Dirac structure


In mathematics a Dirac structure is a geometric structure generalizing both symplectic structures and Poisson structures, and having several applications to mechanics. It is based on the notion of the Dirac bracket constraint introduced by Paul Dirac and was first introduced by Ted Courant and Alan Weinstein.

Linear Dirac structures

Let ' be a real vector space, and ' its dual. A Dirac structure on ' is a linear subspace ' of satisfying
In particular, if ' is finite dimensional, then the second criterion is satisfied if. Similar definitions can be made for vector spaces over other fields.
An alternative definition often used is that satisfies, where orthogonality is with respect to the symmetric bilinear form on given by.

Examples

  1. If is a vector subspace, then is a Dirac structure on, where is the annihilator of ; that is,.
  2. Let be a skew-symmetric linear map, then the graph of is a Dirac structure.
  3. Similarly, if is a skew-symmetric linear map, then its graph is a Dirac structure.

Dirac structures on manifolds

A Dirac structure on a smooth manifold ' is an assignment of a Dirac structure on the tangent space to ' at , for each. That is,
Many authors, in particular in geometry rather than the mechanics applications, require a Dirac structure to satisfy an extra integrability condition as follows:
  • suppose are sections of the Dirac bundle then
In the mechanics literature this would be called a closed or integrable Dirac structure.

Examples


  1. Let be a smooth distribution of constant rank on a manifold ', and for each let, then the union of these subspaces over ' forms a Dirac structure on .
  2. Let be a symplectic form on a manifold, then its graph is a Dirac structure. More generally, this is true for any closed 2-form. If the 2-form is not closed, then the resulting Dirac structure is not closed.
  3. Let be a Poisson structure on a manifold, then its graph is a Dirac structure.
  4. Any submanifold of a Poisson manifold induces a Dirac structure.