The Spencer cohomology and integrability of multisymplectic structures

We study the integrability problem of multisymplectic structures, by identifying them as $G$-structures.

Applying the theory of Spencer cohomology, we give conditions on a multisymplectic form for it to admit a chart in which it has constant coefficients.

This general study allows for a rough classification of multisymplectic structures of constant linear type, depending on the natural action of the stabilizer group.

The theory is illustrated by providing a scheme for proving a Darboux theorem, which is exemplified with several relevant cases.

We also build linear types of multisymplectic forms $\varpi_j$ whose flatness strictly requires a condition of order $j$.

Finally, the corresponding Lie algebras are computed in the case of field theories.