nLab
multivector field

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Definition

          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.

          Properties

          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 (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

          and on p. 6 of

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

          Last revised on July 28, 2018 at 09:01:29. See the history of this page for a list of all contributions to it.