Algebraic Structure of Chiral Anomalies

 Access to full text

 Full (PDF)

1. 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, TopologyII, Les Houches lectures 1983 B.S. de Witt, R. Stora eds., North Holland Publ.Amsterdam1984. 
2. 2) L.D. Faddeev, private communication. 
3. 3) R. Jackiw, R. Rajaraman, P.L. 154B, 305, 1985. 
4. 4) A good example of the complexity involved is given by : L. Alvarez Gaumé, E. Witten, M.P.B.234, 269-330, 1984. 
5. 5) See, for instance O. Alvarez, I.M. Singer, B. Zumino, C.M.P.94, 409, 1984. 
6. M.F. Atiyah, I.M. Singer, P.N.A.S.81, 2597, 1984. Zbl0547.58033MR742394
7. 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. 
8. 7) P. Ginsparg, Lectures in this volume. 
9. 8) a) J. Mañes, R. Stora, B. Zumino, "Algebraic study of chiral anomalies", submitted to C.M.P. ; Zbl0573.53054
10. b) L. Alvarez Gaumé, P. Ginsparg, Ann. Phys.161, 423, 1985. Zbl0579.58038MR793821
11. 9) C. Becchi, A. Rouet, R. Stora : "Ranormalizable theories with symmetry breaking" in : Field theory quantization and statistical mechanics, E. Tirapegui ed., Reidel1981. MR606967
12. 10) R. Stora : "Continuum gauge theories" in : New developments in quantum field theory and statistical mechanics, 1976 Cargèse Lectures, M. Levy, P. Mitter eds, PlenumNew York1977. MR508193
13. 11) M. Dubois Violette, M. Talon, C.M. Viallet, "BRS cohomologies", to appear in C.M.P. Zbl0604.58055
14. 12) E. Witten, "Global aspects of current algebra", N.P.B.223, 422, 1983. MR717915
15. 13) G. Falqui, C. Reina, "BRS cohomology and topological anomalies", Annales IHP to be published. Zbl0613.53055
16. 14) J. Thierry-Mieg, P.L. 147B, 430, 1984. MR767765
17. 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. 
18. 16) L. Baulieu, J. Thierry-Mieg, P.L. 145B, 53, 1984 ; MR757328
19. F. Langouche, T. Schücker, R. Stora, P.L. 145B, 342, 1984; MR760602
20. L. Bonora, P. Pasti, M. Tonin, N.P.B.1985 to appear. 
21. 17) L. Bonora, P. Pasti, M. Tonin, P.L. 155B, 341, 1985 and to appear in NPB; MR794837
22. G. Girardi, R. Grimm, R. Stora, P.L. 156B, 203, 1935); 
23. J. Wess, B. Zumino, in preparation, 
24. 18) H. Itoyama, V.P. Nair, H.C. Ren, IASPrinceton preprint 1985. 
25. 19) G. Moore, P. Nelson, P.R.L.53, 1519, 1934; MR762413
26. G. Moore, P. Nelson, HUTP-84/A076 ; 
27. A. Manohar, G. Moore, P. Nelson, HUTP-84/AC83 ; 
28. L. Alvarez Gaumé, P. Ginsparg, HUTP85/AO15 ; 
29. P. di Vecchia, S. Ferrara, L. Girardello. P.L. 1513B, 199, 1985; MR785131
30. E. Cohen, C. Gomez, N.P.B.254, 235, 1985; MR793133
31. J. Bagger, D. Nemechansky, S. Yankielowicz in Anomalies, Geometry, Topology, Argonne-Chicago1985. 
32. 20) H. Kluberg Stern, J.B. Zuber, P.R.D.12, 467, 1975 ; P.R.D.12, 3159, 1975 ; MR387057
33. S.D. Joglekar, B.W. Lee, Ann. Phys.97, 160, 1976 ; MR428960
34. J. Dixon, 1976-1978, unpublished. 
35. 21) L.D. Faddeev, S.L. Shatashvili, TMФ60, 206, 1984. MR762262
36. 22) K.S. Stelle, P.C. West, N.P.B.140, 285 , 1978 ; 
37. L. Baulieu, M. Bellon, to be published. 
38. 23) M.B. Green, J.H. Schwarz, P.L. 149B, 117, 1984. MR771086
39. 24) S. Cecotti, S. Ferrara, L. Girardello, CERN.TH.4253/85. 
40. 25) J. Thierry-Mieg, unpublished, 1979; 
41. 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. MR722541
42. 26) E.S. Fradkin, G.A. Vilkovisky, P.L. 55B, 224, 1975 ; MR411451
43. I.A. Batalin, G.A. Vilkovisky, P.L. 69B, 309, 1977. 
44. 27) W.A. Bardeen, B. Zumino, N.P.B.244, 421, 1984. MR762380
45. 28) L. Bonora, P. Pasti, M. Tonin, P.L.B. to appear. 
46. 29) S. Ferrara, L. Girardello, O. Piguet, R. Stora, P.L. 157B, 179, 1985. MR797737
47. 30) O. Piguet, K. Sibold, N.P.B.247, 484, 1984. 
48. 31) O. Piguet, M. Schweda, K. Sibold, N.P.B.174, 183, 1980. 
49. 32) B. Zumino in Symposium on Anomalies, Geometry and Topology, Argonne-Chicago, 1985. 
50. 33) a) G. Bandelloni, C. Becchi, A. Blasi, R. Collina, Ann. IHPXXVIII, 225-254 and 255-285, 1978. 
51. b) O. Piguet, A. Rouet, Physics Reports76, 1-77, 1981. 
52. 34) cf. Eq. 6.16 of Ref. 33b). 
53. 35) I am indebted to R. Grimm for stimulating discussions on this subject. 
54. 36) Y. Tanii, TIT/HEP85 ; 
55. P. Howe, P. West, P.L. 156B, 335, 1985. MR794836
56. 37) a) A. Guichardet, "Cohomologie des groupes topologiques et des algèbras de Lie", CEDIC-NATHANParis1980. Zbl0464.22001MR644979
57. b) S. Mac Lane, "Homology", Springer1975. Zbl0328.18009MR1344215
58. c) H. Cartan, S. Eilenberg, "Homclogical algebra", Princeton University Press, 1956. Zbl0933.18001MR77480
59. d) N. Jacobson, "Lie Algebras", Interscience, 1962. Zbl0121.27504MR143793
60. 38) A.G. Reiman, M.A. Semenov-Tjan-Shansky, L.D. Faddeev, in Journal of functional analysis and its applications, 1984-1985 ; 
61. L.D. Faddeev, P.L. 145B, 81, 1984 ; MR757331
62. B. Zumino, N.P.B.253, 477, 1985; and references found in 8a). MR791687
63. 39) I.M. Singer, in Colloque Elie Cartan, Lyon, June 25-29, 1984, to appear in Astérisque, and references therein. 
64. 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. 
65. 41) When an anomaly is present, one can prove a modified Ward identity of the form $\int \frac{\delta \Gamma }{\delta a}\frac{\delta \Gamma }{\delta A}+\frac{\delta \Gamma }{\delta \phi }\frac{\delta \Gamma }{\delta \Phi }+\frac{\delta \Gamma }{\delta \omega }\frac{\delta \Gamma }{\delta \Omega }-\frac{\delta \Gamma }{\delta \overline{\omega }}b-\frac{\partial }{\partial \alpha }\Gamma =0$ where $\alpha$ is a constant odd scalar with dimension $-1$ , $\phi \pi$ charge $-1$ . (C. Becchi, private communication). 
66. 42) C. Becchi, unpublished.