  A variety of local index formulas is constructed for quantum Hamiltonians
with periodic boundary conditions. All dimensions of physical space as well as
many symmetry constraints are covered, notably one-dimensional systems in Class
DIII as well as two- and three-dimensional systems in Class AII. The
constructions are based on several periodic variations of the spectral
localizer and are rooted in the existence of underlying fuzzy tori. For these
latter, a general invariant theory is developed.
