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Completeness in Supergravity Constructions

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Abstract

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component \({\mathcal{H}}\) of a hypersurface {h = 1} defined by a homogeneous cubic polynomial h such that \({-\partial^2h}\) is a complete Riemannian metric on \({\mathcal{H}}\) defines a complete projective special Kähler manifold and any complete projective special Kähler manifold defines a complete quaternionic Kähler manifold of negative scalar curvature. We classify all complete quaternionic Kähler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.

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References

  1. Alekseevskiĭ D.V.: Classification of quaternionic spaces with transitive solvable group of motions. Izv. Akad. Nauk SSSR Ser. Mat. 39(2), 315–362 (1975)

    MathSciNet  MATH  Google Scholar 

  2. Alekseevsky D.V., Cortés V.: Classification of stationary compact homogeneous special pseudo Kähler manifolds of semisimple groups. Proc. London Math. Soc. 81(3), 211–230 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alekseevsky D.V., Cortés V.: Geometric construction of the r-map: from affine special real to special Kähler manifolds. Commun. Math. Phys. 291, 579–590 (2009)

    Article  ADS  MATH  Google Scholar 

  4. Alekseevsky D.V., Cortés V., Devchand C.: Special complex manifolds. J. Geom. Phys. 42, 85–105 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Antoniadis I., Ferrara S., Gava E., Narain K., Taylor T.: Perturbative prepotential and monodromies in N = 2 heterotic superstring. Nucl. Phys. B447, 35–61 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  6. Antoniadis I., Ferrara S., Taylor T.: N = 2 heterotic superstring and its dual theory in five dimensions. Nucl. Phys. B460, 489–505 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  7. Chou A., Kallosh R., Rahmfeld J., Rey S.-J., Shmakova M., Wong W.K.: Critical points and phase transitions in 5d compactifications of M-theory. Nucl. Phys. B508, 147–180 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  8. Cecotti S., Ferrara S., Girardello L.: Geometry of type II superstrings and the moduli of superconformal field theories. Int. J. Mod. Phys. A4, 2475 (1989)

    MathSciNet  ADS  Google Scholar 

  9. Cortés V.: Alekseevskian spaces. J. Diff. Geom. Appl. 6(2), 129–168 (1996)

    Article  MATH  Google Scholar 

  10. Cortés V.: On hyper Kähler manifolds associated to Lagrangian Kähler submanifolds of \({T^*\mathbb{C}^n}\). Trans. Amer. Math. Soc. 350(8), 3193–3205 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cortés, V.: A holomorphic representation formula for parabolic hypersphere. In: Opozda, B., Simon, U., Wiehe, M. (eds.) Proceedings of the International Conference “PDEs, Submanifolds and Affine Differential Geometry” (Warsaw 2000), Banach Center Publications (Polish Academy of Sciences, Institute of Mathematics), Vol. 54, 2000. pp. 11–16. Available at http://arxiv.org/abs/0107037v2 [math.D6], 2001

  12. Cortés V., Mohaupt T.: Special Geometry of Euclidean Supersymmetry III: the local r-map, instantons and black holes. JHEP 07, 066 (2009)

    Article  ADS  Google Scholar 

  13. Cortés V., Mayer C., Mohaupt T., Saueressig F.: Special geometry of Euclidean supersymmetry II: hypermultiplets and the c-map. JHEP 06, 025 (2006)

    ADS  Google Scholar 

  14. Cortés, V., Schäfer, L.: Topological-antitopological fusion equations, pluriharmonic maps and special Kähler manifolds. In: Kowalski, O., Musso, E., Perrone, D. (eds.) Complex, Contact and Symmetric Manifolds, Progress in Mathematics 234, Basel-Boston: Birkhäuser, 2005, pp. 59–74

  15. Cremmer E., Kounnas C., Van Proeyen A., Derendinger J.P., Ferrara S., de Wit B., Girardello L.: Vector multiplets, coupled to N = 2 supergravity: SuperHiggs effect, flat potentials and geometric structure. Nucl. Phys. B250, 325 (1985)

    Google Scholar 

  16. De Jaehger J., de Wit B., Kleijn B., Vandoren S.: Special geometry in hypermultiplets. Nucl. Phys. B514, 553–582 (1998)

    Article  ADS  Google Scholar 

  17. de Wit B.: N = 2 electric-magnetic duality in a chiral background. Nucl. Phys. Proc. Suppl. 49, 191–200 (1996)

    Article  ADS  MATH  Google Scholar 

  18. de Wit B., Rocek M., Vandoren S.: Hypermultiplets, hyperkähler cones and quaternion-Kähler geometry. JHEP 02, 039 (2001)

    Article  Google Scholar 

  19. de Wit B., Saueressig F.: Off-shell N = 2 tensor supermultiplets. JHEP 09, 062 (2006)

    Article  Google Scholar 

  20. de Wit B., Van Proeyen A.: Potentials and symmetries of general gauged N=2 supergravity-Yang-Mills models. Nucl. Phys. B245, 89–117 (1984)

    Article  ADS  Google Scholar 

  21. de Wit B., Van Proeyen A.: Special geometry, cubic polynomials and homogeneous quaternionic spaces. Commun. Math. Phys. 149, 307–333 (1992)

    Article  ADS  MATH  Google Scholar 

  22. Ferrara S., Sabharwal S.: Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces. Nucl. Phys. B332, 317–332 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  23. Freed D.: Special Kähler manifolds. Commun. Math. Phys. 203, 31–52 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Gaiotto D., Moore G.W., Neitzke A.: Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys. 299, 163–224 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Robles-Llana D., Saueressig F., Vandoren S.: String loop corrected hypermultiplet moduli spaces. JHEP 03, 081 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  26. Alexandrov S., Saueressig F., Vandoren S.: Membrane and fivebrane instantons from quaternionic geometry. JHEP 09, 040 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  27. Robles-Llana D., Rocek M., Saueressig F., Theis U., Vandoren S.: Nonperturbative corrections to 4D string effective actions from \({SL(2,\mathbb{Z})}\) duality and supersymmetry. Phys. Rev. Lett. 98, 211602 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  28. Alexandrov S., Pioline B., Saueressig F., Vandoren S.: D-instantons and twistors. JHEP 03, 044 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  29. Hitchin, N.: Quaternionic Kähler moduli spaces. In: Galicki, K., Simanca, S. (eds.) Riemannian Topology and Geometric Structures on Manifolds, Progress in Mathematics 271, Basel-Boston: Birkhäuser, 2009, pp. 49–61

  30. Kachru S., Klemm A., Lerche W., Mayr P., Vafa C.: Nonperturbative results on the point particle limit of N = 2 heterotic string compactifications. Nucl. Phys. B459, 537–558 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  31. Kachru S., Vafa C.: Exact results for N = 2 compactifications of heterotic strings. Nucl. Phys. B450, 69–89 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  32. Louis J., Sonnenschein J., Theisen S., Yankielowicz S.: Non-perturbative properties of heterotic string vacua compactifications. Nucl. Phys. B480, 185–212 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  33. Mayer C., Mohaupt T.: The Kähler cone as cosmic censor. Class. Quant. Grav. 21, 1879–1896 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Morrison D.R., Vafa C.: Compactifications of F-theory on Calabi-Yau threefolds. Nucl. Phys. B473, 74–92 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  35. O’Neill, B.: Semi-Riemannian geometry, With applications to relativity. Pure and Applied Mathematics, 103, New York: Academic Press, 1983

  36. Rocek M., Vafa C., Vandoren S.: Hypermultiplets and topological strings. JHEP 02, 062 (2005)

    ADS  Google Scholar 

  37. Tian, G.: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. In: Mathematical aspects of string theory, Adv. Ser. Math. Phys., Vol. 1, Singapore: World Sci. Publishing, 1987, pp. 629–646

  38. van der Aalst, T.A.F.: A geometric interpretation of the c-map. Master Thesis, University of Utrecht, 2008

  39. Wells, R.O.: Differential analysis on complex manifolds. Graduate Texts in Mathematics 65, New York: Springer, 2008

  40. Witten E.: Phase transitions in M-theory and F-theory. Nucl. Phys. B471, 195–216 (1996)

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Center for Mathematical Physics, University of Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    V. Cortés & X. Han

  2. Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool, L69 7ZL, UK

    T. Mohaupt

Authors
  1. V. Cortés
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  2. X. Han
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  3. T. Mohaupt
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Correspondence to T. Mohaupt.

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Communicated by N. A. Nekrasov

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Cortés, V., Han, X. & Mohaupt, T. Completeness in Supergravity Constructions. Commun. Math. Phys. 311, 191–213 (2012). https://doi.org/10.1007/s00220-012-1443-x

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