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Split states, entropy enigmas, holes and halos

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Abstract

We investigate degeneracies of BPS states of D-branes on compact Calabi-Yau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute explicitly exact indices of various nontrivial D-brane systems, and to clarify the subtle relation of Donaldson-Thomas invariants to BPS indices of stable D6-D2-D0 states, realized in supergravity as “hole halos.” We introduce a convergent generating function for D4 indices in the large CY volume limit, and prove it can be written as a modular average of its polar part, generalizing the fareytail expansion of the elliptic genus. We show polar states are “split” D6-anti-D6 bound states, and that the partition function factorizes accordingly, leading to a refined version of the OSV conjecture. This differs from the original conjecture in several aspects. In particular we obtain a nontrivial measure factor \( g_{\text{top}}^{{ - 2}}{e^{{ - K}}} \) and find factorization requires a cutoff. We show that the main factor determining the cutoff and therefore the error is the existence of “swing states” — D6 states which exist at large radius but do not form stable D6-anti-D6 bound states. We point out a likely breakdown of the OSV conjecture at small g top (in the large background CY volume limit), due to the surprising phenomenon that for sufficiently large background Kähler moduli, a charge Λ Γ supporting single centered black holes of entropy ~ Λ2 S(Γ) also admits two-centered BPS black hole realizations whose entropy grows like Λ3 when Λ → ∞.

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References

  1. A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  2. J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  3. C. Vafa, Black holes and Calabi-Yau threefolds, Adv. Theor. Math. Phys. 2 (1998) 207 [hep-th/9711067] [INSPIRE].

    MATH  MathSciNet  Google Scholar 

  4. S. Ferrara, R. Kallosh and A. Strominger, N = 2 extremal black holes, Phys. Rev. D 52 (1995) 5412 [hep-th/9508072] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  5. A. Strominger, Macroscopic entropy of N = 2 extremal black holes, Phys. Lett. B 383 (1996) 39 [hep-th/9602111] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  6. G. Lopes Cardoso, B. de Wit and T. Mohaupt, Corrections to macroscopic supersymmetric black hole entropy, Phys. Lett. B 451 (1999) 309 [hep-th/9812082] [INSPIRE].

    ADS  Google Scholar 

  7. G. Lopes Cardoso, B. de Wit and T. Mohaupt, Deviations from the area law for supersymmetric black holes, Fortsch. Phys. 48 (2000) 49 [hep-th/9904005] [INSPIRE].

    Article  ADS  Google Scholar 

  8. G. Lopes Cardoso, B. de Wit and T. Mohaupt, Area law corrections from state counting and supergravity, Class. Quant. Grav. 17 (2000) 1007 [hep-th/9910179] [INSPIRE].

    Article  MATH  ADS  Google Scholar 

  9. T. Mohaupt, Black hole entropy, special geometry and strings, Fortsch. Phys. 49 (2001) 3 [hep-th/0007195] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. H. Ooguri, A. Strominger and C. Vafa, Black hole attractors and the topological string, Phys. Rev. D 70 (2004) 106007 [hep-th/0405146] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  11. A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Exact and asymptotic degeneracies of small black holes, JHEP 08 (2005) 021 [hep-th/0502157] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  12. A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Precision counting of small black holes, JHEP 10 (2005) 096 [hep-th/0507014] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  13. D. Shih and X. Yin, Exact black hole degeneracies and the topological string, JHEP 04 (2006) 034 [hep-th/0508174] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  14. G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Asymptotic degeneracy of dyonic N =4 string states and black hole entropy, JHEP 12 (2004) 075 [hep-th/0412287] [INSPIRE].

    ADS  Google Scholar 

  15. 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].

    Article  Google Scholar 

  16. F. Denef, Counting D-brane ground states, talk given at Black holes, topological strings, and invariants of holomorphic submanifolds, Harvard, Cambridge U.S.A. January 31 2006.

  17. F. Denef, A derivation of OSV (refined), talk given at workshop Black holes, black rings and topological strings, Munich Germany April 1 2006.

  18. F. Denef, From OSV to OSV, talk at Strings06, Beijing China 19–24 June 2006.

  19. G. Moore, Split polar attractors, talk at the 4th Simons workshop, http://insti.physics.sunysb.edu/itp/conf/simonswork4/, Stony Brook U.S.A. July 24-August 25 2006.

  20. D. Gaiotto, A. Strominger and X. Yin, From AdS 3 /CFT 2 to black holes/topological strings, JHEP 09 (2007) 050 [hep-th/0602046] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  21. X. Yin, The M 5 brane elliptic genus, talk at Strings06, Beijing China 19–24 June 2006.

  22. J. de Boer, M.C. Cheng, R. Dijkgraaf, J. Manschot and E. Verlinde, A farey tail for attractor black holes, JHEP 11 (2006) 024 [hep-th/0608059] [INSPIRE].

    Article  Google Scholar 

  23. E. Verlinde, A farey tail for N = 2 black holes, talk at Strings06, Beijing China 19–24 June 2006.

  24. C. Beasley et al., Why Z BH = |Z top|2, hep-th/0608021 [INSPIRE].

  25. E.P. Verlinde, Attractors and the holomorphic anomaly, hep-th/0412139 [INSPIRE].

  26. F. Denef, Supergravity flows and D-brane stability, JHEP 08 (2000) 050 [hep-th/0005049] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  27. F. Denef, On the correspondence between D-branes and stationary supergravity solutions of type-II Calabi-Yau compactifications, hep-th/0010222 [INSPIRE].

  28. F. Denef, B.R. Greene and M. Raugas, Split attractor flows and the spectrum of BPS D-branes on the quintic, JHEP 05 (2001) 012 [hep-th/0101135] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  29. B. Bates and F. Denef, Exact solutions for supersymmetric stationary black hole composites, hep-th/0304094 [INSPIRE].

  30. M.-x. Huang, A. Klemm, M. Mariño and A. Tavanfar, Black holes and large order quantum geometry, Phys. Rev. D 79 (2009) 066001 [arXiv:0704.2440] [INSPIRE].

    ADS  Google Scholar 

  31. R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A black hole farey tail, hep-th/0005003 [INSPIRE].

  32. G.W. Moore, Les Houches lectures on strings and arithmetic, hep-th/0401049 [INSPIRE].

  33. P. Kraus and F. Larsen, Partition functions and elliptic genera from supergravity, JHEP 01 (2007) 002 [hep-th/0607138] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  34. R. Dijkgraaf, C. Vafa and E. Verlinde, M-theory and a topological string duality, hep-th/0602087 [INSPIRE].

  35. S. Ashok and M.R. Douglas, Counting flux vacua, JHEP 01 (2004) 060 [hep-th/0307049] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  36. F. Denef and M.R. Douglas, Distributions of flux vacua, JHEP 05 (2004) 072 [hep-th/0404116] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  37. R. Gopakumar and C. Vafa, M theory and topological strings. 1., hep-th/9809187 [INSPIRE].

  38. R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].

  39. A. Klemm, M. Kreuzer, E. Riegler and E. Scheidegger, Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections, JHEP 05 (2005) 023 [hep-th/0410018] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  40. A. Iqbal, N. Nekrasov, A. Okounkov and C. Vafa, Quantum foam and topological strings, JHEP 04 (2008) 011 [hep-th/0312022] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  41. D. Maulik, N. Nekrasov, A. Okounkov and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, I, math.AG/0312059.

  42. D. Maulik, N. Nekrasov, A. Okounkov and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, II, math.AG/0406092.

  43. R. Thomas, Gauge theory on Calabi-Yau manifolds, Ph.D. thesis, University of Oxford, Oxford U.K. (1997).

  44. S. Donaldson and R. Thomas, Gauge theory in higher dimensions, in The geometric universe: science, geometry, and the work of Roger Penrose, S. Huggett et al. eds., Oxford University Press, Oxford U.K. (1998).

    Google Scholar 

  45. R. Thomas, A holomorphic Casson invariant for Calabi-Yau three folds and bundles on K3 fibrations, math.AG/9806111 [INSPIRE].

  46. B. Pioline, Lectures on black holes, topological strings and quantum attractors, Class. Quant. Grav. 23 (2006) S981 [hep-th/0607227] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  47. I. Brunner, M.R. Douglas, A.E. Lawrence and C. Romelsberger, D-branes on the quintic, JHEP 08 (2000) 015 [hep-th/9906200] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  48. J. Gomis, F. Marchesano and D. Mateos, An open string landscape, JHEP 11 (2005) 021 [hep-th/0506179] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  49. D. Gaiotto et al., D4-D0 branes on the quintic, JHEP 03 (2006) 019 [hep-th/0509168] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  50. R. Minasian and G.W. Moore, K theory and Ramond-Ramond charge, JHEP 11 (1997) 002 [hep-th/9710230] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  51. D.S. Freed and E. Witten, Anomalies in string theory with D-branes, Asian J. Math 3 (1999) 819 [hep-th/9907189] [INSPIRE].

    MATH  MathSciNet  Google Scholar 

  52. D. Gaiotto, A. Strominger and X. Yin, The M5-brane elliptic genus: modularity and BPS states, JHEP 08 (2007) 070 [hep-th/0607010] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  53. B.J. Spence, Topological Born-Infeld actions and D-branes, hep-th/9907053 [INSPIRE].

  54. D. Belov and G.W. Moore, Conformal blocks for AdS 5 singletons, hep-th/0412167 [INSPIRE].

  55. A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, hep-th/0604151 [INSPIRE].

  56. N. Nekrasov, Localizing gauge theory, in XIV th international congress on mathematical physics, Lisbon Portugal 2003, J.-C. Zambrini ed., World Scientific, Singapore (2005), pg. 645.

    Google Scholar 

  57. A.A. Tseytlin, Selfduality of Born-Infeld action and Dirichlet three-brane of type IIB superstring theory, Nucl. Phys. B 469 (1996) 51 [hep-th/9602064] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  58. V.V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izvestija 14 (1980) 103 [Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 111].

    Article  MATH  ADS  Google Scholar 

  59. S. Gukov, E. Martinec, G.W. Moore and A. Strominger, Chern-Simons gauge theory and the AdS 3 /CF T 2 correspondence, hep-th/0403225 [INSPIRE].

  60. J. Milnor and D. Husemoller, Symmetric bilinear forms, Ergeb. Math. Grenzgeb., Band 73. MR 58 (1973) #22129, Springer-Verlag, New York U.S.A. (1973).

  61. N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  62. A. Strominger, Massless black holes and conifolds in string theory, Nucl. Phys. B 451 (1995) 96 [hep-th/9504090] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  63. M.R. Douglas, Topics in D geometry, Class. Quant. Grav. 17 (2000) 1057 [hep-th/9910170] [INSPIRE].

    Article  MATH  ADS  Google Scholar 

  64. G.W. Moore, Arithmetic and attractors, hep-th/9807087 [INSPIRE].

  65. G.W. Moore, Attractors and arithmetic, hep-th/9807056 [INSPIRE].

  66. K. Behrndt, D. Lüst and W.A. Sabra, Stationary solutions of N = 2 supergravity, Nucl. Phys. B 510 (1998) 264 [hep-th/9705169] [INSPIRE].

    Article  ADS  Google Scholar 

  67. G. Lopes 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].

    Article  ADS  Google Scholar 

  68. S. Ferrara, G.W. Gibbons and R. Kallosh, Black holes and critical points in moduli space, Nucl. Phys. B 500 (1997) 75 [hep-th/9702103] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  69. M. Shmakova, Calabi-Yau black holes, Phys. Rev. D 56 (1997) 540 [hep-th/9612076] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  70. F. Denef, Quantum quivers and Hall/hole halos, JHEP 10 (2002) 023 [hep-th/0206072] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  71. M.R. Douglas, D-branes, categories and N = 1 supersymmetry, J. Math. Phys. 42 (2001) 2818 [hep-th/0011017] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  72. T. Bridgeland, Spaces of stability conditions, math.AG/0611510.

  73. D. Joyce, Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds, hep-th/0607039 [INSPIRE].

  74. F. Denef, (Dis)assembling special Lagrangians, hep-th/0107152 [INSPIRE].

  75. D. Gaiotto, A. Strominger and X. Yin, New connections between 4D and 5D black holes, JHEP 02 (2006) 024 [hep-th/0503217] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  76. D. Gaiotto, A. Strominger and X. Yin, 5D black rings and 4D black holes, JHEP 02 (2006) 023 [hep-th/0504126] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  77. K. Behrndt, G. Lopes Cardoso and S. Mahapatra, Exploring the relation between 4D and 5D BPS solutions, Nucl. Phys. B 732 (2006) 200 [hep-th/0506251] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  78. M.C. Cheng, More bubbling solutions, JHEP 03 (2007) 070 [hep-th/0611156] [INSPIRE].

    ADS  Google Scholar 

  79. I. Bena and P. Kraus, Microscopic description of black rings in AdS/CFT, JHEP 12 (2004) 070 [hep-th/0408186] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  80. I. Bena and P. Kraus, Microstates of the D1-D5-KK system, Phys. Rev. D 72 (2005) 025007 [hep-th/0503053] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  81. I. Bena, P. Kraus and N.P. Warner, Black rings in Taub-NUT, Phys. Rev. D 72 (2005) 084019 [hep-th/0504142] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  82. H. Elvang, R. Emparan, D. Mateos and H.S. Reall, Supersymmetric 4D rotating black holes from 5D black rings, JHEP 08 (2005) 042 [hep-th/0504125] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  83. P. Berglund, E.G. Gimon and T.S. Levi, Supergravity microstates for BPS black holes and black rings, JHEP 06 (2006) 007 [hep-th/0505167] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  84. V. Balasubramanian, E.G. Gimon and T.S. Levi, Four dimensional black hole microstates: from D-branes to spacetime foam, JHEP 01 (2008) 056 [hep-th/0606118] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  85. I. Bena and N.P. Warner, Bubbling supertubes and foaming black holes, Phys. Rev. D 74 (2006) 066001 [hep-th/0505166] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  86. I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  87. J. de Boer, F. Denef, S. El-Showk, I. Messamah and D. Van den Bleeken, Black hole bound states in AdS 3 × S 2, JHEP 11 (2008) 050 [arXiv:0802.2257] [INSPIRE].

    Article  Google Scholar 

  88. S. Kachru and J. McGreevy, Supersymmetric three cycles and supersymmetry breaking, Phys. Rev. D 61 (2000) 026001 [hep-th/9908135] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  89. M.R. Douglas, B. Fiol and C. Romelsberger, Stability and BPS branes, JHEP 09 (2005) 006 [hep-th/0002037] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  90. E. Witten, BPS bound states of D0-D6 and D0-D8 systems in a B field, JHEP 04 (2002) 012 [hep-th/0012054] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  91. M. Berkooz, M.R. Douglas and R.G. Leigh, Branes intersecting at angles, Nucl. Phys. B 480 (1996) 265 [hep-th/9606139] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  92. C. Vafa, Gas of D-branes and Hagedorn density of BPS states, Nucl. Phys. B 463 (1996) 415 [hep-th/9511088] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  93. J.A. Harvey and G.W. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998) 489 [hep-th/9609017] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  94. E. Witten, On the conformal field theory of the Higgs branch, JHEP 07 (1997) 003 [hep-th/9707093] [INSPIRE].

    Article  ADS  Google Scholar 

  95. R. Dijkgraaf, Fields, strings, matrices and symmetric products, hep-th/9912104 [INSPIRE].

  96. E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  97. L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990) 193.

    Article  MATH  MathSciNet  Google Scholar 

  98. L. Göttsche, Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lect. Notes Math. 1572 (1994) 1, Springer-Verlag, Berlin Germany (1994).

    Article  Google Scholar 

  99. M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2003) 349 [math.QA/0204059].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  100. S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys. B 433 (1995) 501 [hep-th/9406055] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  101. S.H. Katz, A. Klemm and C. Vafa, M theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999) 1445 [hep-th/9910181] [INSPIRE].

    MATH  MathSciNet  Google Scholar 

  102. J.A. Harvey and G.W. Moore, Five-brane instantons and R 2 couplings in N = 4 string theory, Phys. Rev. D 57 (1998) 2323 [hep-th/9610237] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  103. M. Mariño and G.W. Moore, Counting higher genus curves in a Calabi-Yau manifold, Nucl. Phys. B 543 (1999) 592 [hep-th/9808131] [INSPIRE].

    Article  ADS  Google Scholar 

  104. A. Klemm and M. Mariño, Counting BPS states on the enriques Calabi-Yau, Commun. Math. Phys. 280 (2008) 27 [hep-th/0512227] [INSPIRE].

    Article  MATH  ADS  Google Scholar 

  105. R. Lazarsfeld, Positivity in algebraic geometry I, Ergeb. Math. Grenzgeb. 48 (2004) 1, Springer-Verlag, Berlin Germany (2004).

    MathSciNet  Google Scholar 

  106. M.R. Douglas, R. Reinbacher and S.-T. Yau, Branes, bundles and attractors: Bogomolov and beyond, math.AG/0604597 [INSPIRE].

  107. I. Bena, D.-E. Diaconescu and B. Florea, Black string entropy and Fourier-Mukai transform, JHEP 04 (2007) 045 [hep-th/0610068] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  108. M.-x. Huang, A. Klemm and S. Quackenbush, Topological string theory on compact Calabi-Yau: modularity and boundary conditions, Lect. Notes Phys. 757 (2009) 45 [hep-th/0612125] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  109. A. Dabholkar, A. Sen and S.P. Trivedi, Black hole microstates and attractor without supersymmetry, JHEP 01 (2007) 096 [hep-th/0611143] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  110. P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, U.S.A. (1978).

    MATH  Google Scholar 

  111. R. Minasian, G.W. Moore and D. Tsimpis, Calabi-Yau black holes and (0, 4) σ-models, Commun. Math. Phys. 209 (2000) 325 [hep-th/9904217] [INSPIRE].

    MATH  ADS  MathSciNet  Google Scholar 

  112. I.R. Shafarevich, Basic algebraic geometry, volume 1, Springer, Berlin Germany (1994), pg. 245.

    Book  Google Scholar 

  113. E. Cattani, A. Kaplan and P. Deligne, On the locus of Hodge classes, J. Amer. Math. Soc. 8 (1995) 483 [alg-geom/9402009].

    Article  MATH  MathSciNet  Google Scholar 

  114. W. Lerche, P. Mayr and N. Warner, N = 1 special geometry, mixed Hodge variations and toric geometry, hep-th/0208039 [INSPIRE].

  115. L. Martucci, D-branes on general N = 1 backgrounds: superpotentials and D-terms, JHEP 06 (2006) 033 [hep-th/0602129] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  116. R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  117. S. Hosono, M.-H. Saito and A. Takahashi, Relative Lefschetz action and BPS state counting, math.AG/0105148 [INSPIRE].

  118. G. Szegö, Orthogonal polynomials, fourth edition, Amer. Math. Soc. Colloq. Publ. 23, Providence U.S.A. (1975).

  119. H. Ooguri, C. Vafa and E.P. Verlinde, Hartle-Hawking wave-function for flux compactifications, Lett. Math. Phys. 74 (2005) 311 [hep-th/0502211] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  120. M. Günaydin, A. Neitzke and B. Pioline, Topological wave functions and heat equations, JHEP 12 (2006) 070 [hep-th/0607200] [INSPIRE].

    Article  Google Scholar 

  121. S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  122. P. Candelas, X.C. De La Ossa, P.S. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21 [INSPIRE].

    Article  ADS  Google Scholar 

  123. J.A. Harvey and G.W. Moore, Algebras, BPS states and strings, Nucl. Phys. B 463 (1996) 315 [hep-th/9510182] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  124. G.W. Moore, String duality, automorphic forms and generalized Kac-Moody algebras, Nucl. Phys. Proc. Suppl. 67 (1998) 56 [hep-th/9710198] [INSPIRE].

    Article  MATH  ADS  Google Scholar 

  125. C.A.M. Peters, A criterion for flatness of Hodge bundles over curves and geometric applications, Math. Ann. 268 (1984) 1.

    Article  MATH  MathSciNet  Google Scholar 

  126. D.S. Freed, G.W. Moore and G. Segal, The uncertainty of fluxes, Commun. Math. Phys. 271 (2007) 247 [hep-th/0605198] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  127. D.S. Freed, G.W. Moore and G. Segal, Heisenberg groups and noncommutative fluxes, Annals Phys. 322 (2007) 236 [hep-th/0605200] [INSPIRE].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  128. G.W. Moore, Anomalies, Gauss laws and Page charges in M-theory, Comptes Rendus Physique 6 (2005) 251 [hep-th/0409158] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, B-3001, Leuven, Belgium

    Frederik Denef

  2. NHETC and Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Rd, Piscataway, NJ, 08855-0849, USA

    Gregory W. Moore

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  1. Frederik Denef
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  2. Gregory W. Moore
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Correspondence to Frederik Denef.

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ArXiv ePrint: hep-th/0702146

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Denef, F., Moore, G.W. Split states, entropy enigmas, holes and halos. J. High Energ. Phys. 2011, 129 (2011). https://doi.org/10.1007/JHEP11(2011)129

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  • DOI: https://doi.org/10.1007/JHEP11(2011)129

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