Abstract
We establish a relation between time-like, light-like and space-like orbits of the non-compact E 7(7)(\(\mathbb{R} \)) symmetry and discrete E 7(7)(\(\mathbb{Z} \)) invariants. We discuss the Uduality invariant formula for the degeneracy of states \( d (\mathcal{Q}) \) which in the approximation of large occupation numbers reproduces the Bekenstein-Hawking entropy formula for regular black holes with AdS 2 horizon. We explain why the states belonging to light-like orbits, corresponding to classical solutions of “light black holes with null singularity”, decouple from the corresponding index. We also present a separate U-duality invariant formula for the class of light-like orbits specified by discrete E 7(7)(\( \mathbb{Z} \)) invariants. We conclude that the present study of the non-perturbative sector of the theory does not reveal any contradiction with the conjectured all-loop perturbative finiteness of D = 4 \(\mathcal{N} \) = 8 supergravity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Bianchi, S. Ferrara and R. Kallosh, Perturbative and Non-perturbative N = 8 Supergravity, arXiv:0910.3674 [SPIRES].
Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The Ultraviolet Behavior of N = 8 Supergravity at Four Loops, Phys. Rev. Lett. 103 (2009) 081301 [arXiv:0905.2326] [SPIRES].
R. Kallosh, On UV Finiteness of the Four Loop N = 8 Supergravity, JHEP 09 (2009) 116 [arXiv:0906.3495] [SPIRES].
Z. Bern et al., Three-Loop Superfiniteness of N = 8 Supergravity, Phys. Rev. Lett. 98 (2007) 161303 [hep-th/0702112] [SPIRES].
M.B. Green, J.G. Russo and P. Vanhove, Ultraviolet properties of maximal supergravity, Phys. Rev. Lett. 98 (2007) 131602 [hep-th/0611273] [SPIRES].
R. Kallosh, N = 8 Supergravity on the Light Cone, Phys. Rev. D 80 (2009) 105022 [arXiv:0903.4630] [SPIRES].
Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162] [SPIRES].
M. Bianchi, H. Elvang and D.Z. Freedman, Generating Tree Amplitudes in N = 4 SYM and N = 8 SG, JHEP 09 (2008) 063 [arXiv:0805.0757] [SPIRES].
M.B. Green, H. Ooguri and J.H. Schwarz, Decoupling Supergravity from the Superstring, Phys. Rev. Lett. 99 (2007) 041601 [arXiv:0704.0777] [SPIRES].
H.J. Schnitzer, Reggeization of N = 8 Supergravity and N = 4 Yang-Mills Theory II, arXiv:0706.0917 [SPIRES].
J.M. Maldacena, G.W. Moore and A. Strominger, Counting BPS black holes in toroidal type-II string theory, hep-th/9903163 [SPIRES].
D. Shih, A. Strominger and X. Yin, Counting dyons in N = 8 string theory, JHEP 06 (2006) 037 [hep-th/0506151] [SPIRES].
D. Shih and X. Yin, Exact Black Hole Degeneracies and the Topological String, JHEP 04 (2006) 034 [hep-th/0508174] [SPIRES].
B. Pioline, BPS black hole degeneracies and minimal automorphic representations, JHEP 08 (2005) 071 [hep-th/0506228] [SPIRES].
A. Sen, Arithmetic of N = 8 Black Holes, JHEP 02 (2010) 090 [arXiv:0908.0039] [SPIRES].
A. Sen, U-duality Invariant Dyon Spectrum in type-II on T 6, JHEP 08 (2008) 037 [arXiv:0804.0651] [SPIRES].
A. Sen, N = 8 Dyon Partition Function and Walls of Marginal Stability, JHEP 07 (2008) 118 [arXiv:0803.1014] [SPIRES].
R. Kallosh and B. Kol, E 7 Symmetric Area of the Black Hole Horizon, Phys. Rev. D 53 (1996) 5344 [hep-th/9602014] [SPIRES].
C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [SPIRES].
E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [SPIRES].
S. Ferrara and J.M. Maldacena, Branes, central charges and U-duality invariant BPS conditions, Class. Quant. Grav. 15 (1998) 749 [hep-th/9706097] [SPIRES].
S. Ferrara and M. Günaydin, Orbits of exceptional groups, duality and BPS states in string theory, Int. J. Mod. Phys. A 13 (1998) 2075 [hep-th/9708025] [SPIRES].
H. Lü, C.N. Pope and K.S. Stelle, Multiplet structures of BPS solitons, Class. Quant. Grav. 15 (1998) 537 [hep-th/9708109] [SPIRES].
S. Krutelevich, Jordan algebras, exceptional groups, and Bhargava composition, J. Algebra 314 (2007) 924 [math/0411104].
L. Borsten, D. Dahanayake, M.J. Duff and W. Rubens, Black holes admitting a Freudenthal dual, Phys. Rev. D 80 (2009) 026003 [arXiv:0903.5517] [SPIRES].
L. Borsten, D. Dahanayake, M.J. Duff, S. Ferrara, A. Marrani and W. Rubens, Observations on Integral and Continuous U-duality Orbits in N = 8 Supergravity, 1002.4223 [SPIRES].
A. Dabholkar, J.P. Gauntlett, J.A. Harvey and D. Waldram, Strings as Solitons & Black Holes as Strings, Nucl. Phys. B 474 (1996) 85 [hep-th/9511053] [SPIRES].
E. Kiritsis, String theory in a nutshell, Princeton University Press, Princeton U.S.A. (2007) [SPIRES].
M. Bianchi, E. Gava, F. Morales and K.S. Narain, D-strings in unconventional type-I vacuum configurations, Nucl. Phys. B 547 (1999) 96 [hep-th/9811013] [SPIRES].
M. Bianchi and A. Sagnotti, Twist symmetry and open string Wilson lines, Nucl. Phys. B 361 (1991) 519 [SPIRES].
M. Bianchi and A. Sagnotti, On the systematics of open string theories, Phys. Lett. B 247 (1990) 517 [SPIRES].
A. Sen, A Twist in the Dyon Partition Function, arXiv:0911.1563 [SPIRES].
A. Sen, Two Charge System Revisited: Small Black Holes or Horizonless Solutions?, arXiv:0908.3402 [SPIRES].
A. Dabholkar, Exact counting of black hole microstates, Phys. Rev. Lett. 94 (2005) 241301 [hep-th/0409148] [SPIRES].
A. Dabholkar, R. Kallosh and A. Maloney, A stringy cloak for a classical singularity, JHEP 12 (2004) 059 [hep-th/0410076] [SPIRES].
A. Ceresole, S. Ferrara and A. Marrani, 4d/5d Correspondence for the Black Hole Potential and its Critical Points, Class. Quant. Grav. 24 (2007) 5651 [arXiv:0707.0964] [SPIRES].
S. Ferrara and R. Kallosh, Universality of Supersymmetric Attractors, Phys. Rev. D 54 (1996) 1525 [hep-th/9603090] [SPIRES].
G.W. Gibbons, G.T. Horowitz and P.K. Townsend, Higher dimensional resolution of dilatonic black hole singularities, Class. Quant. Grav. 12 (1995) 297 [hep-th/9410073] [SPIRES].
L. Andrianopoli, S. Ferrara, A. Marrani and M. Trigiante, Non-BPS Attractors in 5D and 6D Extended Supergravity, Nucl. Phys. B 795 (2008) 428 [arXiv:0709.3488] [SPIRES].
S. Ferrara, E.G. Gimon and R. Kallosh, Magic supergravities, N = 8 and black hole composites, Phys. Rev. D 74 (2006) 125018 [hep-th/0606211] [SPIRES].
Z. Bern, J.J.M. Carrasco and H. Johansson, Progress on Ultraviolet Finiteness of Supergravity, arXiv:0902.3765 [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 0912.0057
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Bianchi, M., Ferrara, S. & Kallosh, R. Observations on arithmetic invariants and U-duality orbits in \( \mathcal {N} \) = 8 supergravity. J. High Energ. Phys. 2010, 81 (2010). https://doi.org/10.1007/JHEP03(2010)081
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2010)081