Dirac manifolds

Abstract

A Dirac structure on a vector space V V is a subspace of V V with a skew form on it. It is shown that these structures correspond to subspaces of V ⊕ V ∗ V \oplus {V^{\ast }} satisfying a maximality condition, and having the property that a certain symmetric form on V ⊕ V ∗ 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 ) 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 P , which are defined to be smooth subbundles of the bundle T P ⊕ T ∗ P TP \oplus {T^{\ast }}P satisfying pointwise the properties of the linear case. If a bundle L ⊂ T P ⊕ T ∗ P L \subset TP \oplus {T^{\ast }}P defines a Dirac structure on P P , then we call L L a Dirac bundle over P P . A 3 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 ] [B,B] and d Ω d\Omega respectively. A function f f on a Dirac manifold is called admissible if there is a vector field X X such that the pair ( X , d f ) (X,df) is a section of the Dirac bundle L L ; the pair ( X , d f ) (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 Q of a Dirac manifold P P . The induced bracket on admissible functions on Q Q is in fact the Dirac bracket as defined by Dirac for constrained submanifolds.