Symmetries and Reduction of Multisymplectic Manifolds

For zoom details contact Christian Kaiser (kaiser@mpim-bonn.mpg.de).

Multisymplectic manifolds are a straightforward generalization of symplectic manifolds where closed non-degenerate k-forms are considered in place of 2-forms.
A natural theme that arises when dealing with (multi)symplectic structures is investigating the relationship between symmetries (group action preserving the fixed differential form) and reduction.

In this context, by reduction scheme, we mean a procedure to construct a second space of reduced dimension out of a given (multi)-symplectic manifold with symmetries. The so obtained reduced space enjoys the convenient property of embodying the relevant geometric structure of the starting unreduced object.

A well-known result in symplectic geometry, known as Marsden–Weinstein–Meyer theorem, states that the relevant geometric structure of a symplectic manifold can be studied on the level set of a so-called "momentum map".
Sniaticki and Weinstein have further extended this result to encompass singular momentum maps.

The scope of this talk is to review some relevant algebraic structures related to multisymplectic manifolds, namely the higher version of the observables algebra and moment maps, and discuss how the regular and singular reduction schemes extend to the multisymplectic framework.

This talk is based on ongoing joint work with Casey Blacker and Leonid Ryvkin.