Multisymplectic structure and homotopy reduction of General Relativity

Speaker:

Christian Blohmann

Affiliation:

MPIM

Date:

Thu, 08/08/2024 - 12:00 - 13:30

Location:

MPIM Lecture Hall [2]

Parent event:

Physical Math Seminar [3]

In General Relativity, the Hamiltonian formulation of the dynamics and the Poisson bracket of observables depend on the choice of a codimension 1 submanifold as initial time-slice. But such a submanifold is not invariant under diffeomorphisms. As a consequence, Noether's first theorem, which associates to a symmetry a conserved momentum, does not define a homomorphism of Lie algebras. Starting in the 1960s, this issue has led to the development of multisymplectic geometry, eventually replacing the Poisson algebra of momenta with an $L_\infty$-algebra of currents. I will show that:

1.) Noether's map from the symmetry algebra to the conserved currents extends naturally to a homomorphism of $L_\infty$-algebras, called the homotopy momentum map.

2.) There is a homotopy zero locus of the homotopy momentum map generalizing the constraint variety.

3.) The homotopy zero locus is invariant under the identity component of the diffeomorphism group.

We thus obtain the homotopy reduction of General Relativity, a generalization of the usual procedure of imposing constraints and gauge fixing. This is joint work with Janina Bernardy.