multivector field in nLab
Article Images
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
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Contents
Definition
For a smooth manifold and its tangent bundle a multivector field on is an element of the exterior algebra bundle of skew-symmetric tensor powers of sections of .
-
In degree these are simply the smooth functions on .
-
In degree these are simply the tangent vector fields on .
-
In degree these are sometimes called the -vector fields on .
Properties
Hochschild cohomology
In suitable contexts, multivector fields on can be identified with the Hochschild cohomology of the algebra of functions on .
Schouten bracket
There is a canonical bilinear pairing on multivector fields called the Schouten bracket.
Isomorphisms with de Rham complex
Let be a smooth manifold of dimension , which is equipped with an orientation exhibited by a differential form .
Then contraction with induces for all an isomorphism of vector spaces
The transport of the de Rham differential along these isomorphism equips with the structure of a chain complex
The operation 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 -vector fields one has a generalization known as integral sections. Given a -vector field on a manifold , and a point , an integral section is a map such that and
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
- Thomas Willwacher, Damien Calaque, Formality of cyclic cochains (arXiv:0806.4095)
and in section 2 of
- Alberto Cattaneo, Domenico Fiorenza, Riccardo Longoni, On the Hochschild-Kostant-Rosenberg map for graded manifolds, arXiv:math/0503380
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.