Christian Blohmann: Homotopy reduction of the homotopy momentum map of general relativity

Abstract

 In general relativity (and all other field theories with diffeomorphism symmetries), Noether’s theorem does not give rise to an equivariant momentum map. This long-standing fundamental problem has recently found a solution in the setting of multisymplectic geometry, where the map from spacetime vector fields to their Noether currents extends naturally to a morphism of L-infinity algebras, called a homotopy momentum map. The next step is to develop a notion of (premulti)symplectic reduction in this setting, which consists of two steps: 1. restricting to the zero locus of the momenta or charges (imposing constraints) and 2. taking the quotient by the residual symmetry (gauge fixing). In the homotopical setting, the vanishing condition has to be replaced by a cohomological condition in the variational bicomplex. This leads to two simple and reasonable conditions describing the homotopy zero locus. Using diffeological methods, we can then show that the homotopy zero locus, including its singular strata, is invariant under diffeomorphisms. This is joint work with Janina Bernardy.