Abstract
The effective action for fermions moving in external gravitational and gauge fields is analyzed in terms of the corresponding external field propagator. The central object in our approach is the covariant energy-momentum tensor which is extracted from the regular part of the propagator at short distances. It is shown that the Lorentz anomaly, the conformal anomaly and the gauge anomaly can be expressed in terms of the local polynomials which determine the singular part of the propagator. (There are no coordinate anomalies.) Except for the conformal anomaly, for which we give explicit representations only ind<=4, we consider an arbitrary number of dimensions.
Similar content being viewed by others
References
For a review of effective actions on curved space time, see B. De Witt: Phys. Rep.19C, 295 (1975); N. Birrell, P. Davies: Quantum fields in curved space. Cambridge: Cambridge University Press 1982
A. Salam, P.T. Matthews: Phys. Rev.90, 690 (1953); J. Schwinger: Phys. Rev.93, 615 (1954)
S. Coleman, R. Jackiw: Ann. Phys.67, 552 (1971); R. Crewther: Phys. Rev. Lett.28, 1421 (1972)
S. Adler: Phys. Rev.117, 2426 (1969); J. Bell, R. Jackiw: Nuovo Cimento60A, 47 (1969)
L. Alvarez-Gaumé, E. Witten: Nucl. Phys.B234, 269 (1984)
W.A. Bardeen, B. Zumino: Nucl. Phys.B244, 421 (1984)
H. Leutwyler: Phys. Lett.153B, 65 (1985);155B, 469(E) (1985)
H. Leutwyler: Helv. Phys. Acta59, 201 (1986)
K. Fujikawa: Phys. Rev. Lett.42, 1195 (1979);44, 1733 (1980); Phys. Rev.D21, 2848 (1980);D22, 1499(E) (1980);D23, 2262 (1981);D29, 285 (1984)
H. Fritzsch, M. Gell-Mann, H. Leutwyler: Caltech report 68-456 (1974), uppublished
J. Schwinger: Phys. Rev.82, 664 (1951)
L. Alvarez-Gaumé, P. Gimsparg: Nucl. Phys.B243, 449 (1984); Ann. Phys.161, 423 (1985)
J. Gasser, H. Leutwyler: Ann. of Phys.158, 142, Appendix A (1984)
H. Banerjee, R. Banerjee: preprints Saha Institute of Nuclear Physics, SINP-TNP-84/6 and 85/2, Calcutta
H. Leutwyler: Phys. Lett.152B, 78 (1985) and in: Quantum field theory and quantum statistics, essays in honour of E.S. Fradkin. Bristol: Adam Hilger 1986
A calculation of the gravitational anomalies in two dimensions, based on the short distance properties of the propagator was given by F. Langouche: Phys. Lett.148B, 93 (1984). In fact, in isothermal coordinates, the 2-dimensional Weyl determinant can be calculated explicitly: A.M. Polyakov: Phys. Lett.103B, 207, 211 (1981); A.S. Fradkin, A.A. Tseytlin: Phys. Lett.106B, 63 (1981). It was shown in [7] that the determinant can be renormalized in a coordinate invariant manner, such that it is legitimate to perform the calculation in a special coordinate system. The properties of the corresponding energy-momentum tensor are discussed in detail in [7]
B. Zumino: Les Houches Lectures 1983, eds. E. Stora, B. De Witt. North-Holland to be published; R. Stora: Cargèse Lectures 1983, Progress in Gauge Field Theory, ed. H. Lehmann, to be published; B. Zumino, Yong-shi Wu, A. Zee: Nucl. Phys.B239, 477 (1984)
P.B. Gilkey: The index theorem and the heat equation. Boston: Publish or Perish 1974; L. Parker, in: Recent developments in gravitation. Cargèse Lectures 1978, M. Levy, S. Deser, eds. New York: Plenum 1979
M. Lüscher: Ann. Phys.142, 359, Appendix A (1982)
R. Delbourgo, A. Salm: Phys. Lett.40B, 381 (1972); T. Eguchi, P.G.O. Freund: Phys. Rev. Lett.37, 1251 (1976)
O. Alvarez, I.M. Singer, B. Zumino: Commun. Math. Phys.96, 409 (1984)
L. Alvarez-Gaumé, S. Della Pietra, G. Moore: Ann. Phys.163, 288 (1985)
K. Fujikawa, M. Tomiya, O. Yasuda: Z. Phys. C—Particles and Fields28, 289 (1985)
L.N. Chang, H.T. Nieh: Phys. Rev. Lett.53, 21 (1984)
Author information
Authors and Affiliations
Additional information
Work in part supported by Schweizerischer Nationalfonds
Rights and permissions
About this article
Cite this article
Leutwyler, H., Mallik, S. Gravitational anomalies. Z. Phys. C - Particles and Fields 33, 205–226 (1986). https://doi.org/10.1007/BF01411138
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01411138