Journal of Holography Applications in Physics
Hidden Symmetries and Killing Spinor of 5-Dimensional Minimal Gauged Supergravity Solutions
Document Type : Regular article
Author
Department of Theoretical Physics, IFIN-HH, Magurele, Romania; Department of Physics, University of Bucharest, Bucharest, Romania.
Abstract
We investigate the hidden symmetries and existence of Killing spinors of D=5 minimal gauged supergravity solutions which admit a Killing-Maxwell system in the sense of Carter, such as the Chong-Cvetic-Lü-Pope (CCLP) Kerr-dS black hole spacetime and a 5-dimensional minimal gauged supergravity solution endowed with a Sasaki structure deformed by torsion, a limiting case of the generalization of the 5-dimensional toric Sasaki-Einstein La,b,c spacetime. We note that when an electromagnetic tensor is present and an associated Killing-Maxwell system can be constructed in the sense of Carter, the Killing-Maxwell field becomes a PCKY (principal conformal Killing-Yano) tensor or a PGCKY (principal generalized conformal Killing-Yano) tensor, the latter in the presence of torsion. We find some new hidden symmetries of the Chong-Cvetic-Lü-Pope black hole, i.e. we construct two more generalized Stäckel-Killing tensors and the associated generalized Killing-Yano (GKY) tensors. We also explicitly construct a Killing spinor for the specific 5-dimensional minimal gauged supergravity solution that is endowed with a Sasaki structure deformed by torsion mentioned above.
Keywords
- 5-Dimensional Minimal Gauged Supergravity
- Hidden Symmetries
- Killing Spinor
- St\" {a}ckel-Killing Tensor
- Kerr-dS Black Hole
- Killing-Yano Tensor
- Sasaki Deformed by Torsion
Main Subjects
Article PDF
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[22] Rietdijk, R.H., van Holten, J.W., ”Killing tensors and a new geometric duality”,Nucl. Phys. B 472, 427 (1996) https://doi.org/10.1016/0550-3213(96)00206-4
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[30] Ahmedov, H., Aliev, A. N., "Stationary spinning strings and symmetries of classical spacetimes", Phys Lett {\bf B 675}, 257 (2009) https://doi.org/10.1016/j.physletb.2009.03.075
[31] de Sabbata, V., Sivaram, C.: Spin and Torsion in Gravitation, World Scientific (1994) https://doi.org/10.1142/2358
[32] Bismut, J. M., ”A local index theorem for non Kähler manifolds”, Math. Ann. 284, 681 (1989) https://doi.org/10.1007/BF01443359
[33] Agricola, I., Friedrich, T., ”On the holonomy of connections with skew-symmetric torsion”, Math. Ann. 328, 711 (2004) https://doi.org/10.1007/s00208-003-0507-9
[34] Agricola, I.,”The Srni lectures on non-integrable geometries with torsion”, Arch. Math. 42, 584 (2006) https://doi.org/10.48550/arXiv.math/0606705
[35] Kassuba, M., ”Eigenvalue estimates for Dirac operators in geometries with torsion”, Ann. Glob. Anal. Geom. 37, 33 (2010) https://doi.org/10.1007/s10455-009-9172-x
[36] Carter, B., ”Separability of the Killing–Maxwell system underlying the generalized angular momentum constant in the Kerr–Newman black hole metrics”, J. Math. Phys 28, 1535 (1987) https://doi.org/10.1063/1.527509
[37] Carter, B., McLenaghan, R.G., ”Generalized total angular momentum operator for the Dirac equation in curved space-time”, Phys. Rev. D 19, 1093 (1979) https://doi.org/10.1103/PhysRevD.19.1093
[38] Houri, T., Yasui, Y., ”Hidden Symmetry and Exact Solutions in Einstein Gravity”, Prog. Theor. Phys. Suppl. 189, 126 (2011) https://doi.org/10.1143/PTPS.189.126
[39] Chen, W., Lü, H., Pope, C.N., ”General Kerr-NUT-AdS Metrics in All Dimensions” Class. Quant. Grav. 23, 5323 (2006) https://doi.org/10.1088/0264-9381/23/17/013
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[41] Horowitz, G. T., Kunduri, H. K., Lucietti, J., ”Comments on Black Holes in Bubbling Spacetimes”, JHEP 1706, 048 (2017) https://doi.org/10.1007/JHEP06%282017%29048
[42] Chow, D.D.K., ”Higher-dimensional lifts of Killing-Yano forms with torsion”, Class. Quant. Grav. 34, 025005 (2017) https://doi.org/10.1088/1361-6382/34/2/025005
[43] Blazquez-Salcedo, J. L., Kunz, J., Navarro-Lerido, F., Radu, E., ”New black holes in D = 5 minimal gauged supergravity: Deformed boundaries and frozen horizons”, Phys Rev D 97 081502 (2018) DOI: 10.1103/PhysRevD.97.081502
[44] Gauntlett, J. P., Gutowski, J.B., ”All supersymmetric solutions of minimal gauged supergravity in five-dimensions” Phys. Rev. D 68, 105009 (2003); Erratum-ibid. 70 089901 (2004) https://doi.org/10.1103/PhysRevD.68.105009 https://doi.org/10.1103/PhysRevD.70.089901
[45] Gauntlett, J.P., Gutowski, J.B., Hull, C. M., Pakis, S., Reall, H. S., ”All supersymmetric solutions of minimal supergravity in five dimensions”, Class. Quant. Grav. 20, 4587 (2003) https://doi.org/10.1088/0264-9381/20/21/005
[46] Ortin, T.: Gravity and Strings, Cambridge Monographs on Mathematical Physics, Cambridge (2004) https://doi.org/10.1017/CBO9780511616563
[47] Shuster, E., ”Killing spinors and Supersymmetry on AdS”, Nucl. Phys.B 554, 198 (1999) https://doi.org/10.1016/S0550-3213
[48] Houri, T., Takeuchi, H., Yasui, Y., ”A deformation of Sasakian structure in the presence of torsion and supergravity solutions”, Class. Quant. Grav. 30, 135008 (2013) DOI:10.1088/0264-9381/30/13/135008
[49] Gromov, M., Blaine Lawson, H., ”Positive scalar curvature and the Dirac operator on complete riemannian manifolds”, Publications Mathématique de l’Institut des Hautes Études Scientifique 58, 83 (1983) https://doi.org/10.1007/BF02953774
[50] Agricola, I., Becker-Bender, J., Kim, H., ”Twistorial eigenvalue estimates for generalized Dirac operators with torsion”, Adv. Math. 243, 296 (2013) DOI: 10.1016/j.aim.2013.05.001
[2] Frolov, V.P., Kubiznˇa´k, D., ”Hidden Symmetries of Higher-Dimensional Rotating Black Holes”, Phys. Rev. Lett. 98, 011101 (2007), https://doi.org/10.1103/PhysRevLett.98.011101
[3] Kubiznˇa´k, D., Frolov, V.P., ”Hidden Symmetry of Higher Dimensional Kerr-NUTAdS Spacetimes”, Class. Quant. Grav., 24, F1 (2007), https://doi.org/10.1088/0264- 9381/24/3/F01
[4] Frolov, V.P., Krtousˇ, P., Kubiznˇa´k, D., ”Separability of Hamilton-Jacobi and KleinGordon equations in general Kerr-NUT-AdS spacetimes”, JHEP, 0702, 005 (2007) DOI 10.1088/1126-6708/2007/02/005
[5] Oota, T., Yasui, Y., ”Separability of Dirac equation in higher dimensional Kerr-NUT-de Sitter spacetime”, Phys. Lett. B 659, 688 (2008) https://doi.org/10.1016/j.physletb.2007.11.057
[6] Wu, S.-Q., ”Separability of the massive Dirac’s equation in 5-dimensional Myers-Perry black hole geometry and its relation to a rank-three Killing-Yano tensor”, Phys. Rev.D 78, 064052 (2008) https://doi.org/10.1103/PhysRevD.78.064052
[7] Wu, S.-Q., ”Symmetry operators and separability of massive Klein-Gordon and Dirac equations in the general 5-dimensional Kerr-(anti-)de Sitter black hole background”, Class. Quant. Grav. 26, 055001 (2009); Erratum-ibid. 26 , 189801 (2009). https://doi.org/10.1088/0264-9381/26/5/055001; https://doi.org/10.1088/0264-9381/26/18/189801
[8] Krtousˇ, P., Frolov, V.P., Kubiznˇa´k, D., ”Hidden Symmetries of Higher Dimensional Black Holes and Uniqueness of the Kerr-NUT-(A)dS spacetime”, Phys. Rev. D 78, 064022 (2008) https://doi.org/10.48550/arXiv.0804.4705
[9] Chow, D.D.K., ”Symmetries of supergravity black holes”, Class. Quant. Grav. 27, 205009 (2010) DOI 10.1088/0264-9381/27/20/205009
[10] Davis, P., Kunduri, H.K., Lucietti, J., ”Special symmetries of the charged Kerr–AdS black hole of D=5 minimal gauged supergravity”, Phys. Lett. B 628, 275 (2005) https://doi.org/10.1016/j.physletb.2005.09.062
[11] Wu, S. -Q., ”Separability of a modified Dirac equation in a five-dimensional rotating, charged black hole in string theory”, Phys. Rev D 80, 044037 (2009); Erratum-ibid 80 069902. https://doi.org/10.1103/PhysRevD.80.044037; https://doi.org/10.1103/PhysRevD.80.069902
[12] Ahmedov, H., Aliev, A.N., ”Supersymmetry in the spacetime of higherdimensional rotating black holes”, Phys. Rev. D 79, 084019 (2009) https://doi.org/10.1103/PhysRevD.79.084019
[13] Kubiznˇa´k, D., ”Hidden symmetries of higher-dimensional rotating black holes”, PhD thesis (2008). https://doi.org/10.48550/arXiv.0809.2452
[14] Ianuş, S., Vişinescu, M., Vîlcu, G.E., ”Conformal Killing-Yano Tensors on Manifolds with Mixed 3-Structures”, SIGMA 5, 022 (2009) https://doi.org/10.3842/SIGMA.2009.022
[15] Houri, T., Oota, T., Yasui, Y., ”Closed conformal Killing-Yano tensor and Kerr-NUT-de Sitter spacetime uniqueness”, Phys. Lett.B 656, 214 (2007) https://doi.org/10.1016/j.physletb.2007.09.034
[16] Houri, T., Oota, T., Yasui, Y., ”Closed conformal Killing–Yano tensor and the uniqueness of generalized Kerr–NUT–de Sitter spacetime”, Class. Quant. Grav 26, 045015 (2009) DOI 10.1088/0264-9381/26/4/045015
[17] Gibbons, G. W., Rietdijk, R.H., van Holten, J.W., ”SUSY in the sky”, Nucl. Phys. B, 404, 42 (1993) https://doi.org/10.1016/0550-3213
[18] Cariglia, M., ”Quantum mechanics of Yano tensors: Dirac equation in curved spacetime”, Class. Quant. Grav. 21, 1051 (2004) DOI 10.1088/0264-9381/21/4/022
[19] Cotăescu, I.I., Vişinescu, M., ”Symmetries and supersymmetries of the Dirac operators in curved spacetimes”, Progress in General Relativity and Quantum Cosmology Research, Nova Science, New York, 109 (2007) https://doi.org/10.48550/arXiv.hepth/0411016
[20] van Holten, J. W., ”Killing-Yano tensors, non-standard supersymmetries and an index theorem”, Annalen Phys. 9SI, 83 (2000) https://doi.org/10.48550/arXiv.grqc/9910035
[21] Peeters, K., Waldron, A., ”Spinors on manifolds with boundary: APS index theorems with torsion”, JHEP 9902, 024 (1999) https://doi.org/10.1088/1126- 6708/1999/02/024
[22] Rietdijk, R.H., van Holten, J.W., ”Killing tensors and a new geometric duality”,Nucl. Phys. B 472, 427 (1996) https://doi.org/10.1016/0550-3213(96)00206-4
[23] Emparan, R., Reall, H.S., ”Black Holes in Higher Dimensions”, Living Rev. Rel. 11, 6 (2008) https://doi.org/10.12942/lrr-2008-6
[24] Myers, R.C., Perry, M.J., ”Black holes in higher dimensional space-times”, Ann. Phys. (N.Y.) 172, 304 (1986) https://doi.org/10.1016/0003-4916(86)90186-7
[25] Chong, W., Cvetic, M., Lu, H., Pope, C. N.,”General Non-Extremal Rotating Black Holes in Minimal Five-Dimensional Gauged Supergravity”, Phys Rev Lett 95, 161301 (2005) https://doi.org/10.1103/PhysRevLett.95.161301
[26] Kubiznˇa´k, D., Kunduri, H.K., Yasui, Y., ”Generalized Killing–Yano equations in D=5 gauged supergravity”, Phys. Lett. B 678, 240 (2009) https://doi.org/10.1016/j.physletb.2009.06.037
[27] Houri, T., Kubiznˇa´k, D., Warnick, C.M., Yasui, Y., ”Symmetries of the Dirac operator with skew-symmetric torsion”, Class. Quant. Grav. 27, 185019 (2010) DOI: 10.1088/0264-9381/27/18/185019
[28] Kubiznˇa´k, D., Warnick, C. M., Krtou$\check{s}$, P, "Hidden symmetry in the presence of fluxes", Nucl. Phys. B {\bf 844}, 185 (2011) https://doi.org/10.1016/j.nuclphysb.2010.11.001
[29] Houri, T., Kubiznˇa´k, D., Warnick, C.M., Yasui, Y., ”Generalized hidden symmetries and the Kerr-Sen black hole”, JHEP 1007, 055 (2010) https://doi.org/10.1007/JHEP07(2010)055
[30] Ahmedov, H., Aliev, A. N., "Stationary spinning strings and symmetries of classical spacetimes", Phys Lett {\bf B 675}, 257 (2009) https://doi.org/10.1016/j.physletb.2009.03.075
[31] de Sabbata, V., Sivaram, C.: Spin and Torsion in Gravitation, World Scientific (1994) https://doi.org/10.1142/2358
[32] Bismut, J. M., ”A local index theorem for non Kähler manifolds”, Math. Ann. 284, 681 (1989) https://doi.org/10.1007/BF01443359
[33] Agricola, I., Friedrich, T., ”On the holonomy of connections with skew-symmetric torsion”, Math. Ann. 328, 711 (2004) https://doi.org/10.1007/s00208-003-0507-9
[34] Agricola, I.,”The Srni lectures on non-integrable geometries with torsion”, Arch. Math. 42, 584 (2006) https://doi.org/10.48550/arXiv.math/0606705
[35] Kassuba, M., ”Eigenvalue estimates for Dirac operators in geometries with torsion”, Ann. Glob. Anal. Geom. 37, 33 (2010) https://doi.org/10.1007/s10455-009-9172-x
[36] Carter, B., ”Separability of the Killing–Maxwell system underlying the generalized angular momentum constant in the Kerr–Newman black hole metrics”, J. Math. Phys 28, 1535 (1987) https://doi.org/10.1063/1.527509
[37] Carter, B., McLenaghan, R.G., ”Generalized total angular momentum operator for the Dirac equation in curved space-time”, Phys. Rev. D 19, 1093 (1979) https://doi.org/10.1103/PhysRevD.19.1093
[38] Houri, T., Yasui, Y., ”Hidden Symmetry and Exact Solutions in Einstein Gravity”, Prog. Theor. Phys. Suppl. 189, 126 (2011) https://doi.org/10.1143/PTPS.189.126
[39] Chen, W., Lü, H., Pope, C.N., ”General Kerr-NUT-AdS Metrics in All Dimensions” Class. Quant. Grav. 23, 5323 (2006) https://doi.org/10.1088/0264-9381/23/17/013
[40] Krtousˇ, P.„ Kubiznˇa´k, D., Page, D. N., Frolov, V.P., ”Separability of Hamilton-Jacobi and Klein-Gordon Equations in General Kerr-NUT-AdS Spacetimes”, JHEP 02, 004 (2007) https://doi.org/10.1088/1126-6708/2007/02/005
[41] Horowitz, G. T., Kunduri, H. K., Lucietti, J., ”Comments on Black Holes in Bubbling Spacetimes”, JHEP 1706, 048 (2017) https://doi.org/10.1007/JHEP06%282017%29048
[42] Chow, D.D.K., ”Higher-dimensional lifts of Killing-Yano forms with torsion”, Class. Quant. Grav. 34, 025005 (2017) https://doi.org/10.1088/1361-6382/34/2/025005
[43] Blazquez-Salcedo, J. L., Kunz, J., Navarro-Lerido, F., Radu, E., ”New black holes in D = 5 minimal gauged supergravity: Deformed boundaries and frozen horizons”, Phys Rev D 97 081502 (2018) DOI: 10.1103/PhysRevD.97.081502
[44] Gauntlett, J. P., Gutowski, J.B., ”All supersymmetric solutions of minimal gauged supergravity in five-dimensions” Phys. Rev. D 68, 105009 (2003); Erratum-ibid. 70 089901 (2004) https://doi.org/10.1103/PhysRevD.68.105009 https://doi.org/10.1103/PhysRevD.70.089901
[45] Gauntlett, J.P., Gutowski, J.B., Hull, C. M., Pakis, S., Reall, H. S., ”All supersymmetric solutions of minimal supergravity in five dimensions”, Class. Quant. Grav. 20, 4587 (2003) https://doi.org/10.1088/0264-9381/20/21/005
[46] Ortin, T.: Gravity and Strings, Cambridge Monographs on Mathematical Physics, Cambridge (2004) https://doi.org/10.1017/CBO9780511616563
[47] Shuster, E., ”Killing spinors and Supersymmetry on AdS”, Nucl. Phys.B 554, 198 (1999) https://doi.org/10.1016/S0550-3213
[48] Houri, T., Takeuchi, H., Yasui, Y., ”A deformation of Sasakian structure in the presence of torsion and supergravity solutions”, Class. Quant. Grav. 30, 135008 (2013) DOI:10.1088/0264-9381/30/13/135008
[49] Gromov, M., Blaine Lawson, H., ”Positive scalar curvature and the Dirac operator on complete riemannian manifolds”, Publications Mathématique de l’Institut des Hautes Études Scientifique 58, 83 (1983) https://doi.org/10.1007/BF02953774
[50] Agricola, I., Becker-Bender, J., Kim, H., ”Twistorial eigenvalue estimates for generalized Dirac operators with torsion”, Adv. Math. 243, 296 (2013) DOI: 10.1016/j.aim.2013.05.001
)
Volume 5, Issue 2
June 2025Pages 22-43
- Receive Date: 18 December 2024
- Revise Date: 05 March 2025
- Accept Date: 06 March 2025