Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cohomology classes associated to anomalies

Author: Gregory Lambros Peterson
Journal: Trans. Amer. Math. Soc. 340 (1993), 669-704
MSC: Primary 58H99; Secondary 53C07, 53C80, 58G40, 81T50
MathSciNet review: 1129437
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: One of the proposed settings for the description of anomalies in the setting of gauge field theories is a local bicomplex associated to a principal fiber bundle $ G \to P \to M$. It is a bigraded algebra with two differentials which is invariantly defined, i.e. independent of local coordinates of $ M$. I denote it by $ S_M^{ \bullet , \bullet }$. Briefly, $ S_M^{p,q}$ consists of local $ p$-multilinear operators from the gauge algebra into $ q$-forms on $ M$ which depend on a connection $ A$ in a local manner; local means that the operators decrease supports. The gauge algebra is the Lie algebra of the gauge group, which consists of diffeomorphisms of $ P$ that respect the action of $ G$ and cover the identity diffeomorphism of $ M$. In this setting, the anomalies are described as integrals over $ M$ whose integrands can be shown to be representatives of total cohomology classes in $ {H^1}(S_M^{ \bullet , \bullet })$.

The main reason for restricting to a local bicomplex is due to Peetre's theorem. It states that local operators are differential operators over open sets $ U \subset M$. This property is both mathematically natural and required by physical considerations.

This paper explores the computation of the total cohomology of the local bicomplex by beginning with the coordinate description of the differential operators and then determining which of these differential operators can be used to construct invariantly defined objects. What is accomplished is the description of the differential operators which are invariant under the action of the local diffeomorphisms of $ {\mathbb{R}^n}$ and the computation of their total cohomology over open sets $ U \subset M$. The main result is that

$\displaystyle H_d^ \bullet ({(S_U^{ \bullet , \bullet })^{{\operatorname{Diff}_... ...}({\mathbb{R}^n})}}) \simeq H_d^ \bullet (W{(\mathfrak{g})_{[\tfrac{n} {2}]}}),$

where $ {(S_U^{ \bullet , \bullet })^{{{\operatorname{Diff}}_{{\text{loc}}}}({\mathbb{R}^n})}}$ denotes the invariant differential operators over the open set $ U$ and $ W{(\mathfrak{g})_{[\tfrac{n} {2}]}}$ is the Weil algebra of $ \mathfrak{g}$, the Lie algebra of $ G$ truncated at $ [\tfrac{n} {2}]$, the greatest integer less than or equal to half the dimension of $ M$. This shows that the cohomology groups over open sets are nonzero only in the range $ n \leq q \leq n + r$ where $ r$ is the dimension of the Lie algebra $ \mathfrak{g}$ , and in this range they are all finite dimensional. This result is globalized in the special case that the associated fiber bundle $ \operatorname{ad}^\ast\;P$ is trivializable.

References [Enhancements On Off] (What's this?)

  • [1] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London A 308 (1982), 523-615. MR 702806 (85k:14006)
  • [2] M. F. Atiyah and I. M. Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 2597-2600. MR 742394 (86g:58127)
  • [3] L. Bonora and P. Cotta-Ramusino, Some remarks on B. R. S. transformations, anomalies and the cohomology of the Lie algebra of the group of gauge transformations, Comm. Math. Phys. 87 (1983), 589-603. MR 691046 (84g:81030)
  • [4] H. Cartan, Colloque de topologie (espaces fibres), tenu a Bruxelles du 5 au 8 Juin 1950, Centre Belge de Recherches Mathematiques, Thone, 1951.
  • [5] P. Cotta-Ramusino, Anomalies and their cancelation, Fundamental aspects of quantum theory (V. Gorini and A. Frigerio, eds.), Plenum Press, New York and London, 1986, pp. 411-417. MR 873367 (88a:81117)
  • [6] P. Cotta-Ramusino and C. Reina, The action of the group of bundle-automorphisms on the space of connections and the geometry of gauge theories, J. Geom. Phys. 1 (1984), 121-155. MR 828400 (87i:58031)
  • [7] M. De Wilde and P. Lecompte, Cohomology of the Lie algebra of a manifold associated to the Lie derivative of smooth forms, J. Math. Pures Appl. 62 (1983), 197.
  • [8] D. B. Fuks, Cohomology infinite-dimensional Lie algebras, Consultants Bureau, New York, 1986. MR 874337 (88b:17001)
  • [9] W. Greub, S. Halperin, and R. Van Stone, Connections, curvature, and cohomology, Academic Press, 1976. MR 0400275 (53:4110)
  • [10] C. Godbillon, Cohomologies d'algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki, vol. 1972/73, Exposés 418-435, Lecture Notes in Math., vol. 383, Springer-Verlag, 1974.
  • [11] S. Halperin et D. Lehmann, Cohomologies et classes caracteristiques des choux de bruxelles, Univ. of Toronto, preprint. MR 0402772 (53:6586)
  • [12] A. Haefliger, Sur la cohomologie de l'algébre de Lie des champs de vecteurs, Ann. Sci. École Norm. Sup 4 (1976), 503-532. MR 0448367 (56:6674)
  • [13] J. L. Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950), 65. MR 0036511 (12:120g)
  • [14] P. K. Mitter and C. M. Vialette, On the bundle of connections and the gauge orbit manifold in Yang-Mills theory, Comm. Math. Phys. 79 (1981), 457. MR 623962 (83f:81056)
  • [15] M. S. Narasimhan and T. R. Ramadas, Geometry of $ SU(\tau )$ gauge fields, Comm. Math. Phys. 67 (1979), 121. MR 539547 (84k:58050)
  • [16] J. Peetre, Une caracterisation abstraite des opérateurs différentiels, Math. Scand. 7 (1959), 211-218. MR 0112146 (22:3001)
  • [17] R. Schmidt, Local cohomology in gauge theories, Emory Univ., preprint.
  • [18] M. Spivak, A comprehensive introduction to differential geometry, 2nd ed., Publish or Perish, 1979.
  • [19] T. Tsujishita, Continuous cohomology of the Lie algebra of vector fields, Mem. Amer. Math. Soc., vol. 34, no. 253, 1981. MR 634471 (84b:17015)
  • [20] H. Weyl, Classical groups, Princeton Univ. Press, 1939. MR 1488158 (98k:01049)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58H99, 53C07, 53C80, 58G40, 81T50

Retrieve articles in all journals with MSC: 58H99, 53C07, 53C80, 58G40, 81T50

Additional Information

Keywords: Gauge group, gauge algebra, cohomology, invariants, Weil algebra, anomalies
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society