Fractionization of the Witten index

We study the regularization dependence of Witten's index for supersymmetric theories. We show that on a non compact space the index in general depends on how it is regulated. Also there is a dependence on the parameters of the theory. In a theory with an energy gap between the ground state and the first excited state, an integer valued index can be obtained by taking an appropriate limit of the regulators. However we find that this procedure yields a fractional result if the energy spectrum of a theory has a continuum component extending to zero. (orig.)