A computation of Tr(−1)F in supersymmetric gauge theories with matter

We introduce “multi-twisted” boundary conditions (MTBC) for gauge theories in a finite volume. MTBC can be applied for a class of gauge theories with non-trivial centers and allow the existence of some matter multiplets which transform non-trivially under the center group. The Witten index for the supersymmetric extension of the gauge theory can be calculated when global abelian symmetries exist and turns out to be non-zero and independent of the matter representations.