For a smooth manifold and its tangent bundle a multivector field on is an element of the exterior algebra bundle of skew-symmetric tensor powers of sections of .
In degree these are simply the smooth functions on .
In degree these are simply the tangent vector fields on .
In degree these are sometimes called the -vector fields on .
In suitable contexts, multivector fields on can be identified with the Hochschild cohomology of the algebra of functions on .
There is a canonical bilinear pairing on multivector fields called the Schouten bracket.
Isomorphisms with de Rham complex
Let be a smooth manifold of dimension , which is equipped with an orientation exhibited by a differential form .
Then contraction with induces for all an isomorphism of vector spaces
The transport of the de Rham differential along these isomorphism equips with the structure of a chain complex
The operation is a derivation of the Schouten bracket and makes multivectorfields into a BV-algebra.
A more general discussion of this phenomenon in (CattaneoFiorenzaLongoni). Even more generally, see Poincaré duality for Hochschild cohomology.
The isomorphisms between the de Rham complex and the complex of polyvector field is reviewed for instance on p. 3 of
- Thomas Willwacher, Damien Calaque Formality of cyclic cochains (arXiv:0806.4095)
and in section 2 of
and on p. 6 of
- Claude Roger, Gerstenhaber and Batalin-Vilkovisky algebras, Archivum mathematicum, Volume 45 (2009), No. 4 (pdf)
Revised on November 1, 2015 03:31:36
by David Corfield