Abstract
We study properties of D = 4 \( \mathcal{N} \geqslant {2} \) extended supergravities (and related compactifications of superstring theory) and their consistent truncation to the phenomenologically interesting models of \( \mathcal{N} = {1} \) supergravity. This involves a detailed classification of the “degenerations” of the duality groups of type E 7, when the corresponding quartic invariant polynomial built from the symplectic irreducible representation of G4 “degenerates” into a perfect square. With regard to cosmological applications, minimal coupling of vectors in consistent truncation to \( \mathcal{N} = {1} \) from higher-dimensional or \( {\text{higher}} - \mathcal{N} \) theory is non-generic. On the other hand, non-minimal coupling involving vectors coupled to scalars and axions is generic. These features of supergravity, following from the electric-magnetic duality, may be useful in other applications, like stabilization of moduli, and in studies of non-perturbative black-hole solutions of supergravity/string theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Cremmer and B. Julia, The N = 8 supergravity theory. 1. The Lagrangian, Phys. Lett. B 80 (1978) 48 [INSPIRE].
E. Cremmer and B. Julia, The SO(8) supergravity, Nucl. Phys. B 159 (1979) 141 [INSPIRE].
C. Hull and P. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton U.S.A. (1992).
S. Ferrara and R. Kallosh, Creation of matter in the universe and groups of type E 7, JHEP 12 (2011) 096 [arXiv:1110.4048] [INSPIRE].
M.K. Gaillard and B. Zumino, Duality rotations for interacting fields, Nucl. Phys. B 193 (1981) 221 [INSPIRE].
P. Aschieri, S. Ferrara and B. Zumino, Duality rotations in nonlinear electrodynamics and in extended supergravity, Riv. Nuovo Cim. 31 (2008) 625 [arXiv:0807.4039] [INSPIRE].
Andrianopoli L, BertoliniM, Ceresole, A,R. D’Auria, S. Ferrara, et al., N = 2 supergravity and N = 2 super Yang-Mills theory on general scalar manifolds: symplectic covariance, gaugings and the momentum map, J. Geom. Phys. 23 (1997) 111 [hep-th/9605032] [INSPIRE].
E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Yang-Mills theories with local supersymmetry: Lagrangian, transformation laws and superhiggs effect, Nucl. Phys. B 212 (1983) 413 [INSPIRE].
L. Andrianopoli, R. D’Auria and S. Ferrara, Supersymmetry reduction of N extended supergravities in four-dimensions, JHEP 03 (2002) 025 [hep-th/0110277] [INSPIRE].
L. Andrianopoli, R. D’Auria and S. Ferrara, Consistent reduction of N = 2 → N = 1 four-dimensional supergravity coupled to matter, Nucl. Phys. B 628 (2002) 387 [hep-th/0112192] [INSPIRE].
A. Ceresole, R. D’Auria, S. Ferrara and A. Van Proeyen, Duality transformations in supersymmetric Yang-Mills theories coupled to supergravity, Nucl. Phys. B 444 (1995) 92 [hep-th/9502072] [INSPIRE].
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 [hep-th/9509160] [INSPIRE].
E. Witten, Dimensional reduction of superstring models, Phys. Lett. B 155 (1985) 151 [INSPIRE].
A. Cadavid, A. Ceresole, R. D’Auria and S. Ferrara, Eleven-dimensional supergravity compactified on Calabi-Yau threefolds, Phys. Lett. B 357 (1995) 76 [hep-th/9506144] [INSPIRE].
R.B. Brown, Groups of type E 7, J. Reine Angew. Math. 236 (1969) 79.
L. Borsten, D. Dahanayake, M. Duff and W. Rubens, Black holes admitting a Freudenthal dual, Phys. Rev. D 80 (2009) 026003 [arXiv:0903.5517] [INSPIRE].
S. Ferrara and R. Kallosh, Universality of supersymmetric attractors, Phys. Rev. D 54 (1996) 1525 [hep-th/9603090] [INSPIRE].
L. Andrianopoli, R. D’Auria and S. Ferrara, Central extension of extended supergravities in diverse dimensions, Int. J. Mod. Phys. A 12 (1997) 3759 [hep-th/9608015] [INSPIRE].
L. Andrianopoli, R. D’Auria and S. Ferrara, U duality and central charges in various dimensions revisited, Int. J. Mod. Phys. A 13 (1998) 431 [hep-th/9612105] [INSPIRE].
R.S. Garibaldi, Groups of type E 7 over arbitrary fields, Comm. Alg. 29 (2001) 2689 math/9811056.
J. Luciani, Coupling of O(2) supergravity with several vector multiplets, Nucl. Phys. B 132 (1978) 325 [INSPIRE].
K. Meyberg, Eine Theorie der Freudenthalschen Triplesysteme. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 71 (1968) 162.
R. Gilmore, Lie groups, Lie algebras, and some of their applications, Dover Publications, Dover U.K. (2006).
S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York U.S.A. (1978).
A. Marrani, E. Orazi and F. Riccioni, Exceptional reductions, J. Phys. A 44 (2011) 155207 [arXiv:1012.5797] [INSPIRE].
E. Cartan, Oeuvres compl`etes, Editions du Centre National de la Recherche Scientifique, Paris France (1984).
D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Progress toward a theory of supergravity, Phys. Rev. D 13 (1976) 3214 [INSPIRE].
S. Ferrara, J. Scherk and B. Zumino, Algebraic properties of extended supergravity theories, Nucl. Phys. B 121 (1977) 393 [INSPIRE].
S. Ferrara, A. Marrani and A. Yeranyan, On invariant structures of black hole charges, JHEP 02 (2012) 071 [arXiv:1110.4004] [INSPIRE].
S. Ferrara, A. Marrani, E. Orazi, R. Stora and A. Yeranyan, Two-center black holes duality-invariants for STU model and its lower-rank descendants, J. Math. Phys. 52 (2011) 062302 [arXiv:1011.5864] [INSPIRE].
E. Calabi and E. Vesentini, On compact, locally symmetric K¨ahler manifolds, Ann. Math. 71 (1960) 472.
L. Andrianopoli, R. D’Auria, S. Ferrara, A. Marrani and M. Trigiante, Two-centered magical charge orbits, JHEP 04 (2011) 041 [arXiv:1101.3496] [INSPIRE].
M. Günaydin, Lectures on spectrum generating symmetries and u-duality in supergravity, extremal black holes, quantum attractors and harmonic superspace, arXiv:0908.0374 [INSPIRE].
L. Borsten, M. Duff, S. Ferrara, A. Marrani and W. Rubens, Small orbits, Phys. Rev. D 85 (2012) 086002 [arXiv:1108.0424] [INSPIRE].
L. Borsten, M. Duff, S. Ferrara, A. Marrani and W. Rubens, Explicit orbit classification of reducible Jordan algebras and Freudenthal triple systems, arXiv:1108.0908 [INSPIRE].
M. Duff, J.T. Liu and J. Rahmfeld, Four-dimensional string-string-string triality, Nucl. Phys. B 459 (1996) 125 [hep-th/9508094] [INSPIRE].
K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova and W.K. Wong, STU black holes and string triality, Phys. Rev. D 54 (1996) 6293 [hep-th/9608059] [INSPIRE].
L. Andrianopoli, R. D’Auria and S. Ferrara, U invariants, black hole entropy and fixed scalars, Phys. Lett. B 403 (1997) 12 [hep-th/9703156] [INSPIRE].
L. Andrianopoli, R. D’Auria, S. Ferrara and M. Lledó, Super Higgs effect in extended supergravity, Nucl. Phys. B 640 (2002) 46 [hep-th/0202116] [INSPIRE].
S. Ferrara, A. Gnecchi and A. Marrani, D = 4 attractors, effective horizon radius and fake supergravity, Phys. Rev. D 78 (2008) 065003 [arXiv:0806.3196] [INSPIRE].
D. Roest and H. Samtleben, Twin supergravities, Class. Quant. Grav. 26 (2009) 155001 [arXiv:0904.1344] [INSPIRE].
L. Castellani, A. Ceresole, S. Ferrara, R. D’Auria, P. Fré, et al., The complete N = 3 matter coupled supergravity, Nucl. Phys. B 268 (1986) 317 [INSPIRE].
R. Kallosh and B. Kol, E 7 symmetric area of the black hole horizon, Phys. Rev. D 53 (1996) 5344 [hep-th/9602014] [INSPIRE].
L. Andrianopoli, R. D’Auria, S. Ferrara and M. Trigiante, Extremal black holes in supergravity, Lect. Notes Phys. 737 (2008) 661 [hep-th/0611345] [INSPIRE].
M. Günaydin, G. Sierra and P. Townsend, Exceptional supergravity theories and the MAGIC square, Phys. Lett. B 133 (1983) 72 [INSPIRE].
M. Günaydin, G. Sierra and P. Townsend, The geometry of N = 2 Maxwell-Einstein supergravity and Jordan algebras, Nucl. Phys. B 242 (1984) 244 [INSPIRE].
M. Günaydin, G. Sierra and P. Townsend, Gauging the D = 5 Maxwell-Einstein supergravity theories: more on Jordan algebras, Nucl. Phys. B 253 (1985) 573 [INSPIRE].
M. Günaydin, G. Sierra and P. Townsend, More on D = 5 Maxwell-Einstein supergravity: symmetric spaces and kinks, Class. Quant. Grav. 3 (1986) 763 [INSPIRE].
L.K. Hua, On the theory of automorphic functions of a matrix variable. I: geometrical basis, Amer. J. Math. 66 (1944) 470.
C. Bloch and A. Messiah, The canonical form of an antisymmetric tensor and its application to the theory of superconductivity, Nucl. Phys. B 39 (1962) 95.
B. Zumino, Normal forms of complex matrices, J. Math. Phys. 3 (1962) 1055.
S. Ferrara, J. Scherk and B. Zumino, Algebraic properties of extended supergravity theories, Nucl. Phys. B 121 (1977) 393 [INSPIRE].
H. Freudenthal, Beziehungen der E 7 und E8 zur Oktavenebene V, Proc. Konink. Ned. Akad. Wetenschap A 62 (1959) 447.
B.A. Rozenfeld, Geometric interpretation of compact simple Lie groups of the class E, Dokl. Akad. Nauk. SSSR 106 (1956) 600.
J. Tits, Sur certaines classes d’espaces homog`enes de groupes de Lie, Mem. Acad. Roy. Belg. Sci. 29 (1955) 3.
S. Ferrara and A. Marrani, N = 8 non-BPS attractors, fixed scalars and MAGIC supergravities, Nucl. Phys. B 788 (2008) 63 [arXiv:0705.3866] [INSPIRE].
M. Duff, J.T. Liu and J. Rahmfeld, Four-dimensional string-string-string triality, Nucl. Phys. B 459 (1996) 125 [hep-th/9508094] [INSPIRE].
K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova and W.K. Wong, STU black holes and string triality, Phys. Rev. D 54 (1996) 6293 [hep-th/9608059] [INSPIRE].
R. Kallosh, N. Sivanandam and M. Soroush, Exact attractive non-BPS STU black holes, Phys. Rev. D 74 (2006) 065008 [hep-th/0606263] [INSPIRE].
P. Jordan, J. von Neumann and E.P. Wigner, On an algebraic generalization of the quantum mechanical formalism, Annals Math. 35 (1934) 29 [INSPIRE].
N. Jacobson, Structure and representations of Jordan algebras, Ann. Math. Soc. Coll. Publ. 39 (1968).
M. Günaydin, Exceptional realizations of Lorentz group: supersymmetries and leptons, Nuovo Cimento A 29 (1975) 467.
M. Günaydin, C. Piron and H. Ruegg, Moufang plane and octonionic quantum mechanics, Commun. Math. Phys. 61 (1978) 69 [INSPIRE].
S. Cecotti, S. Ferrara and L. Girardello, Geometry of type II superstrings and the moduli of superconformal field theories, Int. J. Mod. Phys. A 4 (1989) 2475 [INSPIRE].
B.L. Cerchiai, S. Ferrara, A. Marrani and B. Zumino, Duality, entropy and ADM mass in supergravity, Phys. Rev. D 79 (2009) 125010 [arXiv:0902.3973] [INSPIRE].
A. Ceresole, S. Ferrara and A. Marrani, Small N = 2 extremal black holes in special geometry, Phys. Lett. B 693 (2010) 366 [arXiv:1006.2007] [INSPIRE].
A. Ceresole, S. Ferrara, A. Gnecchi and A. Marrani, More on N = 8 attractors, Phys. Rev. D 80 (2009) 045020 [arXiv:0904.4506] [INSPIRE].
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] [INSPIRE].
D. Roest and H. Samtleben, Twin supergravities, Class. Quant. Grav. 26 (2009) 155001 [arXiv:0904.1344] [INSPIRE].
S. Ferrara, J.A. Harvey, A. Strominger and C. Vafa, Second quantized mirror symmetry, Phys. Lett. B 361 (1995) 59 [hep-th/9505162] [INSPIRE].
M. G¨unaydin, Lectures on spectrum generating symmetries and U-duality in supergravity, extremal black holes, quantum attractors and harmonic superspace, arXiv:0908.0374 [INSPIRE].
S. Ferrara, A. Marrani and A. Yeranyan, Freudenthal duality and generalized special geometry, Phys. Lett. B 701 (2011) 640 [arXiv:1102.4857] [INSPIRE].
R. D’Auria and S. Ferrara, On fermion masses, gradient flows and potential in supersymmetric theories, JHEP 05 (2001) 034 [hep-th/0103153] [INSPIRE].
B. de Wit, P. Lauwers and A. Van Proeyen, Lagrangians of N = 2 supergravity - Matter systems, Nucl. Phys. B 255 (1985) 569 [INSPIRE].
R. D’Auria, S. Ferrara and P. Fr´e, Special and quaternionic isometries: general couplings in N = 2 supergravity and the scalar potential, Nucl. Phys. B 359 (1991) 705 [INSPIRE].
A. Strominger, Special geometry, Commun. Math. Phys. 133 (1990) 163 [INSPIRE].
L. Castellani, R. D’Auria and S. Ferrara, Special K¨ahler geometry: an intrinsic formulation from N = 2 space-time supersymmetry, Phys. Lett. B 241 (1990) 57 [INSPIRE].
L. Castellani, R. D’Auria and S. Ferrara, Special geometry without special coordinates, Class. Quant. Grav. 7 (1990) 1767 [INSPIRE].
B. Craps, F. Roose, W. Troost and A. Van Proeyen, What is special K¨ahler geometry?, Nucl. Phys. B 503 (1997) 565 [hep-th/9703082] [INSPIRE].
D.V. Alekseevski, Classification of quaternionic spaces with a transitive solvable group of motions, USSR Izvestija 9 (1975) 297.
K. Galicki, A generalization of the momentum mapping construction for quaternionic K¨ahler manifolds, Commun. Math. Phys. 108 (1987) 117 [INSPIRE].
K. Galicki, Geometry of the scalar couplings in N = 2 supergravity models, Class. Quant. Grav. 9 (1992) 27 [INSPIRE].
J. Bagger and E. Witten, Matter couplings in N = 2 supergravity, Nucl. Phys. B 222 (1983) 1 [INSPIRE].
A. Galperin, E. Ivanov and O. Ogievetsky, Harmonic space and quaternionic manifolds, Annals Phys. 230 (1994) 201 [hep-th/9212155] [INSPIRE].
B. de Wit, B. Kleijn and S. Vandoren, Superconformal hypermultiplets, Nucl. Phys. B 568 (2000) 475 [hep-th/9909228] [INSPIRE].
B. de Wit and A. Van Proeyen, Hidden symmetries, special geometry and quaternionic manifolds, Int. J. Mod. Phys. D 3 (1994) 31 [hep-th/9310067] [INSPIRE].
E. Cremmer, B. Julia, J. Scherk, S. Ferrara, L. Girardello, et al., Spontaneous symmetry breaking and Higgs effect in supergravity without cosmological constant, Nucl. Phys. B 147 (1979) 105 [INSPIRE].
R. Kallosh, A. Linde, K.A. Olive and T. Rube, Chaotic inflation and supersymmetry breaking, Phys. Rev. D 84 (2011) 083519 [arXiv:1106.6025] [INSPIRE].
L.J. Dixon, V. Kaplunovsky and J. Louis, On effective field theories describing (2, 2) vacua of the heterotic string, Nucl. Phys. B 329 (1990) 27 [INSPIRE].
T.W. Grimm and J. Louis, The effective action of N = 1 Calabi-Yau orientifolds, Nucl. Phys. B 699 (2004) 387 [hep-th/0403067] [INSPIRE].
T.W. Grimm and J. Louis, The effective action of type IIA Calabi-Yau orientifolds, Nucl. Phys. B 718 (2005) 153 [hep-th/0412277] [INSPIRE].
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional string compactifications with D-branes, orientifolds and fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [INSPIRE].
B. de Wit, F. Vanderseypen and A. Van Proeyen, Symmetry structure of special geometries, Nucl. Phys. B 400 (1993) 463 [hep-th/9210068] [INSPIRE].
A. Ceresole, G. Dall’Agata, S. Ferrara and A. Yeranyan, Universality of the superpotential for D = 4 extremal black holes, Nucl. Phys. B 832 (2010) 358 [arXiv:0910.2697] [INSPIRE].
S. Ferrara and R. Kallosh, Supersymmetry and attractors, Phys. Rev. D 54 (1996) 1514 [hep-th/9602136] [INSPIRE].
R. Kallosh, L. Kofman, A.D. Linde and A. Van Proeyen, Superconformal symmetry, supergravity and cosmology, Class. Quant. Grav. 17 (2000) 4269 [Erratum ibid. 21 (2004) 5017] [hep-th/0006179] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1202.1290
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Ferrara, S., Kallosh, R. & Marrani, A. Degeneration of groups of type E 7 and minimal coupling in supergravity. J. High Energ. Phys. 2012, 74 (2012). https://doi.org/10.1007/JHEP06(2012)074
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2012)074