Marsden–Weinstein reduction on graded symplectic manifolds

Abstract

We extend the Marsden–Weinstein reduction of Hamiltonian systems with symmetry to the category of graded manifolds. We define free, proper, and symplectic actions in the graded setting; especially for the symplectic actions of graded Lie groups, it is possible to speak about momentum mappings. We also examine the reduction of the graded Hamiltonian derivations which determines the dynamics on the reduced phase space for constrained supersystems with symmetry. We apply the general formalism to specific examples, thus constructing graded (complex) projective spaces, and proving that the coadjoint orbits of graded Lie groups are reduced graded symplectic manifolds.