Abstract
We consider dynamics of scalar and vector fields on gravitational backgrounds of the Wess-Zumino-Witten models. For SO(4) and its cosets, we demonstrate full separation of variables for all fields and find a close analogy with a similar separation of vector equations in the backgrounds of the Myers-Perry black holes. For SO(5) and higher groups separation of variables is found only in some subsectors.
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References
E. Witten, NonAbelian Bosonization in Two-Dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].
A.P. Polychronakos and K. Sfetsos, Solving field equations in non-isometric coset CFT backgrounds, Nucl. Phys. B 840 (2010) 534 [arXiv:1006.2386] [INSPIRE].
A.P. Polychronakos and K. Sfetsos, High spin limits and non-Abelian T-duality, Nucl. Phys. B 843 (2011) 344 [arXiv:1008.3909] [INSPIRE].
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, String propagation in a black hole geometry, Nucl. Phys. B 371 (1992) 269 [INSPIRE].
O. Lunin and W. Tian, Scalar fields on λ-deformed cosets, Nucl. Phys. B 938 (2019) 671 [arXiv:1808.02971] [INSPIRE].
O. Lunin and J. Tian, Supergravity excitations of the WZW models, to appear.
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].
M. Walker and R. Penrose, On quadratic first integrals of the geodesic equations for type {22} spacetimes, Commun. Math. Phys. 18 (1970) 265 [INSPIRE].
Z.W. Chong, G.W. Gibbons, H. Lü and C.N. Pope, Separability and Killing tensors in Kerr-Taub-NUT-de Sitter metrics in higher dimensions, Phys. Lett. B 609 (2005) 124 [hep-th/0405061] [INSPIRE].
M. Vasudevan, K.A. Stevens and D.N. Page, Separability of the Hamilton-Jacobi and Klein-Gordon equations in Kerr-de Sitter metrics, Class. Quant. Grav. 22 (2005) 339 [gr-qc/0405125] [INSPIRE].
W. Chen, H. Lü and C.N. Pope, Separability in cohomogeneity-2 Kerr-NUT-AdS metrics, JHEP 04 (2006) 008 [hep-th/0602084] [INSPIRE].
V.P. Frolov and D. Kubiznak, Hidden Symmetries of Higher Dimensional Rotating Black Holes, Phys. Rev. Lett. 98 (2007) 011101 [gr-qc/0605058] [INSPIRE].
D.N. Page, D. Kubiznak, M. Vasudevan and P. Krtous, Complete integrability of geodesic motion in general Kerr-NUT-AdS spacetimes, Phys. Rev. Lett. 98 (2007) 061102 [hep-th/0611083] [INSPIRE].
V.P. Frolov, P. Krtous and D. Kubiznak, Separability of Hamilton-Jacobi and Klein-Gordon Equations in General Kerr-NUT-AdS Spacetimes, JHEP 02 (2007) 005 [hep-th/0611245] [INSPIRE].
P. Krtous, D. Kubiznak, D.N. Page and V.P. Frolov, Killing-Yano Tensors, Rank-2 Killing Tensors, and Conserved Quantities in Higher Dimensions, JHEP 02 (2007) 004 [hep-th/0612029] [INSPIRE].
V.P. Frolov and D. Kubiznak, Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables, Class. Quant. Grav. 25 (2008) 154005 [arXiv:0802.0322] [INSPIRE].
P. Krtous, V.P. Frolov and D. Kubiznak, Hidden Symmetries of Higher Dimensional Black Holes and Uniqueness of the Kerr-NUT-(A)dS spacetime, Phys. Rev. D 78 (2008) 064022 [arXiv:0804.4705] [INSPIRE].
T. Houri, D. Kubiznak, C.M. Warnick and Y. Yasui, Generalized hidden symmetries and the Kerr-Sen black hole, JHEP 07 (2010) 055 [arXiv:1004.1032] [INSPIRE].
D. Kubiznak, C.M. Warnick and P. Krtous, Hidden symmetry in the presence of fluxes, Nucl. Phys. B 844 (2011) 185 [arXiv:1009.2767] [INSPIRE].
M. Cariglia, P. Krtous and D. Kubiznak, Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets, Phys. Rev. D 84 (2011) 024004 [arXiv:1102.4501] [INSPIRE].
M. Cariglia, P. Krtous and D. Kubiznak, Dirac Equation in Kerr-NUT-(A)dS Spacetimes: Intrinsic Characterization of Separability in All Dimensions, Phys. Rev. D 84 (2011) 024008 [arXiv:1104.4123] [INSPIRE].
D. Kubiznak and M. Cariglia, On Integrability of spinning particle motion in higher-dimensional black hole spacetimes, Phys. Rev. Lett. 108 (2012) 051104 [arXiv:1110.0495] [INSPIRE].
V. Frolov, P. Krtous and D. Kubiznak, Black holes, hidden symmetries, and complete integrability, Living Rev. Rel. 20 (2017) 6 [arXiv:1705.05482] [INSPIRE].
D. Kubiznak, H.K. Kunduri and Y. Yasui, Generalized Killing-Yano equations in D = 5 gauged supergravity, Phys. Lett. B 678 (2009) 240 [arXiv:0905.0722] [INSPIRE].
T. Houri, D. Kubiznak, C. Warnick and Y. Yasui, Symmetries of the Dirac Operator with Skew-Symmetric Torsion, Class. Quant. Grav. 27 (2010) 185019 [arXiv:1002.3616] [INSPIRE].
T. Houri, D. Kubiznak, C.M. Warnick and Y. Yasui, Generalized hidden symmetries and the Kerr-Sen black hole, JHEP 07 (2010) 055 [arXiv:1004.1032] [INSPIRE].
Y. Chervonyi and O. Lunin, Killing(-Yano) Tensors in String Theory, JHEP 09 (2015) 182 [arXiv:1505.06154] [INSPIRE].
S.A. Teukolsky, Rotating black holes: separable wave equations for gravitational and electromagnetic perturbations, Phys. Rev. Lett. 29 (1972) 1114 [INSPIRE].
S.A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185 (1973) 635 [INSPIRE].
O. Lunin, Maxwell’s equations in the Myers-Perry geometry, JHEP 12 (2017) 138 [arXiv:1708.06766] [INSPIRE].
V.P. Frolov, P. Krtouš and D. Kubizňák, Separation of variables in Maxwell equations in Plebański-Demiański spacetime, Phys. Rev. D 97 (2018) 101701 [arXiv:1802.09491] [INSPIRE].
P. Krtouš, V.P. Frolov and D. Kubizňák, Separation of Maxwell equations in Kerr-NUT-(A)dS spacetimes, Nucl. Phys. B 934 (2018) 7 [arXiv:1803.02485] [INSPIRE].
R. Cayuso, F. Gray, D. Kubizňák, A. Margalit, R. Gomes Souza and L. Thiele, Principal Tensor Strikes Again: Separability of Vector Equations with Torsion, Phys. Lett. B 795 (2019) 650 [arXiv:1906.10072] [INSPIRE].
T. Houri, N. Tanahashi and Y. Yasui, On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime, Class. Quant. Grav. 37 (2020) 015011 [arXiv:1908.10250] [INSPIRE].
O. Lunin, Excitations of the Myers-Perry Black Holes, JHEP 10 (2019) 030 [arXiv:1907.03820] [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].
I. Affleck, Exact Critical Exponents for Quantum Spin Chains, Nonlinear Sigma Models at θ = π and the Quantum Hall Effect, Nucl. Phys. B 265 (1986) 409 [INSPIRE].
K. Sfetsos, Integrable interpolations: From exact CFTs to non-Abelian T-duals, Nucl. Phys. B 880 (2014) 225 [arXiv:1312.4560] [INSPIRE].
K. Sfetsos and D.C. Thompson, Spacetimes for λ-deformations, JHEP 12 (2014) 164 [arXiv:1410.1886] [INSPIRE].
S. Demulder, K. Sfetsos and D.C. Thompson, Integrable λ-deformations: Squashing Coset CFTs and AdS5 × S5, JHEP 07 (2015) 019 [arXiv:1504.02781] [INSPIRE].
K. Sfetsos, K. Siampos and D.C. Thompson, Generalised integrable λ- and η-deformations and their relation, Nucl. Phys. B 899 (2015) 489 [arXiv:1506.05784] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, Integrable Deformations of Strings on Symmetric Spaces, JHEP 11 (2014) 009 [arXiv:1407.2840] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, An Integrable Deformation of the AdS5 × S5 Superstring, J. Phys. A 47 (2014) 495402 [arXiv:1409.1538] [INSPIRE].
C. Appadu and T.J. Hollowood, β-function of k deformed AdS5 × S5 string theory, JHEP 11 (2015) 095 [arXiv:1507.05420] [INSPIRE].
B. Hoare and A.A. Tseytlin, On integrable deformations of superstring sigma models related to AdSn × Sn supercosets, Nucl. Phys. B 897 (2015) 448 [arXiv:1504.07213] [INSPIRE].
R. Borsato, A.A. Tseytlin and L. Wulff, Supergravity background of λ-deformed model for AdS2 × S2 supercoset, Nucl. Phys. B 905 (2016) 264 [arXiv:1601.08192] [INSPIRE].
Y. Chervonyi and O. Lunin, Supergravity background of the λ-deformed AdS3 × S3 supercoset, Nucl. Phys. B 910 (2016) 685 [arXiv:1606.00394] [INSPIRE].
R. Borsato and L. Wulff, Target space supergeometry of η and λ-deformed strings, JHEP 10 (2016) 045 [arXiv:1608.03570] [INSPIRE].
G. Georgiou, K. Sfetsos and K. Siampos, All-loop anomalous dimensions in integrable λ-deformed σ-models, Nucl. Phys. B 901 (2015) 40 [arXiv:1509.02946] [INSPIRE].
Y. Chervonyi and O. Lunin, Generalized λ-deformations of AdSp × Sp, Nucl. Phys. B 913 (2016) 912 [arXiv:1608.06641] [INSPIRE].
S. Demulder, S. Driezen, A. Sevrin and D.C. Thompson, Classical and Quantum Aspects of Yang-Baxter Wess-Zumino Models, JHEP 03 (2018) 041 [arXiv:1711.00084] [INSPIRE].
O. Lunin and W. Tian, Analytical structure of the generalized λ-deformation, Nucl. Phys. B 929 (2018) 330 [arXiv:1711.02735] [INSPIRE].
C. Appadu, T.J. Hollowood, D. Price and D.C. Thompson, Quantum Anisotropic Sigma and Lambda Models as Spin Chains, J. Phys. A 51 (2018) 405401 [arXiv:1802.06016] [INSPIRE].
K. Yano, Some Remarks on Tensor Fields and Curvature, Ann. Math. 55 (1952) 328.
S.-i. Tachibana, On conformal Killing tensor in a Riemannian space, Tohoku Math. J. 21 (1969) 56.
T. Kashiwada, On conformal Killing tensor, Nat. Sci. Rep. Ochanomizu Univ. 19 (1968) 67 [INSPIRE].
R.C. Myers and M.J. Perry, Black Holes in Higher Dimensional Space-Times, Annals Phys. 172 (1986) 304 [INSPIRE].
I. Bars and K. Sfetsos, Generalized duality and singular strings in higher dimensions, Mod. Phys. Lett. A 7 (1992) 1091 [hep-th/9110054] [INSPIRE].
I. Bars and K. Sfetsos, A Superstring theory in four curved space-time dimensions, Phys. Lett. B 277 (1992) 269 [hep-th/9111040] [INSPIRE].
I.T. Ivanov, B.-b. Kim and M. Roček, Complex structures, duality and WZW models in extended superspace, Phys. Lett. B 343 (1995) 133 [hep-th/9406063] [INSPIRE].
K. Murata and J. Soda, A Note on separability of field equations in Myers-Perry spacetimes, Class. Quant. Grav. 25 (2008) 035006 [arXiv:0710.0221] [INSPIRE].
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Lunin, O., Tian, J. Separation of variables in the WZW models. J. High Energ. Phys. 2021, 114 (2021). https://doi.org/10.1007/JHEP06(2021)114
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DOI: https://doi.org/10.1007/JHEP06(2021)114