Fermions on non-trivial topologies

An exact expression for the Green function of a purely fermionic system moving on the manifold R ×ΣD−1, where ΣD−1 is a (D−1)-torus, is found. This expression involves the bosonic analog of χn=einθ corresponding to the irreducible representation for the nth class of homotopy and in the fermionic case for D=2 and 3, χn is a measure of the statistics of the particles. For higher dimensions (D≥4), there is no analogue interpretation however this could, presumably, indicate a generation of mass as in quantum field theories at finite temperature.