×

The evaluation map in field theory, sigma-models and strings. I. (English) Zbl 0645.53071

This, the first in a series of two papers, presents a systematic analysis of anomalies in field theories (the second part [the authors, ibid. 114, 381-437 (1988)] is devoted to sigma-models) from a unifying point of view. The latter is based on the evaluation map defined by the group of all automorphisms of a principal fibre bundle P acting on the bundle itself. All anomalies (gauge, gravitational, Lorentz) are generated by pulling back suitable forms on P via this evaluation map, allowing a straightforward computation both with and without a background connection. The authors discuss the “gauge” interpretation of gravitational anomalies (relevant for the equivalence between gravitational and Lorentz anomalies) and the topological significance of gauge anomalies in the general case (i.e. for any compact base manifold and any compact structure group).
Furthermore it is shown that, in general, “local” anomalies are generated by forms on P which are the pullback of forms defined on the universal bundle associated with P. This identification of locality and universality underlies the authors’ discussion of the compatibility between locality and the family’s index theorem approach to anomalies. The final subject of the paper is the anomaly cancellation mechanism in 10-dimensional field theories inspired by the Green-Schwarz superstring [M. Green and J. Schwarz, Phys. Lett., B 149, 117-122 (1984)]. The necessary geometrical constraints are discussed. A discussion of the same subject in the framework of sigma-models is contained in Part II of the series.
Reviewer: H.Rumpf

MSC:

53C80 Applications of global differential geometry to the sciences
81T60 Supersymmetric field theories in quantum mechanics
55R99 Fiber spaces and bundles in algebraic topology
81T99 Quantum field theory; related classical field theories
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bonora, L., Cotta-Ramusino, P., Rinaldi, M., Stasheff, J.: The evaluation map in field theory, sigma models and strings-II, preprint CERN-TH. 4750/87 · Zbl 0655.53069
[2] Atiyah, M. F., Singer, I. M.: Dirac operators coupled to vector potentials. Proc-Natl. Acad. Sci.81, 2597 (1984) · Zbl 0547.58033 · doi:10.1073/pnas.81.8.2597
[3] Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funkts. Anal. Prilozh.19, 37 (1985) · Zbl 0603.32016
[4] Mañes, J., Stora, R., Zumino, B.: Algebraic study of chiral anomalies. Commun. Math. Phys.102, 157 (1985) · Zbl 0573.53054 · doi:10.1007/BF01208825
[5] Bonora, L., Cotta-Ramusino, P.: Some remarks on BRS transformations, anomalies and the cohomology of the Lie algebra of the group of gauge transformations. Commun. Math. Phys.87, 589 (1983) · Zbl 0521.53064 · doi:10.1007/BF01208267
[6] Milnor, J.: Infinite dimensional Lie groups. In: Relativity, groups and topology II. Les Houches 82; Witt B.De, Stora R., (eds.). Amsterdam: North Holland 1984 · Zbl 0594.22009
[7] Atiyah, M. F., Bott, R.: Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. A308, 523 (1982) · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017
[8] Trautman, A.: On groups of gauge transformations. In: Geometrical and topological methods in gauge theories. Lecture Notes in Physics, Vol.129. Harnad, J., Shnider, S. (eds.). Berlin, Heidelberg, New York: Springer 1980 · Zbl 0447.53061
[9] Kobayashi, S., Nomizu, K.: Foundation of differential geometry, Vols. I, II. New York, London: Inter science 1963, 1969 · Zbl 0119.37502
[10] Trautman, A.: On gauge transformations and symmetries. Bull. Acad. Pol. Sci. Phys. Astron.,XXVII-No 1 (1979) · Zbl 0429.53046
[11] Trautman, A.: Geometrical aspects of gauge configurations. Acta Phys. Austriaca [Supp]XXIII, 401 (1981)
[12] Lecomte, P.: Homomorphisme de Chern-Weil et automorphismes infinitésimaux des fibrés vectoriels. C. R. Acad. Sci. Paris,T294, Sér A, 369 (1982) · Zbl 0492.55012
[13] Chern, S. S.: Complex manifolds without potential theory. Berlin, Heidelberg, New York: Springer 2nd ed 1979 · Zbl 0444.32004
[14] Bourguignon, J. P.: Une stratification de l’espace des structures Riemanniennes. Compos. Mat.30, 1 (1975) · Zbl 0301.58015
[15] Palais, R.: Foundations of global non linear analysis. New York: Benjamin 1968 · Zbl 0164.11102
[16] Michor, P. W.: Manifolds of differentiable mappings. Orpington (U.K.): Shiva 1980 · Zbl 0433.58001
[17] Mitter, P., Viallet, C.: On the bundle of connections and the gauge orbits manifolds in Yang-Mills theory. Commun. Math. Phys.79, 457 (1981) · Zbl 0474.58004 · doi:10.1007/BF01209307
[18] Omori, H.: Infinite Lie transformation groups. Lecture Notes in Mathematics, Vol.427, Berlin, Heidelberg, New York: Springer 1975 · Zbl 0328.58005
[19] Abbati, M. C., Cirelli, R., Maniá, A., Michor, P. W.: Smoothness of the action of gauge transformation group on connections. J. Math. Phys.27, 2469 (1986) · Zbl 0618.58040 · doi:10.1063/1.527404
[20] Stora, R.: Algebraic structure and topological origin of anomalies. In: Progress in gauge fields theory. Hooft, G.’t, Jaffe, A., Lehmann, H., Mitter, P. K., Singer, I. M., Stora, R. (eds.). New York: Plenum Press 1984
[21] Bardeen, W., Zumino, B.: Consistent and covariant anomalies in gauge and gravitational theories. Nucl. Phys.B244, 421 (1984) · doi:10.1016/0550-3213(84)90322-5
[22] Bonora, L., Cotta-Ramusino, P.: ABJ anomalies and superfield formalism in gauge theories. Phys. Lett.B107, 87 (1981)
[23] Stasheff, J.: The De Rham bar construction as a setting for the Zumino, Fadde’ev, etc. descent equation. In: Proc. of the symposium on anomalies, geometry and topology. Bardeen, W., White, A. (eds.). Singapore: World Scientific 1985 · Zbl 0647.55011
[24] Bonora, L., Cotta-Ramusino, P.: Consistent and covariant anomalies and local cohomology. Phys. Rev.D33, 3055 (1986) · Zbl 0521.53064
[25] Greub, W., Halperin, S., Vanstone, R.: Connections, curvature and cohomology, Vol. II. New York London: Academic Press 1973 · Zbl 0335.57001
[26] Dieudonné, J.: Éléments d’analyse, Vol.1-9, Paris: Gauthier-Villars
[27] Langouche, F., Schücker, T., Stora, R.: Gravitational anomalies of the Adler-Bardeen type, Phys. Lett.145B, 342 (1984)
[28] Bott, R., Tu, L.: Differential forms in algebraic topology. Berlin, Heidelberg, New York: Springer 1982 · Zbl 0496.55001
[29] Sullivan, D.: Infinitesimal calculations in topology. Publ. Math. I.H.E.S.47, 269 (1977) · Zbl 0374.57002
[30] Haefliger, A.: Rational homotopy of the space of sections of a nilpotent bundle. Trans. Am. Math. Soc.273, 609 (1982) · Zbl 0508.55019 · doi:10.1090/S0002-9947-1982-0667163-8
[31] Narasimhan, M., Ramanan, S.: Existence of universal connections Am. J. Math.83, 563 (1961),85, 223 (1963) · Zbl 0114.38203 · doi:10.2307/2372896
[32] Dubois-Violette, M., Talon, M., Viallet, C. M.: B.R.S. algebras. Analysis of the consistency equations in gauge theories. Commun. Math. Phys.102, 105 (1985) · Zbl 0604.58055 · doi:10.1007/BF01208822
[33] Peetre, J.: Une caractérisation abstraite des opérateurs differentiels. Math. Scand.7, 211 (1959),8, 116 (1960)
[34] De Wilde, M., Lecomte, P.: Cohomology of the Lie-algebra of smooth vector fields of a manifold associated to the Lie derivative of smooth forms. J. Math. Pures Appl.62, 197 (1983) · Zbl 0481.58032
[35] Flato, M., Lichnerowitz, A.: Cohomologie des représentations définies par la dérivation de Lie á valeurs dans les forms, de l’algèbre de Lie des champs de vecteurs d’une variété différentiable. C. R. Acad. Sci.,T291, sér, A, 331 (1980)
[36] Husemoller, D.: Fibre bundles, Berlin, Heidelberg, New York: Springer Verlag, 2nd ed (1979) · Zbl 0202.22903
[37] Alvarez, O., Singer, I. M., Zumino, B.: Gravitational anomalies and the family’s index theorem. Commun. Math. Phys.96, 409 (1984) · Zbl 0587.58042 · doi:10.1007/BF01214584
[38] Singer, I. M.: Families of Dirac operators with applications to physics. Séminaire Elie Cartan, Astérisque; to be published
[39] Reiman, A. G., Semyonov-Tjan-Shansky, M. A., Faddeev, L. D.: Quantum anomalies and cycles of the gauge group. J. Funct. Anal. Appl. (in Russian) (1984)
[40] Shlafly, R.: Universal connections. Invent. Math.59, 59 (1980) · Zbl 0431.53028 · doi:10.1007/BF01390314
[41] Palais, R.: Seminar on the Atiyah index theorem. Princeton N.J., Princeton University Press 1965 · Zbl 0137.17002
[42] Atiyah, M. F., Singer, I. M.: The index of elliptic operators I, III, IV. Ann. Math.87, 484 (1968);87, 546;92, 119 (1970) · Zbl 0164.24001 · doi:10.2307/1970715
[43] Dabrowski, L., Percacci, R.: Spinors and diffeomorphisms. Commun. Math. Phys.106, 691 (1986) · Zbl 0605.53042 · doi:10.1007/BF01463403
[44] Alvarez-Gaumé, L., Witten, E.: Gravitational anomalies. Nucl. Phys.B234, 269 (1984) · doi:10.1016/0550-3213(84)90066-X
[45] Green, M., Schwarz, J.: Anomaly cancellation is supersymmetric D = 10 gauge theory and superstrings. Phys. Lett.149B, 117 (1984)
[46] Candelas, P., Horowitz, G., Strominger, A., Witten, E.: Vacuum configurations for superstrings. Nucl. Phys.B258, 46 (1985) · doi:10.1016/0550-3213(85)90602-9
[47] Chapline, G. F., Manton, N. S.: Unification of Yang-Mills theory and supergravity in ten dimensions. Phys. Lett.120B, 105 (1983)
[48] Bonora, L., Cotta-Ramusino, P.: Some remarks on anomaly cancellation in field theories derived from superstrings. Phys. Lett.169B, 187 (1986)
[49] Asorey, M., Mitter, P. K.: Cohomology of the Yang-Mills gauge orbit space and dimensional reduction. Ann. Inst. H. Poincaré45, 61 (1986) · Zbl 0596.55003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.