An Introduction to Gravitational Anomalies

  • Luis Alvarez-Gaumé
Part of the NATO ASI Series book series (NSSB, volume 115)


The study of anomalies in global and gauge currents in Quantum Field Theory has had a remarkable number of important applications during the 70’s. In the original version of the anomaly1 one considers a massless fermion triangle diagram with one axial current and two vector currents. Requiring the vector currents to be conserved, one finds that the axial current is not conserved therefore leading to a breakdown of chiral symmetry in the presence of gauge fields coupled to conserved vector currents. This breakdown of chiral symmetry led to the understanding of π° decay and to the resolution of the u (1) problem.2 The anomaly has also been instrumental in posing constraints to insure the mathematical consistency of gauge theories coupled to chiral currents. If one considers a theory with gauge fields coupled for instance to left handed currents, one must look at a fermion triangle diagram with V-A currents at each vertex. Again, this diagram is anomalous, and unless the anomalies cancel when summing over all the fermion species running around the loop, one finds that the V-A currents are not conserved, implying that gauge invariance is broken and thus the anomaly renders the theory inconsistent. The anomaly cancellation condition has proven to be very useful in constraining the particle content of unified gauge theories.3 More recently4, the anomaly has also been shown to be useful in analyzing the spectrum of massless fermions in confining theories. In the context of low energy chiral theories, the Wess-Zumino lagrangian5 has recently played a central role in showing that the soliton solutions of certain models6 can be identified with baryons.7 This recent development has in turn shed new light into our understanding of chiral anomalies.


Dirac Operator Gauge Field Energy Momentum Tensor Anomaly Cancellation Gravitational Anomaly 
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  1. 1.
    S. Adler, Phys. Rev. 177 (1969) 2426, and in “Lectures in Elementary Particles and Quantum Field Theory”, ed. S. Deser et al. (M.I.T. Press, 1970)ADSCrossRefGoogle Scholar
  2. J. Bell and R. Jackiw, Nuovo Cimento 60A (1969) 47; R. Jackiw, in “Lectures on Current Algebra and its Applications” (Princeton University Press, 1972)ADSGoogle Scholar
  3. S. L. Adler and W. Bardeen, Phys. Rev. 182 (1969) 1517ADSCrossRefGoogle Scholar
  4. W. A. Bardeen, Phys. Rev. 184 (1969) 1848ADSCrossRefGoogle Scholar
  5. R. W. Brown, C. C. Shi, and B. L. Young, Phys. Rev. 186 (1969) 1491ADSCrossRefGoogle Scholar
  6. J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95MathSciNetADSGoogle Scholar
  7. A. Zee, Phys. Rev. Lett. 29 (1972) 1198.ADSCrossRefGoogle Scholar
  8. 2.
    J. Steinberg, Phys. Rev. 76 (1949) 1180ADSCrossRefGoogle Scholar
  9. J. Schwinger, Phys. Rev. 82 (1951) 664MathSciNetADSMATHCrossRefGoogle Scholar
  10. L. Rosenberg, Phys. Rev. 129 (1963) 7786Google Scholar
  11. R. Jackiw and K. Johnson, Phys. Rev. 182 (1969) 1459ADSCrossRefGoogle Scholar
  12. S. Adler and D. G. Boulware, Phys. Rev. 184 (1969), 1740ADSCrossRefGoogle Scholar
  13. S. L. Adler, B. W. Lee, S. B. Treiman, and A. Zee, Phys. Rev. D4 (1971) 3497ADSGoogle Scholar
  14. R. Aviv and A. Zee, Phys. Rev. D5 (1972) 2372ADSGoogle Scholar
  15. M. V. Terentiev, J.E.T.P. Letters 14 (1971) 140Google Scholar
  16. A. M. Belevin, A. M. Polyakov, A. S. Schwarz and Yn. S. Tyupkin, Phys. Lett. 59B (1975) 85ADSGoogle Scholar
  17. G. ’t Hooft, Phys. Rev. Lett. 37. (1976) 8, Phys. Rev. D14 (1976) 3432ADSCrossRefGoogle Scholar
  18. C. Callan, R. Dashen, and D. J. Gross, Phys. Lett. 63B (1976) 334ADSGoogle Scholar
  19. R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172.ADSCrossRefGoogle Scholar
  20. 3.
    P. J. Gross and R. Jackiw, Phys. Rev. D6 (1972) 477ADSGoogle Scholar
  21. C. Bouchiat, J. Iliopoulos and Ph. Meyer, Phys. Lett. 38B (1972) 519ADSGoogle Scholar
  22. H. Georgi and S. L. Glashow, Phys. Rev. D6 (1972) 429.ADSGoogle Scholar
  23. 4.
    G. ’t Hooft, in “Recent Developments in Gauge Theories”, G. ’t Hooft et al. eds. (Plenum Press,##New York, 1980)Google Scholar
  24. A. A. Ansel’m, J.E.T.P. Lett. 32: (1980) 138ADSGoogle Scholar
  25. A. Zee, Phys. Lett. 95B (1980) 290ADSGoogle Scholar
  26. Y. Frishman, A. Schwimmer, T. Banks, and S. S. Yankielowicz, Nucl. Phys. B177 (1981) 157ADSCrossRefGoogle Scholar
  27. S. Colemand and B. Grossman, Nucl. Phys. B203 (1982) 205ADSCrossRefGoogle Scholar
  28. G. R. Farrar, Phys. Lett. 96B (1980) 273ADSGoogle Scholar
  29. S. Weinberg, Phys. Lett. 102B (1981) 401ADSGoogle Scholar
  30. C. H. Albright, Phys. Rev. D24 (1981) 1969ADSGoogle Scholar
  31. I. Bars, Phys. Lett. 109B (1982) 73ADSGoogle Scholar
  32. T. Banks, S. Yankielowicz, and A. Schwimmer, Phys. Lett. 96B (1980) 67MathSciNetADSGoogle Scholar
  33. A. Schwimmer, Nucl. Phys. B198 (1982) 269.MathSciNetADSCrossRefGoogle Scholar
  34. 5.
    J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95.MathSciNetADSGoogle Scholar
  35. 6.
    J. H. R. Skyrme, Proc. Roy. Soc. (London) A260 (1961) 127.MathSciNetADSGoogle Scholar
  36. 7.
    A. D. Balachandran, Y. P. Nair, S. G. Raicev, and A. Stern, Phys. Rev. Lett. 49 (1982) 1182 and Syracuse University Preprint (1982). E. Witten, “Global Aspects of Current Algebra” and Current Algebra, Baryons and Quark Confinement”, Princeton Preprints.CrossRefGoogle Scholar
  37. 8.
    P. H. Frampton and T. W. Kephart, Phys. Rev. Lett. 50 (1983) 1343, 1347; P. K. Townsend and G. Sierra (L.P.T.E.N.S. Preprint, 1983), Y. Matsuki and A. Hill (OSV Preprints, 1983).ADSCrossRefGoogle Scholar
  38. 9.
    L. Alvarez-Gaume and E. Witten, Harvard Prepirnt HUTP-83/A039.Google Scholar
  39. 10.
    B. Zumino, W. Y. Shi and A. Zee (Univ. of Washington Preprint, 1983); B. Zumino, Lectures at the Les Houches Summer School, August 1983; R. Stora and B. Zumino, in preparation, and R. Stora, Lectures at Les Houches Summer School.Google Scholar
  40. 11.
    M. F. Atiyah and I. M. Singer, paper in preparation. Quillen in preparation.Google Scholar
  41. 12.
    O. Alvarez, I. M. Singer, and B. Zumino, in preparation, L. Alvarez-Gaumé and P. Ginsparg, in preparation.Google Scholar
  42. 13.
    R. Jackiw and C. Rebbi, Phys. Rev. D14 (1976) 517MathSciNetADSGoogle Scholar
  43. N. K. Nielsen, H. Römer, and B. Schroer, Phys. Lett. 70B (1977) 445.ADSGoogle Scholar
  44. 14.
    R. Delbourgo and A. Salam, Phys. Lett. 40B (1972) 381ADSGoogle Scholar
  45. T. Eguchi and P. Freund, Phys. Rev. Lett. 37 (1976) 1251MathSciNetADSCrossRefGoogle Scholar
  46. S. W. Hawking and C. N. Pope, Nucl. Phys. B146 (1978) 381MathSciNetADSCrossRefGoogle Scholar
  47. M. J. Perry, Nucl. Phys. B143 (1978) 114MathSciNetADSCrossRefGoogle Scholar
  48. N. K. Nielsen, H. Römer, and B. Shroer, Nucl. Phys. B136 (1978) 475ADSCrossRefGoogle Scholar
  49. N. K. Nielsen, M. T. Grisaru, H. Römer, and P. V. Nieuwenhuisen, Nucl. Phys. B140 (1978) 477ADSCrossRefGoogle Scholar
  50. A. J. Hanson and H. Römer, Phys. Lett. 80B (1978) 58ADSGoogle Scholar
  51. R. Critchley, Phys. Lett. 78B (1978) 410ADSGoogle Scholar
  52. S. M. Christensen and M. J. Duff, Phys. Lett. 76B (1978) 571, Nucl. Phys. B154 (1979) 301ADSGoogle Scholar
  53. T. Eguchi, P. B. Gilkey, and A. J. Hanson, Phys. Rep. 66 (1980) 213.MathSciNetADSCrossRefGoogle Scholar
  54. 15.
    K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195, 44 (1980) 1733, Phys. Rev. D21 (1980) 2848; D22 (1980) 1499 (E); D23 (1981) 2262; M. B. Einhorn and D. R. T. Jones, U.M.-Th. 83-3 PreprintADSCrossRefGoogle Scholar
  55. A. Balachandran et al., Phys. Rev. D25 (1982) 2713.ADSGoogle Scholar
  56. 16.
    E. Witten, Nucl. Phys. B202 (1982) 253.MathSciNetADSCrossRefGoogle Scholar
  57. 17.
    T. Parker, unpublished and private communication.Google Scholar
  58. 18.
    D. Friedan and P. Windey, paper in preparation.Google Scholar
  59. 19.
    L. Alvarez-Gaume, Comm. Math. Phys. 90 (1983) 161.MathSciNetADSMATHCrossRefGoogle Scholar
  60. 20.
    L. Alvarez-Gaume, Harvard Preprint HUTP-83/A035.Google Scholar
  61. 21.
    M. F. Atiyah and R. Bott, Ann. Math. 86 (1967) 374; 88 (1968) 451.MathSciNetMATHCrossRefGoogle Scholar
  62. 22.
    E. Getzler, Harvard Preprint.Google Scholar
  63. 23.
    T. Eguchi, P. B. Gilkey, and A. J. Hanson, Phys. Rep. 66 (1980) 213.MathSciNetADSCrossRefGoogle Scholar
  64. 24.
    E. Witten, Phys. Lett. 117B (1982) 324.MathSciNetADSGoogle Scholar
  65. 25.
    See, for example, H. Georgi, “Lie Algebras and Particle Physics”, (Benjamin Publ, 1981).Google Scholar
  66. 26.
    R. Stora and T. Schücker have outlined a general method to derive the Wess-Zumino consistency conditions in this case (Private communication).Google Scholar
  67. 27.
    J. S. Schwinger, Phys. Rev. 82 (1951) 664.MathSciNetADSMATHCrossRefGoogle Scholar
  68. 28.
    P. K. Townsend, University of Texas (Austin) Preprint, 1983.Google Scholar
  69. 29.
    E. Cremmer, B. Julia, and J. Scherk, Phys. Lett. 76B (1978) 409ADSGoogle Scholar
  70. E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141.MathSciNetADSCrossRefGoogle Scholar
  71. 30.
    W. Nahm, Nucl. Phys. B135 (1978) 149ADSCrossRefGoogle Scholar
  72. M. B. Green and J. H. Schwartz, Phys. Lett. 109B (1982) 444, 122B (1983) 143ADSGoogle Scholar
  73. J. H. Schwartz and P. West, Phys. Lett. 126B (1983) 301; P. S. Howe and P. West, King’s College preprint (1983).ADSGoogle Scholar
  74. 31.
    M. B. Green and J. H. Schwartz, Nucl. Phys. B181 502, B198 (1982) 252, 441; Phys. Lett. 109B (1982) 444ADSGoogle Scholar
  75. J. H. Schwartz, Phys. Reports 89 (1982) 223.ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1984

Authors and Affiliations

  • Luis Alvarez-Gaumé
    • 1
  1. 1.Lyman Laboratory of PhysicsHarvard UniversityCambridgeUSA

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