September 93

Currents on Grassmann algebras,

R. Coquereaux, E. Ragoucy

Journal of Geometry and Physics 15 (1995) 333-352

Centre de Physique Theorique, CPT, CNRS Luminy - Case 907, 13288 Marseille, France

Laboratoire de Physique Theorique, ENSLAPP, B.P. 110, 74941, Annecy-le-Vieux, France

Abstract : We define currents on a Grassmann algebra Gr(N) with N generators as distributions on its exterior algebra (using the symmetric wedge product). We interpret the currents in terms of -graded Hochschild cohomology and closed currents in terms of cyclic cocycles (they are particular multilinear forms on Gr(N)). An explicit construction of the vector space of closed currents of degree p on Gr(N) is given by using Berezin integration.

Keywords :

Grassmann algebras, super-algebras, supersymmetries, supermanifolds, Hochschild and cyclic cohomologies, De Rham currents.

October 1993
CPT - 93 / P.2935

ENSLAPP - 440/93

• Introduction
• Differential forms over Grassmann algebras and Grassmannian currents
  • Classical analogies
  • Grassmann algebras (a compendium of basic results and notations)
  • Derivations and differential forms over Gr(N)
• Hochschild and cyclic cohomologies for Grassmann algebras
  • Hochschild cohomology
  • Cyclic cohomology (results)
  • Closed currents, divergence-free Grassmann- valued contravariant symmetric tensors and cyclic cocycles
  • The hierarchy of cocycles
  • The dimension of the space of closed currents of degree p on Gr(N)
  • The case p=1
  • An explicit example: N=2, p=2
• The action of operators S and B in the Grassmannian case
• Some combinatorial identities
• References
• About this document ...