For $X$ a smooth manifold and $T X$ its tangent bundle a multivector field on $X$ is an element of the exterior algebra bundle $\wedge^\bullet_{C^\infty(X)}(\Gamma(T X))$ of skew-symmetric tensor powers of sections of $T X$.
In degree $0$ these are simply the smooth functions on $X$.
In degree $1$ these are simply the tangent vector fields on $X$.
In degree $p$ these are sometimes called the $p$-vector fields on $X$.
In suitable contexts, multivector fields on $X$ can be identified with the Hochschild cohomology $HH^\bullet(C(X), C(X))$ of the algebra of functions on $X$.
There is a canonical bilinear pairing on multivector fields called the Schouten bracket.
Let $X$ be a smooth manifold of dimension $d$, which is equipped with an orientation exhibited by a differential form $\omega \in \Omega^d(X)$.
Then contraction with $\omega$ induces for all $0 \leq n \leq d$ an isomorphism of vector spaces
The transport of the de Rham differential along these isomorphism equips $T^\bullet_{poly}$ with the structure of a chain complex
The operation $div_\omega$ 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
and in section 2 of
and on p. 6 of
Last revised on November 1, 2015 at 03:31:36. See the history of this page for a list of all contributions to it.