  Motivated by the smallness of the cosmological constant we investigate
whether it is possible to have vanishing one-loop heterotic string partition
functions for six-dimensional non-supersymmetric toroidal orbifolds. A
straightforward way to realize this presents itself, when each orbifold sector
separately preserves some Killing spinors, but none of them survives in all
sectors combined. By applying some representation theory to the abstract finite
point groups underlying toroidal orbifolds it turns out, that this is never
possible. This leads to a nonexistence proof of locally but not globally
supersymmetric orbifolds.
