Skip to main content
Download PDF

Integral invariants in maximally supersymmetric Yang-Mills theories

Journal of High Energy Physics volume 2011, Article number: 21 (2011) Cite this article

Abstract

Integral invariants in maximally supersymmetric Yang-Mills theories are discussed in spacetime dimensions 4 ≤ D ≤ 10 for SU(k) gauge groups. It is shown that, in addition to the action, there are three special invariants in all dimensions. Two of these, the single-and double-trace F 4 invariants, are of Chern-Simons type in D = 9, 10 and BPS type in D ≤ 8, while the third, the double-trace of two derivatives acting on F 4, can be expressed in terms of a gauge-invariant super-D -form in all dimensions. We show that the super-ten-forms for D = 10 F 4 invariants have interesting cohomological properties and we also discuss some features of other invariants, including the single-trace d 2 F 4, which has a special form in D = 10. The implications of these results for ultra-violet divergences are discussed in the framework of algebraic renormalisation.

References

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

    ADS  Article  Google Scholar 

  2. 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 [SPIRES].

    ADS  Article  Google Scholar 

  3. 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].

    MathSciNet  ADS  Article  Google Scholar 

  4. 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].

    MathSciNet  ADS  Google Scholar 

  5. 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 [SPIRES].

    ADS  Google Scholar 

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

    Google Scholar 

  7. S.J. Gates Jr., Ectoplasm has no topology: the prelude, hep-th/9709104 [SPIRES].

  8. 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] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  9. N. Berkovits and P.S. Howe, The cohomology of superspace, pure spinors and invariant integrals, JHEP 06 (2008) 046 [arXiv:0803.3024] [SPIRES].

    MathSciNet  ADS  Article  Google Scholar 

  10. L. Bonora, P. Pasti and M. Tonin, Superspace formulation of 10-D SUGRA+SYM theory a la Green-Schwarz, Phys. Lett. B 188 (1987) 335 [SPIRES].

    ADS  Google Scholar 

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

    MathSciNet  ADS  Article  Google Scholar 

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

    MathSciNet  ADS  Article  Google Scholar 

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

  14. 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] [SPIRES].

    MathSciNet  ADS  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

  16. L. Bonora, P. Pasti and M. Tonin, Chiral anomalies in higher dimensional supersymmetric theories, Nucl. Phys. B 286 (1987) 150 [SPIRES].

    MathSciNet  ADS  Article  Google Scholar 

  17. P.S. Howe, O. Raetzel and E. Sezgin, On brane actions and superembeddings, JHEP 08 (1998) 011 [hep-th/9804051] [SPIRES].

    MathSciNet  ADS  Article  Google Scholar 

  18. M. Movshev and A. Schwarz, Supersymmetric deformations of maximally supersymmetric gauge theories. I, arXiv:0910.0620 [SPIRES].

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

    MathSciNet  ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  21. S. Deser and U. Lindström, Extended supersymmetry invariants by dimensional reduction, Phys. Lett. B 90 (1980) 68 [SPIRES].

    ADS  Google Scholar 

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

    MathSciNet  ADS  MATH  Article  Google Scholar 

  23. P. Breitenlohner, D. Maison and K.S. Stelle, Anomalous dimensions and the Adler-Bardeen theorem in supersymmetric Yang-Mills theories, Phys. Lett. B 134 (1984) 63 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  24. N. Marcus and A. Sagnotti, The ultraviolet behavior of N =4 Yang-Mills and the power counting of extended superspace, Nucl. Phys. B 256 (1985) 77 [SPIRES].

    ADS  Article  Google Scholar 

  25. Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. 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] [SPIRES].

    ADS  Article  Google Scholar 

  26. Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The complete four-loop four-point amplitude in N =4 super-Yang-Mills theory, Phys. Rev. D 82 (2010) 125040 [arXiv:1008.3327] [SPIRES].

    ADS  Google Scholar 

  27. N. Berkovits, M.B. Green, J.G. Russo and P. Vanhove, Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory, JHEP 11 (2009) 063 [arXiv:0908.1923] [SPIRES].

    MathSciNet  ADS  Article  Google Scholar 

  28. J. Bjornsson and M.B. Green, 5 loops in 24/5 dimensions, JHEP 08 (2010) 132 [arXiv:1004.2692] [SPIRES].

    MathSciNet  ADS  Article  Google Scholar 

  29. P.J. Heslop and P.S. Howe, A note on composite operators in N =4 SYM, Phys. Lett. B 516 (2001) 367 [hep-th/0106238] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  30. G. Arutyunov, S. Frolov and A.C. Petkou, Operator product expansion of the lowest weight CPOs in N =4 SYM(4) at strong coupling, Nucl. Phys. B 586 (2000) 547 [Erratum ibid. B 609 (2001) 539] [hep-th/0005182] [SPIRES].

    MathSciNet  ADS  Article  Google Scholar 

  31. B. Eden, A.C. Petkou, C. Schubert and E. Sokatchev, Partial non-renormalisation of the stress-tensor four-point function in N =4 SYM and AdS/CFT, Nucl. Phys. B 607 (2001) 191 [hep-th/0009106] [SPIRES].

    MathSciNet  ADS  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A EI, Max Planck Institut für Gravitationsphysik, Am Mühlenberg 1, D-14476, Potsdam, Germany

    G. Bossard

  2. Centre de Physique Théorique (UMR 7644), Ecole Polytechnique, 91128, Palaiseau Cedex, France

    G. Bossard

  3. Department of Mathematics, King’s College, University of London, Strand, London, WC2R 2LS, U.K.

    P. S. Howe

  4. Theoretical Physics, Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120, Uppsala, Sweden

    U. Lindström

  5. Theoretical Physics Group, Imperial College London, Prince Consort Road, London, SW7 2AZ, U.K.

    K. S. Stelle

  6. George and Cynthia Mitchell Institute for Fundamental Physics and Astronomy, Texas A &M University, College Station, TX, 77843, U.S.A.

    L. Wulff

Authors
  1. G. Bossard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. S. Howe
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. U. Lindström
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. K. S. Stelle
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. L. Wulff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. S. Howe.

Additional information

ArXiv ePrint:1012.3142

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.

Reprints and Permissions

About this article

Cite this article

Bossard, G., Howe, P.S., Lindström, U. et al. Integral invariants in maximally supersymmetric Yang-Mills theories. J. High Energ. Phys. 2011, 21 (2011). https://doi.org/10.1007/JHEP05(2011)021

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP05(2011)021

Keywords

  • Extended Supersymmetry
  • Superspaces
  • Gauge Symmetry
  • Field Theories in Higher Dimensions