  We develop a toolbox for manipulating arrays of Rydberg atoms prepared in
high-dimensional hydrogen-like manifolds in the regime of linear Stark and
Zeeman effect. We exploit the SO(4) symmetry to characterize the action of
static electric and magnetic fields as well as microwave and optical fields on
the well-structured manifolds of states with principal quantum number $n$. This
enables us to construct generalized large-spin Heisenberg models for which we
develop state-preparation and readout schemes. Due to the available large
internal Hilbert space, these models provide a natural framework for the
quantum simulation of Quantum Field Theories, which we illustrate for the case
of the sine-Gordon and massive Schwinger models. Moreover, these
high-dimensional manifolds also offer the opportunity to perform quantum
information processing operations for qudit-based quantum computing, which we
exemplify with an entangling gate and a state-transfer protocol for the states
in the neighborhood of the circular Rydberg level.
