In some situations (e.g. molecular dynamics situations, Euclidean QFT, cosmology), a common trick to eliminate boundary effects or to provide an infrared cutoff is to use periodic boundary conditions. Therefore, I am trying to understand the implications of the global torus topology on the system's dynamics.

What is the effect of the global topology of the space manifold (or base space) on, e.g., the symmetries, the conservation laws and the equations of motion? (for fields, including gauge fields, or even classical point particles and/or usual QM see this).

I am particularly interested in spacetimes of the kind $T^n\times\mathbb{R}$, where $T^n$ is the flat n-dimensional torus (space) and time is just the usual thing (i.e. not "thermal" fields, where time lives on the circle $T^1$, see e.g. this and this example with fermions).

Specific question: Are there dedicated references (reviews, books) that discuss the Noether theorem, conservation laws and, generally, the dynamics of a system on the torus? (or manifolds other than the usual Galielaian or Minkowski space).

Concrete example: The flat torus is not symmetric under rotations, so there should be no conservation of angular momentum for a system on the torus. However, a point particle on the flat torus "looks like" a particle on the plane (locally) and the Noether theorem seems to apply. Where does the global topology (that for the torus is defined by boundary conditions, see this question) enter the Noether theorem (or conservation laws)?



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