  Given a vector field on a manifold M, we define a globally conserved quantity
to be a differential form whose Lie derivative is exact. Integrals of conserved
quantities over suitable submanifolds are constant under time evolution, the
Kelvin circulation theorem being a well-known special case. More generally,
conserved quantities are well-behaved under transgression to spaces of maps
into M.
  We focus on the case of multisymplectic manifolds and Hamiltonian vector
fields. We show that in the presence of a Lie group of symmetries admitting a
homotopy co-momentum map, one obtains a whole family of globally conserved
quantities. This extends a classical result in symplectic geometry. We carry
this out in a general setting, considering several variants of the notion of
globally conserved quantity.
