Special geometry of Euclidean supersymmetry IV: the local c-map

Open Access
Regular Article - Theoretical Physics

Abstract

We consider timelike and spacelike reductions of 4D, \( \mathcal{N}=2 \) Minkowskian and Euclidean vector multiplets coupled to supergravity and the maps induced on the scalar geometry. In particular, we investigate (i) the (standard) spatial c-map, (ii) the temporal c-map, which corresponds to the reduction of the Minkowskian theory over time, and (iii) the Euclidean c-map, which corresponds to the reduction of the Euclidean theory over space. In the last two cases we prove that the target manifold is para-quaternionic Kähler.

In cases (i) and (ii) we construct two integrable complex structures on the target manifold, one of which belongs to the quaternionic and para-quaternionic structure, respectively. In case (iii) we construct two integrable para-complex structures, one of which belongs to the para-quaternionic structure.

In addition we provide a new global construction of the spatial, temporal and Euclidean c-maps, and separately consider a description of the target manifold as a fibre bundle over a projective special Kähler or para-Kähler base.

Keywords

Differential and Algebraic Geometry Supergravity Models 

References

  1. [1]
    V. Cortés, C. Mayer, T. Mohaupt and F. Saueressig, Special geometry of Euclidean supersymmetry. 1. Vector multiplets, JHEP 03 (2004) 028 [hep-th/0312001] [INSPIRE].CrossRefADSGoogle Scholar
  2. [2]
    V. Cortés, C. Mayer, T. Mohaupt and F. Saueressig, Special geometry of euclidean supersymmetry. II. Hypermultiplets and the c-map, JHEP 06 (2005) 025 [hep-th/0503094] [INSPIRE].CrossRefADSGoogle Scholar
  3. [3]
    V. Cortés and T. Mohaupt, Special Geometry of Euclidean Supersymmetry III: The Local r-map, instantons and black holes, JHEP 07 (2009) 066 [arXiv:0905.2844] [INSPIRE].CrossRefADSGoogle Scholar
  4. [4]
    G. Sierra and P.K. Townsend, An introduction to N = 2 rigid supersymmetry, Preprint LPTENS 83/26 (1983), lectures given at The 19th Karpacz Winter School on Theoretical Physics, Karpacz, Poland, 14–28 February 1983.Google Scholar
  5. [5]
    S.J. Gates Jr., Superspace Formulation of New Nonlinear σ-models, Nucl. Phys. B 238 (1984) 349 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  6. [6]
    B. de Wit and A. Van Proeyen, Potentials and Symmetries of General Gauged N = 2 Supergravity: Yang-Mills Models, Nucl. Phys. B 245 (1984) 89 [INSPIRE].CrossRefADSGoogle Scholar
  7. [7]
    A. Strominger, Special geometry, Commun. Math. Phys. 133 (1990) 163 [INSPIRE].MATHMathSciNetCrossRefADSGoogle Scholar
  8. [8]
    L. Castellani, R. D’Auria and S. Ferrara, Special geometry without special coordinates, Class. Quant. Grav. 7 (1990) 1767 [INSPIRE].MATHMathSciNetCrossRefADSGoogle Scholar
  9. [9]
    R. D’Auria, S. Ferrara and P. Fré, Special and quaternionic isometries: General couplings in N = 2 supergravity and the scalar potential, Nucl. Phys. B 359 (1991) 705 [INSPIRE].CrossRefADSGoogle Scholar
  10. [10]
    B. Craps, F. Roose, W. Troost and A. Van Proeyen, The Definitions of special geometry, hep-th/9606073 [INSPIRE].
  11. [11]
    L. Andrianopoli et al., N=2 supergravity and N = 2 super Yang-Mills theory on general scalar manifolds: Symplectic covariance, gaugings and the momentum map, J. Geom. Phys. 23 (1997) 111 [hep-th/9605032] [INSPIRE].MATHMathSciNetCrossRefADSGoogle Scholar
  12. [12]
    D.S. Freed, Special Kähler manifolds, Commun. Math. Phys. 203 (1999) 31 [hep-th/9712042] [INSPIRE].MATHMathSciNetCrossRefADSGoogle Scholar
  13. [13]
    D.V. Alekseevsky, V. Cortés and C. Devchand, Special complex manifolds, J. Geom. Phys. 42 (2002) 85 [math/9910091] [INSPIRE].MATHMathSciNetCrossRefADSGoogle Scholar
  14. [14]
    L. Álvarez-Gaumé and D.Z. Freedman, Geometrical Structure and Ultraviolet Finiteness in the Supersymmetric σ-model, Commun. Math. Phys. 80 (1981) 443 [INSPIRE].CrossRefADSGoogle Scholar
  15. [15]
    J. Bagger and E. Witten, Matter Couplings in N = 2 Supergravity, Nucl. Phys. B 222 (1983) 1 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  16. [16]
    J. De Jaegher, B. de Wit, B. Kleijn and S. Vandoren, Special geometry in hypermultiplets, Nucl. Phys. B 514 (1998) 553 [hep-th/9707262] [INSPIRE].CrossRefADSGoogle Scholar
  17. [17]
    M. Günaydin, G. Sierra and P.K. Townsend, The Geometry of N = 2 Maxwell-Einstein Supergravity and Jordan Algebras, Nucl. Phys. B 242 (1984) 244 [INSPIRE].CrossRefADSGoogle Scholar
  18. [18]
    D.V. Alekseevsky, V. Cortés, C. Devchand and A. Van Proeyen, Flows on quaternionic Kähler and very special real manifolds, Commun. Math. Phys. 238 (2003) 525 [hep-th/0109094] [INSPIRE].MATHCrossRefADSGoogle Scholar
  19. [19]
    V. Cortés, Special Kähler manifolds: A Survey, math/0112114 [INSPIRE].
  20. [20]
    T. Mohaupt and O. Vaughan, Developments in special geometry, J. Phys. Conf. Ser. 343 (2012) 012078 [arXiv:1112.2873] [INSPIRE].CrossRefADSGoogle Scholar
  21. [21]
    D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, (2012).Google Scholar
  22. [22]
    B. de Wit and A. Van Proeyen, Special geometry, cubic polynomials and homogeneous quaternionic spaces, Commun. Math. Phys. 149 (1992) 307 [hep-th/9112027] [INSPIRE].MATHCrossRefADSGoogle Scholar
  23. [23]
    D.V. Alekseevsky and V. Cortés, Geometric construction of the r-map: from affine special real to special Kähler manifolds, Comm. Math. Phys. 291 (2009) 579 [arXiv:0811.1658] [INSPIRE].MATHMathSciNetCrossRefADSGoogle Scholar
  24. [24]
    S. Cecotti, S. Ferrara and L. Girardello, Geometry of Type II Superstrings and the Moduli of Superconformal Field Theories, Int. J. Mod. Phys. A 4 (1989) 2475 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  25. [25]
    S. Ferrara and S. Sabharwal, Quaternionic Manifolds for Type II Superstring Vacua of Calabi-Yau Spaces, Nucl. Phys. B 332 (1990) 317 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  26. [26]
    G.W. Gibbons, M.B. Green and M.J. Perry, Instantons and seven-branes in type IIB superstring theory, Phys. Lett. B 370 (1996) 37 [hep-th/9511080] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  27. [27]
    W. Sabra and O. Vaughan, 10D to 4D Euclidean Supergravity over a Calabi-Yau three-fold, arXiv:1503.05095 [INSPIRE].
  28. [28]
    P. Libermann, Sur les structures presque paracomplexes, C.R. Acad. Sci. Paris Sér I Math. 234 (1952) 2517.MATHMathSciNetGoogle Scholar
  29. [29]
    P. Libermann, Sur le problème déquivalence de certaine structures infinitésimales, Ann. Mat. Pura Appl. 36 (1954) 27.MATHMathSciNetCrossRefGoogle Scholar
  30. [30]
    V. Cruceanu, P. Fortuny and P.M. Gadea, A survey of paracomplex geometry, Rocky Mountain J. Math. 26 (1996) 83.MATHMathSciNetCrossRefGoogle Scholar
  31. [31]
    M. Roček, C. Vafa and S. Vandoren, Hypermultiplets and topological strings, JHEP 02 (2006) 062 [hep-th/0512206] [INSPIRE].ADSGoogle Scholar
  32. [32]
    B. de Wit and F. Saueressig, Off-shell N = 2 tensor supermultiplets, JHEP 09 (2006) 062 [hep-th/0606148] [INSPIRE].CrossRefGoogle Scholar
  33. [33]
    D.V. Alekseevsky, V. Cortés and T. Mohaupt, Conification of Kähler and hyper-Kähler manifolds, Commun. Math. Phys. 324 (2013) 637 [arXiv:1205.2964] [INSPIRE].MATHCrossRefADSGoogle Scholar
  34. [34]
    A. Haydys, Hyper-Kähler and quaternionic Kähler manifolds with S 1 -symmetries, J. Geom. Phys. 58 (2008) 293 [arXiv:0706.4473].MATHMathSciNetCrossRefADSGoogle Scholar
  35. [35]
    S. Alexandrov, D. Persson and B. Pioline, Wall-crossing, Rogers dilogarithm and the QK/HK correspondence, JHEP 12 (2011) 027 [arXiv:1110.0466] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  36. [36]
    N. Hitchin, On the Hyperkähler/Quaternion Kähler Correspondence, Commun. Math. Phys. 324 (2013) 77 [INSPIRE].MATHMathSciNetCrossRefADSGoogle Scholar
  37. [37]
    N. Hitchin, The hyperholomorphic line bundle, Algebraic and Complex Geometry, A. Frühbis-Krüger et al. eds., Springer Proc. Math. Stat. 71 (2014) 209 [arXiv:1306.4241].
  38. [38]
    O. Macia and A. Swann, Twist geometry of the c-map, Commun. Math. Phys. 336 (2015) 1329 [arXiv:1404.0785] [INSPIRE].MATHMathSciNetCrossRefADSGoogle Scholar
  39. [39]
    O. Macia and A. Swann, Elementary deformations and the hyperKähler-quaternionic Kähler correspondence, arXiv:1404.1169 [INSPIRE].
  40. [40]
    D. Robles-Llana, F. Saueressig and S. Vandoren, String loop corrected hypermultiplet moduli spaces, JHEP 03 (2006) 081 [hep-th/0602164] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  41. [41]
    D.V. Alekseevsky, V. Cortés, M. Dyckmanns and T. Mohaupt, Quaternionic Kähler metrics associated with special Kähler manifolds, J. Geom. Phys. 92 (2015) 271 [arXiv:1305.3549] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  42. [42]
    N. Hitchin, Quaternionic Kähler moduli spaces, Prog. Math. 271 (2008) 49, K. Galicki and S.R. Simanca eds., Birkhäuser.Google Scholar
  43. [43]
    M. Günaydin, A. Neitzke, O. Pavlyk and B. Pioline, Quasi-conformal actions, quaternionic discrete series and twistors: SU(2, 1) and G 2(2), Commun. Math. Phys. 283 (2008) 169 [arXiv:0707.1669] [INSPIRE].MATHCrossRefADSGoogle Scholar
  44. [44]
    D. Alekseevsky and V. Cortés, The twistor spaces of a para-quaternionic Kähler manifold, Osaka J. Math. 45 (2008) 215.MATHMathSciNetGoogle Scholar
  45. [45]
    V. Cortés, T. Mohaupt and H. Xu, Completeness in supergravity constructions, Commun. Math. Phys. 311 (2012) 191 [arXiv:1101.5103] [INSPIRE].MATHCrossRefADSGoogle Scholar
  46. [46]
    V. Cortés, M. Dyckmanns and D. Lindemann, Classification of complete projective special real surfaces, arXiv:1302.4570 [INSPIRE].
  47. [47]
    V. Cortés, M. Nardmann and S. Suhr, Completeness of hyperbolic centroaffine hypersurfaces, to appear in Comm. Anal. Geom. (2014), arXiv:1407.3251.
  48. [48]
    T. Mohaupt and O. Vaughan, The Hesse potential, the c-map and black hole solutions, JHEP 07 (2012) 163 [arXiv:1112.2876] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  49. [49]
    D. Errington, T. Mohaupt and O. Vaughan, Non-extremal black hole solutions from the c-map, JHEP 05 (2015) 052 [arXiv:1408.0923] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  50. [50]
    P. Dempster, D. Errington and T. Mohaupt, Nernst branes from special geometry, JHEP 05 (2015) 079 [arXiv:1501.07863] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  51. [51]
    N.J. Hitchin, The Moduli space of complex Lagrangian submanifolds, Asian J. Math. 3 (1999) 77 [math/9901069] [INSPIRE].MATHMathSciNetGoogle Scholar
  52. [52]
    O. Baues and V. Cortés, Realisation of special Kähler manifolds as parabolic spheres, Proc. Am. Math. Soc. 129 (2001) 2403 [math/9911079] [INSPIRE].MATHCrossRefGoogle Scholar
  53. [53]
    O. Baues and V. Cortés, Proper Affine Hyperspheres which fiber over Projective Special Kähler Manifolds, Asian J. Math. 7 (2003) 115 [math/0205308] [INSPIRE].MATHMathSciNetGoogle Scholar
  54. [54]
    V. Cortés, M.-A. Lawn and L. Schäfer, Affine hyperspheres associated to special para-Kähler manifolds, Int. J. Geom. Methods Mod. Phys. 3 (2006) 995.MATHMathSciNetCrossRefGoogle Scholar
  55. [55]
    H. Ooguri, A. Strominger and C. Vafa, Black hole attractors and the topological string, Phys. Rev. D 70 (2004) 106007 [hep-th/0405146] [INSPIRE].MathSciNetADSGoogle Scholar
  56. [56]
    G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Black hole partition functions and duality, JHEP 03 (2006) 074 [hep-th/0601108] [INSPIRE].CrossRefGoogle Scholar
  57. [57]
    G.L. Cardoso, B. de Wit and S. Mahapatra, BPS black holes, the Hesse potential and the topological string, JHEP 06 (2010) 052 [arXiv:1003.1970] [INSPIRE].CrossRefADSGoogle Scholar
  58. [58]
    G.L. Cardoso, B. de Wit and S. Mahapatra, Deformations of special geometry: in search of the topological string, JHEP 09 (2014) 096 [arXiv:1406.5478] [INSPIRE].CrossRefADSGoogle Scholar
  59. [59]
    S. Ferrara and O. Macia, Real symplectic formulation of local special geometry, Phys. Lett. B 637 (2006) 102 [hep-th/0603111] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  60. [60]
    S. Ferrara and O. Macia, Observations on the Darboux coordinates for rigid special geometry, JHEP 05 (2006) 008 [hep-th/0602262] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  61. [61]
    M. Günaydin, A. Neitzke, B. Pioline and A. Waldron, BPS black holes, quantum attractor flows and automorphic forms, Phys. Rev. D 73 (2006) 084019 [hep-th/0512296] [INSPIRE].ADSGoogle Scholar
  62. [62]
    V. Cortés and A. Alekseevsky, Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type, Am. Math. Soc. Transl. 213 (2005) 33.Google Scholar
  63. [63]
    M. Krahe, Para-pluriharmonic maps and twistor spaces, in Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys. 16 (2010) 497, European Mathematical Society, Zürich.Google Scholar
  64. [64]
    V. Cortés, J. Louis, P. Smyth and H. Triendl, On certain Káhler quotients of quaternionic Káhler manifolds, Commun. Math. Phys. 317 (2013) 787 [arXiv:1111.0679] [INSPIRE].MATHCrossRefADSGoogle Scholar
  65. [65]
    S. Kobayashi and K. Nomitzu, Foundations of Differential Geometry Volume II, Interscience Publishers, (1969).Google Scholar
  66. [66]
    V. Cortés, A holomorphic representation formula for parabolic hyperspheres, Banach Center Publ. 57 (2002) 11 [math/0107037] [INSPIRE].CrossRefGoogle Scholar
  67. [67]
    B. de Wit, V. Kaplunovsky, J. Louis and D. Lüst, Perturbative couplings of vector multiplets in N = 2 heterotic string vacua, Nucl. Phys. B 451 (1995) 53 [hep-th/9504006] [INSPIRE].CrossRefADSGoogle Scholar
  68. [68]
    B. de Wit, N=2 electric-magnetic duality in a chiral background, Nucl. Phys. Proc. Suppl. 49 (1996) 191 [hep-th/9602060] [INSPIRE].MATHCrossRefADSGoogle Scholar
  69. [69]
    M. Krahe, Die c-Abbildung nach Ferrara und Sabharwal: Von spezieller zu quaternionisher Kählergeometrie, Diplomarbeit, Universität Bonn (2001).Google Scholar
  70. [70]
    U. Theis and S. Vandoren, Instantons in the double tensor multiplet, JHEP 09 (2002) 059 [hep-th/0208145] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  71. [71]
    U. Theis and P. Van Nieuwenhuizen, Ward identities for N = 2 rigid and local supersymmetry in Euclidean space, Class. Quant. Grav. 18 (2001) 5469 [hep-th/0108204] [INSPIRE].MATHCrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • V. Cortés
    • 1
  • P. Dempster
    • 2
  • T. Mohaupt
    • 3
  • O. Vaughan
    • 1
  1. 1.Department of Mathematics and Center for Mathematical PhysicsUniversity of HamburgHamburgGermany
  2. 2.School of Physics & Astronomy and Center for Theoretical PhysicsSeoul National UniversitySeoulKorea
  3. 3.Department of Mathematical SciencesUniversity of LiverpoolLiverpoolU.K.

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