Abstract

These lecture notes are based on a series of lectures given at the school on “Geometric and Topological Methods for Quantum Field Theory”, in Villa de Leyva, Colombia. We present a basic introduction to Dirac manifolds, recalling the original context in which they were defined, their main features, and briefly mentioning more recent developments.

Introduction

Phase spaces of classical mechanical systems are commonly modeled by symplectic manifolds. It often happens that the dynamics governing the system's evolution are constrained to particular submanifolds of the phase space, e.g. level sets of conserved quantities (typically associated with symmetries of the system, such as momentum maps), or submanifolds resulting from constraints in the possible configurations of the system, etc. Any submanifold C of a symplectic manifold M inherits a presymplectic form (i.e. a closed 2-form, possibly degenerate), given by the pullback of the ambient symplectic form to C. It may be desirable to treat C in its own right, which makes presymplectic geometry the natural arena for the study of constrained systems; see e.g. [23, 25].

In many situations, however, phase spaces are modeled by more general objects: Poisson manifolds (see e.g. [35]). A Poisson structure on a manifold M is a bivector field π ϵ Γ(Λ2TM) such that the skew-symmetric bracket {f, g} ≔ π(df, dg) on C∞(M) satisfies the Jacobi identity. Just as for symplectic phase spaces, there are natural examples of systems on Poisson phase spaces which are constrained to submanifolds. The present notes address the following motivating questions: what kind of geometric structure is inherited by a submanifold C of a Poisson manifold M?