Abstract
The separability of the Hamilton-Jacobi equation has a well-known connection to the existence of Killing vectors and rank-two Killing tensors. This paper combines this connection with the detailed knowledge of the compactification metrics of consistent truncations on spheres. The fact that both the inverse metric of such compactifications, as well as the rank-two Killing tensors can be written in terms of bilinears of Killing vectors on the underlying “round metric,” enables us to perform a detailed analyses of the separability of the Hamilton-Jacobi equation for consistent truncations. We introduce the idea of a separating isometry and show that when a consistent truncation, without reduction gauge vectors, has such an isometry, then the Hamilton-Jacobi equation is always separable. When gauge vectors are present, the gauge group is required to be an abelian subgroup of the separating isometry to not impede separability. We classify the separating isometries for consistent truncations on spheres, Sn, for n = 2, …, 7, and exhibit all the corresponding Killing tensors. These results may be of practical use in both identifying when supergravity solutions belong to consistent truncations and generating separable solutions amenable to scalar probe calculations. Finally, while our primary focus is the Hamilton-Jacobi equation, we also make some remarks about separability of the wave equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. 174 (1968) 1559 [INSPIRE].
B. Carter, Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations, Commun. Math. Phys. 10 (1968) 280 [INSPIRE].
Y. Chervonyi and O. Lunin, Killing(-Yano) Tensors in String Theory, JHEP 09 (2015) 182 [arXiv:1505.06154] [INSPIRE].
B. de Wit, H. Nicolai and N.P. Warner, The Embedding of Gauged N = 8 Supergravity Into d = 11 Supergravity, Nucl. Phys. B 255 (1985) 29.
B. de Wit and H. Nicolai, The Consistency of the S7 Truncation in D = 11 Supergravity, Nucl. Phys. B 281 (1987) 211 [INSPIRE].
H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistent nonlinear K K reduction of 11 − D supergravity on AdS7 × S4 and selfduality in odd dimensions, Phys. Lett. B 469 (1999) 96 [hep-th/9905075] [INSPIRE].
H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistency of the AdS7 × S4 reduction and the origin of selfduality in odd dimensions, Nucl. Phys. B 581 (2000) 179 [hep-th/9911238] [INSPIRE].
A. Khavaev, K. Pilch and N.P. Warner, New vacua of gauged N = 8 supergravity in five-dimensions, Phys. Lett. B 487 (2000) 14 [hep-th/9812035] [INSPIRE].
K. Pilch and N.P. Warner, N = 2 supersymmetric RG flows and the IIB dilaton, Nucl. Phys. B 594 (2001) 209 [hep-th/0004063] [INSPIRE].
M. Cvetič, H. Lü, C.N. Pope, A. Sadrzadeh and T.A. Tran, Consistent SO(6) reduction of type IIB supergravity on S5 , Nucl. Phys. B 586 (2000) 275 [hep-th/0003103] [INSPIRE].
K. Lee, C. Strickland-Constable and D. Waldram, Spheres, generalised parallelisability and consistent truncations, Fortsch. Phys. 65 (2017) 1700048 [arXiv:1401.3360] [INSPIRE].
A. Baguet, O. Hohm and H. Samtleben, Consistent Type IIB Reductions to Maximal 5D Supergravity, Phys. Rev. D 92 (2015) 065004 [arXiv:1506.01385] [INSPIRE].
A. Guarino, D.L. Jafferis and O. Varela, String Theory Origin of Dyonic N = 8 Supergravity and Its Chern-Simons Duals, Phys. Rev. Lett. 115 (2015) 091601 [arXiv:1504.08009] [INSPIRE].
A. Guarino and O. Varela, Consistent \( \mathcal{N} \) = 8 truncation of massive IIA on S6, JHEP 12 (2015) 020 [arXiv:1509.02526] [INSPIRE].
M. Cvetič et al., Embedding AdS black holes in ten-dimensions and eleven-dimensions, Nucl. Phys. B 558 (1999) 96 [hep-th/9903214] [INSPIRE].
M. Cvetič, H. Lü and C.N. Pope, Consistent Kaluza-Klein sphere reductions, Phys. Rev. D 62 (2000) 064028 [hep-th/0003286] [INSPIRE].
H. Nastase and D. Vaman, On the nonlinear KK reductions on spheres of supergravity theories, Nucl. Phys. B 583 (2000) 211 [hep-th/0002028] [INSPIRE].
H. Samtleben and O. Sarıoglu, Consistent S3 reductions of six-dimensional supergravity, Phys. Rev. D 100 (2019) 086002 [arXiv:1907.08413] [INSPIRE].
I. Bena, D. Turton, R. Walker and N.P. Warner, Integrability and Black-Hole Microstate Geometries, JHEP 11 (2017) 021 [arXiv:1709.01107] [INSPIRE].
R. Walker, D1-D5-P superstrata in 5 and 6 dimensions: separable wave equations and prepotentials, JHEP 09 (2019) 117 [arXiv:1906.04200] [INSPIRE].
D.R. Mayerson, R.A. Walker and N.P. Warner, Microstate Geometries from Gauged Supergravity in Three Dimensions, JHEP 10 (2020) 030 [arXiv:2004.13031] [INSPIRE].
E.G. Kalnins and J. Willard Miller, Killing tensors and variable separation for hamilton-jacobi and helmholtz equations, SIAM J. Math. Anal. 11 (1980) (1011).
T. Sumitomo and K. Tandai, Killing tensor fields on the standard sphere and spectra of so(n + 1)/(so(n − 1) × so(2)) and o(n + 1)/(o(n − 1) × o(2)), Osaka J. Math. 20 (1983) 51.
M. Takeuchi, Killing tensor fields on spaces of constant curvature, Tsukuba J. Math. 7 (1983) 233.
G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 (1986) (2693).
P. Stäckel, Über die Bewegung eines Punktes in einer einfachen Mannigfaltigkeit, Math. Ann. 42 (1893) 537.
T. Levi-Civita, Sulla integrazione della equazione di Hamilton-Jacobi per separazione di variabili, Math. Ann. 59 (1904) 383.
L.P. Eisenhart, Separable systems of Stäckel, Annals Math. 35 (1934) 284.
L. Eisenhart, Riemannian Geometry, Princeton Landmarks in Mathematics and Physics, Princeton University Press (1997).
S. Benenti, C. Chanu and G. Rastelli, Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schroedinger equation. I. The completeness and Robertson conditions, J. Math. Phys. 43 (2002) 5183.
E.G. Kalnins, J.M. Kress and W. Miller, Separation of Variables and Superintegrability, IOP Publishing (2018) 2053 [DOI].
S. Benenti, Separability in riemannian manifolds, SIGMA (2016) [arXiv:1512.07833].
E.G. Kalnins and W. Miller, Separation of variables on n-dimensional riemannian manifolds. i. the n-sphere sn and euclidean n-space rn, J. Math. Phys. 27 (1986) 1721.
E.G. Kalnins and J. Willard Miller, Killing tensors and nonorthogonal variable separation for Hamilton-Jacobi equations, SIAM J. Math. Anal. 12 (1981) 617.
B. de Wit and H. Nicolai, Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions, JHEP 05 (2013) 077 [arXiv:1302.6219] [INSPIRE].
O. Hohm and H. Samtleben, Consistent Kaluza-Klein Truncations via Exceptional Field Theory, JHEP 01 (2015) 131 [arXiv:1410.8145] [INSPIRE].
H. Godazgar, M. Godazgar, O. Krüger and H. Nicolai, Consistent 4-form fluxes for maximal supergravity, JHEP 10 (2015) 169 [arXiv:1507.07684] [INSPIRE].
K. Schöbel, An Algebraic Geometric Approach to Separation of Variables, Springer Fachmedien Wiesbaden (2015) [DOI].
R.G. McLenaghan, R. Milson and R.G. Smirnov, Killing tensors as irreducible representations of the general linear group, Compt. Rend. Math. 339 (2004) 621.
R. Delong, Killing Tensors and the Hamilton-Jacobi Equation, University of Minnesota (1982).
N. Bobev, F.F. Gautason, K. Pilch, M. Suh and J. van Muiden, Holographic interfaces in \( \mathcal{N} \) = 4 SYM: Janus and J-folds, JHEP 05 (2020) 134 [arXiv:2003.09154] [INSPIRE].
E. D’Hoker, J. Estes and M. Gutperle, Exact half-BPS Type IIB interface solutions. I. Local solution and supersymmetric Janus, JHEP 06 (2007) 021 [arXiv:0705.0022] [INSPIRE].
C.N. Pope and N.P. Warner, A Dielectric flow solution with maximal supersymmetry, JHEP 04 (2004) 011 [hep-th/0304132] [INSPIRE].
E. D’Hoker, J. Estes, M. Gutperle and D. Krym, Janus solutions in M-theory, JHEP 06 (2009) 018 [arXiv:0904.3313] [INSPIRE].
N. Bobev, K. Pilch and N.P. Warner, Supersymmetric Janus Solutions in Four Dimensions, JHEP 06 (2014) 058 [arXiv:1311.4883] [INSPIRE].
R. Slansky, Group Theory for Unified Model Building, Phys. Rept. 79 (1981) 1 [INSPIRE].
R. Feger, T.W. Kephart and R.J. Saskowski, LieART 2.0 – a mathematica application for lie algebras and representation theory, Comput. Phys. Commun. 257 (2020) 107490.
N. Vilenkin, Special Functions and the Theory of Group Representations, Translations of mathematical monographs, American Mathematical Soc. (1978).
D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].
K. Pilch and N.P. Warner, N = 1 supersymmetric renormalization group flows from IIB supergravity, Adv. Theor. Math. Phys. 4 (2002) 627 [hep-th/0006066] [INSPIRE].
A. Brandhuber and K. Sfetsos, An N = 2 gauge theory and its supergravity dual, Phys. Lett. B 488 (2000) 373 [hep-th/0004148] [INSPIRE].
B. de Wit and H. Nicolai, N = 8 Supergravity, Nucl. Phys. B 208 (1982) 323 [INSPIRE].
E. Cremmer and B. Julia, The SO(8) Supergravity, Nucl. Phys. B 159 (1979) 141.
O. Varela, Complete D = 11 embedding of SO(8) supergravity, Phys. Rev. D 97 (2018) 045010 [arXiv:1512.04943] [INSPIRE].
O. Krüger, Non-linear uplift Ansätze for the internal metric and the four-form field-strength of maximal supergravity, JHEP 05 (2016) 145 [arXiv:1602.03327] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
N.P. Warner, Some New Extrema of the Scalar Potential of Gauged N = 8 Supergravity, Phys. Lett. B 128 (1983) 169 [INSPIRE].
H. Nicolai and N.P. Warner, The SU(3) X U(1) Invariant Breaking of Gauged N = 8 Supergravity, Nucl. Phys. B 259 (1985) 412 [INSPIRE].
M. Benna, I. Klebanov, T. Klose and M. Smedback, Superconformal Chern-Simons Theories and AdS4/CFT3 Correspondence, JHEP 09 (2008) 072 [arXiv:0806.1519] [INSPIRE].
I. Klebanov, T. Klose and A. Murugan, AdS4/CFT3 Squashed, Stretched and Warped, JHEP 03 (2009) 140 [arXiv:0809.3773] [INSPIRE].
C.-h. Ahn and J. Paeng, Three-dimensional SCFTs, supersymmetric domain wall and renormalization group flow, Nucl. Phys. B 595 (2001) 119 [hep-th/0008065] [INSPIRE].
C.-h. Ahn and K. Woo, Supersymmetric domain wall and RG flow from 4-dimensional gauged N = 8 supergravity, Nucl. Phys. B 599 (2001) 83 [hep-th/0011121] [INSPIRE].
R. Corrado, K. Pilch and N.P. Warner, An N = 2 supersymmetric membrane flow, Nucl. Phys. B 629 (2002) 74 [hep-th/0107220] [INSPIRE].
K. Pilch, A. Tyukov and N.P. Warner, \( \mathcal{N} \) = 2 Supersymmetric Janus Solutions and Flows: From Gauged Supergravity to M-theory, JHEP 05 (2016) 005 [arXiv:1510.08090] [INSPIRE].
D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Continuous distributions of D3-branes and gauged supergravity, JHEP 07 (2000) 038 [hep-th/9906194] [INSPIRE].
M. Cvetič, S.S. Gubser, H. Lü and C.N. Pope, Symmetric potentials of gauged supergravities in diverse dimensions and Coulomb branch of gauge theories, Phys. Rev. D 62 (2000) 086003 [hep-th/9909121] [INSPIRE].
A. Khavaev and N.P. Warner, A Class of N = 1 supersymmetric RG flows from five-dimensional N = 8 supergravity, Phys. Lett. B 495 (2000) 215 [hep-th/0009159] [INSPIRE].
C.N. Gowdigere and N.P. Warner, Holographic Coulomb branch flows with N = 1 supersymmetry, JHEP 03 (2006) 049 [hep-th/0505019] [INSPIRE].
M. Cvetič, H. Lü and C.N. Pope, Geometry of the embedding of supergravity scalar manifolds in D = 11 and D = 10, Nucl. Phys. B 584 (2000) 149 [hep-th/0002099] [INSPIRE].
I. Bena, P. Heidmann and D. Turton, AdS2 holography: mind the cap, JHEP 12 (2018) 028 [arXiv:1806.02834] [INSPIRE].
I. Bena, P. Heidmann, R. Monten and N.P. Warner, Thermal Decay without Information Loss in Horizonless Microstate Geometries, SciPost Phys. 7 (2019) 063 [arXiv:1905.05194] [INSPIRE].
I. Bena, F. Eperon, P. Heidmann and N.P. Warner, The Great Escape: Tunneling out of Microstate Geometries, JHEP 04 (2021) 112 [arXiv:2005.11323] [INSPIRE].
E.J. Martinec and N.P. Warner, The Harder They Fall, the Bigger They Become: Tidal Trapping of Strings by Microstate Geometries, JHEP 04 (2021) 259 [arXiv:2009.07847] [INSPIRE].
N. Bobev, E. Malek, B. Robinson, H. Samtleben and J. van Muiden, Kaluza-Klein Spectroscopy for the Leigh-Strassler SCFT, JHEP 04 (2021) 208 [arXiv:2012.07089] [INSPIRE].
E. Malek and H. Samtleben, Kaluza-Klein Spectrometry for Supergravity, Phys. Rev. Lett. 124 (2020) 101601 [arXiv:1911.12640] [INSPIRE].
E. Malek and H. Samtleben, Kaluza-Klein Spectrometry from Exceptional Field Theory, Phys. Rev. D 102 (2020) 106016 [arXiv:2009.03347] [INSPIRE].
I.R. Klebanov, S.S. Pufu and F.D. Rocha, The Squashed, Stretched, and Warped Gets Perturbed, JHEP 06 (2009) 019 [arXiv:0904.1009] [INSPIRE].
A. Guarino, E. Malek and H. Samtleben, Stable Nonsupersymmetric Anti-de Sitter Vacua of Massive IIA Supergravity, Phys. Rev. Lett. 126 (2021) 061601 [arXiv:2011.06600] [INSPIRE].
S.S. Pufu, I.R. Klebanov, T. Klose and J. Lin, Green’s Functions and Non-Singlet Glueballs on Deformed Conifolds, J. Phys. A 44 (2011) 055404 [arXiv:1009.2763] [INSPIRE].
A. Houppe and N.P. Warner, Supersymmetry and Superstrata in Three Dimensions, arXiv:2012.07850 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2102.12493
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Pilch, K., Walker, R. & Warner, N.P. Separability in consistent truncations. J. High Energ. Phys. 2021, 8 (2021). https://doi.org/10.1007/JHEP07(2021)008
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2021)008