Algebraic Structure of Chiral Anomalies
Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 36 (1986), Exposé no. 3, 34 p.
@article{RCP25_1986__36__27_0,
     author = {Stora, R.},
     title = {Algebraic {Structure} of {Chiral} {Anomalies}},
     journal = {Les rencontres physiciens-math\'ematiciens de Strasbourg -RCP25},
     note = {talk:3},
     publisher = {Institut de Recherche Math\'ematique Avanc\'ee - Universit\'e Louis Pasteur},
     volume = {36},
     year = {1986},
     language = {en},
     url = {http://www.numdam.org/item/RCP25_1986__36__27_0/}
}
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PB  - Institut de Recherche Mathématique Avancée - Université Louis Pasteur
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Stora, R. Algebraic Structure of Chiral Anomalies. Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 36 (1986), Exposé no. 3, 34 p. http://www.numdam.org/item/RCP25_1986__36__27_0/

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 742394 | Zbl 0547.58033

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 0573.53054

b) L. Alvarez Gaumé, P. Ginsparg, Ann. Phys. 161, 423, 1985. | MR 793821 | Zbl 0579.58038

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 606967

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 508193

11) M. Dubois Violette, M. Talon, C.M. Viallet, "BRS cohomologies", to appear in C.M.P. | Zbl 0604.58055

12) E. Witten, "Global aspects of current algebra", N.P.B. 223, 422, 1983. | MR 717915

13) G. Falqui, C. Reina, "BRS cohomology and topological anomalies", Annales IHP to be published. | Zbl 0613.53055

14) J. Thierry-Mieg, P.L. 147B, 430, 1984. | MR 767765

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 757328

F. Langouche, T. Schücker, R. Stora, P.L. 145B, 342, 1984; | MR 760602

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 794837

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 762413

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 785131

E. Cohen, C. Gomez, N.P.B. 254, 235, 1985; | MR 793133

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 387057

S.D. Joglekar, B.W. Lee, Ann. Phys. 97, 160, 1976 ; | MR 428960

J. Dixon, 1976-1978, unpublished.

21) L.D. Faddeev, S.L. Shatashvili, TMФ 60, 206, 1984. | MR 762262

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 771086

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 722541

26) E.S. Fradkin, G.A. Vilkovisky, P.L. 55B, 224, 1975 ; | MR 411451

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

27) W.A. Bardeen, B. Zumino, N.P.B. 244, 421, 1984. | MR 762380

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 797737

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 794836

37) a) A. Guichardet, "Cohomologie des groupes topologiques et des algèbras de Lie", CEDIC-NATHAN Paris 1980. | MR 644979 | Zbl 0464.22001

b) S. Mac Lane, "Homology", Springer 1975. | MR 1344215 | Zbl 0328.18009

c) H. Cartan, S. Eilenberg, "Homclogical algebra", Princeton University Press, 1956. | MR 77480 | Zbl 0933.18001

d) N. Jacobson, "Lie Algebras", Interscience, 1962. | MR 143793 | Zbl 0121.27504

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 757331

B. Zumino, N.P.B. 253, 477, 1985; and references found in 8a). | MR 791687

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 δΓ δaδΓ δA+δΓ δϕδΓ δΦ+δΓ δωδΓ δΩ-δΓ δω ¯b- αΓ=0 where α is a constant odd scalar with dimension -1 , ϕπ charge -1 . (C. Becchi, private communication).

42) C. Becchi, unpublished.