Algebraic Structure of Chiral Anomalies

Stora, R.

Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Conférences de : C. Bardos, L. Baulieu, H.J. Borchers, M. Dubois-Violette, P. Pansu et R. Stora, Tome 36 (1986), Exposé no. 3, 34 p.

1) A thorough review of and bibliography on the early period can for instance be found in : R. Jackiw: "Topological investigations of quantized gauge theories" in: Relativity, Groups, Topology II, Les Houches lectures 1983 B.S. de Witt, R. Stora eds., North Holland Publ. Amsterdam 1984.

2) L.D. Faddeev, private communication.

3) R. Jackiw, R. Rajaraman, P.L. 154B, 305, 1985.

4) A good example of the complexity involved is given by : L. Alvarez Gaumé, E. Witten, M.P.B. 234, 269-330, 1984.

5) See, for instance O. Alvarez, I.M. Singer, B. Zumino, C.M.P. 94, 409, 1984.

M.F. Atiyah, I.M. Singer, P.N.A.S. 81, 2597, 1984. | MR | Zbl

6) Typically, the family's index treatment is related to the topology of the field configuration space whereas the algebraic treatment based on locality is not.

7) P. Ginsparg, Lectures in this volume.

8) a) J. Mañes, R. Stora, B. Zumino, "Algebraic study of chiral anomalies", submitted to C.M.P. ; | Zbl

b) L. Alvarez Gaumé, P. Ginsparg, Ann. Phys. 161, 423, 1985. | MR | Zbl

9) C. Becchi, A. Rouet, R. Stora : "Ranormalizable theories with symmetry breaking" in : Field theory quantization and statistical mechanics, E. Tirapegui ed., Reidel 1981. | MR

10) R. Stora : "Continuum gauge theories" in : New developments in quantum field theory and statistical mechanics, 1976 Cargèse Lectures, M. Levy, P. Mitter eds, Plenum New York 1977. | MR

11) M. Dubois Violette, M. Talon, C.M. Viallet, "BRS cohomologies", to appear in C.M.P. | Zbl

12) E. Witten, "Global aspects of current algebra", N.P.B. 223, 422, 1983. | MR

13) G. Falqui, C. Reina, "BRS cohomology and topological anomalies", Annales IHP to be published. | Zbl

14) J. Thierry-Mieg, P.L. 147B, 430, 1984. | MR

15) I am indebted to P. Van Nieuwenhuizen for pointing out and discussing the article by D. Sullivan, IHES Pub. n°47, p.269, 1977, whose reading led in particular to the algebraic interpretation of the Faddeev Popov ghost which can be found in the appendix.

16) L. Baulieu, J. Thierry-Mieg, P.L. 145B, 53, 1984 ; | MR

F. Langouche, T. Schücker, R. Stora, P.L. 145B, 342, 1984; | MR

L. Bonora, P. Pasti, M. Tonin, N.P.B. 1985 to appear.

17) L. Bonora, P. Pasti, M. Tonin, P.L. 155B, 341, 1985 and to appear in NPB; | MR

G. Girardi, R. Grimm, R. Stora, P.L. 156B, 203, 1935);

J. Wess, B. Zumino, in preparation,

18) H. Itoyama, V.P. Nair, H.C. Ren, IAS Princeton preprint 1985.

19) G. Moore, P. Nelson, P.R.L. 53, 1519, 1934; | MR

G. Moore, P. Nelson, HUTP-84/A076 ;

A. Manohar, G. Moore, P. Nelson, HUTP-84/AC83 ;

L. Alvarez Gaumé, P. Ginsparg, HUTP 85/AO15 ;

P. Di Vecchia, S. Ferrara, L. Girardello. P.L. 1513B, 199, 1985; | MR

E. Cohen, C. Gomez, N.P.B. 254, 235, 1985; | MR

J. Bagger, D. Nemechansky, S. Yankielowicz in Anomalies, Geometry, Topology, Argonne-Chicago 1985.

20) H. Kluberg Stern, J.B. Zuber, P.R.D. 12, 467, 1975 ; P.R.D. 12, 3159, 1975 ; | MR

S.D. Joglekar, B.W. Lee, Ann. Phys. 97, 160, 1976 ; | MR

J. Dixon, 1976-1978, unpublished.

21) L.D. Faddeev, S.L. Shatashvili, TMФ 60, 206, 1984. | MR

22) K.S. Stelle, P.C. West, N.P.B. 140, 285 , 1978 ;

L. Baulieu, M. Bellon, to be published.

23) M.B. Green, J.H. Schwarz, P.L. 149B, 117, 1984. | MR

24) S. Cecotti, S. Ferrara, L. Girardello, CERN.TH.4253/85.

25) J. Thierry-Mieg, unpublished, 1979;

L. Baulieu, J. Thierry-Mieg, N.P.B. 228, 259, 1983 ; Although both BRS and BRS are used in these papers, BRS alone is sufficient if one does not want to restrict oneself to linear gauge functions. | MR

26) E.S. Fradkin, G.A. Vilkovisky, P.L. 55B, 224, 1975 ; | MR

I.A. Batalin, G.A. Vilkovisky, P.L. 69B, 309, 1977.

27) W.A. Bardeen, B. Zumino, N.P.B. 244, 421, 1984. | MR

28) L. Bonora, P. Pasti, M. Tonin, P.L.B. to appear.

29) S. Ferrara, L. Girardello, O. Piguet, R. Stora, P.L. 157B, 179, 1985. | MR

30) O. Piguet, K. Sibold, N.P.B. 247, 484, 1984.

31) O. Piguet, M. Schweda, K. Sibold, N.P.B. 174, 183, 1980.

32) B. Zumino in Symposium on Anomalies, Geometry and Topology, Argonne-Chicago, 1985.

33) a) G. Bandelloni, C. Becchi, A. Blasi, R. Collina, Ann. IHP XXVIII, 225-254 and 255-285, 1978.

b) O. Piguet, A. Rouet, Physics Reports 76, 1-77, 1981.

34) cf. Eq. 6.16 of Ref. 33b).

35) I am indebted to R. Grimm for stimulating discussions on this subject.

36) Y. Tanii, TIT/HEP 85 ;

P. Howe, P. West, P.L. 156B, 335, 1985. | MR

37) a) A. Guichardet, "Cohomologie des groupes topologiques et des algèbras de Lie", CEDIC-NATHAN Paris 1980. | MR | Zbl

b) S. Mac Lane, "Homology", Springer 1975. | MR | Zbl

c) H. Cartan, S. Eilenberg, "Homclogical algebra", Princeton University Press, 1956. | MR | Zbl

d) N. Jacobson, "Lie Algebras", Interscience, 1962. | MR | Zbl

38) A.G. Reiman, M.A. Semenov-Tjan-Shansky, L.D. Faddeev, in Journal of functional analysis and its applications, 1984-1985 ;

L.D. Faddeev, P.L. 145B, 81, 1984 ; | MR

B. Zumino, N.P.B. 253, 477, 1985; and references found in 8a). | MR

39) I.M. Singer, in Colloque Elie Cartan, Lyon, June 25-29, 1984, to appear in Astérisque, and references therein.

40) G. Girardi and R. Grimm have just informed me they have constructed a consistent superspace geometry [namely, a correct system of constraints] which allows to incorporate (in a manifestly supersymmetric way) Chern Simons terms into the curvature of the super two form appearing in the Sohnius West multiplet.

41) When an anomaly is present, one can prove a modified Ward identity of the form $\int \frac{\delta \Gamma }{\delta a}\frac{\delta \Gamma }{\delta A}+\frac{\delta \Gamma }{\delta \varphi }\frac{\delta \Gamma }{\delta \Phi }+\frac{\delta \Gamma }{\delta \omega }\frac{\delta \Gamma }{\delta \Omega }-\frac{\delta \Gamma }{\delta \overline{\omega }}b-\frac{\partial }{\partial \alpha }\Gamma =0$ where $\alpha$ is a constant odd scalar with dimension $-1$ , $\varphi \pi$ charge $-1$ . (C. Becchi, private communication).

42) C. Becchi, unpublished.