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multivector field

Definition

For $X$ a smooth manifold and $TX$ its tangent bundle a multivector field on $X$ is an element of the exterior algebra bundle ${\wedge }_{C^{\mathrm{\infty }}(X)}^{•}(\Gamma (TX))$ 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 ${\mathrm{HH}}^{•}(C(X),C(X))$ 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 $\omega \in {\Omega }^d(X)$.

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

$$\omega (-):T_{\mathrm{poly}}^n(X)\stackrel{\simeq }{\to }{\Omega }^{(d-n)}(X).$$\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

$$\begin{array}{ccc}{\Omega }^n(X)& \stackrel{d_{\mathrm{dR}}}{\to }& {\Omega }^{n+1}(X)\\ {\downarrow }^{\simeq }& & {\downarrow }^{\simeq }\\ T_{\mathrm{poly}}^{d-n}& \stackrel{{\mathrm{div}}_{\omega }}{\to }& T_{\mathrm{poly}}^{d-n-1}\end{array},$$\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

• Alberto Cattaneo, Domenico Fiorenza, Riccardo Longoni, On the Hochschild-Kostant-Rosenberg map for graded manifolds (pdf)

and on p. 6 of

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