Lectures on black holes, topological strings and quantum attractors

Classical and Quantum Gravity

Volumn 23, Issue 21, 2006, Pages

  Pioline, Boris ^a,b  

^a LPTHE   (France)

^b ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33750610508     PISSN: 02649381     EISSN: 13616382     Source Type: Journal    
DOI: 10.1088/0264-9381/23/21/S05     Document Type: Conference Paper

References (174)

1. Hughes S A 2005 Trust but verify: The case for astrophysical black holes ECONF C 0507252 L006 (Preprint hep-ph/0511217)
2. Banks T 2003 A critique of pure string theory: heterodox opinions of diverse dimensions Preprint hep-th/0306074
3. Townsend P K 1997 Black holes Preprint gr-qc/9707012
4. Damour T 2004 The entropy of black holes: a primer Preprint hep-th/0401160
5. Hawking S W 1975 Particle creation by black holes Commun. Math. Phys. 43 199-220
6. Unruh W G 1976 Notes on black hole evaporation Phys. Rev. D 14 870
7. Wald R 2001 Black hole thermodynamics Living Rev. Rel. 4 http://www.livingreviews.org/lrr-2001-6
8. Strominger A 1999 Ads(2) quantum gravity and string theory J. High Energy Phys. JHEP01(1999)007 (Preprint hep-th/9809027)
9. Cvetic M and Youm D 1996 Dyonic BPS saturated black holes of heterotic string on a six torus Phys. Rev. D 53 584-8 (Preprint hep-th/9507090)
10. Cvetic M and Youm D 1996 All the static spherically symmetric black holes of heterotic string on a six torus Nucl. Phys. B 472 249-67 (Preprint hep-th/9512127)
11. Witten E and Olive D I 1978 Supersymmetry algebras that include topological charges Phys. Lett. B 78 97
12. Larsen F and Wilczek F 1996 Internal structure of black holes Phys. Lett. B 375 37-42 (Preprint hep-th/9511064)
13. Tripathy P K and Trivedi S P 2006 Non-supersymmetric attractors in string theory J. High Energy Phys. JHEP03(2006)022 (Preprint hep-th/0511117)
14. Goldstein K, Iizuka N, Jena R P and Trivedi S P 2005 Non-supersymmetric attractors Phys. Rev. D 72 124021 (Preprint hep-th/0507096)
15. Kallosh R 2005 New attractors J. High Energy Phys. JHEP12(2005)022 (Preprint hep-th/0510024)
16. Kallosh R, Sivanandam N and Soroush M 2006 The non-BPS black hole attractor equation J. High Energy Phys. JHEP03(2006)060 (Preprint hep-th/0602005)
17. Sahoo B and Sen A 2006 Higher derivative corrections to non-supersymmetric extremal black holes in n ≤ 2 supergravity Preprint hep-th/0603149
18. Arkani-Hamed N, Motl L, Nicolis A and Vafa C 2006 The string landscape, black holes and gravity as the weakest force Preprint hep-th/0601001
19. Kaura P and Misra A 2006 On the existence of non-supersymmetric black hole attractors for two-parameter Calabi-Yau's and attractor equations Preprint hep-th/0607132
20. Kats Y, Motl L and Padi M 2006 Higher-order corrections to mass-charge relation of extremal black holes Preprint hep-th/0606100
21. Maldacena J M 1996 Black holes in string theory Preprint hep-th/9607235
22. Peet A W 2000 Tasi lectures on black holes in string theory Preprint hep-th/0008241
23. David J R, Mandal G and Wadia S R 2002 Microscopic formulation of black holes in string theory Phys. Rep. 369 549-686 (Preprint hep-th/0203048)
24. Mathur S D 2006 The quantum structure of black holes Class. Quantum Grav. 23 R115 (Preprint hep-th/0510180)
25. Strominger A and Vafa C 1996 Microscopic origin of the Bekenstein-Hawking entropy Phys. Lett. B 379 99-104 (Preprint hep-th/9601029)
26. Maldacena J M and Strominger A 1996 Statistical entropy of four-dimensional extremal black holes Phys. Rev. Lett. 77 428-9 (Preprint hep-th/9603060)
27. Johnson C V, Khuri R R and Myers R C 1996 Entropy of 4d extremal black holes Phys. Lett. B 378 78-86 (Preprint hep-th/9603061)
28. Maldacena J M, Strominger A and Witten E 1997 Black hole entropy in M-theory J. High Energy Phys. JHEP12(1997)002 (Preprint hep-th/9711053)
29. Dijkgraaf R, Verlinde H L and Verlinde E P 1997 Counting dyons in string theory Nucl. Phys. B 484 543-61 (Preprint hep-th/9607026)
30. Cvetic M and Tseytlin A A 1996 Solitonic strings and BPS saturated dyonic black holes Phys. Rev. D 53 5619-33 (Preprint hep-th/9512031)
31. Cardoso G L, de Wit B, Kappeli J and Mohaupt T 2004 Asymptotic degeneracy of dyonic N ≤ 4 string states and black hole entropy Preprint hep-th/0412287
32. Shih D, Strominger A and Yin X 2005 Recounting dyons in string theory Preprint hep-th/0505094
33. Jatkar D P and Sen A 2006 Dyon spectrum in chl models J. High Energy Phys. JHEP04(2006)018 (Preprint hep-th/0510147)
34. David J R, Jatkar D P and Sen A 2006 Product representation of dyon partition function in chl models Preprint hep-th/0602254
35. Dabholkar A and Nampuri S 2006 Spectrum of dyons and black holes in chl orbifolds using borcherds lift Preprint hep-th/0603066
36. Gaiotto D 2005 Re-recounting dyons in n ≤ 4 string theory Preprint hep-th/0506249
37. Ceresole A, D'Auria R and Ferrara S 1996 The symplectic structure of n ≤ 2 supergravity and its central extension Nucl. Phys. Proc. Suppl. 46 67-74 (Preprint hep-th/9509160)
38. Fre P 1997 Supersymmetry and first order equations for extremal states: monopoles, hyperinstantons, black holes and p-branes Nucl. Phys. Proc. Suppl. 57 52-64 (Preprint hep-th/9701054)
39. Moore G W 1998 Arithmetic and attractors Preprint hep-th/9807087
40. Mohaupt T 2001 Black hole entropy, special geometry and strings Fortschr. Phys. 49 3-161 (Preprint hep-th/0007195)
41. Hori K, Katz S, Klemm A, Pandharipande R, Thomas R, Vafa C, Vakil R and Zaslow E 2003 Mirror Symmetry (Clay Mathematics Monographs vol 1) (Providence, RI: American Mathematical Society) With a preface by Vafa
42. Huebscher M, Meessen P and Ortin T 2006 Supersymmetric solutions of n ≤ 2 d ≤ 4 sugra: the whole ungauged shebang Preprint hep-th/0606281
43. Ferrara S, Kallosh R and Strominger A 1995 extremal black holes Phys. Rev. D 52 5412-6 (Preprint hep-th/9508072)
44. Ferrara S and Kallosh R 1996 Universality of supersymmetric attractors Phys. Rev. D 54 1525-34 (Preprint hep-th/9603090)
45. Ferrara S, Gibbons G W and Kallosh R 1997 Black holes and critical points in moduli space Nucl. Phys. B 500 75-93 (Preprint hep-th/9702103)
46. Denef F 2000 Supergravity flows and d-brane stability J. High Energy Phys. JHEP08(2000)050 (Preprint hep-th/0005049)
47. Ooguri H, Vafa C and Verlinde E P 2005 Hartle-Hawking wave-function for flux compactifications Preprint hep-th/0502211
48. Lopes Cardoso G, de Wit B, Kappeli J and Mohaupt T 2006 Black hole partition functions and duality J. High Energy Phys. JHEP03(2006)074 (Preprint hep-th/0601108)
49. Gunaydin M, Sierra G and Townsend P K 1984 The geometry of Maxwell-Einstein supergravity and Jordan algebras Nucl. Phys. B 242 244
50. McCrimmon K 1978 Jordan algebras and their applications Bull. Am. Math. Soc. 84 612-27
51. Jordan P, von Neumann J and Wigner E P 1934 On an algebraic generalization of the quantum mechanical formalism Ann. Math. 35 29-64
52. Gunaydin M, Sierra G and Townsend P K 1983 Exceptional supergravity theories and the magic square Phys. Lett. B 133 72
53. Ferrara S, Gimon E G and Kallosh R 2006 Magic supergravities, n ≤ 8 and black hole composites Preprint hep-th/0606211
54. Pioline B 2005 BPS black hole degeneracies and minimal automorphic representations J. High Energy Phys. JHEP0508(2005)071 (Preprint hep-th/0506228)
55. Etingof D, Kazhdan P and Polishchuk A 2000 When is the Fourier transform of an elementary function elementary? Preprint math.AG/0003009
56. Pioline B 2003 Cubic free field theory Preprint hep-th/0302043
57. Duff M J 2006 String triality, black hole entropy and Cayley's hyperdeterminant Preprint hep-th/0601134
58. Kallosh R and Linde A 2006 Strings, black holes and quantum information Phys. Rev. D 73 104033 (Preprint hep-th/0602061)
59. Levay P 2006 Stringy black holes and the geometry of entanglement Phys. Rev. D 74 024030 (Preprint hep-th/0603136)
60. 7 symmetric area of the black hole horizon Phys. Rev. D 53 5344-8 (Preprint hep-th/9602014)
61. Gaiotto D, Strominger A and Yin X 2006 New connections between 4d and 5d black holes J. High Energy Phys. JHEP02(2006)024 (Preprint hep-th/0503217)
62. Witten E 1991 Mirror manifolds and topological field theory Preprint hep-th/9112056
63. Marino M 2005 Chern-Simons theory and topological strings Rev. Mod. Phys. 77 675-720 (Preprint hep-th/0406005)
64. Marino M 2004 Les Houches lectures on matrix models and topological strings Preprint hep-th/0410165
65. Neitzke A and Vafa C 2004 Topological strings and their physical applications Preprint hep-th/0410178
66. Vonk M 2005 A mini-course on topological strings Preprint hep-th/0504147
67. Cordes S, Moore G W and Ramgoolam S 1995 Lectures on 2-d Yang-Mills theory, equivariant cohomology and topological field theories Nucl. Phys. Proc. Suppl. 41 184-244 (Preprint hep-th/9411210)
68. Bershadsky M, Cecotti S, Ooguri H and Vafa C 1994 Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes Commun. Math. Phys. 165 311-428 (Preprint hep-th/9309140)
69. Gopakumar R and Vafa C 1998 M-theory and topological strings. I Preprint hep-th/9809187
70. Gopakumar R and Vafa C 1998 Preprint M-theory and topological strings. II Preprint hep-th/9812127
71. Dabholkar A, Denef F, Moore G W and Pioline B 2005 Precision counting of small black holes J. High Energy Phys. JHEP10(2005)096 (Preprint hep-th/0507014)
72. Grisaru M T, van de Ven A E M and Zanon D 1986 Four loop divergences for the n ≤ 1 supersymmetric nonlinear sigma model in two-dimensions Nucl. Phys. B 277 409
73. Candelas P, De La Ossa X C, Green P S and Parkes L 1991 A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory Nucl. Phys. B 359 21-74
74. 4 couplings in M- and type II theories on Calabi-Yau spaces Nucl. Phys. B 507 571-88 (Preprint hep-th/9707013)
75. Marino M and Moore G W 1999 Counting higher genus curves in a Calabi-Yau manifold Nucl. Phys. B 543 592-614 (Preprint hep-th/9808131)
76. Faber C and Pandharipande R 1998 Hodge integrals and Gromov-Witten theory Preprint math.AG/9810173
77. Dabholkar A, Denef F, Moore G W and Pioline B 2005 Exact and asymptotic degeneracies of small black holes J. High Energy Phys. JHEP08(2005)021 (Preprint hep-th/0502157)
78. Thomas R P 1997 Gauge theories on Calabi-Yau manifolds (available as of July 2006 at http://www.ma.ic.ac.uk/ rpwt/thesis.pdf)
79. Nekrasov N A, Ooguri H and Vafa C 2004 S-duality and topological strings Preprint hep-th/0403167
80. Kapustin A 2004 Gauge theory, topological strings and S-duality J. High Energy Phys. JHEP09(2004)034 (Preprint hep-th/0404041)
81. Maulik D, Nekrasov N A, Okounkov A and Pandharipande R 2003 Gromov-Witten theory and Donaldson-Thomas theory Preprint math.AG/0312059
82. Maulik D, Nekrasov N A, Okounkov A and Pandharipande R 2004 Gromov-Witten theory and Donaldson-Thomas theory: II Preprint math.AG/0406092
83. Okounkov A and Pandharipande R 2005 The local Donaldson-Thomas theory of curves Preprint math.AG/0512573
84. Verlinde E 2004 Attractors and the holomorphic anomaly Preprint hep-th/0412139
85. Witten E 1993 Quantum background independence in string theory Preprint hep-th/9306122
86. Gerasimov A A and Shatashvili S L 2004 Towards integrability of topological strings: I. Three-forms on Calabi-Yau manifolds J. High Energy Phys. JHEP11(2004)074 (Preprint hep-th/0409238)
87. Gunaydin M, Neitzke A and Pioline B 2006 Topological wave functions and heat equations Preprint hep-th/0607200
88. D'Hoker E and Phong D H 2005 Complex geometry and supergeometry Preprint hep-th/0512197
89. Berkovits N 2002 ICTP lectures on covariant quantization of the superstring Preprint hep-th/0209059
90. Antoniadis I, Gava E, Narain K S and Taylor T R 1994 Topological amplitudes in string theory Nucl. Phys. B 413 162-84 (Preprint hep-th/9307158)
91. Antoniadis I, Gava E, Narain K S and Taylor T R 1995 N ≤ 2 type II heterotic duality and higher derivative F terms Nucl. Phys. B 455 109-30 (Preprint hep-th/9507115)
92. Lopes Cardoso G, de Wit B and Mohaupt T 1999 Corrections to macroscopic supersymmetric black-hole entropy Phys. Lett. B 451 309-16 (Preprint hep-th/9812082)
93. Lopes Cardoso G, de Wit B and Mohaupt T 2000 Deviations from the area law for supersymmetric black holes Fortschr. Phys. 48 49-64 (Preprint hep-th/9904005)
94. Lopes Cardoso G, de Wit B and Mohaupt T 2000 Macroscopic entropy formulae and non-holomorphic corrections for supersymmetric black holes Nucl. Phys. B 567 87-110 (Preprint hep-th/9906094)
95. Lopes Cardoso G, de Wit B and Mohaupt T 2000 Area law corrections from state counting and supergravity Class. Quantum Grav. 17 1007-15 (Preprint hep-th/9910179)
96. Wald R M 1993 Black hole entropy in the Noether charge Phys. Rev. D 48 3427-31 (Preprint gr-qc/9307038)
97. Jacobson T, Kang G and Myers R C 1994 On black hole entropy Phys. Rev. D 49 6587-98 (Preprint gr-qc/9312023)
98. Jacobson T, Kang G and Myers R C 1995 Increase of black hole entropy in higher curvature gravity Phys. Rev. D 52 3518-28 (Preprint gr-qc/9503020)
99. Kraus P and Larsen F 2005 Microscopic black hole entropy in theories with higher derivatives Preprint hep-th/0506176
100. Ooguri H, Strominger A and Vafa C 2004 Black hole attractors and the topological string Phys. Rev. D 70 106007 (Preprint hep-th/0405146)
101. Sen A 2005 Black hole entropy function and the attractor mechanism in higher derivative gravity J. High Energy Phys. JHEP09(2005)038 (Preprint hep-th/0506177)
102. Vafa C 2004 Two dimensional Yang-Mills, black holes and topological strings Preprint hep-th/0406058
103. Aganagic M, Ooguri H, Saulina N and Vafa C 2004 Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings Preprint hep-th/0411280
104. Gross D J and Taylor W 1993 Two-dimensional QCD is a string theory Nucl. Phys. B 400 181-210 (Preprint hep-th/9301068)
105. Dijkgraaf R, Gopakumar R, Ooguri H and Vafa C 2005 Baby universes in string theory Preprint hep-th/0504221
106. Shih D and Yin X 2006 Exact black hole degeneracies and the topological string J. High Energy Phys. JHEP04(2006)034 (Preprint hep-th/0508174)
107. Gaiotto D, Strominger A and Yin X 2006 From AdS(3)/CFT(2) to black holes/topological strings Preprint hep-th/0602046
108. Denef F and Moore G W (to appear)
109. 2 Preprint hep-th/0608021
110. de Boer J, Cheng M C N, Dijkgraaf R, Manschot J and Verlinde E 2006 A Farey tail for attractor black holes Preprint hep-th/0608059
111. Dabholkar A 2004 Exact counting of black hole microstates Preprint hep-th/0409148
112. Dabholkar A and Harvey J A 1989 Nonrenormalization of the superstring tension Phys. Rev. Lett. 63 478
113. Dabholkar A, Gibbons G W, Harvey J A and Ruiz Ruiz F 1990 Superstrings and solitons Nucl. Phys. B 340 33-55
114. Dijkgraaf R, Maldacena J M, Moore G W and Verlinde E 2000 A black hole Farey tail Preprint hep-th/0005003
115. 2 couplings in N ≤ 4 string theory Phys. Rev. D 57 2323-8 (Preprint hep-th/9610237)
116. Sen A 1995 Extremal black holes and elementary string states Mod. Phys. Lett. A 10 2081-94 (Preprint hep-th/9504147)
117. Sen A 1998 Black holes and elementary string states in N ≤ 2 supersymmetric string theories J. High Energy Phys. JHEP02(1998)011 (Preprint hep-th/9712150)
118. Sen A 2004 How does a fundamental string stretch its horizon? Preprint hep-th/0411255
119. Dabholkar A, Kallosh R and Maloney A 2004 A stringy cloak for a classical singularity J. High Energy Phys. JHEP12(2004)059 (Preprint hep-th/0410076)
120. Hubeny V, Maloney A and Rangamani M 2004 String-corrected black holes Preprint hep-th/0411272
121. Sen A 2005 Black holes and the spectrum of half-BPS states in n ≤ 4 supersymmetric string theory Adv. Theor. Math. Phys. 9 527-58 (Preprint hep-th/0504005)
122. Sen A 2005 Black holes, elementary strings and holomorphic anomaly J. High Energy Phys. JHEP07(2005)063 (Preprint hep-th/0502126)
123. Gunaydin M, Neitzke A, Pioline B and Waldron A 2006 BPS black holes, quantum attractor flows and automorphic forms Phys. Rev. D 73 084019 (Preprint hep-th/0512296)
124. Kastrup H A and Thiemann T 1993 Canonical quantization of spherically symmetric gravity in Ashtekar's selfdual representation Nucl. Phys. B 399 211-58 (Preprint gr-qc/9310012)
125. Kuchar K V 1994 Geometrodynamics of Schwarzschild black holes Phys. Rev. D 50 3961-81 (Preprint gr-qc/9403003)
126. Cavaglia M, de Alfaro V and Filippov A T 1995 Hamiltonian formalism for black holes and quantization Int. J. Mod. Phys. D 4 661-72 (Preprint gr-qc/9411070)
127. Hollmann H 1996 Group theoretical quantization of Schwarzschild and Taub-NUT Phys. Lett. B 388 702-6 (Preprint gr-qc/9609053)
128. Hollmann H 1996 A harmonic space approach to spherically symmetric quantum gravity Preprint gr-qc/9610042
129. Breitenlohner P, Hollmann H and Maison D 1998 Quantization of the Reissner-Nordström black hole Phys. Lett. B 432 293-7 (Preprint gr-qc/9804030)
130. Mandal G 2005 Fermions from half-BPS supergravity J. High Energy Phys. JHEP08(2005)052 (Preprint hep-th/0502104)
131. Maoz L and Rychkov V S 2005 Geometry quantization from supergravity: the case of bubbling AdS J. High Energy Phys. JHEP08(2005)096 (Preprint hep-th/0508059)
132. Grant L, Maoz L, Marsano J, Papadodimas K and Rychkov V S 2005 Minisuperspace quantization of bubbling AdS and free fermion droplets J. High Energy Phys. (Preprint hep-th/0505079)
133. Biswas I, Gaiotto D, Lahiri S and Minwalla S 2006 Supersymmetric states of n ≤ 4 Yang-Mills from giant gravitons Preprint hep-th/0606087
134. Mandal G and Suryanarayana N V 2006 Counting 1/8-BPS dual-giants Preprint hep-th/0606088
135. Rychkov V S 2006 D1-d5 black hole microstate counting from supergravity J. High Energy Phys. JHEP08(2006)063 (Preprint hep-th/0512053)
136. Maldacena J M, Michelson J and Strominger A 1999 Anti-de Sitter fragmentation J. High Energy Phys. JHEP02(1999)011 (Preprint hep-th/9812073)
137. Pioline B and Troost J 2005 Schwinger pair production in AdS(2) J. High Energy Phys. JHEP03(2005)043 (Preprint hep-th/0501169)
138. Breitenlohner P, Gibbons G W and Maison D 1988 Four-dimensional black holes from Kaluza-Klein theories Commun. Math. Phys. 120 295
139. Hull C M and Julia B L 1998 Duality and moduli spaces for time-like reductions Nucl. Phys. B 534 250-60 (Preprint hep-th/9803239)
140. Damour T, Henneaux M and Nicolai H 2003 Cosmological billiards Class. Quantum Grav. 20 R145-200 (Preprint hep-th/0212256)
141. Pioline B and Waldron A 2003 Quantum cosmology and conformal invariance Phys. Rev. Lett. 90 031302 (Preprint hep-th/0209044)
142. de Alfaro V, Fubini S and Furlan G 1976 Conformal invariance in quantum mechanics Nuovo Cimento A 34 569
143. Obers N A and Pioline B 1999 U-duality and M-theory Phys. Rep. 318 113-225 (Preprint hep-th/9809039)
144. Sekiguchi J 1987 Remarks on real nilpotent orbits of a symmetric pair J. Math. Soc. Japan 39 127-38
145. Vergne M 1995 Instantons et correspondance de Kostant-Sekiguchi C. R. Acad. Sci. Paris I 320 901-6
146. Cecotti S, Ferrara S and Girardello L 1989 Geometry of type II superstrings and the moduli of superconformal field theories Int. J. Mod. Phys. A 4 2475
147. Ferrara S and Sabharwal S 1990 Quaternionic manifolds for type II superstring Vacua of Calabi-Yau spaces Nucl. Phys. B 332 317
148. Cortes V, Mayer C, Mohaupt T and Saueressig F 2005 Special geometry of Euclidean supersymmetry: II. Hypermultiplets and the C-map J. High Energy Phys. (Preprint hep-th/0503094)
149. Gunaydin M, Neitzke A, Pioline B and Waldron A Quantum attractors (to appear)
150. Giddings S B and Strominger A 1989 Baby universes, third quantization and the cosmological constant Nucl. Phys. B 321 481
151. de Wit B, Rocek M and Vandoren S 2001 Hypermultiplets, hyperKaehler cones and quaternion-Kaehler geometry J. High Energy Phys. JHEP02(2001)039 (Preprint hep-th/0101161)
152. Atiyah M F, Hitchin N J and Singer I M 1978 Self-duality in four-dimensional Riemannian geometry Proc. R. Soc. Lond. Ser. A 362 425-61
153. Salamon S M 1986 Differential geometry of quaternionic manifolds Ann. Sci. de l'Icole Normale Supérieure 4, 19 31-55
154. Baston R J 1992 Quaternionic complexes J. Geom. Phys. 8 29-52
155. Cecotti S 1989 Homogeneous Kahler manifolds and t algebras in n ≤2 supergravity and superstrings Commun. Math. Phys. 124 23-55
156. Alekseevski D V 1975 Classification of quaternionic spaces with transitive solvable group of motions Izv. Akad. Nauk SSSR Ser. Mat. 39 315-62, 472
157. Gunaydin M, Koepsell K and Nicolai H 2001 Conformal and quasiconformal realizations of exceptional Lie groups Commun. Math. Phys. 221 57-76 (Preprint hep-th/0008063)
158. Pioline B and Waldron A 2003 Automorphic forms: a physicist's survey Preprint hep-th/0312068
159. Gross B H and Wallach N R 1996 On quaternionic discrete series representations, and their continuations J. Reine Angew. Math. 481 73-123
160. Kirillov A A 1999 Merits and demerits of the orbit method Bull. Am. Math. Soc. (N. S.) 36 433-88
161. Günaydin M, Neitzke A, Pioline B, Vandoren S, Rocek M and Waldron A (to appear)
162. Gunaydin M, Neitzke A, Pavlyk O, Pioline B and Waldron A Quasiconformal, minimal representations and their spherical vectors (to appear)
163. Ferrara S and Gunaydin M 1998 Orbits of exceptional groups, duality and BPS states in string theory Int. J. Mod. Phys. A 13 2075-88 (Preprint hep-th/9708025)
164. Biedenharn L C, Cusson R Y, Han M Y and Weaver O L 1972 Hadronic Regge sequences as primitive realizations of sl(3, r) symmetry Phys. Lett. B 42 257-60
165. 4 couplings, the fundamental membrane and exceptional theta correspondences J. High Energy Phys. JHEP03(2001)036 (Preprint hep-th/0102123)
166. Kazhdan D, Pioline B and Waldron A 2002 Minimal representations, spherical vectors, and exceptional theta series: I Commun. Math. Phys. 226 1-40 (Preprint hep-th/0107222)
167. Pioline B and Waldron A 2004 The automorphic membrane J. High Energy Phys. JHEP06(2004)009 (Preprint hep-th/0404018)
168. 8(8) Adv. Theor. Math. Phys. 5 923-46 (Preprint hep-th/0109005)
169. Gunaydin M and Pavlyk O 2005 Minimal unitary realizations of exceptional U-duality groups and their subgroups as quasiconformal groups J. High Energy Phys. JHEP01(2005)019 (Preprint hep-th/0409272)
170. Gunaydin M and Pavlyk O 2006 A unified approach to the minimal unitary realizations of noncompact groups and supergroups Preprint hep-th/0604077
171. Anselmi D and Fre P 1994 Topological sigma models in four-dimensions and triholomorphic maps Nucl. Phys. B 416 255-300 (Preprint hep-th/9306080)
172. Wallach N R 2003 Generalized Whittaker vectors for holomorphic and quaternionic representations Comment. Math. Helv. 78 266-307
173. 2 Duke Math. J. 115 105-69
174. Gunaydin M 2005 Unitary realizations of U-duality groups as conformal and quasiconformal groups and extremal black holes of supergravity theories AIP Conf. Proc. 767 268-87 (Preprint hep-th/0502235)