Abstract
In these lecture notes, we review some recent developments on the relation between the macroscopic entropy of four-dimensional BPS black holes and the microscopic counting of states beyond the thermodynamical, large charge limit. After a brief overview of charged black holes in supergravity and string theory, we give an extensive introduction to special and very special geometry, attractor flows and topological string theory, including holomorphic anomalies. We then expose the Ooguri-Strominger-Vafa (OSV) conjecture which relates microscopic degeneracies to the topological string amplitude, and review precision tests of this formula on “small” black holes. Finally, motivated by a holographic interpretation of the OSV conjecture, we give a systematic approach to the radial quantization of BPS black holes (i.e. quantum attractors). This suggests the existence of a one-parameter generalization of the topological string amplitude and provides a general framework for constructing automorphic partition functions for black hole degeneracies in theories with sufficient degree of symmetry.
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References
S. A. Hughes, “Trust but verify: The case for astrophysical black holes,” ECONF C0507252 (2005) L006, hep-ph/0511217.
T. Banks, “A critique of pure string theory: Heterodox opinions of diverse dimensions,” hep-th/0306074.
A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy,” Phys. Lett. B379 (1996) 99–104, hep-th/9601029.
H. Ooguri, A. Strominger, and C. Vafa, “Black hole attractors and the topological string,” Phys. Rev. D70 (2004) 106007, hep-th/0405146.
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.
A. Dabholkar, F. Denef, G. W. Moore, and B. Pioline, “Precision counting of small black holes,” JHEP 10 (2005) 096, hep-th/0507014.
H. Ooguri, C. Vafa, and E. P. Verlinde, “Hartle-Hawking wave-function for flux compactifications,” hep-th/0502211.
B. Pioline, “BPS black hole degeneracies and minimal automorphic representations,” JHEP 0508 (2005) 071, hep-th/0506228.
M. Gunaydin, A. Neitzke, and B. Pioline, “Topological wave functions and heat equations,” hep-th/0607200.
M. Gunaydin, A. Neitzke, B. Pioline, and A. Waldron, “Bps black holes, quantum attractor flows and automorphic forms,” Phys. Rev. D73 (2006) 084019, hep-th/0512296.
A. Neitzke, B. Pioline, and S. Vandoren, “Twistors and black holes,” hep-th/0701214.
M. Gunaydin, A. Neitzke, B. Pioline, and A. Waldron, “Quantum attractor flows,” arXiv:0707.0267 [hep-th].
M. Gunaydin, A. Neitzke, O. Pavlyk, and B. Pioline, “Quasi-conformal actions, quaternionic discrete series and twistors: su(2, 1) and g2(2),” arXiv:0707.1669 [hep-th].
P. K. Townsend, “Black holes,” gr-qc/9707012.
T. Damour, “The entropy of black holes: A primer,” hep-th/0401160.
S. W. Hawking, “Particle creation by black holes,” Commun. Math. Phys. 43 (1975) 199–220.
W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D14 (1976) 870.
R. Wald, “Black hole thermodynamics,” Living Rev. Relativity 4 (2001) http://www.livingreviews.org/lrr-2001-6.
A. Strominger, “Ads(2) quantum gravity and string theory,” JHEP 01 (1999) 007, hep-th/9809027.
M. Cvetic and D. Youm, “Dyonic bps saturated black holes of heterotic string on a six torus,” Phys. Rev. D53 (1996) 584–588, hep-th/9507090.
M. Cvetic and D. Youm, “All the static spherically symmetric black holes of heterotic string on a six torus,” Nucl. Phys. B472 (1996) 249–267, hep-th/9512127.
E. Witten and D. I. Olive, “Supersymmetry algebras that include topological charges,” Phys. Lett. B78 (1978) 97.
F. Larsen and F. Wilczek, “Internal structure of black holes,” Phys. Lett. B375 (1996) 37–42, hep-th/9511064.
P. K. Tripathy and S. P. Trivedi, “Non-supersymmetric attractors in string theory,” JHEP 03 (2006) 022, hep-th/0511117.
K. Goldstein, N. Iizuka, R. P. Jena, and S. P. Trivedi, “Non-supersymmetric attractors,” Phys. Rev. D72 (2005) 124021, hep-th/0507096.
R. Kallosh, “New attractors,” JHEP 12 (2005) 022, hep-th/0510024.
R. Kallosh, N. Sivanandam, and M. Soroush, “The non-bps black hole attractor equation,” JHEP 03 (2006) 060, hep-th/0602005.
B. Sahoo and A. Sen, “Higher derivative corrections to non-supersymmetric extremal black holes in n=2 supergravity,” hep-th/0603149.
N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, “The string landscape, black holes and gravity as the weakest force,” hep-th/0601001.
P. Kaura and A. Misra, “On the existence of non-supersymmetric black hole attractors for two-parameter calabi-yau’s and attractor equations,” hep-th/0607132.
Y. Kats, L. Motl, and M. Padi, “Higher-order corrections to mass-charge relation of extremal black holes,” hep-th/0606100.
J. M. Maldacena, “Black holes in string theory,” hep-th/9607235.
A. W. Peet, “Tasi lectures on black holes in string theory,” hep-th/0008241.
J. R. David, G. Mandal, and S. R. Wadia, “Microscopic formulation of black holes in string theory,” Phys. Rept. 369 (2002) 549–686, hep-th/0203048.
S. D. Mathur, “The quantum structure of black holes,” Class. Quant. Grav. 23 (2006) R115, hep-th/0510180.
J. M. Maldacena and A. Strominger, “Statistical entropy of four-dimensional extremal black holes,” Phys. Rev. Lett. 77 (1996) 428–429, hep-th/9603060.
C. V. Johnson, R. R. Khuri, and R. C. Myers, “Entropy of 4d extremal black holes,” Phys. Lett. B378 (1996) 78–86, hep-th/9603061.
J. M. Maldacena, A. Strominger, and E. Witten, “Black hole entropy in M-theory,” JHEP 12 (1997) 002, hep-th/9711053.
R. Dijkgraaf, H. L. Verlinde, and E. P. Verlinde, “Counting dyons in N=4 string theory,” Nucl. Phys. B484 (1997) 543–561, hep-th/9607026.
M. Cvetic and A. A. Tseytlin, “Solitonic strings and bps saturated dyonic black holes,” Phys. Rev. D53 (1996) 5619–5633, hep-th/9512031.
G. L. Cardoso, B. de Wit, J. Kappeli, and T. Mohaupt, “Asymptotic degeneracy of dyonic N=4 string states and black hole entropy,” hep-th/0412287.
D. Shih, A. Strominger, and X. Yin, “Recounting dyons in N=4 string theory,” hep-th/0505094.
D. P. Jatkar and A. Sen, “Dyon spectrum in chl models,” JHEP 04 (2006) 018, hep-th/0510147.
J. R. David, D. P. Jatkar, and A. Sen, “Product representation of dyon partition function in chl models,” hep-th/0602254.
A. Dabholkar and S. Nampuri, “Spectrum of dyons and black holes in chl orbifolds using borcherds lift,” hep-th/0603066.
D. Gaiotto, “Re-recounting dyons in N=4 string theory,” hep-th/0506249.
A. Ceresole, R. D’Auria, and S. Ferrara, “The symplectic structure of n=2 supergravity and its central extension,” Nucl. Phys. Proc. Suppl. 46 (1996) 67–74, hep-th/9509160.
P. Fre, “Supersymmetry and first order equations for extremal states: Monopoles, hyperinstantons, black holes and p- branes,” Nucl. Phys. Proc. Suppl. 57 (1997) 52–64, hep-th/9701054.
G. W. Moore, “Arithmetic and attractors,” hep-th/9807087.
T. Mohaupt, “Black hole entropy, special geometry and strings,” Fortsch. Phys. 49 (2001) 3–161, hep-th/0007195.
K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror symmetry, vol. 1 of Clay Mathematics Monographs. American Mathematical Society, Providence, RI, 2003. With a preface by Vafa.
M. Huebscher, P. Meessen, and T. Ortin, “Supersymmetric solutions of n=2 d=4 sugra: the whole ungauged shebang,” hep-th/0606281.
S. Ferrara, R. Kallosh, and A. Strominger, “N=2 extremal black holes,”Phys. Rev. D52 (1995) 5412–5416, hep-th/9508072.
S. Ferrara and R. Kallosh, “Universality of supersymmetric attractors,” Phys. Rev. D54 (1996) 1525–1534, hep-th/9603090.
S. Ferrara, G. W. Gibbons, and R. Kallosh, “Black holes and critical points in moduli space,” Nucl. Phys. B500 (1997) 75–93, hep-th/9702103.
F. Denef, “Supergravity flows and d-brane stability,” JHEP 08 (2000) 050, hep-th/0005049.
K. Behrndt et al., “Classical and quantum n=2 supersymmetric black holes,” Nucl. Phys. B488 (1997) 236–260, hep-th/9610105.
G. Lopes Cardoso, B. de Wit, J. Kappeli, and T. Mohaupt, “Black hole partition functions and duality,” JHEP 03 (2006) 074, hep-th/0601108.
M. Gunaydin, G. Sierra, and P. K. Townsend, “The geometry of N=2 Maxwell-Einstein supergravity and Jordan algebras,” Nucl. Phys. B242 (1984) 244.
K. McCrimmon, “Jordan algebras and their applications,” Bull. Amer. Math. Soc. 84 (1978), no. 4, 612–627.
P. Jordan, J. von Neumann, and E. P. Wigner, “On an algebraic generalization of the quantum mechanical formalism,” Annals Math. 35 (1934) 29–64.
M. Gunaydin, G. Sierra, and P. K. Townsend, “Exceptional supergravity theories and the magic square,” Phys. Lett. B133 (1983) 72.
S. Ferrara, E. G. Gimon, and R. Kallosh, “Magic supergravities, n=8 and black hole composites,” hep-th/0606211.
S. Ferrara, J. A. Harvey, A. Strominger, and C. Vafa, “Second quantized mirror symmetry,” Phys. Lett. B361 (1995) 59–65, hep-th/9505162.
S. Ferrara and M. Porrati, “The manifolds of scalar background fields in Zn orbifolds,” Phys. Lett. B216 (1989) 289.
D. Etingof P., Kazhdan and A. Polishchuk, “When is the Fourier transform of an elementary function elementary?,” math.AG/0003009.
B. Pioline, “Cubic free field theory,” hep-th/0302043.
M. J. Duff, “String triality, black hole entropy and cayley’s hyperdeterminant,” hep-th/0601134.
R. Kallosh and A. Linde, “Strings, black holes, and quantum information,” Phys. Rev. D73 (2006) 104033, hep-th/0602061.
P. Levay, “Stringy black holes and the geometry of entanglement,” Phys. Rev. D74 (2006) 024030, hep-th/0603136.
R. Kallosh and B. Kol, “E7Symmetric area of the black hole horizon,” Phys. Rev. D53 (1996) 5344–5348, hep-th/9602014.
D. Gaiotto, A. Strominger, and X. Yin, “New connections between 4d and 5d black holes,” JHEP 02 (2006) 024, hep-th/0503217.
E. Witten, “Mirror manifolds and topological field theory,” hep-th/9112056.
M. Marino, “Chern-simons theory and topological strings,” Rev. Mod. Phys. 77 (2005) 675–720, hep-th/0406005.
M. Marino, “Les houches lectures on matrix models and topological strings,” hep-th/0410165.
A. Neitzke and C. Vafa, “Topological strings and their physical applications,” hep-th/0410178.
M. Vonk, “A mini-course on topological trings,” hep-th/0504147.
S. Cordes, G. W. Moore, and S. Ramgoolam, “Lectures on 2-d yang-mills theory, equivariant cohomology and topological field theories,” Nucl. Phys. Proc. Suppl. 41 (1995) 184–244, hep-th/9411210.
M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, “Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes,” Commun. Math. Phys. 165 (1994) 311–428, hep-th/9309140.
R. Gopakumar and C. Vafa, “M-theory and topological strings. I,” hep-th/9809187.
R. Gopakumar and C. Vafa, “M-theory and topological strings. II,” hep-th/9812127.
M. T. Grisaru, A. E. M. van de Ven, and D. Zanon, “Four loop divergences for the $\mathrm{n}=1$ supersymmetric nonlinear sigma model in two-dimensions,” Nucl. Phys. B277 (1986) 409.
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. B359 (1991) 21–74.
I. Antoniadis, S. Ferrara, R. Minasian, and K. S. Narain, “R4 couplings in M- and type II theories on Calabi-Yau spaces,” Nucl. Phys. B507 (1997) 571–588, hep-th/9707013.
M. Marino and G. W. Moore, “Counting higher genus curves in a Calabi-Yau manifold,” Nucl. Phys. B543 (1999) 592–614, hep-th/9808131.
C. Faber and R. Pandharipande, “Hodge integrals and Gromov-Witten theory,” math.AG/9810173.
R. P. Thomas, “Gauge theories on Calabi-Yau manifolds,” 1997. Available as of July 2006 at http://www.ma.ic.ac.uk/˜rpwt/thesis.pdf.
N. A. Nekrasov, H. Ooguri, and C. Vafa, “S-duality and topological strings,” hep-th/0403167.
A. Kapustin, “Gauge theory, topological strings, and S-duality,” JHEP 09 (2004) 034, hep-th/0404041.
D. Maulik, N. A. Nekrasov, A. Okounkov, and R. Pandharipande, “Gromov-Witten theory and Donaldson-Thomas theory,” math.AG/0312059.
D. Maulik, N. A. Nekrasov, A. Okounkov, and R. Pandharipande, “Gromov-Witten theory and Donaldson-Thomas theory, II,” math.AG/0406092.
A. Okounkov and R. Pandharipande, “The local Donaldson-Thomas theory of curves,” math.AG/0512573.
E. Verlinde, “Attractors and the holomorphic anomaly,” hep-th/0412139.
E. Witten, “Quantum background independence in string theory,” hep-th/9306122.
A. A. Gerasimov and S. L. Shatashvili, “Towards integrability of topological strings. i: Three- forms on calabi-yau manifolds,” JHEP 11 (2004) 074, hep-th/0409238.
E. D’Hoker and D. H. Phong, “Complex geometry and supergeometry,” hep-th/0512197.
N. Berkovits, “Ictp lectures on covariant quantization of the superstring,” hep-th/0209059.
I. Antoniadis, E. Gava, K. S. Narain, and T. R. Taylor, “Topological amplitudes in string theory,” Nucl. Phys. B413 (1994) 162–184, hep-th/9307158.
I. Antoniadis, E. Gava, K. S. Narain, and T. R. Taylor, “N=2 type II heterotic duality and higher derivative F terms,” Nucl. Phys. B455 (1995) 109–130, hep-th/9507115.
G. Lopes Cardoso, B. de Wit, and T. Mohaupt, “Corrections to macroscopic supersymmetric black-hole entropy,” Phys. Lett. B451 (1999) 309–316, hep-th/9812082.
G. Lopes Cardoso, B. de Wit, and T. Mohaupt, “Deviations from the area law for supersymmetric black holes,” Fortsch. Phys. 48 (2000) 49–64, hep-th/9904005.
G. Lopes Cardoso, B. de Wit, and T. Mohaupt, “Macroscopic entropy formulae and non-holomorphic corrections for supersymmetric black holes,” Nucl. Phys. B567 (2000) 87–110, hep-th/9906094.
G. Lopes Cardoso, B. de Wit, and T. Mohaupt, “Area law corrections from state counting and supergravity,” Class. Quant. Grav. 17 (2000) 1007–1015, hep-th/9910179.
R. M. Wald, “Black hole entropy in the Noether charge,” Phys. Rev. D48 (1993) 3427–3431, gr-qc/9307038.
T. Jacobson, G. Kang, and R. C. Myers, “On black hole entropy,” Phys. Rev. D49 (1994) 6587–6598, gr-qc/9312023.
T. Jacobson, G. Kang, and R. C. Myers, “Increase of black hole entropy in higher curvature gravity,” Phys. Rev. D52 (1995) 3518–3528, gr-qc/9503020.
P. Kraus and F. Larsen, “Microscopic black hole entropy in theories with higher derivatives,” JHEP 09 (2005) 034, hep-th/0506176.
A. Sen, “Black hole entropy function and the attractor mechanism in higher derivative gravity,” JHEP 09 (2005) 038, hep-th/0506177.
C. Vafa, “Two dimensional Yang-Mills, black holes and topological strings,” hep-th/0406058.
M. Aganagic, H. Ooguri, N. Saulina, and C. Vafa, “Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings,” hep-th/0411280.
D. J. Gross and I. Taylor, Washington, “Two-dimensional qcd is a string theory,” Nucl. Phys. B400 (1993) 181–210, hep-th/9301068.
R. Dijkgraaf, R. Gopakumar, H. Ooguri, and C. Vafa, “Baby universes in string theory,” hep-th/0504221.
D. Shih and X. Yin, “Exact black hole degeneracies and the topological string,” JHEP 04 (2006) 034, hep-th/0508174.
D. Gaiotto, A. Strominger, and X. Yin, “From ads(3)/cft(2) to black holes/topological strings,” hep-th/0602046.
F. Denef and G. W. Moore, “Split states, entropy enigmas, holes and halos,” hep-th/0702146.
C. Beasley et al., “Why ZBH=|Ztop|2,” hep-th/0608021.
J. de Boer, M. C. N. Cheng, R. Dijkgraaf, J. Manschot, and E. Verlinde, “A Farey tail for attractor black holes,” hep-th/0608059.
A. Dabholkar, “Exact counting of black hole microstates,” Phys. Rev. Lett. 94 (2005) 241301, hep-th/0409148.
A. Dabholkar and J. A. Harvey, “Nonrenormalization of the superstring tension,” Phys. Rev. Lett. 63 (1989) 478.
A. Dabholkar, G. W. Gibbons, J. A. Harvey, and F. Ruiz Ruiz, “Superstrings and solitons,” Nucl. Phys. B340 (1990) 33–55.
R. Dijkgraaf, J. M. Maldacena, G. W. Moore, and E. Verlinde, “A black hole Farey tail,” hep-th/0005003.
J. A. Harvey and G. W. Moore, “Fivebrane instantons and R2 couplings in N=4 string theory,” Phys. Rev. D57 (1998) 2323–2328, hep-th/9610237.
A. Sen, “Extremal black holes and elementary string states,” Mod. Phys. Lett. A10 (1995) 2081–2094, hep-th/9504147.
A. Sen, “Black holes and elementary string states in N=2 supersymmetric string theories,” JHEP 02 (1998) 011, hep-th/9712150.
A. Sen, “How does a fundamental string stretch its horizon?,” hep-th/0411255.
A. Dabholkar, R. Kallosh, and A. Maloney, “A stringy cloak for a classical singularity,” JHEP 12 (2004) 059, hep-th/0410076.
V. Hubeny, A. Maloney, and M. Rangamani, “String-corrected black holes,” JHEP 05 (2005) 035, hep-th/0411272.
A. Sen, “Black holes and the spectrum of half-bps states in n=4 supersymmetric string theory,” Adv. Theor. Math. Phys. 9 (2005) 527–558, hep-th/0504005.
A. Sen, “Black holes, elementary strings and holomorphic anomaly,” JHEP 07 (2005) 063, hep-th/0502126.
P. Kraus and F. Larsen, “Holographic gravitational anomalies,” JHEP 01 (2006) 022, hep-th/0508218.
P. Kraus and F. Larsen, “Partition functions and elliptic genera from supergravity,” JHEP 01 (2007) 002, hep-th/0607138.
Kraus, P.: Lectures on Black Holes and the $AdS_3/CFT_2$ Correspondence Lect. Notes Phys. 755 x–xx (2008).
H. A. Kastrup and T. Thiemann, “Canonical quantization of spherically symmetric gravity in Ashtekar’s selfdual representation,” Nucl. Phys. B399 (1993) 211–258, gr-qc/9310012.
K. V. Kuchar, “Geometrodynamics of Schwarzschild black holes,” Phys. Rev. D50 (1994) 3961–3981, gr-qc/9403003.
M. Cavaglia, V. de Alfaro, and A. T. Filippov, “Hamiltonian formalism for black holes and quantization,” Int. J. Mod. Phys. D4 (1995) 661–672, gr-qc/9411070.
H. Hollmann, “Group theoretical quantization of Schwarzschild and Taub-NUT,” Phys. Lett. B388 (1996) 702–706, gr-qc/9609053.
H. Hollmann, “A harmonic space approach to spherically symmetric quantum gravity,” gr-qc/9610042.
P. Breitenlohner, H. Hollmann, and D. Maison, “Quantization of the Reissner-Nordström black hole,” Phys. Lett. B432 (1998) 293–297, gr-qc/9804030.
G. Mandal, “Fermions from half-bps supergravity,” JHEP 08 (2005) 052, hep-th/0502104.
L. Maoz and V. S. Rychkov, “Geometry quantization from supergravity: The case of ’bubbling ads’,” JHEP 08 (2005) 096, hep-th/0508059.
V. S. Rychkov, “D1-d5 black hole microstate counting from supergravity,” JHEP 01 (2006) 063, hep-th/0512053.
L. Grant, L. Maoz, J. Marsano, K. Papadodimas, and V. S. Rychkov, “Minisuperspace quantization of ’bubbling ads’ and free fermion droplets,” JHEP 08 (2005) 025, hep-th/0505079.
I. Biswas, D. Gaiotto, S. Lahiri, and S. Minwalla, “Supersymmetric states of n=4 yang-mills from giant gravitons,” hep-th/0606087.
G. Mandal and N. V. Suryanarayana, “Counting 1/8-bps dual-giants,” JHEP 03 (2007) 031, hep-th/0606088.
J. M. Maldacena, J. Michelson, and A. Strominger, “Anti-de Sitter fragmentation,” JHEP 02 (1999) 011, hep-th/9812073.
B. Pioline and J. Troost, “Schwinger pair production in ads(2),” JHEP 03 (2005) 043, hep-th/0501169.
P. Breitenlohner, G. W. Gibbons, and D. Maison, “Four-dimensional black holes from Kaluza-Klein theories,” Commun. Math. Phys. 120 (1988) 295.
C. M. Hull and B. L. Julia, “Duality and moduli spaces for time-like reductions,” Nucl. Phys. B534 (1998) 250–260, hep-th/9803239.
S. Ferrara and S. Sabharwal, “Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces,” Nucl. Phys. B332 (1990) 317.
T. Damour, M. Henneaux, and H. Nicolai, “Cosmological billiards,” Class. Quant. Grav. 20 (2003) R145–R200, hep-th/0212256.
B. Pioline and A. Waldron, “Quantum cosmology and conformal invariance,” Phys. Rev. Lett. 90 (2003) 031302, hep-th/0209044.
V. de Alfaro, S. Fubini, and G. Furlan, “Conformal invariance in quantum mechanics,” Nuovo Cim. A34 (1976) 569.
W. Kinnersley, “Generation of stationary Einstein-Maxwell fields,” J. Math. Phys. 14 (1973), no. 5 651–653.
A. A. Kirillov, “Merits and demerits of the orbit method,” Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 4, 433–488.
S. Cecotti, S. Ferrara, and L. Girardello, “Geometry of type II superstrings and the moduli of superconformal field theories,” Int. J. Mod. Phys. A4 (1989) 2475.
J. Bagger and E. Witten, “Matter couplings in N=2 supergravity,” Nucl. Phys. B222 (1983) 1.
M. Gutperle and M. Spalinski, “Supergravity instantons for N=2 hypermultiplets,” Nucl. Phys. B598 (2001) 509–529, hep-th/0010192.
K. Behrndt, I. Gaida, D. Lust, S. Mahapatra, and T. Mohaupt, “From type iia black holes to t-dual type iib d-instantons in n=2, d=4 supergravity,” Nucl. Phys. B508 (1997) 659–699, hep-th/9706096.
M. de Vroome and S. Vandoren, “Supergravity description of spacetime instantons,” hep-th/0607055.
A. Swann, “Hyper-Kähler and quaternionic Kähler geometry,” Math. Ann. 289 (1991), no. 3, 421–450.
S. M. Salamon, “Quaternionic Kähler manifolds,” Invent. Math. 67 (1982), no. 1, 143–171.
M. Rocek, C. Vafa, and S. Vandoren, “Hypermultiplets and topological strings,” JHEP 02 (2006) 062, hep-th/0512206.
S. B. Giddings and A. Strominger, “Baby universes, third quantization and the cosmological constant,” Nucl. Phys. B321 (1989) 481.
E. Witten, “Constraints on supersymmetry breaking,” Nucl. Phys. B202 (1982) 253.
L. Alvarez-Gaume, “Supersymmetry and the atiyah-singer index theorem,” Commun. Math. Phys. 90 (1983) 161.
D. Friedan and P. Windey, “Supersymmetric derivation of the atiyah-singer index and the chiral anomaly,” Nucl. Phys. B235 (1984) 395.
J. P. Gauntlett, “Low-energy dynamics of supersymmetric solitons,” Nucl. Phys. B400 (1993) 103–125, hep-th/9205008.
J. P. Gauntlett, “Low-energy dynamics of n=2 supersymmetric monopoles,” Nucl. Phys. B411 (1994) 443–460, hep-th/9305068.
S. M. Salamon, “Differential geometry of quaternionic manifolds,” Annales Scientifiques de l’École Normale Supérieure Sr. 4, 19 (1986) 31–55.
R. J. Baston, “Quaternionic complexes,” J. Geom. Phys. 8 (1992), no. 1-4, 29–52.
M. F. Atiyah, N. J. Hitchin, and I. M. Singer, “Self-duality in four-dimensional Riemannian geometry,” Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461.
D. Anselmi and P. Fre, “Topological sigma models in four-dimensions and triholomorphic maps,” Nucl. Phys. B416 (1994) 255–300, hep-th/9306080.
S. Cecotti, “Homogeneous kahler manifolds and t algebras in n=2 supergravity and superstrings,” Commun. Math. Phys. 124 (1989) 23–55.
M. Gunaydin, K. Koepsell, and H. Nicolai, “Conformal and quasiconformal realizations of exceptional Lie groups,” Commun. Math. Phys. 221 (2001) 57–76, hep-th/0008063.
B. Pioline and A. Waldron, “Automorphic forms: A physicist’s survey,” hep-th/0312068.
B. H. Gross and N. R. Wallach, “On quaternionic discrete series representations, and their continuations,” J. Reine Angew. Math. 481 (1996) 73–123.
M. G. Eastwood and M. L. Ginsberg, “Duality in twistor theory,” Duke Math. J. 48 (1981), no. 1, 177–196.
L. C. Biedenharn, R. Y. Cusson, M. Y. Han, and O. L. Weaver, “Hadronic Regge sequences as primitive realizations of sl(3,r) symmetry,” Phys. Lett. B42 (1972) 257–260.
B. Pioline, H. Nicolai, J. Plefka, and A. Waldron, “R4 couplings, the fundamental membrane and exceptional theta correspondences,” JHEP 03 (2001) 036, hep-th/0102123.
B. Pioline and A. Waldron, “The automorphic membrane,” JHEP 06 (2004) 009, hep-th/0404018.
M. Gunaydin, K. Koepsell, and H. Nicolai, “The minimal unitary representation of E8(8),” Adv. Theor. Math. Phys. 5 (2002) 923–946, hep-th/0109005.
D. Kazhdan and G. Savin, “The smallest representation of simply laced groups,” in Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989), vol. 2 of Israel Math. Conf. Proc., pp. 209–223. Weizmann, Jerusalem, 1990.
D. Kazhdan, B. Pioline, and A. Waldron, “Minimal representations, spherical vectors, and exceptional theta series. I,” Commun. Math. Phys. 226 (2002) 1–40, hep-th/0107222.
M. Gunaydin and O. Pavlyk, “Minimal unitary realizations of exceptional U-duality groups and their subgroups as quasiconformal groups,” JHEP 01 (2005) 019, hep-th/0409272.
M. Gunaydin and O. Pavlyk, “A unified approach to the minimal unitary realizations of noncompact groups and supergroups,” hep-th/0604077.
S. Ferrara and M. Gunaydin, “Orbits of exceptional groups, duality and BPS states in string theory,” Int. J. Mod. Phys. A13 (1998) 2075–2088, hep-th/9708025.
M. Gunaydin, “Unitary realizations of U-duality groups as conformal and quasiconformal groups and extremal black holes of supergravity theories,” AIP Conf. Proc. 767 (2005) 268–287, hep-th/0502235.
N. R. Wallach, “Generalized Whittaker vectors for holomorphic and quaternionic representations,” Comment. Math. Helv. 78 (2003), no. 2, 266–307.
D. Kazhdan and A. Polishchuk, “Minimal representations: spherical vectors and automorphic functionals,” in Algebraic groups and arithmetic, pp. 127–198. Tata Inst. Fund. Res., Mumbai, 2004.
A. Neitzke, B. Pioline, and S. Vandoren. To appear.
D. Shih, A. Strominger, and X. Yin, “Counting dyons in N=8 string theory,” hep-th/0506151.
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Pioline, B. (2008). Lectures on Black Holes, Topological Strings, and Quantum Attractors (2.0). In: Supersymmetric Mechanics - Vol. 3. Lecture Notes in Physics, vol 755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79523-0_6
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