Abstract
We construct a two-dimensional crystal melting model which reproduces the BPS index of D2-D0 states bound to a non-compact D4-brane on an arbitrary toric CalabiYau singularity. The crystalline structure depends on the toric divisor wrapped by the D4-brane. The molten crystals are in one-to-one correspondence with the torus fixed points of the moduli space of the quiver gauge theory on D-branes. The F- and D-term constraints of the gauge theory are regarded as a generalization of the ADHM constraints on instantons. We also show in several examples that our model is consistent with the wall-crossing formula for the BPS index.
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M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, Progr. Math. 244 (2006) 597 [hep-th/0309208] [INSPIRE].
B. Szendröi, Non-commutative Donaldson-Thomas theory and the conifold, Geom. Topol. 12 (2008) 1171 [arXiv:0705.3419] [INSPIRE].
S. Mozgovoy and M. Reineke, On the noncommutative Donaldson-Thomas invariants arising from brane tilings, arXiv:0809.0117 [INSPIRE].
H. Ooguri and M. Yamazaki, Crystal melting and toric Calabi-Yau manifolds, Commun. Math. Phys. 292 (2009) 179 [arXiv:0811.2801] [INSPIRE].
M. Aganagic and K. Schaeffer, Wall crossing, quivers and crystals, JHEP 10 (2012) 153 [arXiv:1006.2113] [INSPIRE].
W.-y. Chuang and D.L. Jafferis, Wall crossing of BPS states on the conifold from Seiberg duality and pyramid partitions, Commun. Math. Phys. 292 (2009) 285 [arXiv:0810.5072] [INSPIRE].
W.-y. Chuang and G. Pan, BPS state counting in local obstructed curves from quiver theory and Seiberg duality, J. Math. Phys. 51 (2010) 052305 [arXiv:0908.0360] [INSPIRE].
P. Sulkowski, Wall-crossing, free fermions and crystal melting, Commun. Math. Phys. 301 (2011) 517 [arXiv:0910.5485] [INSPIRE].
T. Dimofte and S. Gukov, Refined, motivic and quantum, Lett. Math. Phys. 91 (2010) 1 [arXiv:0904.1420] [INSPIRE].
A. Iqbal, N. Nekrasov, A. Okounkov and C. Vafa, Quantum foam and topological strings, JHEP 04 (2008) 011 [hep-th/0312022] [INSPIRE].
H. Ooguri and M. Yamazaki, Emergent Calabi-Yau geometry, Phys. Rev. Lett. 102 (2009) 161601 [arXiv:0902.3996] [INSPIRE].
T. Nishinaka and S. Yamaguchi, Statistical model and BPS D4-D2-D0 counting, JHEP 05 (2011) 072 [arXiv:1102.2992] [INSPIRE].
T. Nishinaka and Y. Yoshida, A note on statistical model for BPS D4-D2-D0 states, Phys. Lett. B 711 (2012) 132 [arXiv:1108.4326] [INSPIRE].
H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lectures Series, American Mathematical Society (1999).
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [INSPIRE].
S. Franco, A. Hanany, K.D. Kennaway, D. Vegh and B. Wecht, Brane dimers and quiver gauge theories, JHEP 01 (2006) 096 [hep-th/0504110] [INSPIRE].
S. Franco et al., Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [INSPIRE].
A. Hanany and D. Vegh, Quivers, tilings, branes and rhombi, JHEP 10 (2007) 029 [hep-th/0511063] [INSPIRE].
K.D. Kennaway, Brane tilings, Int. J. Mod. Phys. A 22 (2007) 2977 [arXiv:0706.1660] [INSPIRE].
M. Yamazaki, Brane tilings and their applications, Fortschr. Phys. 56 (2008) 555 [arXiv:0803.4474] [INSPIRE].
S. Franco and A.M. Uranga, Dynamical SUSY breaking at meta-stable minima from D-branes at obstructed geometries, JHEP 06 (2006) 031 [hep-th/0604136] [INSPIRE].
N. Broomhead, Dimer models and Calabi-Yau algebras, arXiv:0901.4662 [INSPIRE].
F. Fucito, J.F. Morales and R. Poghossian, Multi instanton calculus on ALE spaces, Nucl. Phys. B 703 (2004) 518 [hep-th/0406243] [INSPIRE].
S. Fujii and S. Minabe, A combinatorial study on quiver varieties, math.AG/0510455 [INSPIRE].
R. Dijkgraaf and P. Sulkowski, Instantons on ALE spaces and orbifold partitions, JHEP 03 (2008) 013 [arXiv:0712.1427] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
M. Cirafici, A.-K. Kashani-Poor and R.J. Szabo, Crystal melting on toric surfaces, J. Geom. Phys. 61 (2011) 2199 [arXiv:0912.0737] [INSPIRE].
R.J. Szabo, Crystals, instantons and quantum toric geometry, Acta Phys. Polon. Supp. 4 (2011) 461 [arXiv:1102.3861] [INSPIRE].
M. Cirafici and R.J. Szabo, Curve counting, instantons and McKay correspondences, J. Geom. Phys. 72 (2013) 54 [arXiv:1209.1486] [INSPIRE].
W.-y. Chuang, D.-E. Diaconescu, J. Manschot, G.W. Moore and Y. Soibelman, Geometric engineering of (framed) BPS states, arXiv:1301.3065 [INSPIRE].
T. Maeda, T. Nakatsu, K. Takasaki and T. Tamakoshi, Five-dimensional supersymmetric Yang-Mills theories and random plane partitions, JHEP 03 (2005) 056 [hep-th/0412327] [INSPIRE].
T. Maeda, T. Nakatsu, K. Takasaki and T. Tamakoshi, Free fermion and Seiberg-Witten differential in random plane partitions, Nucl. Phys. B 715 (2005) 275 [hep-th/0412329] [INSPIRE].
T. Maeda, T. Nakatsu, Y. Noma and T. Tamakoshi, Gravitational quantum foam and supersymmetric gauge theories, Nucl. Phys. B 735 (2006) 96 [hep-th/0505083] [INSPIRE].
T. Maeda and T. Nakatsu, Amoebas and instantons, Int. J. Mod. Phys. A 22 (2007) 937 [hep-th/0601233] [INSPIRE].
T. Nakatsu and K. Takasaki, Melting crystal, quantum torus and Toda hierarchy, Commun. Math. Phys. 285 (2009) 445 [arXiv:0710.5339] [INSPIRE].
S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS operators in gauge theories: quivers, syzygies and plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].
Y. Noma, T. Nakatsu and T. Tamakoshi, Plethystics and instantons on ALE spaces, hep-th/0611324 [INSPIRE].
D. Forcella, A. Hanany and A. Zaffaroni, Baryonic generating functions, JHEP 12 (2007) 022 [hep-th/0701236] [INSPIRE].
A. Butti, D. Forcella, A. Hanany, D. Vegh and A. Zaffaroni, Counting chiral operators in quiver gauge theories, JHEP 11 (2007) 092 [arXiv:0705.2771] [INSPIRE].
H. Ooguri and C. Vafa, Two-dimensional black hole and singularities of CY manifolds, Nucl. Phys. B 463 (1996) 55 [hep-th/9511164] [INSPIRE].
A. Hanany and A.M. Uranga, Brane boxes and branes on singularities, JHEP 05 (1998) 013 [hep-th/9805139] [INSPIRE].
B. Feng, Y.-H. He, K.D. Kennaway and C. Vafa, Dimer models from mirror symmetry and quivering amoebae, Adv. Theor. Math. Phys. 12 (2008) 489 [hep-th/0511287] [INSPIRE].
Y. Imamura, Anomaly cancellations in brane tilings, JHEP 06 (2006) 011 [hep-th/0605097] [INSPIRE].
A.D. King, Moduli of representations of finite dimensional algebras, Q. J. Math. 45 (1994) 515.
K. Larjo, Consistently melting crystals, arXiv:0902.0614 [INSPIRE].
A. Ishii and K. Ueda, A note on consistency conditions on dimer models, RIMS Kokyuroku Bessatsu B 24 (2011) 143 [arXiv:1012.5449].
D.R. Gulotta, Properly ordered dimers, R-charges and an efficient inverse algorithm, JHEP 10 (2008) 014 [arXiv:0807.3012] [INSPIRE].
S. Franco and D. Vegh, Moduli spaces of gauge theories from dimer models: proof of the correspondence, JHEP 11 (2006) 054 [hep-th/0601063] [INSPIRE].
A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].
S. Shadchin, On F-term contribution to effective action, JHEP 08 (2007) 052 [hep-th/0611278] [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
Y. Yoshida, Localization of vortex partition functions in \( \mathcal{N} \) = (2, 2) super Yang-Mills theory, arXiv:1101.0872 [INSPIRE].
A. Hanany, C.P. Herzog and D. Vegh, Brane tilings and exceptional collections, JHEP 07 (2006) 001 [hep-th/0602041] [INSPIRE].
A. King, Instantons and holomorphic bundles on the blown-up plane, Ph.D. Thesis, Oxford University, Oxford U.K. (1989).
H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blow-up. I. A quiver description, arXiv:0802.3120 [INSPIRE].
T. Nishinaka and S. Yamaguchi, Wall-crossing of D4-D2-D0 and flop of the conifold, JHEP 09 (2010) 026 [arXiv:1007.2731] [INSPIRE].
T. Nishinaka, Multiple D4-D2-D0 on the conifold and wall-crossing with the flop, JHEP 06 (2011) 065 [arXiv:1010.6002] [INSPIRE].
T. Nishinaka and S. Yamaguchi, Affine SU(N) algebra from wall-crossings, arXiv:1107.4762 [INSPIRE].
T. Nishinaka, T. Okada, T. Okazaki and S. Yamaguchi, Evidence for duality of conifold from fundamental string, JHEP 10 (2011) 051 [arXiv:1107.3660] [INSPIRE].
D.L. Jafferis and G.W. Moore, Wall crossing in local Calabi Yau manifolds, arXiv:0810.4909 [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings — I, hep-th/9809187 [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings — II, hep-th/9812127 [INSPIRE].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].
M. Aganagic, H. Ooguri, N. Saulina and C. Vafa, Black holes, q-deformed 2d Yang-Mills and non-perturbative topological strings, Nucl. Phys. B 715 (2005) 304 [hep-th/0411280] [INSPIRE].
M. Aganagic, D. Jafferis and N. Saulina, Branes, black holes and topological strings on toric Calabi-Yau manifolds, JHEP 12 (2006) 018 [hep-th/0512245] [INSPIRE].
L. Griguolo, D. Seminara, R.J. Szabo and A. Tanzini, Black holes, instanton counting on toric singularities and q-deformed two-dimensional Yang-Mills theory, Nucl. Phys. B 772 (2007) 1 [hep-th/0610155] [INSPIRE].
H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994) 365.
H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [INSPIRE].
R. Kenyon and J.-M. Schlenker, Rhombic embeddings of planar quad-graphs, Trans. Am. Math. Soc. 357 (2005) 3443 [math-ph/0305057].
B. Feng, A. Hanany and Y.-H. He, Phase structure of D-brane gauge theories and toric duality, JHEP 08 (2001) 040 [hep-th/0104259] [INSPIRE].
Y. Imamura, K. Kimura and M. Yamazaki, Anomalies and O-plane charges in orientifolded brane tilings, JHEP 03 (2008) 058 [arXiv:0801.3528] [INSPIRE].
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Nishinaka, T., Yamaguchi, S. & Yoshida, Y. Two-dimensional crystal melting and D4-D2-D0 on toric Calabi-Yau singularities. J. High Energ. Phys. 2014, 139 (2014). https://doi.org/10.1007/JHEP05(2014)139
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DOI: https://doi.org/10.1007/JHEP05(2014)139