Abstract
We construct the first eleven-dimensional supergravity solutions, which are regular, have no smearing and possess only SO(2, 4) × SO(3) × U(1) R isometry. They are dual to four-dimensional field theories with \( \mathcal{N} \) = 2 superconformal symmetry. We utilise the Toda frame of self-dual four-dimensional Euclidean metrics with SU(2) rotational symmetry. They are obtained by transforming the Atiyah-Hitchin instanton under SL(2, \( \mathbb{R} \)) and are expressed in terms of theta functions. The absence of any extra U(1) symmetry, even asymptotically, renders inapplicable the electrostatic description of our solution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [INSPIRE].
M.V. Saveliev, Integro-differential non-linear equations and continual Lie algebras,Commun. Math. Phys. 121 (1989) 283.
M.V. Saveliev and A.M. Vershik, New examples of continuum graded Lie algebras, Phys. Lett. A 143 (1990) 121.
R.S. Ward, Einstein-Weyl spaces and SU(∞) Toda fields, Class. Quant. Grav. 7 (1990) L95 [INSPIRE].
D. Gaiotto and J.M. Maldacena, The gravity duals of N = 2 superconformal field theories, JHEP 10 (2012) 189 [arXiv:0904.4466] [INSPIRE].
R.A. Reid-Edwards and B. Stefanski Jr., On type IIA geometries dual to N = 2 SCFTs, Nucl. Phys. B 849 (2011) 549 [arXiv:1011.0216] [INSPIRE].
A. Donos and J. Simon, The electrostatic view on M-theory LLM geometries, JHEP 01 (2011) 067 [arXiv:1010.3101] [INSPIRE].
O. Aharony, L. Berdichevsky and M. Berkooz, 4d N = 2 superconformal linear quivers with type IIA duals, JHEP 08 (2012) 131 [arXiv:1206.5916] [INSPIRE].
M.F. Atiyah and N.J. Hitchin, Low energy scattering of non-Abelian monopoles, Phys. Lett. A 107 (1985) 21 [INSPIRE].
M.F. Atiyah and N.J. Hitchin, The geometry and dynamics of magnetic monopoles, Princeton University Press, Princeton U.S.A. (1988) [INSPIRE].
G.H. Halphen, Sur un système d’équations différentielles, C. R. Acad. Sc. Paris 92 (1881) 1001.
G.H. Halphen, Sur certains systèmes d’équations différentielles, C. R. Acad. Sc. Paris 92 (1881) 1004.
D. Olivier, Complex coordinates and Kähler potential for the Atiyah-Hitchin metric, Gen. Rel. Grav. 23 (1991) 1349 [INSPIRE].
D. Finley and J.K. McIver, Solutions of the sDiff(2)Toda equation with SU(2) symmetry, Class. Quant. Grav. 27 (2010) 145001 [arXiv:1001.1753] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Classification of gravitational instanton symmetries, Commun. Math. Phys. 66 (1979) 291 [INSPIRE].
C.P. Boyer and J.D. Finley, Killing vectors in selfdual, Euclidean Einstein spaces, J. Math. Phys. 23 (1982) 1126 [INSPIRE].
J.D. Gegenberg and A. Das, Stationary Riemannian space-times with self-dual curvature, Gen. Rel. Grav. 16 (1984) 817.
G.W. Gibbons and P. Ruback, The hidden symmetries of multicenter metrics, Commun. Math. Phys. 115 (1988) 267 [INSPIRE].
E.T. Newman, L. Tamubrino and T.J. Unti, Empty space generalization of the Schwarzschild metric, J. Math. Phys. 4 (1963) 915 [INSPIRE].
S.W. Hawking, Gravitational instantons, Phys. Lett. A 60 (1977) 81 [INSPIRE].
T. Eguchi and A.J. Hanson, Selfdual solutions to Euclidean gravity, Annals Phys. 120 (1979) 82 [INSPIRE].
T. Eguchi and A.J. Hanson, Gravitational instantons, Gen. Rel. Grav. 11 (1979) 315 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Gravitational multi-instantons, Phys. Lett. B 78 (1978) 430 [INSPIRE].
G.W. Gibbons and C.N. Pope, The positive action conjecture and asymptotically Euclidean metrics in quantum gravity, Commun. Math. Phys. 66 (1979) 267 [INSPIRE].
T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, gauge theories and differential geometry, Phys. Rept. 66 (1980) 213 [INSPIRE].
F. Bourliot, J. Estes, P.M. Petropoulos and P. Spindel, Gravitational instantons, self-duality and geometric flows, Phys. Rev. D 81 (2010) 104001 [arXiv:0906.4558] [INSPIRE].
F. Bourliot, J. Estes, P.M. Petropoulos and P. Spindel, G3-homogeneous gravitational instantons, Class. Quant. Grav. 27 (2010) 105007 [arXiv:0912.4848] [INSPIRE].
G.W. Gibbons and P.J. Ruback, The hidden symmetries of multicenter metrics, Commun. Math. Phys. 115 (1988) 267 [INSPIRE].
G. Darboux, Mémoire sur la théorie des coordonnées curvilignes et des systèmes orthogonaux, Ann. Ec. Normale Supér. 7 (1878) 101.
J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
L.A. Takhtajan, A simple example of modular forms as tau functions for integrable equations, Theor. Math. Phys. 93 (1992) 1308 [Teor. Mat. Fiz. 93 (1992) 330] [INSPIRE].
A.J. Maciejewski and J.M. Strelcyn, On the algebraic non-integrability of the Halphen system, Phys. Lett. A 201 (1995) 161.
N.S. Manton, A remark on the scattering of BPS monopoles, Phys. Lett. B 110 (1982) 54 [INSPIRE].
G.W. Gibbons and N.S. Manton, Classical and quantum dynamics of BPS monopoles, Nucl. Phys. B 274 (1986) 183 [INSPIRE].
V.A. Belinsky, G.W. Gibbons, D.N. Page and C.N. Pope, Asymptotically Euclidean Bianchi IX metrics in quantum gravity, Phys. Lett. B 76 (1978) 433 [INSPIRE].
P.M. Petropoulos and P. Vanhove, Gravity, strings, modular and quasimodular forms, Ann. Math. Blaise Pascal 19 (2012) 379 [arXiv:1206.0571] [INSPIRE].
J.P. Serre, Cours d’arithméthique, PUF, Paris France (1972).
N. Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, Springer (1993).
I. Bakas and K. Sfetsos, Toda fields of SO(3) hyperKähler metrics and free field realizations, Int. J. Mod. Phys. A 12 (1997) 2585 [hep-th/9604003] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1308.6583
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
Petropoulos, P.M., Sfetsos, K. & Siampos, K. Gravity duals of \( \mathcal{N} \) = 2 superconformal field theories with no electrostatic description. J. High Energ. Phys. 2013, 118 (2013). https://doi.org/10.1007/JHEP11(2013)118
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2013)118