nLab multivector field

Definition

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 $X$.

Properties

Hochschild cohomology

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 $X$.

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 $0 \leq n \leq d$ an isomorphism of vector spaces

$$\omega(-) : T^n_{poly}(X) \stackrel{\simeq}{\to} \Omega^{(d-n)}(X) \,.$$

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

$$\array{ \Omega^{n}(X) &\stackrel{d_{dR}}{\to}& \Omega^{n+1}(X) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ T^{d-n}_{poly} &\stackrel{div_\omega}{\to}& T^{d-n-1}_{poly} } \,,$$

The operation $div_\omega$ is a derivation of the Schouten bracket and makes multivectorfields into a BV-algebra.

A more general discussion of this phenomenon in (Cattaneo–Fiorenza–Longoni). Even more generally, see Poincaré duality for Hochschild cohomology.

Integral sections

Just as for vector fields one has the notion of an integral curve, so for $k$-vector fields one has a generalization known as integral sections. Given a $k$-vector field $X$ on a manifold $M$, and a point $p\in M$, an integral section is a map $\phi: U\subset \mathbb{R}^k \to M$ such that $\phi(0)=p$ and

$$\phi_* (x) (\partial_{x^{\alpha}} \vert_x ) = X_{\alpha} (\phi(x))$$

See e.g. Section 3.1 in de León et al. 2015 and the references cited therein for more.

Related concepts

• bivector

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, arXiv:math/0503380

and on p. 6 of

• Claude Roger, Gerstenhaber and Batalin-Vilkovisky algebras, Archivum mathematicum, Volume 45 (2009), No. 4 (pdf)

An exposition of multivector fields and their use in Lagrangian and Hamiltonian theories:

• Manuel De León, Modesto Salgado, Silvia Vilarino-Fernández. Methods of differential geometry in classical field theories: k-symplectic and k-cosymplectic approaches. World Scientific, 2015. (arXiv:1409.5604).