Abstract
We study the perturbative quantisation of \( \mathcal{N} = 8 \) supergravity in a formulation where its E 7(7) symmetry is realised off-shell. Relying on the cancellation of SU(8) current anomalies we show that there are no anomalies for the non-linearly realised E 7(7) either; this result extends to all orders in perturbation theory. As a consequence, the \( {\mathfrak{e}_{7(7)}} \) Ward identities can be consistently implemented and imposed at all orders in perturbation theory, and therefore potential divergent counterterms must in particular respect the full non-linear E 7(7) symmetry.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Cremmer and B. Julia, The SO(8) supergravity, Nucl. Phys. B 159 (1979) 141 [SPIRES].
B. de Wit and H. Nicolai, \( \mathcal{N} = 8 \) supergravity, Nucl. Phys. B 208 (1982) 323 [SPIRES].
R.E. Kallosh, Counterterms in extended supergravities, Phys. Lett. B 99 (1981) 122 [SPIRES].
P.S. Howe, K.S. Stelle and P.K. Townsend, Superactions, Nucl. Phys. B 191 (1981) 445 [SPIRES].
Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson, D.A. Kosower and R. Roiban, Three-loop superfiniteness of \( \mathcal{N} = 8 \) supergravity, Phys. Rev. Lett. 98 (2007) 161303 [hep-th/0702112] [SPIRES].
Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The ultraviolet behavior of N = 8 supergravity at four loops, Phys. Rev. Lett. 103 (2009) 081301 [arXiv:0905.2326] [SPIRES].
G. Bossard, P.S. Howe and K.S. Stelle, A note on the UV behaviour of maximally supersymmetric Yang-Mills theories, Phys. Lett. B 682 (2009) 137 [arXiv:0908.3883] [SPIRES].
M.B. Green, J.G. Russo and P. Vanhove, Automorphic properties of low energy string amplitudes in various dimensions, Phys. Rev. D 81 (2010) 086008 [arXiv:1001.2535] [SPIRES].
B. Pioline, R 4 couplings and automorphic unipotent representations, JHEP 03 (2010) 116 [arXiv:1001.3647] [SPIRES].
M. Henneaux and C. Teitelboim, Dynamics of chiral (selfdual) P forms, Phys. Lett. B 206 (1988) 650 [SPIRES].
C. Hillmann, E 7(7) invariant Lagrangian of D = 4 N = 8 supergravity, JHEP 04 (2010) 010 [arXiv:0911.5225] [SPIRES].
J.H. Schwarz and A. Sen, Duality symmetric actions, Nucl. Phys. B 411 (1994) 35 [hep-th/9304154] [SPIRES].
M.K. Gaillard and B. Zumino, Duality rotations for interacting fields, Nucl. Phys. B 193 (1981) 221 [SPIRES].
O. Piguet and S.P. Sorella, Algebraic renormalization: perturbative renormalization, symmetries and anomalies, Lect. Notes Phys. M28 (1995) 1 [SPIRES].
P. di Vecchia, S. Ferrara and L. Girardello, Anomalies of hidden local chiral symmetries in σ-modelS and extended supergravities, Phys. Lett. B 151 (1985) 199 [SPIRES].
B. de Wit and M.T. Grisaru, Compensating fields and anomalies, in Essays in Honor of 60th birthday of E.S. Fradkin, Quantum field theory and quantum statistics, 2 (1985) 411 [SPIRES].
N. Marcus, Composite anomalies in supegravity, Phys. Lett. B 157 (1985) 383 [SPIRES].
S. Weinberg, The quantum theory of fields. Vol. 1: Foundations, Cambridge University Press, Cambridge U.K. (1995) [SPIRES].
S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications, Cambridge University Press, Cambridge U.K. (1996) [SPIRES].
M.B. Green, J.G. Russo and P. Vanhove, String theory dualities and supergravity divergences, JHEP 06 (2010) 075 [arXiv:1002.3805] [SPIRES].
J.M. Drummond, P.J. Heslop, P.S. Howe and S.F. Kerstan, Integral invariants in N = 4 SYM and the effective action for coincident D-branes, JHEP 08 (2003) 016 [hep-th/0305202] [SPIRES].
J. Broedel and L.J. Dixon, R 4 counterterm and E 7(7) symmetry in maximal supergravity, JHEP 05 (2010) 003 [arXiv:0911.5704] [SPIRES].
H. Elvang and M. Kiermaier, Stringy KLT relations, global symmetries and E 7(7) violation, JHEP 10 (2010) 108 [arXiv:1007.4813] [SPIRES].
M.B. Green, H. Ooguri and J.H. Schwarz, Decoupling supergravity from the superstring, Phys. Rev. Lett. 99 (2007) 041601 [arXiv:0704.0777] [SPIRES].
L. Alvarez-Gaumé and É. Witten, Gravitational anomalies, Nucl. Phys. B 234 (1984) 269 [SPIRES].
O. Alvarez, I.M. Singer and B. Zumino, Gravitational anomalies and the family’s index theorem, Commun. Math. Phys. 96 (1984) 409 [SPIRES].
E.S. Fradkin and G.A. Vilkovisky, Quantization of relativistic systems with constraints: equivalence of canonical and covariant formalisms in quantum theory of gravitational field, CERN-TH-2332 (1977) [SPIRES].
U. Lindström, N.K. Nielsen, M. Roςcek and P. van Nieuwenhuizen, The supersymmetric regularized path-integral measure in x space, Phys. Rev. D 37 (1988) 3588 [SPIRES].
R. Kallosh and M. Soroush, Explicit action of E 7(7) on N = 8 supergravity fields, Nucl. Phys. B 801 (2008) 25 [arXiv:0802.4106] [SPIRES].
G.W. Gibbons and S.W. Hawking, Classification of gravitational instanton symmetries, Commun. Math. Phys. 66 (1979) 291 [SPIRES].
M.B. Green and M. Gutperle, Effects of D-instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [SPIRES].
P. Ramond, Field theory: A modern primer, Front. Phys. 51 (1981) 1 [SPIRES].
R.A. Bertlmann, Anomalies in quantum field theory, Clarendon, Oxford U.K. (1996) [SPIRES].
M.F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1 [SPIRES].
G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in the antifield formalism. 1. General theorems, Commun. Math. Phys. 174 (1995) 57 [hep-th/9405109] [SPIRES].
J.A. Dixon, Cohomology and renormalization of gauge theories. 2, (1979) HUTMP78/B64 [SPIRES].
L. Bonora and P. Cotta-Ramusino, Some remarks on BRS transformations, anomalies and the cohomology of the Lie algebra of the group of gauge transformations, Commun. Math. Phys. 87 (1983) 589 [SPIRES].
L. Baulieu, Perturbative gauge theories, Phys. Rept. 129 (1985) 1 [SPIRES].
J. Manes, R. Stora and B. Zumino, Algebraic study of chiral anomalies, Commun. Math. Phys. 102 (1985) 157 [SPIRES].
M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University Press, Princeton U.S.A. (1992) [SPIRES]
B. de Wit and J.W. van Holten, Covariant quantization of gauge theories with open gauge algebra, Phys. Lett. B 79 (1978) 389 [SPIRES].
I.A. Batalin and G.A. Vilkovisky, Quantization of gauge theories with linearly dependent generators, Phys. Rev. D 28 (1983) 2567 [Erratum ibid. D 30 (1984) 508] [SPIRES].
K.S. Stelle and P.C. West, Matter coupling and BRS transformations with auxiliary fields in supergravity, Nucl. Phys. B 140 (1978) 285 [SPIRES].
L. Baulieu and M.P. Bellon, A simple algebraic construction of the symmetries of supergravity, Phys. Lett. B 161 (1985) 96 [SPIRES].
L. Baulieu, G. Bossard and S.P. Sorella, Shadow fields and local supersymmetric gauges, Nucl. Phys. B 753 (2006) 273 [hep-th/0603248] [SPIRES].
D. Zwanziger, Renormalization in the Coulomb gauge and order parameter for confinement in QCD, Nucl. Phys. B 518 (1998) 237 [SPIRES].
A. Andrasi and J.C. Taylor, Cancellation of energy-divergences in Coulomb gauge QCD, Eur. Phys. J. C 41 (2005) 377 [hep-th/0503099] [SPIRES].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1007.5472
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Bossard, G., Hillmann, C. & Nicolai, H. E7(7) symmetry in perturbatively quantised \( \mathcal{N} = 8 \) supergravity. J. High Energ. Phys. 2010, 52 (2010). https://doi.org/10.1007/JHEP12(2010)052
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2010)052