Séminaire Physique mathématique ICJ

Multisymplectic observables and higher Courant algebroids

by Dr Antonio Miti (MPI Bonn)

Fokko du Cloux (Bâtiment Braconnier)

Fokko du Cloux (Bâtiment Braconnier)


Multisimplectic manifolds are a straightforward generalization of symplectic manifolds where one considers closed non-degenerate k-forms $\omega$ in place of 2-forms. Recent works by Rogers and Zambon showed how one could associate to such a geometric structure two higher algebraic structures: an $L_{\infty}$-algebra of observables and an $L_{\infty}$- algebra of sections of the higher Courant algebroid twisted by $\omega$. 

The scope of this talk is to report on joint work with Marco Zambon (arXiv:2209.05836). Our main result is proving the existence of an $L_{\infty}$-embedding between the above two $L_\infty$-algebras generalizing a construction already found by Rogers around 2012 valid for multisymplectic 3-forms only. Moreover, we display explicit formulae for the sought morphism involving the Bernoulli numbers. Although this construction is essentially algebraic, it also admits a geometric interpretation when declined to the particular case of pre-quantizable symplectic forms. The latter case provides some evidence that this construction may be related to the higher analogue of geometric quantization for integral multisymplectic forms.