.D. King, Moduli of representations of finite dimensional algebras, Q. J. Math. 45 (1994) 515.
A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, Prog. Math. 244 (2006) 597 [hep-th/0309208] [INSPIRE].
A. Iqbal, N. Nekrasov, A. Okounkov and C. Vafa, Quantum foam and topological strings, JHEP 04 (2008) 011 [hep-th/0312022] [INSPIRE].
K. Nagao and H. Nakajima, Counting invariant of perverse coherent sheaves and its wall-crossing, arXiv:0809.2992 [INSPIRE].
S. Mozgovoy and M. Reineke, On the noncommutative Donaldson-Thomas invariants arising from brane tilings, arXiv:0809.0117 [INSPIRE].
M. Yamazaki, Crystal Melting and Wall Crossing Phenomena, Int. J. Mod. Phys. A 26 (2011) 1097 [arXiv:1002.1709] [INSPIRE].
Sequence A000293 at the OEIS, https://oeis.org/A000293.
D. Galakhov and M. Yamazaki, Quiver Yangian and Supersymmetric Quantum Mechanics, Commun. Math. Phys. 396 (2022) 713 [arXiv:2008.07006] [INSPIRE].
S. Franco, 4d Crystal Melting, Toric Calabi-Yau 4-Folds and Brane Brick Models, arXiv:2311.04404 [INSPIRE].
R. Kenyon, A. Okounkov and S. Sheffield, Dimers and amoebae, math-ph/0311005 [INSPIRE].
R. Dijkgraaf, D. Orlando and S. Reffert, Quantum Crystals and Spin Chains, Nucl. Phys. B 811 (2009) 463 [arXiv:0803.1927] [INSPIRE].
T. Procházka, \( \mathcal{W} \)-symmetry, topological vertex and affine Yangian, JHEP 10 (2016) 077 [arXiv:1512.07178] [INSPIRE].
C. Closset, J. Guo and E. Sharpe, B-branes and supersymmetric quivers in 2d, JHEP 02 (2018) 051 [arXiv:1711.10195] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, (0, 2) trialities, JHEP 03 (2014) 076 [arXiv:1310.0818] [INSPIRE].
Acknowledgments
We would like to thank Masahito Yamazaki for the initial collaboration on this project, and also Alexei Morozov, Alexander Popolitov and Nikita Tselousov for helpful discussions. We would also like to thank Alexander Popolitov for his invaluable help with handling computer hardware issues. The work of DG is supported by the Russian Science Foundation (Grant No.20-12-00195). The work of WL is supported by NSFC No. 11875064, No. 12275334, No. 11947302, and the Max-Planck Partnergruppen fund; WL is also grateful for the support and hospitality of the Max Planck Institute for Gravitational Physics, the Kavli Institute for Theoretical Physics, Perimeter Institute, and the Issac Newton Institute for Mathematical Sciences (during Program “Black holes: bridges between number theory and holographic quantum information” with EPSRC Grant ER/R014604/1) when part of this work was carried out.
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Galakhov, D., Li, W. Charging solid partitions. J. High Energ. Phys. 2024, 43 (2024). https://doi.org/10.1007/JHEP01(2024)043
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DOI: https://doi.org/10.1007/JHEP01(2024)043