Consistent Kaluza-Klein sphere reductions

M. Cvetič; H. Lü; C. N. Pope

doi:10.1103/PhysRevD.62.064028

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Consistent Kaluza-Klein sphere reductions

M. Cvetič and H. Lü

C. N. Pope

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104

Center for Theoretical Physics, Texas A&M University, College Station, Texas 77843

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Phys. Rev. D 62, 064028 – Published 28 August, 2000

DOI: https://doi.org/10.1103/PhysRevD.62.064028

Abstract

We study the circumstances under which a Kaluza-Klein reduction on an n-sphere, with a massless truncation that includes all the Yang-Mills fields of $SO (n + 1),$ can be consistent at the full non-linear level. We take as the starting point a theory comprising a p-form field strength and (possibly) a dilaton, coupled to gravity in the higher dimension D. We show that aside from the previously studied cases with $(D, p) = (11, 4)$ and (10,5) (associated with the $S^{4}$ and $S^{7}$ reductions of $D = 11$ supergravity, and the $S^{5}$ reduction of type IIB supergravity), the only other possibilities that allow consistent reductions are for $p = 2,$ reduced on $S^{2},$ and for $p = 3,$ reduced on $S^{3}$ or $S^{D - 3} .$ We construct the fully non-linear Kaluza-Klein Ansätze in all these cases. In particular, we obtain $D = 3,$ $N = 8,$ $SO (8)$ and $D = 7,$ $N = 2,$ $SO (4)$ gauged supergravities from $S^{7}$ and $S^{3}$ reductions of $N = 1$ supergravity in $D = 10.$

References (33)

M.J. Duff and C.N. Pope, in Supersymmetry and Supergravity 82, edited by S. Ferrara, J.G. Taylor, and P. van Nieuwenhuizen (World Scientific, Singapore, 1983).
P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B, 233 (1980).
B. de Wit and H. Nicolai, Nucl. Phys. B208, 323 (1982).
B. de Wit and H. Nicolai, Nucl. Phys. B281, 211 (1987).
H. Nastase, D. Vaman, and P. van Nieuwenhuizen, Phys. Lett. B 469, 96 (1999).
H. Nastase, D. Vaman, and P. van Nieuwenhuizen, “Consistency of the ${AdS}_{7} \times S_{4}$ reduction and the origin of self-duality in odd dimensions,” hep-th/9911238.
M. Cvetič, M.J. Duff, P. Hoxha, J.T. Liu, H. Lü, J.X. Lu, R. Martinez-Acosta, C.N. Pope, H. Sati, and T.A. Tran, Nucl. Phys. B558, 96 (1999).
P. Kraus, F. Larsen, and S.P. Trivedi, J. High Energy Phys. 03, 003 (1991).
J.G. Russo and K. Sfetsos, Adv. Theor. Math. Phys. 3, 131 (1991).
M. Cvetič and S.S. Gubser, J. High Energy Phys. 04, 024 (1999).
M. Cvetič and D. Youm, Nucl. Phys. B499, 253 (1997).
H. Lü and C.N. Pope, Phys. Lett. B 467, 67 (1999).
H. Lü, C.N. Pope, and T.A. Tran, Phys. Lett. B 475, 261 (2000).
M. Cvetič, H. Lü, and C.N. Pope, Nucl. Phys. B574, 761 (2000).
M. Cvetič, H. Lü, and C.N. Pope, Phys. Rev. Lett. 83, 5226 (1999).
M. Cvetič, S.S. Gubser, H. Lü, and C.N. Pope, Phys. Rev. D (to be published), hep-th/9906221.
M. Cvetič, H. Lü, C.N. Pope, and A. Sadrzadeh, Phys. Rev. D 62, 046005 (2000).
M. Cvetič, H. Lü, C.N. Pope, A. Sadrzadeh, and T.A. Tran, “Consistent $SO (6)$ reduction of type IIB supergravity,” hep-th/0003103.
E. Cremmer, in Superspace and Supergravity, edited by S.W. Hawking and M. Rocek (Cambridge University Press, Cambridge, England, 1981), p. 267.
B. Julia, in Superspace and Supergravity, edited by S.W. Hawking and M. Rocek (Cambridge University Press, Cambridge, England, 1981), p. 331.
E. Cremmer, B. Julia, H. Lü, and C.N. Pope, Nucl. Phys. B523, 73 (1998).
E. Cremmer, B. Julia, H. Lü, and C.N. Pope, “Higher-dimensional origin of $D = 3$ coset symmetries,” hep-th/9909099.
H. Nastase and D. Vaman, “On the non-linear KK reductions on spheres of supergravity theories,” hep-th/0002028.
H. Lü, C.N. Pope, E. Sezgin, and K.S. Stelle, Nucl. Phys. B456, 669 (1995).
M.J. Duff, R.R. Khuri, and J.X. Lu, Phys. Rep. 259, 213 (1995).
S. Mizoguchi and N. Ohta, Phys. Lett. B 441, 123 (1998).
A. Sen, Int. J. Mod. Phys. A 9, 3707 (1994).
A. Sen, Nucl. Phys. B434, 179 (1995).
A.H. Chamseddine and W.A. Sabra, Phys. Lett. B 476, 415 (2000).
M. Cvetič, J.T. Liu, H. Lü, and C.N. Pope, Nucl. Phys. B560, 230 (1999).
A. Achúcarro and P.K. Townsend, Phys. Lett. B 180, 89 (1986).
C.N. Pope and K.S. Stelle, Phys. Lett. B 198, 151 (1987).
C.G. Callan, D. Friedan, E.J. Martinec, and M.J. Perry, Nucl. Phys. B262, 593 (1985).

[c1] M.J. Duff and C.N. Pope, in Supersymmetry and Supergravity 82, edited by S. Ferrara, J.G. Taylor, and P. van Nieuwenhuizen (World Scientific, Singapore, 1983).

[c2] P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B, 233 (1980).

[c3] B. de Wit and H. Nicolai, Nucl. Phys. B208, 323 (1982).

[c4] B. de Wit and H. Nicolai, Nucl. Phys. B281, 211 (1987).

[c5] H. Nastase, D. Vaman, and P. van Nieuwenhuizen, Phys. Lett. B 469, 96 (1999).

[c6] H. Nastase, D. Vaman, and P. van Nieuwenhuizen, “Consistency of the ${AdS}_{7} \times S_{4}$ reduction and the origin of self-duality in odd dimensions,” hep-th/9911238.

[c7] M. Cvetič, M.J. Duff, P. Hoxha, J.T. Liu, H. Lü, J.X. Lu, R. Martinez-Acosta, C.N. Pope, H. Sati, and T.A. Tran, Nucl. Phys. B558, 96 (1999).

[c8] P. Kraus, F. Larsen, and S.P. Trivedi, J. High Energy Phys. 03, 003 (1991).

[c9] J.G. Russo and K. Sfetsos, Adv. Theor. Math. Phys. 3, 131 (1991).

[c10] M. Cvetič and S.S. Gubser, J. High Energy Phys. 04, 024 (1999).

[c11] M. Cvetič and D. Youm, Nucl. Phys. B499, 253 (1997).

[c12] H. Lü and C.N. Pope, Phys. Lett. B 467, 67 (1999).

[c13] H. Lü, C.N. Pope, and T.A. Tran, Phys. Lett. B 475, 261 (2000).

[c14] M. Cvetič, H. Lü, and C.N. Pope, Nucl. Phys. B574, 761 (2000).

[c15] M. Cvetič, H. Lü, and C.N. Pope, Phys. Rev. Lett. 83, 5226 (1999).

[c16] M. Cvetič, S.S. Gubser, H. Lü, and C.N. Pope, Phys. Rev. D (to be published), hep-th/9906221.

[c17] M. Cvetič, H. Lü, C.N. Pope, and A. Sadrzadeh, Phys. Rev. D 62, 046005 (2000).

[c18] M. Cvetič, H. Lü, C.N. Pope, A. Sadrzadeh, and T.A. Tran, “Consistent $SO (6)$ reduction of type IIB supergravity,” hep-th/0003103.

[c19] E. Cremmer, in Superspace and Supergravity, edited by S.W. Hawking and M. Rocek (Cambridge University Press, Cambridge, England, 1981), p. 267.

[c20] B. Julia, in Superspace and Supergravity, edited by S.W. Hawking and M. Rocek (Cambridge University Press, Cambridge, England, 1981), p. 331.

[c21] E. Cremmer, B. Julia, H. Lü, and C.N. Pope, Nucl. Phys. B523, 73 (1998).

[c22] E. Cremmer, B. Julia, H. Lü, and C.N. Pope, “Higher-dimensional origin of $D = 3$ coset symmetries,” hep-th/9909099.

[c23] H. Nastase and D. Vaman, “On the non-linear KK reductions on spheres of supergravity theories,” hep-th/0002028.

[c24] H. Lü, C.N. Pope, E. Sezgin, and K.S. Stelle, Nucl. Phys. B456, 669 (1995).

[c25] M.J. Duff, R.R. Khuri, and J.X. Lu, Phys. Rep. 259, 213 (1995).

[c26] S. Mizoguchi and N. Ohta, Phys. Lett. B 441, 123 (1998).

[c27] A. Sen, Int. J. Mod. Phys. A 9, 3707 (1994).

[c28] A. Sen, Nucl. Phys. B434, 179 (1995).

[c29] A.H. Chamseddine and W.A. Sabra, Phys. Lett. B 476, 415 (2000).

[c30] M. Cvetič, J.T. Liu, H. Lü, and C.N. Pope, Nucl. Phys. B560, 230 (1999).

[c31] A. Achúcarro and P.K. Townsend, Phys. Lett. B 180, 89 (1986).

[c32] C.N. Pope and K.S. Stelle, Phys. Lett. B 198, 151 (1987).

[c33] C.G. Callan, D. Friedan, E.J. Martinec, and M.J. Perry, Nucl. Phys. B262, 593 (1985).

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