The (pre)multisymplectic geometry of the De Donder–Weyl formalism for field
theories is further developed for a variety of field theories including a
scalar field theory from the canonical Klein-Gordon action, the electric and
magnetic Carrollian scalar field theories, bosonic string theory from the
Nambu-Goto action, and $2+1$ gravity as a Chern-Simons theory. The Lagrangians
for the scalar field theories and for $2+1$ Chern-Simons gravity are found to
be singular in the De Donder–Weyl sense while the Nambu-Goto Lagrangian is
found to be regular. Furthermore, the constraint structure of the
premultisymplectic phase spaces of singular field theories is explained and
applied to these theories. Finally, it is studied how symmetries are developed
on the multisymplectic phase spaces in the presence of constraints.
