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 = e inθ corresponding to the irreducible representation for the n-th 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. Typeset using REVTEX E-mail: [email protected] 1 The scattering of fermions by vortices is a problem that has important applications in particle physics [1] and in the realistic description of the Aharonov-Bohm effect [2]. Although this topic was partially discussed in the literature and new corrections to the AB cross section have been found [3], as far as we know, there are no studies trying to understand the connection between these processes and the possibility of summing over the homotopy classes as in the bosonic case [4], [5]. If we have a multiply connected manifold with one vortex, the propagation amplitude for a spinless particle is