nLab multivector field



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



For XX a smooth manifold and TXT X its tangent bundle a multivector field on XX is an element of the exterior algebra bundle C (X) (Γ(TX))\wedge^\bullet_{C^\infty(X)}(\Gamma(T X)) of skew-symmetric tensor powers of sections of TXT X.

  • In degree 00 these are simply the smooth functions on XX.

  • In degree 11 these are simply the tangent vector fields on XX.

  • In degree pp these are sometimes called the pp-vector fields on XX.


Hochschild cohomology

In suitable contexts, multivector fields on XX can be identified with the Hochschild cohomology HH (C(X),C(X))HH^\bullet(C(X), C(X)) of the algebra of functions on XX.

Schouten bracket

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

Isomorphisms with de Rham complex

Let XX be a smooth manifold of dimension dd, which is equipped with an orientation exhibited by a differential form ωΩ d(X)\omega \in \Omega^d(X).

Then contraction with ω\omega induces for all 0nd0 \leq n \leq d an isomorphism of vector spaces

ω():T poly n(X)Ω (dn)(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 poly T^\bullet_{poly} with the structure of a chain complex

Ω n(X) d dR Ω n+1(X) T poly dn div ω T poly dn1, \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 ωdiv_\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 kk-vector fields one has a generalization known as integral sections. Given a kk-vector field XX on a manifold MM, and a point pMp\in M, an integral section is a map ϕ:U kM\phi: U\subset \mathbb{R}^k \to M such that ϕ(0)=p\phi(0)=p and

ϕ *(x)( x α| x)=X α(ϕ(x)) \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.


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.