Dirac Manifolds

Abstract

A Dirac structure on a vector space $V$ is a subspace of $V$ with a skew form on it. It is shown that these structures correspond to subspaces of $V \oplus {V^{\ast }}$ satisfying a maximality condition, and having the property that a certain symmetric form on $V \oplus {V^{\ast }}$ vanishes when restricted to them. Dirac structures on a vector space are analyzed in terms of bases, and a generalized Cayley transformation is defined which takes a Dirac structure to an element of $O(V)$. Finally a method is given for passing a Dirac structure on a vector space to a Dirac structure on any subspace. Dirac structures on vector spaces are generalized to smooth Dirac structures on a manifold $P$, which are defined to be smooth subbundles of the bundle $TP \oplus {T^{\ast }}P$ satisfying pointwise the properties of the linear case. If a bundle $L \subset TP \oplus {T^{\ast }}P$ defines a Dirac structure on $P$, then we call $L$ a Dirac bundle over $P$. A $3$-tensor is defined on Dirac bundles whose vanishing is the integrability condition of the Dirac structure. The basic examples of integrable Dirac structures are Poisson and presymplectic manifolds; in these cases the Dirac bundle is the graph of a bundle map, and the integrability tensors are $[B,B]$ and $d\Omega$ respectively. A function $f$ on a Dirac manifold is called admissible if there is a vector field $X$ such that the pair $(X,df)$ is a section of the Dirac bundle $L$; the pair $(X,df)$ is called an admissible section. The set of admissible functions is shown to be a Poisson algebra. A process is given for passing Dirac structures to a submanifold $Q$ of a Dirac manifold $P$. The induced bracket on admissible functions on $Q$ is in fact the Dirac bracket as defined by Dirac for constrained submanifolds.