# Contents

## Definition

For $X$ a smooth manifold and $TX$ its tangent bundle a multivector field on $X$ is an element of the exterior algebra bundle ${\wedge }_{{C}^{\infty }\left(X\right)}^{•}\left(\Gamma \left(TX\right)\right)$ of skew-symmetric tensor powers of sections of $TX$.

• 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 $X$.

## Properties

### Hochschild cohomology

In suitable contexts, multivector fields on $X$ can be identified with the Hochschild cohomology ${\mathrm{HH}}^{•}\left(C\left(X\right),C\left(X\right)\right)$ of the algebra of functions on $X$.

### Schouten bracket

There is a canonical bilinear pairing on multivector fields called the Schouten bracket.

### Isomorphisms with de Rham complex

Let $X$ be a smooth manifold of dimension $d$, which is equipped with an orientation exhibited by a differential form $\omega \in {\Omega }^{d}\left(X\right)$.

Then contraction with $\omega$ induces for all $0\le n\le d$ an isomorphism of vector spaces

$\omega \left(-\right):{T}_{\mathrm{poly}}^{n}\left(X\right)\stackrel{\simeq }{\to }{\Omega }^{\left(d-n\right)}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$\omega(-) : T^n_{poly}(X) \stackrel{\simeq}{\to} \Omega^{(d-n)}(X) \,.

The transport of the de Rham differential along these isomorphism equips ${T}_{\mathrm{poly}}^{•}$ with the structure of a chain complex

$\begin{array}{ccc}{\Omega }^{n}\left(X\right)& \stackrel{{d}_{\mathrm{dR}}}{\to }& {\Omega }^{n+1}\left(X\right)\\ {↓}^{\simeq }& & {↓}^{\simeq }\\ {T}_{\mathrm{poly}}^{d-n}& \stackrel{{\mathrm{div}}_{\omega }}{\to }& {T}_{\mathrm{poly}}^{d-n-1}\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ \Omega^{n}(X) &\stackrel{d_{dR}}{\to}& \Omega^{n+1}(X) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ T^{d-n}_{poly} &\stackrel{div_\omega}{\to}& T^{d-n-1}_{poly} } \,,

The operation ${\mathrm{div}}_{\omega }$ 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.

## References

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

• C. Roger, Gerstenhaber and Batalin-Vilkovisky algebras (ps)

Revised on March 4, 2011 12:05:37 by Urs Schreiber (131.211.232.235)