Abstract
Freudenthal duality can be defined as an anti-involutive, non-linear map acting on symplectic spaces. It was introduced in four-dimensional Maxwell-Einstein theories coupled to a non-linear sigma model of scalar fields. In this short review, I will consider its relation to the U-duality Lie groups of type E 7 in extended supergravity theories, and comment on the relation between the Hessian of the black hole entropy and the pseudo-Euclidean, rigid special (pseudo)Kähler metric of the pre-homogeneous spaces associated to the U-orbits.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Breitenlohner, G. W. Gibbons and D. Maison, “Four-dimensional black holes from Kaluza-Klein theories,” Commun. Math. Phys. 120, 295 (1988).
A. Papapetrou, “A static solution of the equations of the gravitational field for an arbitrary charge distribution,” Proc. R. Irish Acad. A 51, 191 (1947)
S. D. Majumdar, “A class of exact solutions of Einstein’s field equations,” Phys. Rev. 72, 930 (1947).
S. Ferrara, G. W. Gibbons and R. Kallosh, “Black holes and critical points in moduli space,” Nucl. Phys. B 500, 75 (1997) [hep-th/9702103].
L. Borsten, D. Dahanayake, M. J. Duff and W. Rubens, “Black holes admitting a Freudenthal dual,” Phys. Rev. D 80, 026003 (2009) [arXiv:0903.5517 [hep-th]].
S. Ferrara, A. Marrani and A. Yeranyan, “Freudenthal duality and generalized special geometry,” Phys. Lett. B 701, 640 (2011) [arXiv:1102.4857 [hep-th]].
L. Borsten, M. J. Duff, S. Ferrara and A. Marrani, “Freudenthal dual Lagrangians,” Class. Quant. Grav. 30, 235003 (2013) [arXiv:1212.3254 [hep-th]].
S. Ferrara, R. Kallosh and A. Strominger, “N= 2 extremal black holes,” Phys. Rev. D52, 5412 (1995) [hepth/9508072]
A. Strominger, “Macroscopic entropy of N= 2 extremal black holes,” Phys. Lett. B383, 39 (1996) [hep-th/9602111]
S. Ferrara and R. Kallosh, “Supersymmetry and attractors,” Phys. Rev. D54, 1514 (1996) [hep-th/9602136]
S. Ferrara and R. Kallosh, “Universality of supersymmetric attractors,” Phys. Rev. D54, 1525 (1996) [hep-th/9603090].
S. W. Hawking, “Gravitational radiation from colliding black holes,” Phys. Rev. Lett. 26, 1344 (1971)
J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D7, 2333 (1973).
P. Galli, P. Meessen and T. Ortín, “The Freudenthal gauge symmetry of the black holes of N= 2, d = 4 supergravity,” JHEP 1305, 011 (2013) [arXiv:1211.7296 [hep-th]].
J. J. Fernandez-Melgarejo and E. Torrente-Lujan, “N= 2 Sugra BPS Multi-center solutions, quadratic prepotentials and Freudenthal transformations,” JHEP 1405, 081 (2014) [arXiv:1310.4182 [hep-th]].
A. Marrani, C.-X. Qiu, S.-Y. D. Shih, A. Tagliaferro and B. Zumino, “Freudenthal gauge theory,” JHEP 1303, 132 (2013) [arXiv:1208.0013 [hep-th]].
R. B. Brown, “Groups of type E 7,” J. Reine Angew. Math. 236, 79 (1969).
K. Meyberg, “Eine Theorie der Freudenthalschen Triplesysteme. I, II,” Nederl. Akad. Wetensch. Proc. Ser. A71, 162 (1968).
R. S. Garibaldi, “Groups of type E 7 over arbitrary fields,” Commun. Algebra 29, 2689 (2001), [math/9811056 [math.AG]].
S. Krutelevich, “Jordan algebras, exceptional groups, and higher composition laws,” [arXiv:math/0411104]
S. Krutelevich, “Jordan algebras, exceptional groups, and Bhargava composition,” J. Algebra 314, 924 (2007).
S. Ferrara, R. Kallosh and A. Marrani, “Degeneration of groups of type E 7 and minimal coupling in supergravity,” JHEP 1206, 074 (2012) [arXiv:1202.1290[hep-th]].
A. Marrani, E. Orazi and F. Riccioni, “Exceptional reductions,” J. Phys. A44, 155207 (2011) [arXiv:1012.5797 [hep-th]].
E. Cremmer and B. Julia, “The N= 8 supergravity theory. 1. The Lagrangian,” Phys. Lett. B 80, 48 (1978)
E. Cremmer and B. Julia, “The SO(8) supergravity,” Nucl. Phys. B159, 141 (1979).
C. Hull and P. K. Townsend, “Unity of superstring dualities,” Nucl. Phys. B438, 109 (1995) [hepth/9410167].
M. Günaydin, “Lectures on spectrum generating symmetries and U-duality in supergravity, extremal black holes, quantum attractors and harmonic superspace,” Springer Proc. Phys. 134, 31 (2010) [arXiv:0908.0374 [hep-th]].
L. Borsten, M. J. Duff, S. Ferrara, A. Marrani and W. Rubens, “Small orbits,” Phys. Rev. D85, 086002 (2012) [arXiv:1108.0424 [hep-th]].
L. Borsten, M. J. Duff, S. Ferrara, A. Marrani and W. Rubens, “Explicit orbit classification of reducible Jordan algebras and Freudenthal triple systems,” Commun. Math. Phys. 325, 17 (2014) [arXiv:1108.0908 [math.RA]].
M. Günaydin, G. Sierra and P. K. Townsend, “Exceptional supergravity theories and the magic square,” Phys. Lett. B133, 72 (1983)
M. Günaydin, G. Sierra and P. K. Townsend, “The geometry of N= 2 Maxwell-Einstein supergravity and Jordan algebras,” Nucl. Phys. B242, 244 (1984).
M. J. Duff, J. T. Liu and J. Rahmfeld, “Four-dimensional string-string-string triality,” Nucl. Phys. B459, 125 (1996) [hep-th/9508094]
K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova and W. K. Wong, “STU black holes and string triality,” Phys. Rev. D54, 6293 (1996) [hep-th/9608059].
L. Andrianopoli, R. D’Auria and S. Ferrara, “U invariants, black hole entropy and fixed scalars,” Phys. Lett. B403, 12 (1997) [hep-th/9703156].
S. Ferrara, A. Marrani and A. Gnecchi, “d = 4 attractors, effective horizon radius and fake supergravity,” Phys. Rev. D 78, 065003 (2008) [arXiv:0806.3196 [hep-th]].
D. Roest and H. Samtleben, “Twin supergravities,” Class. Quant. Grav. 26, 155001 (2009) [arXiv:0904.1344 [hep-th]].
S. Garibaldi and R. Guralnick, Simple Groups Stabilizing Polynomials, Forum of Mathematics 3 (3), Pi (2015) [arXiv:1309.6611 [math.GR]].
S. Ferrara, A. Marrani, E. Orazi and M. Trigiante, “Dualities near the horizon,” JHEP 1311, 056 (2013) [arXiv:1305.2057 [hep-th]].
T. Kimura, Introduction to Prehomogeneous Vector Spaces, Translations of Math. Monographs 215 (AMS, Providence, 2003).
D. S. Freed, “Special Kähler manifolds,” Commun. Math. Phys. 203, 31 (1999) [arXiv:hep-th/9712042].
I. Yokota, “Subgroup SU(8)/Z 2 of compact simple Lie group E 7 and non-compact simple Lie group E 7(7) of type E 7,” Math. J. OkoyamaUniv. 24, 53 (1982).
P. Aschieri, S. Ferrara and B. Zumino, “Duality rotations in nonlinear electrodynamics and in extended supergravity,” Riv. Nuovo Cim. 31, 625 (2008) [arXiv:0807.4039 [hep-th]].
M. Sato and T. Kimura, “A classification of irreducible prehomogeneous vector spaces and their relative invariants,” Nagoya Math. J. 65, 1 (1977).
R. W. Richardson, “Conjugacy classes in parabolic subgroups of semisimple algebraic groups,” Bull. London Math. Soc. 6, 21 (1974).
E. Vinberg, “The classification of nilpotent elements of graded Lie algebras,” Soviet Math. Dokl. 16 (6), 1517 (1975).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the author in English.
Rights and permissions
About this article
Cite this article
Marrani, A. Freudenthal duality in gravity: from groups of type E 7 to pre-homogeneous spaces. P-Adic Num Ultrametr Anal Appl 7, 322–331 (2015). https://doi.org/10.1134/S207004661504007X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S207004661504007X