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Communications in Mathematical Physics

, Volume 361, Issue 1, pp 155–204 | Cite as

Calabi–Yau Volumes and Reflexive Polytopes

  • Yang-Hui He
  • Rak-Kyeong Seong
  • Shing-Tung Yau
Open Access
Article

Abstract

We study various geometrical quantities for Calabi–Yau varieties realized as cones over Gorenstein Fano varieties, obtained as toric varieties from reflexive polytopes in various dimensions. Focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the Sasaki–Einstein base of the corresponding Calabi–Yau cone are calculated. By doing so, we conjecture new bounds for the Sasaki–Einstein volume with respect to various topological quantities of the corresponding toric varieties. We give interpretations about these volume bounds in the context of associated field theories via the AdS/CFT correspondence.

References

  1. 1.
    Batyrev, V.V.: Toroidal Fano 3-folds. Math. USSR Izv. 19, 13 (1982)Google Scholar
  2. 2.
    Batyrev, V.V., Borisov, L.A.: On Calabi–Yau complete intersections in toric varieties. arXiv:alg-geom/9412017
  3. 3.
    Kreuzer M., Skarke H.: On the classification of reflexive polyhedra. Commun. Math. Phys. 185, 495–508 (1997) arXiv:hep-th/9512204 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Kreuzer M., Skarke H.: Classification of reflexive polyhedra in three-dimensions. Adv. Theor. Math. Phys. 2, 847–864 (1998) arXiv:hep-th/9805190 CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Kreuzer M., Skarke H.: Complete classification of reflexive polyhedra in four-dimensions. Adv. Theor. Math. Phys. 4, 1209–1230 (2002) arXiv:hep-th/0002240 CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Kreuzer M., Skarke H.: PALP: a package for analyzing lattice polytopes with applications to toric geometry. Comput. Phys. Commun. 157, 87–106 (2004) arXiv:math/0204356 ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Batyrev V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3, 493–545 (1994) arXiv:alg-geom/9310003 zbMATHMathSciNetGoogle Scholar
  8. 8.
    Batyrev, V.V.: On the Classification of Toric Fano 4-folds. ArXiv Mathematics e-prints (Jan., 1998). arXiv:math/9801107
  9. 9.
  10. 10.
    Braun, A.P., Walliser, N.-O.: A new offspring of PALP. arXiv:1106.4529
  11. 11.
    Altman R., Gray J., He Y.-H., Jejjala V., Nelson B.D.: A Calabi–Yau Database: threefolds constructed from the Kreuzer–Skarke list. JHEP 02, 158 (2015) arXiv:1411.1418 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    “Toric Calabi–Yau database.” http://www.rossealtman.com/
  13. 13.
    Candelas P., de la Ossa X., Katz S.H.: Mirror symmetry for Calabi–Yau hypersurfaces in weighted P**4 and extensions of Landau–Ginzburg theory. Nucl. Phys. B 450, 267–292 (1995) arXiv:hep-th/9412117 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Maldacena J.M.: The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113–1133 (1999) arXiv:hep-th/9711200 CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Morrison D.R., Plesser M.R.: Nonspherical horizons. 1. Adv. Theor. Math. Phys. 3, 1–81 (1999) arXiv:hep-th/9810201 CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Acharya B.S., Figueroa-O’Farrill J.M., Hull C.M., Spence B.J.: Branes at conical singularities and holography. Adv. Theor. Math. Phys. 2, 1249–1286 (1999) arXiv:hep-th/9808014 CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Witten E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998) arXiv:hep-th/9802150 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Klebanov I.R., Witten E.: Superconformal field theory on three-branes at a Calabi–Yau singularity. Nucl. Phys. B 536, 199–218 (1998) arXiv:hep-th/9807080 ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Martelli D., Sparks J.: Toric geometry, Sasaki–Einstein manifolds and a new infinite class of AdS/CFT duals. Commun. Math. Phys. 262, 51–89 (2006) arXiv:hep-th/0411238 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Benvenuti S., Franco S., Hanany A., Martelli D., Sparks J.: An infinite family of superconformal quiver gauge theories with Sasaki–Einstein duals. JHEP 06, 064 (2005) arXiv:hep-th/0411264 ADSCrossRefMathSciNetGoogle Scholar
  21. 21.
    Benvenuti S., Kruczenski M.: From Sasaki–Einstein spaces to quivers via BPS geodesics: L**p,q|r. JHEP 04, 033 (2006) arXiv:hep-th/0505206 ADSCrossRefMathSciNetGoogle Scholar
  22. 22.
    Butti A., Forcella D., Zaffaroni A.: The dual superconformal theory for L**pqr manifolds. JHEP 09, L018 (2005) arXiv:hep-th/0505220 ADSGoogle Scholar
  23. 23.
    Hanany A., Kazakopoulos P., Wecht B.: A new infinite class of quiver gauge theories. JHEP 08, 054 (2005) arXiv:hep-th/0503177 ADSCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hanany, A., Kennaway, K.D.: Dimer models and toric diagrams. arXiv:hep-th/0503149
  25. 25.
    Franco S., Hanany A., Kennaway K.D., Vegh D., Wecht B.: Brane dimers and quiver gauge theories. JHEP 01, 096 (2006) arXiv:hep-th/0504110 ADSCrossRefMathSciNetGoogle Scholar
  26. 26.
    Gubser S.S.: Einstein manifolds and conformal field theories. Phys. Rev. D 59, 025006 (1999) arXiv:hep-th/9807164 ADSCrossRefMathSciNetGoogle Scholar
  27. 27.
    Henningson M., Skenderis K.: The holographic Weyl anomaly. JHEP 07, 023 (1998) arXiv:hep-th/9806087 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Martelli D., Sparks J., Yau S.-T.: The geometric dual of a-maximisation for Toric Sasaki–Einstein manifolds. Commun. Math. Phys. 268, 39–65 (2006) arXiv:hep-th/0503183 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Intriligator K.A., Wecht B.: The exact superconformal R symmetry maximizes a. Nucl. Phys. B 667, 183–200 (2003) arXiv:hep-th/0304128 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Martelli D., Sparks J., Yau S.-T.: Sasaki–Einstein manifolds and volume minimisation. Commun. Math. Phys. 280, 611–673 (2008) arXiv:hep-th/0603021 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Benvenuti S., Feng B., Hanany A., He Y.-H.: Counting BPS operators in gauge theories: quivers, syzygies and plethystics. JHEP 11, 050 (2007) arXiv:hep-th/0608050 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Feng B., Hanany A., He Y.-H.: Counting gauge invariants: the plethystic program. JHEP 03, 090 (2007) arXiv:hep-th/0701063 ADSCrossRefMathSciNetGoogle Scholar
  33. 33.
    Aharony O., Bergman O., Jafferis D.L., Maldacena J.: N = 6 superconformal Chern–Simons-matter theories, M2-branes and their gravity duals. JHEP 10, 091 (2008) arXiv:0806.1218 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Martelli D., Sparks J.: Moduli spaces of Chern–Simons quiver gauge theories and AdS(4)/CFT(3). Phys. Rev. D 78, 126005 (2008) arXiv:0808.0912 ADSCrossRefMathSciNetGoogle Scholar
  35. 35.
    Hanany A., Zaffaroni A.: Tilings, Chern–Simons theories and M2 branes. JHEP 10, 111 (2008) arXiv:0808.1244 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Hanany A., Vegh D., Zaffaroni A.: Brane tilings and M2 branes. JHEP 03, 012 (2009) arXiv:0809.1440 ADSCrossRefMathSciNetGoogle Scholar
  37. 37.
    Franco S., Ghim D., Lee S., Seong R.-K., Yokoyama D.: 2d (0,2) quiver gauge theories and D-branes. JHEP 09, 072 (2015) arXiv:1506.03818 CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Franco S., Lee S., Seong R.-K.: Brane brick models, toric Calabi–Yau 4-folds and 2d (0,2) quivers. JHEP 02, 047 (2016) arXiv:1510.01744 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Franco, S., Lee, S., Seong, R.-K., Vafa, C.: Brane brick models in the mirror. arXiv:1609.01723
  40. 40.
    Garcia-Compean H., Uranga A.M.: Brane box realization of chiral gauge theories in two-dimensions. Nucl. Phys. B 539, 329–366 (1999) arXiv:hep-th/9806177 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Mohri K.: D-branes and quotient singularities of Calabi–Yau fourfolds. Nucl. Phys. B 521, 161–182 (1998) arXiv:hep-th/9707012 ADSCrossRefzbMATHGoogle Scholar
  42. 42.
    Franco, S., Lee, S., Seong, R.-K., Vafa, C.: Quadrality for supersymmetric matrix models. arXiv:1612.06859
  43. 43.
    Butti A., Zaffaroni A.: R-charges from toric diagrams and the equivalence of a-maximization and Z-minimization. JHEP 11, 019 (2005) arXiv:hep-th/0506232 ADSCrossRefMathSciNetGoogle Scholar
  44. 44.
    Butti A., Zaffaroni A.: From toric geometry to quiver gauge theory: the equivalence of a-maximization and Z-minimization. Fortsch. Phys. 54, 309–316 (2006) arXiv:hep-th/0512240 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Herzog C.P., Klebanov I.R., Pufu S.S., Tesileanu T.: Multi-matrix models and tri-Sasaki Einstein spaces. Phys. Rev. D 83, 046001 (2011) arXiv:1011.5487 ADSCrossRefGoogle Scholar
  46. 46.
    Martelli D., Sparks J.: The large N limit of quiver matrix models and Sasaki–Einstein manifolds. Phys. Rev. D 84, 046008 (2011) arXiv:1102.5289 ADSCrossRefGoogle Scholar
  47. 47.
    Hanany A., Seong R.-K.: Brane tilings and reflexive polygons. Fortsch. Phys. 60, 695–803 (2012) arXiv:1201.2614 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Feng B., He Y.-H., Kennaway K.D., Vafa C.: Dimer models from mirror symmetry and quivering amoebae. Adv. Theor. Math. Phys. 12, 489–545 (2008) arXiv:hep-th/0511287 CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    Nill, B.: Gorenstein toric Fano varieties. ArXiv Mathematics e-prints (May, 2004). arXiv:math/0405448
  50. 50.
    Doran C.F., Whitcher U.A.: From polygons to string theory. Math. Mag. 85, 343–360 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Skarke, H.: How to classify reflexive Gorenstein cones. In: Rebhan, A., Katzarkov, L., Knapp, J., Rashkov, R., Scheidegger, E., (ed.) Strings, Gauge Fields, and the Geometry Behind: The Legacy of Maximilian Kreuzer, pp. 443–458 (2012). arXiv:1204.1181
  52. 52.
    Fulton W.: Introduction to Toric Varieties. Annals of Mathematics Studies. Princeton University Press, Princeton (1993)Google Scholar
  53. 53.
    Cox, D., Little, J., Schenck, H.: Toric Varieties. Graduate Studies in Mathematics. American Mathematical Society, Providence (2011)Google Scholar
  54. 54.
    Ewald G.: On the classification of toric fano varieties. Discrete Comput. Geom. 3, 49–54 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    Watanabe, K., Watanabe, M.: The classification of fano 3-folds with torus embeddings. Tokyo J. Math. 05, 37–48 (1982)Google Scholar
  56. 56.
    Stein, W., et al.: Sage Mathematics Software (Version 7.5.1). The Sage Development TeamGoogle Scholar
  57. 57.
    He Y.-H., Lee S.-J., Lukas A.: Heterotic models from vector bundles on toric Calabi–Yau manifolds. JHEP 05, 071 (2010) arXiv:0911.0865 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
  59. 59.
    Candelas P., Davies R.: New Calabi–Yau manifolds with small hodge numbers. Fortsch. Phys. 58, 383–466 (2010) arXiv:0809.4681 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  60. 60.
    Anderson L.B., Gao X., Gray J., Lee S.-J.: Multiple fibrations in Calabi–Yau geometry and string dualities. JHEP 10, 105 (2016) arXiv:1608.07555 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  61. 61.
    Johnson S.B., Taylor W.: Calabi–Yau threefolds with large h 2,1. JHEP 10, 23 (2014) arXiv:1406.0514 ADSCrossRefGoogle Scholar
  62. 62.
    Gray J., Haupt A.S., Lukas A.: Topological invariants and fibration structure of complete intersection Calabi–Yau four-folds. JHEP 09, 093 (2014) arXiv:1405.2073 ADSCrossRefGoogle Scholar
  63. 63.
    Candelas, P., Constantin, A., Mishra C.: Calabi–Yau threefolds with small hodge numbers. arXiv:1602.06303
  64. 64.
    He, Y.-H., Jejjala, V., Pontiggia, L.: Patterns in Calabi–Yau distributions. arXiv:1512.01579
  65. 65.
    Futaki A., Ono H., Wang G.: Transverse Kahler geometry of Sasaki manifolds and toric Sasaki–Einstein manifolds. J. Differ. Geom. 83, 585–636 (2009) arXiv:math/0607586 CrossRefzbMATHGoogle Scholar
  66. 66.
    Kenyon R.: Local statistics of lattice dimers. Ann. Inst. Henri Poincare Sect. Phys. Theor. 33, 591–618 (1997) arXiv:math/0105054 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  67. 67.
    Kenyon, R.: An introduction to the dimer model. ArXiv Mathematics e-prints (Oct., 2003). arXiv:math/0310326
  68. 68.
    Seiberg N.: Electric–magnetic duality in supersymmetric non-Abelian gauge theories. Nucl. Phys. B 435, 129–146 (1995) arXiv:hep-th/9411149 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  69. 69.
    Feng B., Hanany A., He Y.-H.: D-brane gauge theories from toric singularities and toric duality. Nucl. Phys. B 595, 165–200 (2001) arXiv:hep-th/0003085 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  70. 70.
    Bagger J., Lambert N.: Modeling multiple M2’s. Phys. Rev. D 75, 045020 (2007) arXiv:hep-th/0611108 ADSCrossRefMathSciNetGoogle Scholar
  71. 71.
    Gustavsson A.: Algebraic structures on parallel M2-branes. Nucl. Phys. B 811, 66–76 (2009) arXiv:0709.1260 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  72. 72.
    Martelli D., Sparks J.: Notes on toric Sasaki–Einstein seven-manifolds and AdS(4) / CFT(3). JHEP 11, 016 (2008) arXiv:0808.0904 ADSCrossRefMathSciNetGoogle Scholar
  73. 73.
    Franco S., Lee S., Seong R.-K.: Brane brick models and 2d (0, 2) triality. JHEP 05, 020 (2016) arXiv:1602.01834 ADSCrossRefGoogle Scholar
  74. 74.
    Gadde A., Gukov S., Putrov P.: (0, 2) trialities. JHEP 03, 076 (2014) arXiv:1310.0818 ADSCrossRefGoogle Scholar
  75. 75.
    Hori, K., Vafa, C.: Mirror symmetry. arXiv:hep-th/0002222
  76. 76.
    Batyrev, V., Schaller, K.: Stringy Chern classes of singular toric varieties and their applications. arXiv:1607.04135
  77. 77.
    Dixon L.J., Harvey J.A., Vafa C., Witten E.: Strings on orbifolds. Nucl. Phys. B 261, 678–686 (1985)ADSCrossRefMathSciNetGoogle Scholar
  78. 78.
    Göttsche L.: The betti numbers of the hilbert scheme of points on a smooth projective surface. Math. Ann. 286, 193–208 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  79. 79.
    Long C., McAllister L., McGuirk P.: Heavy tails in Calabi–Yau moduli spaces. JHEP 10, 187 (2014) arXiv:1407.0709 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  80. 80.
    Diaz R., Robins S.: The ehrhart polynomial of a lattice n-simplex. Electron. Res. Announc. Am. Math. Soc. 2, 1–6 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  81. 81.
    Bergman A., Herzog C.P.: The volume of some nonspherical horizons and the AdS/CFT correspondence. JHEP 01, 030 (2002) arXiv:hep-th/0108020 ADSCrossRefGoogle Scholar
  82. 82.
    Stanley R.P.: Hilbert functions of graded algebras. Adv. Math. 28, 57–83 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  83. 83.
    Futaki A., Ono H., Sano Y.: Hilbert series and obstructions to asymptotic semistability. Adv. Math. 226, 254–284 (2011) arXiv:0811.1315 CrossRefzbMATHMathSciNetGoogle Scholar
  84. 84.
    Hanany A., Vegh D.: Quivers, tilings, branes and rhombi. JHEP 10, 029 (2007) arXiv:hep-th/0511063 ADSCrossRefMathSciNetGoogle Scholar
  85. 85.
    Tachikawa Y.: Five-dimensional supergravity dual of a-maximization. Nucl. Phys. B 733, 188–203 (2006) arXiv:hep-th/0507057 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  86. 86.
    Kapustin A., Willett B., Yaakov I.: Exact results for Wilson loops in superconformal Chern–Simons theories with matter. JHEP 03, 089 (2010) arXiv:0909.4559 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  87. 87.
    Jafferis D.L.: The exact superconformal R-symmetry extremizes Z. JHEP 05, 159 (2012) arXiv:1012.3210 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  88. 88.
    Closset C., Dumitrescu T.T., Festuccia G., Komargodski Z., Seiberg N.: Contact terms, unitarity, and F-maximization in three-dimensional superconformal theories. JHEP 10, 053 (2012) arXiv:1205.4142 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  89. 89.
    Lee S.: Superconformal field theories from crystal lattices. Phys. Rev. D 75, 101901 (2007) arXiv:hep-th/0610204 ADSCrossRefMathSciNetGoogle Scholar
  90. 90.
    Amariti A., Franco S.: Free energy vs Sasaki–Einstein volume for infinite families of M2-brane theories. JHEP 09, 034 (2012) arXiv:1204.6040 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  91. 91.
    Lee S., Yokoyama D.: Geometric free energy of toric AdS 4/CFT 3 models. JHEP 03, 103 (2015) arXiv:1412.8703 CrossRefGoogle Scholar
  92. 92.
    Bishop R., Crittenden R.: Geometry of Manifolds. Pure and Applied Mathematics. Academic Press, London (1964)zbMATHGoogle Scholar
  93. 93.
    Gulotta D.R.: Properly ordered dimers, R-charges, and an efficient inverse algorithm. JHEP 10, 014 (2008) arXiv:0807.3012 ADSCrossRefzbMATHMathSciNetGoogle Scholar

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Merton College, University of OxfordOxfordUK
  2. 2.Department of MathematicsCity, University of LondonLondonUK
  3. 3.School of PhysicsNanKai UniversityTianjinPeople’s Republic of China
  4. 4.Angstrom Laboratory, Department of Physics and AstronomyUppsala UniversityUppsalaSweden
  5. 5.Jefferson Physical Laboratory, Center of Mathematical Sciences and Applications, Department of MathematicsHarvard UniversityCambridgeUSA

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