Fermions on Non-Trivial Topologies

Abstract

An exact expression for the Green function of a purely fermionic system moving on the manifold $\Re \times Σ^{D-1}$, where $Σ^{D-1}$ is a $(D-1)$-torus, is found. This expression involves the bosonic analog of $χ_n = e^{inθ}$ corresponding to the irreducible representation for the n-th class of homotopy and in the fermionic case for D=2 and 3, $χ_n$ is a measure of the statistics of the particles. For higher dimensions ($D \geq 4$), there is no analogue interpretation however this could, presumably, indicate a generation of mass as in quantum field theories at finite temperature.