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multivector field

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

  • C. Roger, Gerstenhaber and Batalin-Vilkovisky algebras (ps)

Revised on March 4, 2011 12:05:37 by Urs Schreiber (131.211.232.235)