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Crystal melting on toric surfaces. (English) Zbl 1226.81122

Summary: We study the relationship between the statistical mechanics of crystal melting and instanton counting in \(\mathcal N=4\) supersymmetric \(U(1)\) gauge theory on toric surfaces. We argue that, in contrast to their six-dimensional cousins, the two problems are related but not identical. We develop a vertex formalism for the crystal partition function, which calculates a generating function for the dimension 0 and 1 subschemes of the toric surface, and describe the modifications required to obtain the corresponding gauge theory partition function.

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
81T20 Quantum field theory on curved space or space-time backgrounds
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
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[1] Aganagic, M.; Klemm, A.; Marino, M.; Vafa, C., The topological vertex, Comm. Math. Phys., 254, 425-478 (2005) · Zbl 1114.81076
[2] Li, J.; Liu, C.-C. M.; Liu, K.; Zhou, J., A mathematical theory of the topological vertex, Geom. Topol., 13, 1, 527-621 (2009), arXiv:math/0408426 [math.AG] · Zbl 1184.14084
[3] Okounkov, A.; Reshetikhin, N.; Vafa, C., Quantum Calabi-Yau and classical crystals, Progr. Math., 244, 597-618 (2006) · Zbl 1129.81080
[4] Iqbal, A.; Nekrasov, N.; Okounkov, A.; Vafa, C., Quantum foam and topological strings, J. High Energy Phys., 04, 011 (2008) · Zbl 1246.81338
[5] Nekrasov, N., Instanton partition functions and M-theory, Japan J. Math., 46, 63-93 (2009) · Zbl 1181.81095
[6] Maulik, D.; Nekrasov, N.; Okounkov, A.; Pandharipande, R., Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math., 142, 5, 1263-1285 (2006), arXiv:math/0312059 [math.AG] · Zbl 1108.14046
[7] D. Maulik, A. Oblomkov, A. Okounkov, R. Pandharipande, Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds. arXiv:0809.3976; D. Maulik, A. Oblomkov, A. Okounkov, R. Pandharipande, Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds. arXiv:0809.3976 · Zbl 1232.14039
[8] Vafa, C.; Witten, E., A strong coupling test of \(S\)-duality, Nuclear Phys., B431, 3-77 (1994) · Zbl 0964.81522
[9] N.A. Nekrasov, Localizing gauge theories, Prepared for 14th International Congress on Mathematical Physics, ICMP 2003, Lisbon, Portugal, 28 July-2 August 2003.; N.A. Nekrasov, Localizing gauge theories, Prepared for 14th International Congress on Mathematical Physics, ICMP 2003, Lisbon, Portugal, 28 July-2 August 2003. · Zbl 1192.81235
[10] Fucito, F.; Morales, J. F.; Poghossian, R., Instanton on toric singularities and black hole countings, J. High Energy Phys., 12, 073 (2006) · Zbl 1226.81262
[11] Griguolo, L.; Seminara, D.; Szabo, R. J.; Tanzini, A., Black holes, instanton counting on toric singularities and \(q\)-deformed two-dimensional Yang-Mills theory, Nuclear Phys., B772, 1-24 (2007) · Zbl 1117.83064
[12] Bonelli, G.; Tanzini, A., Topological gauge theories on local spaces and black hole entropy countings, Adv. Theor. Math. Phys., 12, 6, 1429-1446 (2008), arXiv:0706.2633 [hep-th] · Zbl 1153.83352
[13] Dijkgraaf, R.; Sulkowski, P., Instantons on ALE spaces and orbifold partitions, J. High Energy Phys., 03, 013 (2008), arXiv:0712.1427 [hep-th]
[14] E. Gasparim, C.-C.M. Liu, The Nekrasov conjecture for toric surfaces. arXiv:0808.0884; E. Gasparim, C.-C.M. Liu, The Nekrasov conjecture for toric surfaces. arXiv:0808.0884 · Zbl 1194.14066
[15] M. Kool, Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces. arXiv:0906.3393; M. Kool, Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces. arXiv:0906.3393 · Zbl 1331.14023
[16] U. Bruzzo, R. Poghossian, A. Tanzini, Poincaré polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces. arXiv:0909.1458; U. Bruzzo, R. Poghossian, A. Tanzini, Poincaré polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces. arXiv:0909.1458 · Zbl 1216.81114
[17] Nakajima, H., Homology of moduli spaces of instantons on ALE spaces. I, J. Differential Geom., 40, 1, 105-127 (1994) · Zbl 0822.53019
[18] S. Fujii, S. Minabe, A combinatorial study on quiver varieties. arXiv:math/0510455; S. Fujii, S. Minabe, A combinatorial study on quiver varieties. arXiv:math/0510455 · Zbl 1376.14015
[19] Cirafici, M.; Sinkovics, A.; Szabo, R. J., Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory, Nuclear Phys., B809, 452-518 (2009), arXiv:0803.4188 [hep-th] · Zbl 1192.81309
[20] Hartshorne, R., (Algebraic Geometry. Algebraic Geometry, Graduate Texts in Mathematics, vol. 52 (1977), Springer-Verlag: Springer-Verlag New York) · Zbl 0367.14001
[21] Martin-Deschamps, M.; Perrin, D., Sur la classification des courbes gauches, Astérisque, 184-185 (1990)
[22] Dürr, M.; Kabanov, A.; Okonek, C., Poincaré invariants, Topology, 46, 3, 225-294 (2007), arXiv:math/0408131 [math.AG] · Zbl 1120.14034
[23] Dijkgraaf, R.; Moore, G. W.; Verlinde, E. P.; Verlinde, H. L., Elliptic genera of symmetric products and second quantized strings, Comm. Math. Phys., 185, 197-209 (1997) · Zbl 0872.32006
[24] Fogarty, J., Algebraic families on an algebraic surface, Amer. J. Math., 90, 511-521 (1968) · Zbl 0176.18401
[25] J. Stoppa, R.P. Thomas, Hilbert schemes and stable pairs: GIT and derived category wall crossings. arXiv:0903.1444; J. Stoppa, R.P. Thomas, Hilbert schemes and stable pairs: GIT and derived category wall crossings. arXiv:0903.1444 · Zbl 1243.14009
[26] Losev, A.; Moore, G. W.; Nekrasov, N.; Shatashvili, S., Four-dimensional avatars of two-dimensional RCFT, Nuclear Phys. Proc. Suppl., 46, 130-145 (1996) · Zbl 0957.81710
[27] Nekrasov, N.; Schwarz, A. S., Instantons on noncommutative \(R^{\ast \ast} 4\) and (2, 0) superconformal six dimensional theory, Comm. Math. Phys., 198, 689-703 (1998) · Zbl 0923.58062
[28] Kronheimer, P. B.; Nakajima, H., Yang-Mills instantons on ALE gravitational instantons, Math. Ann., 288, 2, 263-307 (1990) · Zbl 0694.53025
[29] Donaldson, S. K.; Kronheimer, P. B., (The Geometry of Four-Manifolds. The Geometry of Four-Manifolds, Oxford Mathematical Monographs (1990), Oxford University Press: Oxford University Press New York), Oxford Science Publications · Zbl 0820.57002
[30] Okonek, C.; Schneider, M.; Spindler, H., (Vector Bundles on Complex Projective Spaces. Vector Bundles on Complex Projective Spaces, Progress in Mathematics, vol. 3 (1980), Birkhäuser: Birkhäuser Boston, MA) · Zbl 0438.32016
[31] Harris, J.; Morrison, I., (Moduli of Curves. Moduli of Curves, Graduate Texts in Mathematics, vol. 187 (1998), Springer-Verlag: Springer-Verlag New York) · Zbl 0913.14005
[32] Bott, R.; Tu, L. W., (Differential Forms in Algebraic Topology. Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, vol. 82 (1982), Springer-Verlag: Springer-Verlag New York) · Zbl 0496.55001
[33] Griffiths, P.; Harris, J., (Principles of Algebraic Geometry. Principles of Algebraic Geometry, Wiley Classics Library (1994), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York), Reprint of the 1978 original · Zbl 0836.14001
[34] Göttsche, L., The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann., 286, 1-3, 193-207 (1990) · Zbl 0679.14007
[35] Ellingsrud, G.; Strømme, S. A., On the homology of the Hilbert scheme of points in the plane, Invent. Math., 87, 2, 343-352 (1987) · Zbl 0625.14002
[36] Atiyah, M. F.; Bott, R., The moment map and equivariant cohomology, Topology, 23, 1, 1-28 (1984) · Zbl 0521.58025
[37] Edidin, D.; Graham, W., Localization in equivariant intersection theory and the Bott residue formula, Amer. J. Math., 120, 3, 619-636 (1998) · Zbl 0980.14004
[38] Eisenbud, D.; Harris, J., (The Geometry of Schemes. The Geometry of Schemes, Graduate Texts in Mathematics, vol. 197 (2000), Springer-Verlag: Springer-Verlag New York) · Zbl 0960.14002
[39] Fulton, W., (Introduction to Toric Varieties. Introduction to Toric Varieties, Annals of Mathematics Studies, vol. 131 (1997), Princeton University Press)
[40] Hartshorne, R., Connectedness of the Hilbert scheme, Publ. Math. Inst. Hautes Études Sci., 29, 1, 7-48 (1966) · Zbl 0171.41502
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