# nLab multivector field

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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.

## References

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

• 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).

Last revised on April 18, 2024 at 16:44:53. See the history of this page for a list of all contributions to it.