in particular, the anomalies coming from fermions and the half-level chern–simons terms will cancel if and only if the number of majorana fermions plus twice the chern–simons level is even. in the case n=1, this statement is the half-integer quantization condition in formula_8 supersymmetric chern–simons gauge theories presented in the chern-simons coefficient in supersymmetric yang-mills chern-simons theories. when n=2 this contribution to the partition function was found in formula_9 and 3 gauge theories in branes and supersymmetry breaking in three dimensional gauge theories.