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.


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)

Last revised on March 3, 2023 at 19:10:56. See the history of this page for a list of all contributions to it.