Fermions on non-trivial topologies

An exact expression for the Green function of a purely fermionic system moving on the manifold R x Sigma(D-1), where Sigma(D-1) is a (D - 1)-torus, is found. This expression involves the bosonic analog of chi(n) = e(in theta) corresponding to the irreducible representation for the nth class of homotopy and in the fermionic case for D = 2 and 3, chi(n) is a measure of the statistics of the particles. For higher dimensions (D greater than or equal to 4), there is no analogue interpretation however this could, presumably, indicate a generation of mass as in quantum field theories at finite temperature. (C) 2000 Elsevier Science B.V. All rights reserved.