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Communications in Mathematical Physics

, Volume 325, Issue 3, pp 1139–1170 | Cite as

Two-Sphere Partition Functions and Gromov–Witten Invariants

  • Hans Jockers
  • Vijay Kumar
  • Joshua M. Lapan
  • David R. Morrison
  • Mauricio Romo
Article

Abstract

Many \({\mathcal{N}=(2,2)}\) two-dimensional nonlinear sigma models with Calabi–Yau target spaces admit ultraviolet descriptions as \({\mathcal{N}=(2,2)}\) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories—recently computed via localization by Benini et al. and Doroud et al.—yields the exact Kähler potential on the quantum Kähler moduli space for Calabi–Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov–Witten invariants for any such Calabi–Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kähler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in α′. We compute these quantities for the quintic and for Rødland’s Pfaffian Calabi–Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi–Yau threefold in \({\mathbb{P}^{7}}\) , recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi–Yau is currently known. We derive predictions for its Gromov–Witten invariants and verify that our predictions satisfy nontrivial geometric checks.

Keywords

Partition Function Modulus Space Complete Intersection Toric Variety Gauge Linear Sigma Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Dixon, L.J.: Some world-sheet properties of superstring compactifications, on orbifolds and otherwise. In: Superstrings, Unified Theories, and Cosmology 1987, G. Furlan et al., eds., Singapore, New Jersey, Hong Kong: World Scientific, 1988, pp. 67–126Google Scholar
  2. 2.
    Lerche W., Vafa C., Warner N.P.: Chiral rings in N = 2 superconformal theories. Nucl. Phys. B 324, 427–474 (1989)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Candelas P., Lynker M., Schimmrigk R.: Calabi–Yau manifolds in weighted \({{\mathbb{P}}_4}\) . Nucl. Phys. B 341, 383–402 (1990)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Aspinwall P.S., Lütken C.A., Ross G.G.: Construction and couplings of mirror manifolds. Phys. Lett. B 241, 373–380 (1990)ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    Greene B.R., Plesser M.R.: Duality in Calabi–Yau moduli space. Nucl. Phys. B 338, 15–37 (1990)ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Candelas P., De La Ossa X.C., Green P.S., Parkes L.: A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B 359, 21–74 (1991)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gromov M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dine M., Seiberg N., Wen X.G., Witten E.: Nonperturbative effects on the string world sheet (II). Nucl. Phys. B 289, 319–363 (1987)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    Witten E.: Topological sigma models. Commun. Math. Phys. 118, 411 (1988)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Candelas P., de la Ossa X., Font A., Katz S., Morrison D.R.: Mirror symmetry for two parameter models-I. Nucl. Phys. B 416, 481–562 (1994)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Candelas P., Font A., Katz S., Morrison D.R.: Mirror symmetry for two parameter models-II. Nucl. Phys. B 429, 626–674 (1994)ADSCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hosono S., Klemm A., Theisen S., Yau S.-T.: Mirror symmetry, mirror map and applications to Calabi–Yau hypersurfaces. Commun. Math. Phys. 167, 301–350 (1995)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Hosono S., Klemm A., Theisen S., Yau S.-T.: Mirror symmetry, mirror map and applications to complete intersection Calabi–Yau spaces. Nucl. Phys. B 433, 501–554 (1995)ADSCrossRefMathSciNetGoogle Scholar
  14. 14.
    Givental A.B.: Equivariant Gromov–Witten invariants. Int. Math. Res. Not 1996, 613–663 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lian B.H., Liu K., Yau S.-T.: Mirror principle. I. Asian J. Math. 1, 729–763 (1997)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Witten E.: Phases of N =  2 theories in two dimensions. Nucl. Phys. B 403, 159–222 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Morrison D.R., Plesser M.R.: Summing the instantons: quantum cohomology and mirror symmetry in toric varieties. Nucl. Phys. B 440, 279–354 (1995)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Hori K., Tong D.: Aspects of non-abelian gauge dynamics in two-dimensional \({\mathcal{N}=(2,2)}\) theories. JHEP 0705, 079 (2007)ADSCrossRefMathSciNetGoogle Scholar
  19. 19.
    Donagi R., Sharpe E.: GLSMs for partial flag manifolds. J. Geom. Phys. 58, 1662–1692 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hori, K.: Duality in two-dimensional (2,2) supersymmetric non-Abelian gauge theories, http://arxiv.org/abs/1104.2853v1 [hep-th], 2011
  21. 21.
    Jockers H., Kumar V., Lapan J.M., Morrison D.R., Romo M.: Nonabelian 2D gauge theories for determinantal Calabi–Yau varieties. JHEP 1211, 166 (2012)ADSCrossRefMathSciNetGoogle Scholar
  22. 22.
    Benini, F., Cremonesi, S.: Partition functions of \({\mathcal{N}=(2,2)}\) gauge theories on S 2 and vortices. http://arxiv.org/abs/1206.2356v2 [hep-th] 2012
  23. 23.
    Doroud, N., Gomis, J., Le Floch, B., Lee, S.: Exact results in D = 2 supersymmetric gauge theories. http://arxiv.org/abs/1206.2606v2 [hep-th], 2012
  24. 24.
    Böhm, J.: Mirror symmetry and tropical geometry. http://arxiv.org/abs/0708.4402v1 [math.AG], 2007
  25. 25.
    Böhm, J.: A framework for tropical mirror symmetry. http://arxiv.org/abs/1103.2673v1 [math.AG], 2011
  26. 26.
    de Wit B., Van Proeyen A.: Potentials and symmetries of general gauged N=2 supergravity: Yang–Mills models. Nucl. Phys. B 245, 89–117 (1984)ADSCrossRefGoogle Scholar
  27. 27.
    Bryant, R.L., Griffiths, P.A.: Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle. In: Arithmetic and geometry, Vol. II (Boston, MA), Progr. Math., Vol. 36, Boston, MA: Birkhäuser Boston, 1983, pp. 77–102Google Scholar
  28. 28.
    Strominger A.: Special geometry. Commun. Math. Phys. 133, 163–180 (1990)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Candelas P., de la Ossa X.: Moduli space of Calabi–Yau manifolds. Nucl. Phys. B 355, 455–481 (1991)ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311–428 (1994)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Craps B., Roose F., Troost W., Van Proeyen A.: What is special Kähler geometry? Nucl. Phys. B 503, 565–613 (1997)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Freed D.S.: Special Kähler manifolds. Commun. Math. Phys. 203, 31–52 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Carlson J., Green M., Griffiths P., Harris J.: Infinitesimal variations of Hodge structure. I. Comp. Math. 50, 109–205 (1983)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Morrison, D.R.: Mathematical aspects of mirror symmetry. In: Complex algebraic geometry (Park City, UT, 1993), IAS/Park City Math. Ser., Vol. 3, Providence, RI: Amer. Math. Soc., 1997, pp. 265–327Google Scholar
  35. 35.
    Moore G.W., Witten E.: Self-duality, Ramond–Ramond fields, and K-theory. JHEP 0005, 032 (2000)ADSCrossRefMathSciNetGoogle Scholar
  36. 36.
    Mukai, S.: On the moduli space of bundles on K3 surfaces. I. In: Vector bundles on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math., Vol. 11, Bombay: Tata Inst. Fund. Res., 1987, pp. 341–413Google Scholar
  37. 37.
    Brunner I., Douglas M.R., Lawrence A.E., Römelsberger C.: D-branes on the quintic. JHEP 0008, 015 (2000)ADSCrossRefGoogle Scholar
  38. 38.
    Mayr P.: Phases of supersymmetric D-branes on Kahler manifolds and the McKay correspondence. JHEP 0101, 018 (2001)ADSCrossRefMathSciNetGoogle Scholar
  39. 39.
    Kontsevich M., Manin Y.: Gromov–Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Ruan Y., Tian G.: A mathematical theory of quantum cohomology. J. Diff. Geom. 42, 259–367 (1995)zbMATHMathSciNetGoogle Scholar
  41. 41.
    Kontsevich, M.: Enumeration of rational curves via torus actions. In: The moduli space of curves (Texel Island, 1994), Progr. Math., Vol. 129, Boston, MA: Birkhäuser Boston, 1995, pp. 335–368Google Scholar
  42. 42.
    Morrison D.R.: Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians. J. Am. Math. Soc. 6, 223–247 (1993)CrossRefzbMATHGoogle Scholar
  43. 43.
    Deligne, P.: Local behavior of Hodge structures at infinity. In: Mirror symmetry, II. AMS/IP Stud. Adv. Math., Vol. 1, Providence, RI: Amer. Math. Soc., 1997, pp. 683–699Google Scholar
  44. 44.
    Morrison, D.R.: Compactifications of moduli spaces inspired by mirror symmetry. In: Journées de Géométrie Algébrique d’Orsay (Juillet 1992), Astérisque, Vol. 218. Paris: Société Mathématique de France, 1993, pp. 243–271Google Scholar
  45. 45.
    Grisaru M.T., van de Ven A., Zanon D.: Four loop beta function for the N=1 and N=2 supersymmetric nonlinear sigma model in two-dimensions. Phys. Lett. B 173, 423 (1986)ADSCrossRefMathSciNetGoogle Scholar
  46. 46.
    Aspinwall P.S., Morrison D.R.: Topological field theory and rational curves. Commun. Math. Phys. 151, 245–262 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Cecotti S., Vafa C.: Topological antitopological fusion. Nucl. Phys. B 367, 359–461 (1991)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Festuccia G., Seiberg N.: Rigid supersymmetric theories in curved superspace. JHEP 1106, 114 (2011)ADSCrossRefMathSciNetGoogle Scholar
  49. 49.
    Witten E.: Topological quantum field theory. Commun. Math. Phys. 117, 353 (1988)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    Losev A., Nekrasov N., Shatashvili S.L.: Issues in topological gauge theory. Nucl. Phys. B 534, 549–611 (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Losev, A., Nekrasov, N., Shatashvili, S.: Testing Seiberg–Witten solution. In: Strings, branes and dualities (Cargèse, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 520. Dordrecht: Kluwer Acad. Publ., 1999, pp. 359–372Google Scholar
  52. 52.
    Nekrasov N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004)CrossRefMathSciNetGoogle Scholar
  53. 53.
    Pestun V.: Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. Commun. Math. Phys. 313, 71–129 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Kapustin A., Willett B., Yaakov I.: Exact results for Wilson loops in superconformal Chern–Simons theories with matter. JHEP 1003, 089 (2010)ADSCrossRefMathSciNetGoogle Scholar
  55. 55.
    Hama N., Hosomichi K., Lee S.: SUSY gauge theories on squashed three-spheres. JHEP 1105, 014 (2011)ADSCrossRefMathSciNetGoogle Scholar
  56. 56.
    Pasquetti S.: Factorisation of N = 2 theories on the squashed 3-sphere. JHEP 1204, 120 (2012)ADSCrossRefMathSciNetGoogle Scholar
  57. 57.
    Beem, C., Dimofte, T., Pasquetti, S.: Holomorphic Blocks in Three Dimensions. http://arxiv.org/abs/1211.1986v1 [hep-th], 2012
  58. 58.
    Zamolodchikov A.: Irreversibility of the flux of the renormalization group in a 2D field theory. JETP Lett. 43, 730–732 (1986)ADSMathSciNetGoogle Scholar
  59. 59.
    Periwal V., Strominger A.: Kähler geometry of the space of N=2 superconformal field theories. Phys. Lett. B 235, 261 (1990)ADSCrossRefMathSciNetGoogle Scholar
  60. 60.
    Cecotti S., Vafa C.: Exact results for supersymmetric sigma models. Phys. Rev. Lett. 68, 903–906 (1992)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  61. 61.
    Rødland E.A.: The Pfaffian Calabi–Yau, its mirror, and their link to the Grassmannian G(2,7). Comp. Math. 122, 135–149 (2000)CrossRefGoogle Scholar
  62. 62.
    Tjøtta E.N.: Quantum cohomology of a Pfaffian Calabi–Yau variety: verifying mirror symmetry predictions. Comp. Math. 126, 79–89 (2001)CrossRefGoogle Scholar
  63. 63.
    Batyrev V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Alg. Geom. 3, 493–535 (1994)zbMATHMathSciNetGoogle Scholar
  64. 64.
    Batyrev, V.V., Borisov, L.A.: On Calabi–Yau complete intersections in toric varieties. In: Higher-dimensional complex varieties (Trento, 1994), Berlin: de Gruyter, 1996, pp. 39–65Google Scholar
  65. 65.
    Batyrev V.V., van Straten D.: Generalized hypergeometric functions and rational curves on Calabi–Yau complete intersections in toric varieties. Commun. Math. Phys. 168, 493–533 (1995)ADSCrossRefzbMATHGoogle Scholar
  66. 66.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants and multidimensional determinants. Boston, MA: Birkhäuser Boston, 1994Google Scholar
  67. 67.
    Batyrev V.V., Ciocan-Fontanine I., Kim B., van Straten D.: Conifold transitions and mirror symmetry for Calabi–Yau complete intersections in Grassmannians. Nucl. Phys. B 514, 640–666 (1998)ADSCrossRefzbMATHGoogle Scholar
  68. 68.
    Gulliksen T.H., Negård O.G.: Un complexe résolvant pour certains idéaux déterminantiels. C. R. Acad. Sci. Paris Sér. A–B 274, A16–A18 (1972)Google Scholar
  69. 69.
    Gross M., Popescu S.: Calabi–Yau threefolds and moduli of abelian surfaces. I. Comp. Math 127, 169–228 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  70. 70.
    Bertin M.-A.: Examples of Calabi–Yau 3-folds of \({\mathbb{P}^7}\) with \({\rho=1}\) . Can. J. Math. 61, 1050–1072 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  71. 71.
    Kapustka M., Kapustka G.: A cascade of determinantal Calabi–Yau threefolds. Math. Nachr. 283, 1795–1809 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  72. 72.
    Namikawa Y.: Deformation theory of Calabi–Yau threefolds and certain invariants of singularities. J. Alg. Geom. 6, 753–776 (1997)zbMATHMathSciNetGoogle Scholar
  73. 73.
    Ciocan-Fontanine I., Kim B., Sabbah C.: The abelian/nonabelian correspondence and Frobenius manifolds. Invent. Math. 171, 301–343 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  74. 74.
    Givental, A.: Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture. In: Topics in singularity theory, Amer. Math. Soc. Transl. Ser. 2, Vol. 180, Providence, RI: Amer. Math. Soc., 1997, pp. 103–115Google Scholar
  75. 75.
    Kim B.: Quantum hyperplane section theorem for homogeneous spaces. Acta Math. 183, 71–99 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  76. 76.
    Clemens C.H.: Double solids. Adv. Math. 47, 107–230 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  77. 77.
    Friedman R.: Simultaneous resolution of threefold double points. Math. Ann. 274, 671–689 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  78. 78.
    Reid M.: The moduli space of 3-folds with K = 0 may nevertheless be irreducible. Math. Ann. 278, 329–334 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  79. 79.
    Candelas P., Green P.S., Hübsch T.: Rolling among Calabi–Yau vacua. Nucl. Phys. B 330, 49 (1990)ADSCrossRefGoogle Scholar
  80. 80.
    Greene B.R., Morrison D.R., Strominger A.: Black hole condensation and the unification of string vacua. Nucl. Phys. B 451, 109–120 (1995)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  81. 81.
    Li A.-M., Ruan Y.: Symplectic surgery and Gromov–Witten invariants of Calabi–Yau 3-folds. Invent. Math. 145, 151–218 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  82. 82.
    Libgober A., Teitelbaum J.: Lines on Calabi–Yau complete intersections, mirror symmetry, and Picard–Fuchs equations. Int. Math. Res. Not 1993, 29–39 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  83. 83.
    Hosono, S., Takagi, H.: Mirror symmetry and projective geometry of Reye congruences I. J. Alg. Geom. toappear, doi: 10.1090/s1056-3911-2013-00618-9.2013
  84. 84.
    Morrison, D.R., Beyond the Kähler cone. In: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry, M. Teicher, ed., Israel Math. Conf. Proc., Vol. 9, Ramat-Gan: Bar-Ilan University, 1996, pp. 361–376Google Scholar
  85. 85.
    Coates T., Corti A., Iritani H., Tseng H.-H.: Computing genus-zero twisted Gromov–Witten invariants. Duke Math. J. 147, 377–438 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  86. 86.
    Bayer A., Cadman C.: Quantum cohomology of \({[\mathbb{C}^N/\mu_r]}\) . Comp. Math. 146, 1291–1322 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  87. 87.
    Bouchard V., Klemm A., Mariño M., Pasquetti S.: Remodeling the B-model. Commun. Math. Phys. 287, 117–178 (2009)ADSCrossRefzbMATHGoogle Scholar
  88. 88.
    Horja, R.P.: Hypergeometric functions and mirror symmetry in toric varieties. http://arxiv.org/abs/math/9912109v3 [math A6], 2000
  89. 89.
    Hori, K., Vafa, C.: Mirror symmetry. http://arxiv.org/abs/hep-th/0002222v3, 2000
  90. 90.
    Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: John Willey & Sons, Inc., 1994Google Scholar
  91. 91.
    Fulton, W.: Intersection theory. Ergeb. Math. Grenzgeb. (3), Vol. 2, Berlin: Springer, 1984Google Scholar
  92. 92.
    Bott, R., Tu, L.W.: Differential forms in algebraic topology. Graduate Texts in Mathematics, Vol. 82, New York: Springer, 2010Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Hans Jockers
    • 1
  • Vijay Kumar
    • 2
  • Joshua M. Lapan
    • 3
  • David R. Morrison
    • 4
    • 5
  • Mauricio Romo
    • 6
  1. 1.Bethe Center for Theoretical Physics, Physikalisches InstitutUniversität BonnBonnGermany
  2. 2.KITPUniversity of CaliforniaSanta BarbaraUSA
  3. 3.Department of PhysicsMcGill UniversityMontrealCanada
  4. 4.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA
  5. 5.Department of PhysicsUniversity of CaliforniaSanta BarbaraUSA
  6. 6.Kavli IPMU (WPI)The University of TokyoKashiwaJapan

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