Multisymplectic observables and higher Courant algebroids

Abstract

Multisimplectic manifolds are a straightforward generalization of symplectic manifolds where one considers closed non-degenerate k-forms 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∞L_{\infty}-algebra of observables and an L∞L_{\infty}- algebra of sections of the higher Courant algebroid twisted by ω\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∞L_{\infty}-embedding between the above two L∞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.

Full text

Description
Multisimplectic manifolds are a straightforward generalization of symplectic manifolds where one considers closed non-degenerate k-forms 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 -algebra of observables and an - algebra of sections of the higher Courant algebroid twisted by .
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 -
embedding between the above two -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.
Séminaire Physique mathématique ICJ
Multisymplectic observables and higher Courant algebroids
by Dr Antonio Miti (MPI Bonn)

Friday Dec 9, 2022, 2:00 PM → 3:00 PM Europe/Paris
Fokko du Cloux (Bâtiment Braconnier)
ω
L
∞
L
∞
ω
L
∞
L
∞