Canonical Lifts in Multisymplectic De Donder-Weyl Hamiltonian Field Theories

In this paper, 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, Einstein-Cartan gravity in $3+1$ dimensions, and the Polyakov bosonic string 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.