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Invariants and divergences in half-maximal supergravity theories

Abstract

The invariants in half-maximal supergravity theories in D = 4, 5 are discussed in detail up to dimension eight (e.g. R 4). In D = 4, owing to the anomaly in the rigid SL(2, \( \mathbb{R} \)) duality symmetry, the restrictions on divergences need careful treatment. In pure \( \mathcal{N}=4 \) supergravity, this anomalous symmetry still implies duality invariance of candidate counterterms at three loops. Provided one makes the additional assumption that there exists a full 16-supercharge off-shell formulation of the theory, counterterms at L ≥ 2 loops would also have to be writable as full-superspace integrals. At the three-loop order such a duality-invariant full-superspace integral candidate counterterm exists, but its duality invariance is marginal in the sense that the full-superspace counter-Lagrangian is not itself duality-invariant. We show that such marginal invariants are not allowable as counterterms in a 16-supercharge off-shell formalism. It is not possible to draw the same conclusion when vector multiplets are present because of the appearance of F 4 terms in the SL(2, \( \mathbb{R} \)) anomaly. In D = 5 there is no one-loop anomaly in the shift invariance of the dilaton, and we argue that this implies finiteness at two loops, again subject to the assumption that 16 supercharges can be preserved off-shell.

References

  1. Z. Bern, J. Carrasco, L.J. Dixon, H. Johansson, D. Kosower and R. Roiban, Three-Loop Superfiniteness of N = 8 Supergravity, Phys. Rev. Lett. 98 (2007) 161303 [hep-th/0702112] [INSPIRE].

    ADS  Article  Google Scholar 

  2. Z. Bern, 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] [INSPIRE].

    ADS  Article  Google Scholar 

  3. R. Kallosh, Counterterms in extended supergravities, Phys. Lett. B 99 (1981) 122 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  4. P.S. Howe, K. Stelle and P. Townsend, Superactions, Nucl. Phys. B 191 (1981) 445 [INSPIRE].

    ADS  Article  Google Scholar 

  5. G. Bossard, P. Howe and K. Stelle, The Ultra-violet question in maximally supersymmetric field theories, Gen. Rel. Grav. 41 (2009) 919 [arXiv:0901.4661] [INSPIRE].

    MathSciNet  ADS  MATH  Article  Google Scholar 

  6. Z. Bern, L.J. Dixon, D. Dunbar, M. Perelstein and J. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162] [INSPIRE].

    ADS  Article  Google Scholar 

  7. C. Hillmann, E 7(7) invariant Lagrangian of D = 4 \( \mathcal{N}=8 \) supergravity, JHEP 04 (2010) 010 [arXiv:0911.5225] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  8. G. Bossard, C. Hillmann and H. Nicolai, E 7(7) symmetry in perturbatively quantised N = 8 supergravity, JHEP 12 (2010) 052 [arXiv:1007.5472] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  9. H. Elvang and M. Kiermaier, Stringy KLT relations, global symmetries and E 7(7) violation, JHEP 10 (2010) 108 [arXiv:1007.4813] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  10. G. Bossard, P. Howe and K. Stelle, On duality symmetries of supergravity invariants, JHEP 01 (2011) 020 [arXiv:1009.0743] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  11. J. Drummond, P. Heslop, P. Howe and S. Kerstan, Integral invariants in N = 4 SYM and the effective action for coincident D-branes, JHEP 08 (2003) 016 [hep-th/0305202] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  12. G. Bossard, P. Howe, K. Stelle and P. Vanhove, The vanishing volume of D = 4 superspace, Class. Quant. Grav. 28 (2011) 215005 [arXiv:1105.6087] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  13. N. Beisert, H. Elvang, D.Z. Freedman, M. Kiermaier, A. Morales and S. Stieberger, E 7(7) constraints on counterterms in N = 8 supergravity, Phys. Lett. B 694 (2010) 265 [arXiv:1009.1643] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  14. M.B. Green, J.G. Russo and P. Vanhove, String theory dualities and supergravity divergences, JHEP 06 (2010) 075 [arXiv:1002.3805] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  15. M.B. Green, S.D. Miller, J.G. Russo and P. Vanhove, Eisenstein series for higher-rank groups and string theory amplitudes, Commun. Num. Theor. Phys. 4 (2010) 551 [arXiv:1004.0163] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  16. H. Elvang, D.Z. Freedman and M. Kiermaier, A simple approach to counterterms in N = 8 supergravity, JHEP 11 (2010) 016 [arXiv:1003.5018] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  17. J. Drummond, P. Heslop and P. Howe, A Note on N = 8 counterterms, arXiv:1008.4939 [INSPIRE].

  18. Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Absence of Three-Loop Four-Point Divergences in N = 4 Supergravity, Phys. Rev. Lett. 108 (2012) 201301 [arXiv:1202.3423] [INSPIRE].

    ADS  Article  Google Scholar 

  19. P. Tourkine and P. Vanhove, An R 4 non-renormalisation theorem in N = 4 supergravity, Class. Quant. Grav. 29 (2012) 115006 [arXiv:1202.3692] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  20. P. Tourkine and P. Vanhove, One-loop four-graviton amplitudes in N = 4 supergravity models, Phys. Rev. D 87 (2013) 045001 [arXiv:1208.1255] [INSPIRE].

    ADS  Google Scholar 

  21. Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Ultraviolet Cancellations in Half-Maximal Supergravity as a Consequence of the Double-Copy Structure, Phys. Rev. D 86 (2012) 105014 [arXiv:1209.2472] [INSPIRE].

    ADS  Google Scholar 

  22. R. Kallosh, E 7(7) Symmetry and Finiteness of N = 8 Supergravity, JHEP 03 (2012) 083 [arXiv:1103.4115] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  23. R. Kallosh, N = 8 Counterterms and E 7(7) Current Conservation, JHEP 06 (2011) 073 [arXiv:1104.5480] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  24. P.S. Howe and U. Lindström, Higher order invariants in extended supergravity, Nucl. Phys. B 181 (1981) 487 [INSPIRE].

    ADS  Article  Google Scholar 

  25. R. Kallosh, On Absence of 3-loop Divergence in N = 4 Supergravity, Phys. Rev. D 85 (2012) 081702 [arXiv:1202.4690] [INSPIRE].

    ADS  Google Scholar 

  26. S. Ferrara, R. Kallosh and A. Van Proeyen, Conjecture on Hidden Superconformal Symmetry of N = 4 Supergravity, Phys. Rev. D 87 (2013) 025004 [arXiv:1209.0418] [INSPIRE].

    ADS  Google Scholar 

  27. N. Marcus, Composite anomalies in supergravity, Phys. Lett. B 157 (1985) 383 [INSPIRE].

    ADS  Google Scholar 

  28. P.S. Howe, G. Papadopoulos and K. Stelle, Quantizing the N = 2 super σ-model in two-dimensions, Phys. Lett. B 174 (1986) 405 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  29. M. Fischler, Finiteness calculations for O(4) through O(8) extended supergravity and O(4) supergravity coupled to selfdual O(4) matter, Phys. Rev. D 20 (1979) 396 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  30. W. Siegel and M. Roček, On off-shell supermultiplets, Phys. Lett. B 105 (1981) 275 [INSPIRE].

    ADS  Google Scholar 

  31. V.O. Rivelles and J. Taylor, Off-shell no go theorems for higher dimensional supersymmetries and supergravities, Phys. Lett. B 121 (1983) 37 [INSPIRE].

    ADS  Google Scholar 

  32. P.S. Howe, H. Nicolai and A. Van Proeyen, Auxiliary fields and a superspace lagrangian for linearized ten-dimensional supergravity, Phys. Lett. B 112 (1982) 446 [INSPIRE].

    ADS  Google Scholar 

  33. A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev, N = 3 supersymmetric gauge theory, Phys. Lett. B 151 (1985) 215 [INSPIRE].

    ADS  Google Scholar 

  34. P.S. Howe, K. Stelle and P.C. West, N = 1D = 6 harmonic superspace, Class. Quant. Grav. 2 (1985) 815 [INSPIRE].

    MathSciNet  ADS  MATH  Article  Google Scholar 

  35. A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev, Unconstrained N = 2 Matter, Yang-Mills and Supergravity Theories in Harmonic Superspace, Class. Quant. Grav. 1 (1984) 469 [INSPIRE].

    ADS  Article  Google Scholar 

  36. A. Karlhede, U. Lindström and M. Roček, Selfinteracting tensor multiplets in N = 2 superspace, Phys. Lett. B 147 (1984) 297 [INSPIRE].

    ADS  Google Scholar 

  37. M. Henneaux and C. Teitelboim, Dynamics of chiral (selfdual) p forms, Phys. Lett. B 206 (1988) 650 [INSPIRE].

    ADS  Google Scholar 

  38. E. Sokatchev, Light cone harmonic superspace and its applications, Phys. Lett. B 169 (1986) 209 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  39. F. Delduc, S. Kalitsyn and E. Sokatchev, Learning the abc of light cone harmonic space, Class. Quant. Grav. 6 (1989) 1561 [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  40. A. Galperin, P.S. Howe and K. Stelle, The Superparticle and the Lorentz group, Nucl. Phys. B 368 (1992) 248 [hep-th/9201020] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  41. G. Bossard, P. Howe and K. Stelle, Anomalies and divergences in N = 4 supergravity, Phys. Lett. B 719 (2013) 424 [arXiv:1212.0841] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  42. P.S. Howe, Supergravity in superspace, Nucl. Phys. B 199 (1982) 309 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  43. P.S. Howe, A superspace approach to extended conformal supergravity, Phys. Lett. B 100 (1981) 389 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  44. J. Gates, S.J. and R. Grimm, Consequences of conformally covariant constraints for N > 4 superspace, Phys. Lett. B 133 (1983) 192 [INSPIRE].

  45. L. Brink and P.S. Howe, The N = 8 supergravity in superspace, Phys. Lett. B 88 (1979) 268 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  46. A.A. Rosly, Super Yang-Mills constraints as integrability conditions, in proceedings of International Seminar on Group Theoretical Methods in Physics (in Russian), Zvenigorod, USSR (1982), M.A. Markov ed., Nauka, Moscow (1983), vol. 1, p. 263 [English translation: in Group Theoretical Methods in Physics, M.A. Markov, V.I. Man’ko and A.E. Shabad eds., Harwood Academic Publishers, London, U.K. (1987), vol. 3, p. 587].

  47. P.S. Howe and G. Hartwell, A Superspace survey, Class. Quant. Grav. 12 (1995) 1823 [INSPIRE].

    MathSciNet  ADS  MATH  Article  Google Scholar 

  48. A. Galperin, E. Ivanov, V. Ogievetsky and E. Sokatchev, N = 2 supergravity in superspace: different versions and matter couplings, Class. Quant. Grav. 4 (1987) 1255 [INSPIRE].

    MathSciNet  ADS  MATH  Article  Google Scholar 

  49. S. Kuzenko, U. Lindström, M. Roček and G. Tartaglino-Mazzucchelli, 4D N = 2 Supergravity and Projective Superspace, JHEP 09 (2008) 051 [arXiv:0805.4683] [INSPIRE].

    ADS  Article  Google Scholar 

  50. G. Hartwell and P.S. Howe, (N,p,q) harmonic superspace, Int. J. Mod. Phys. A 10 (1995) 3901 [hep-th/9412147] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  51. K.A. Intriligator, Bonus symmetries of N = 4 super Yang-Mills correlation functions via AdS duality, Nucl. Phys. B 551 (1999) 575 [hep-th/9811047] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  52. P. Heslop and P. Howe, Aspects of N = 4 SYM, JHEP 01 (2004) 058 [hep-th/0307210] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  53. V. Dobrev and V. Petkova, All Positive Energy Unitary Irreducible Representations of Extended Conformal Supersymmetry, Phys. Lett. B 162 (1985) 127 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  54. I. Antoniadis, S. Hohenegger, K. Narain and E. Sokatchev, Harmonicity in N = 4 supersymmetry and its quantum anomaly, Nucl. Phys. B 794 (2008) 348 [arXiv:0708.0482] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  55. S. de Haro, A. Sinkovics and K. Skenderis, On a supersymmetric completion of the R4 term in 2B supergravity, Phys. Rev. D 67 (2003) 084010 [hep-th/0210080] [INSPIRE].

    ADS  Google Scholar 

  56. P.S. Howe and P.C. West, The Complete N = 2, D = 10 Supergravity, Nucl. Phys. B 238 (1984) 181 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  57. N. Berkovits and P.S. Howe, Ten-dimensional supergravity constraints from the pure spinor formalism for the superstring, Nucl. Phys. B 635 (2002) 75 [hep-th/0112160] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  58. T. Voronov, Geometric integration theory on supermanifolds, Sov. Sci. Rev. C 9 (1992) 1.

    Google Scholar 

  59. S.J. Gates Jr., Ectoplasm has no topology: The Prelude, hep-th/9709104 [INSPIRE].

  60. S.J. Gates Jr., M.T. Grisaru, M.E. Knutt-Wehlau and W. Siegel, Component actions from curved superspace: Normal coordinates and ectoplasm, Phys. Lett. B 421 (1998) 203 [hep-th/9711151] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  61. L. Bonora, P. Pasti and M. Tonin, Superspace formulation of 10 − D sugra+SYM theory à la Green-Schwarz, Phys. Lett. B 188 (1987) 335 [INSPIRE].

    ADS  Article  Google Scholar 

  62. M. Cederwall, B.E. Nilsson and D. Tsimpis, The Structure of maximally supersymmetric Yang-Mills theory: Constraining higher order corrections, JHEP 06 (2001) 034 [hep-th/0102009] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  63. M. Cederwall, B.E. Nilsson and D. Tsimpis, Spinorial cohomology and maximally supersymmetric theories, JHEP 02 (2002) 009 [hep-th/0110069] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  64. P. Howe and D. Tsimpis, On higher order corrections in M-theory, JHEP 09 (2003) 038 [hep-th/0305129] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  65. N. Berkovits and P. Howe, The Cohomology of superspace, pure spinors and invariant integrals, JHEP 06 (2008) 046 [arXiv:0803.3024] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  66. P.S. Howe, Pure spinors lines in superspace and ten-dimensional supersymmetric theories, Phys. Lett. B 258 (1991) 141 [Addendum ibid. B 259 (1991) 511] [INSPIRE].

  67. P.S. Howe, Pure spinors, function superspaces and supergravity theories in ten-dimensions and eleven-dimensions, Phys. Lett. B 273 (1991) 90 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  68. N. Berkovits, ICTP lectures on covariant quantization of the superstring, hep-th/0209059 [INSPIRE].

  69. F. Brandt, Supersymmetry algebra cohomology I: Definition and general structure, J. Math. Phys. 51 (2010) 122302 [arXiv:0911.2118] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  70. F. Brandt, Supersymmetry algebra cohomology III: Primitive elements in four and five dimensions, J. Math. Phys. 52 (2011) 052301 [arXiv:1005.2102] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  71. M.V. Movshev, A. Schwarz and R. Xu, Homology of Lie algebra of supersymmetries, arXiv:1011.4731 [INSPIRE].

  72. M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D 59 (1999) 046006 [hep-th/9808061] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  73. M.T. Grisaru, Two Loop Renormalizability of Supergravity, Phys. Lett. B 66 (1977) 75 [INSPIRE].

    ADS  Google Scholar 

  74. S. Deser, J. Kay and K. Stelle, Renormalizability Properties of Supergravity, Phys. Rev. Lett. 38 (1977) 527 [INSPIRE].

    ADS  Article  Google Scholar 

  75. S. Deser and J. Kay, Three loop counterterms for extended supergravity, Phys. Lett. B 76 (1978) 400 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  76. S.M. Kuzenko and G. Tartaglino-Mazzucchelli, Different representations for the action principle in 4D N = 2 supergravity, JHEP 04 (2009) 007 [arXiv:0812.3464] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  77. I. Antoniadis, S. Hohenegger and K. Narain, N = 4 Topological Amplitudes and String Effective Action, Nucl. Phys. B 771 (2007) 40 [hep-th/0610258] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  78. O. Alvarez, I. Singer and B. Zumino, Gravitational anomalies and the familys index theorem, Commun. Math. Phys. 96 (1984) 409 [INSPIRE].

    MathSciNet  ADS  MATH  Article  Google Scholar 

  79. Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  80. Z. Bern and L. J. Dixon, private communication.

  81. J. Carrasco, R. Kallosh, R. Roiban and A. Tseytlin, On the U(1) duality anomaly and the S-matrix of N = 4 supergravity, arXiv:1303.6219 [INSPIRE].

  82. 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 [INSPIRE].

    ADS  Google Scholar 

  83. 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 (1987) 411.

  84. T. Pugh, E. Sezgin and K. Stelle, D = 7/D = 6 Heterotic Supergravity with Gauged R-Symmetry, JHEP 02 (2011) 115 [arXiv:1008.0726] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  85. O. Piguet and S.P. Sorella, Algebraic renormalization: Perturbative renormalization, symmetries and anomalies, Lect. Notes Phys. M 28 (1995) 1.

    MathSciNet  Google Scholar 

  86. J.A. Harvey and G.W. Moore, Five-brane instantons and R 2 couplings in N = 4 string theory, Phys. Rev. D 57 (1998) 2323 [hep-th/9610237] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  87. A. Gregori, E. Kiritsis, C. Kounnas, N. Obers, P. Petropoulos and B. Pioline, R 2 corrections and nonperturbative dualities of N = 4 string ground states, Nucl. Phys. B 510 (1998) 423 [hep-th/9708062] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  88. O. Yasuda, Nonrenormalization theorem for the Green-Schwarz counterterm and the low-energy effective action, Phys. Lett. B 218 (1989) 455 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  89. E. Kiritsis, N.A. Obers and B. Pioline, Heterotic/type-II triality and instantons on K(3), JHEP 01 (2000) 029 [hep-th/0001083] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  90. E. Fradkin and A.A. Tseytlin, One loop infinities in dimensionally reduced supergravities, Phys. Lett. B 137 (1984) 357 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  91. P. Howe, U. Lindström and L. Wulff, D = 10 supersymmetric Yang-Mills theory at α ′4, JHEP 07 (2010) 028 [arXiv:1004.3466] [INSPIRE].

    ADS  Article  Google Scholar 

  92. E. Bergshoeff, M. Rakowski and E. Sezgin, Higher derivative super Yang-Mills theories, Phys. Lett. B 185 (1987) 371 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  93. P. Koerber and A. Sevrin, The NonAbelian D-brane effective action through order α ′4, JHEP 10 (2002) 046 [hep-th/0208044] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  94. M.B. Green and P. Vanhove, Duality and higher derivative terms in M-theory, JHEP 01 (2006) 093 [hep-th/0510027] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  95. E. Sokatchev and B. Zupnik, unpublished.

  96. P.S. Howe and K. Stelle, The ultraviolet properties of supersymmetric field theories, Int. J. Mod. Phys. A 4 (1989) 1871 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  97. S.J. Gates Jr., K. Stelle and P.C. West, Algebraic origins of superspace constraints in supergravity, Nucl. Phys. B 169 (1980) 347 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  98. K. Stelle and P.C. West, Algebraic derivation of N = 2 supergravity constraints, Phys. Lett. B 90 (1980) 393 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  99. W. Siegel and S.J. Gates Jr., Superfield Supergravity, Nucl. Phys. B 147 (1979) 77 [INSPIRE].

    ADS  Article  Google Scholar 

  100. M.T. Grisaru and W. Siegel, Supergraphity. Part 1. Background field formalism, Nucl. Phys. B 187 (1981) 149 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  101. M.T. Grisaru and W. Siegel, Supergraphity. 2. Manifestly Covariant Rules and Higher Loop Finiteness, Nucl. Phys. B 201 (1982) 292 [Erratum ibid. B 206 (1982) 496] [INSPIRE].

  102. P.S. Howe, K. Stelle and P. Townsend, Miraculous Ultraviolet Cancellations in Supersymmetry Made Manifest, Nucl. Phys. B 236 (1984) 125 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  103. L. Baulieu and M.P. Bellon, A simple algebraic construction of the symmetries of supergravity, Phys. Lett. B 161 (1985) 96 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  104. A. Blasi, V. Lemes, N. Maggiore, S. Sorella, A. Tanzini, O.S. Ventura and L.C.Q. Vilar, Perturbative β-function of N = 2 super Yang-Mills theories, JHEP 05 (2000) 039 [hep-th/0004048] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  105. V. Lemes, M. Sarandy, S. Sorella, O. Ventura and L. Vilar, An Algebraic criterion for the ultraviolet finiteness of quantum field theories, J. Phys. A 34 (2001) 9485 [hep-th/0103110] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  106. J.H. Schwarz and A. Sen, Duality symmetric actions, Nucl. Phys. B 411 (1994) 35 [hep-th/9304154] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  107. E. Sokatchev, An off-shell formulation of N = 4 supersymmetric Yang-Mills theory in twistor harmonic superspace, Phys. Lett. B 217 (1989) 489 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  108. P.S. Howe and U. Lindström, The supercurrent in five-dimensions, Phys. Lett. B 103 (1981) 422 [INSPIRE].

    ADS  Google Scholar 

  109. E. D’Hoker and D. Phong, Two-loop superstrings VI: Non-renormalization theorems and the 4-point function, Nucl. Phys. B 715 (2005) 3 [hep-th/0501197] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  110. A.A. Tseytlin, Heterotic type-I superstring duality and low-energy effective actions, Nucl. Phys. B 467 (1996) 383 [hep-th/9512081] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  111. G. Bossard, P. Howe, U. Lindström, K. Stelle and L. Wulff, Integral invariants in maximally supersymmetric Yang-Mills theories, JHEP 05 (2011) 021 [arXiv:1012.3142] [INSPIRE].

    ADS  Article  Google Scholar 

  112. P.S. Howe and M. Leeming, Harmonic superspaces in low dimensions, Class. Quant. Grav. 11 (1994) 2843 [hep-th/9408062] [INSPIRE].

    MathSciNet  ADS  MATH  Article  Google Scholar 

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Bossard, G., Howe, P.S. & Stelle, K.S. Invariants and divergences in half-maximal supergravity theories. J. High Energ. Phys. 2013, 117 (2013). https://doi.org/10.1007/JHEP07(2013)117

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Keywords

  • Extended Supersymmetry
  • Field Theories in Higher Dimensions
  • Supergravity Models
  • Renormalization Regularization and Renormalons