Canonical lifts in multisymplectic De Donder-Weyl Hamiltonian field theories

We define canonical lifts of vector fields to the multisymplectic multimomentum bundles of De Donder-Weyl Hamiltonian (first-order) field theories and to the appropriate premultisymplectic embedded constraint submanifolds on which singular field theories are studied. These new canonical lifts are used to study the so-called natural Noether symmetries present in both regular and singular Hamiltonian field theories along with their associated conserved quantities obtained from Noether's theorem. The Klein-Gordon field, the Polyakov bosonic string, and Einstein-Cartan gravity in 3+1 dimensions are analyzed in depth as applications of these concepts; as a peripheral result obtained in the analysis of the bosonic string, we provide a new geometrical interpretation of the well-known Virasoro constraint.