A brief introduction to Dirac manifolds

摘要: Abstract These lecture notes are based on a series of lectures given at the school “Geometric and Topological Methods for Quantum Field Theory”, in Villa de Leyva, Colombia. We present basic introduction to Dirac manifolds, recalling original context which they were defined, their main features, briefly mentioning more recent developments. Introduction Phase spaces classical mechanical systems commonly modeled by symplectic manifolds. It often happens that dynamics governing system's evolution constrained particular submanifolds phase space, e.g. level sets conserved quantities (typically associated with symmetries system, such as momentum maps), or resulting from constraints possible configurations etc. Any submanifold C manifold M inherits presymplectic form (i.e. closed 2-form, possibly degenerate), pullback ambient . may be desirable treat its own right, makes geometry natural arena study systems; see [23, 25]. In many situations, however, general objects: Poisson manifolds (see [35]). A structure is bivector field π ϵ Γ(Λ 2 TM ) skew-symmetric bracket { f, g } ≔ π( df, dg ∞ ( satisfies Jacobi identity. Just spaces, there examples submanifolds. The address following motivating questions: what kind geometric inherited ?