Diffeological groupoid symmetries and constraints of field theories

When the vacuum Einstein equations are cast in the form of an initial value problem, the initial data lie in the cotangent bundle of the manifold of Riemannian metrics on the Cauchy hypersurface S, which carries its natural symplectic structure. As for every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike for gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. I will show that the Poisson bracket relations among the constraints correspond to those among the constant sections of a Lie algebroid that integrates to a diffeological groupoid obtained by localizing the groupoid of diffeomorphisms between Lorentz manifolds to space-like embeddings of S. In a second step, I show that the constraints come from a generalized momentum map associated to the groupoid action. This construction also applies to other Lagrangian field theories.