Download PDF

Indefinite theta functions for counting attractor backgrounds

  • Gabriel Lopes Cardoso
  • Michele Cirafici
  • Suresh Nampuri
Open Access
Article
First Online:
Received:
Accepted:

Abstract

In this note, we employ indefinite theta functions to regularize canonical partition functions for single-center dyonic BPS black holes. These partition functions count dyonic degeneracies in the Hilbert space of four-dimensional toroidally compactified heterotic string theory, graded by electric and magnetic charges. The regularization is achieved by viewing the weighted sums of degeneracies as sums over charge excitations in the near-horizon attractor geometry of an arbitrarily chosen black hole background, and eliminating the unstable modes. This enables us to rewrite these sums in terms of indefinite theta functions. Background independence is then implemented by using the transformation property of indefinite theta functions under elliptic transformations, while modular transformations are used to make contact with semi-classical results in supergravity.

Keywords

Supersymmetry and Duality Black Holes in String Theory Extended Supersymmetry Black Holes 
Download to read the full article text

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    H. Ooguri, A. Strominger and C. Vafa, Black hole attractors and the topological string, Phys. Rev. D 70 (2004) 106007 [hep-th/0405146] [INSPIRE].ADSMathSciNetGoogle Scholar
  2. [2]
    D. Shih and X. Yin, Exact black hole degeneracies and the topological string, JHEP 04 (2006) 034 [hep-th/0508174] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    G.L. Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Black hole partition functions and duality, JHEP 03 (2006) 074 [hep-th/0601108] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    G.L. Cardoso, J.R. David, B. de Wit and S. Mahapatra, The mixed black hole partition function for the STU model, JHEP 12 (2008) 086 [arXiv:0810.1233] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Precision counting of small black holes, JHEP 10 (2005) 096 [hep-th/0507014] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Counting dyons in N = 4 string theory, Nucl. Phys. B 484 (1997) 543 [hep-th/9607026] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    D. Shih, A. Strominger and X. Yin, Recounting Dyons in N = 4 string theory, JHEP 10 (2006) 087 [hep-th/0505094] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    J.R. David and A. Sen, CHL Dyons and Statistical Entropy Function from D1-D5 System, JHEP 11 (2006) 072 [hep-th/0605210] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    A. Dabholkar, S. Murthy and D. Zagier, Quantum Black Holes, Wall Crossing and Mock Modular Forms, arXiv:1208.4074 [INSPIRE].
  11. [11]
    G.L. Cardoso, M. Cirafici, R. Jorge and S. Nampuri, Indefinite theta functions and black hole partition functions, JHEP 02 (2014) 019 [arXiv:1309.4428] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    S. Zwegers, Mock Theta Functions, arXiv:0807.4834 [INSPIRE].
  13. [13]
    J. Manschot, Stability and duality in N = 2 supergravity, Commun. Math. Phys. 299 (2010) 651 [arXiv:0906.1767] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    J. Manschot, Wall-crossing of D4-branes using flow trees, Adv. Theor. Math. Phys. 15 (2011) 1 [arXiv:1003.1570] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    S. Alexandrov, J. Manschot and B. Pioline, D3-instantons, Mock Theta Series and Twistors, JHEP 04 (2013) 002 [arXiv:1207.1109] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    S. Ferrara and A. Van Proeyen, A theorem on N = 2 special Kähler product manifolds, Class. Quant. Grav. 6 (1989) L243 [INSPIRE].CrossRefMATHGoogle Scholar
  17. [17]
    P.S. Aspinwall, Compactification, geometry and duality: N = 2, hep-th/0001001 [INSPIRE].
  18. [18]
    S. Ferrara, R. Kallosh and A. Strominger, N = 2 extremal black holes, Phys. Rev. D 52 (1995) 5412 [hep-th/9508072] [INSPIRE].ADSMathSciNetGoogle Scholar
  19. [19]
    S. Ferrara and R. Kallosh, Supersymmetry and attractors, Phys. Rev. D 54 (1996) 1514 [hep-th/9602136] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  20. [20]
    S. Ferrara and R. Kallosh, Universality of supersymmetric attractors, Phys. Rev. D 54 (1996) 1525 [hep-th/9603090] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  21. [21]
    K. Behrndt et al., Classical and quantum N = 2 supersymmetric black holes, Nucl. Phys. B 488 (1997) 236 [hep-th/9610105] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  22. [22]
    G.L. Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Stationary BPS solutions in N = 2 supergravity with R 2 interactions, JHEP 12 (2000) 019 [hep-th/0009234] [INSPIRE].CrossRefMATHGoogle Scholar
  23. [23]
    G.L. Cardoso, B. de Wit and T. Mohaupt, Macroscopic entropy formulae and nonholomorphic corrections for supersymmetric black holes, Nucl. Phys. B 567 (2000) 87 [hep-th/9906094] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  24. [24]
    G.L. Cardoso and A. Veliz-Osorio, On the σ-model of deformed special geometry, Nucl. Phys. B 872 (2013) 228 [arXiv:1212.4364] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    D.P. Jatkar and A. Sen, Dyon spectrum in CHL models, JHEP 04 (2006) 018 [hep-th/0510147] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    A. Dabholkar, D. Gaiotto and S. Nampuri, Comments on the spectrum of CHL dyons, JHEP 01 (2008) 023 [hep-th/0702150] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    M.C.N. Cheng and E. Verlinde, Dying Dyons Don’t Count, JHEP 09 (2007) 070 [arXiv:0706.2363] [INSPIRE].ADSMathSciNetGoogle Scholar
  28. [28]
    D. Gaiotto, A. Strominger and X. Yin, The M5-Brane Elliptic Genus: Modularity and BPS States, JHEP 08 (2007) 070 [hep-th/0607010] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    J. de Boer, M.C.N. Cheng, R. Dijkgraaf, J. Manschot and E. Verlinde, A Farey Tail for Attractor Black Holes, JHEP 11 (2006) 024 [hep-th/0608059] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  30. [30]
    L. Göttsche and D. Zagier, Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b + = 1, alg-geom/9612020 [INSPIRE].
  31. [31]
    B. de Wit, BPS black holes, Nucl. Phys. Proc. Suppl. 171 (2007) 16 [arXiv:0704.1452] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    B. Eynard and M. Mariño, A holomorphic and background independent partition function for matrix models and topological strings, J. Geom. Phys. 61 (2011) 1181 [arXiv:0810.4273] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    R. Akhoury and A. Comtet, Anomalous Behavior of the Witten Index: Exactly Soluble Models, Nucl. Phys. B 246 (1984) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    A. Giveon, N. Itzhaki and J. Troost, Lessons on Black Holes from the Elliptic Genus, arXiv:1401.3104 [INSPIRE].
  35. [35]
    S.K. Ashok, E. Dell’Aquila and J. Troost, Higher Poles and Crossing Phenomena from Twisted Genera, JHEP 08 (2014) 087 [arXiv:1404.7396] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    F. Denef, TASI lectures on complex structures, arXiv:1104.0254 [INSPIRE].

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Gabriel Lopes Cardoso
    • 1
  • Michele Cirafici
    • 1
  • Suresh Nampuri
    • 2
  1. 1.Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior TécnicoUniversidade de LisboaLisboaPortugal
  2. 2.National Institute for Theoretical Physics (NITheP), School of Physics and Center for Theoretical PhysicsUniversity of WitwatersrandJohannesburgSouth Africa

Personalised recommendations