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We derive the general supertrace formula for a system with $N$ chiralsuperfields and one nilpotent chiral superfield in global and localsupersymmetry. The nilpotent multiplet is realized by taking thescalar-decoupling limit of a chiral superfield breaking supersymmetryspontaneously. As we show, however, the modified formula is not simply relatedto the scalar-decoupling limit of the supertrace in linearly-realizedsupersymmetry. We also show that the supertrace formula reduces to that of alinearly realized supersymmetric theory with a decoupled sGoldstino if theGoldstino is the fermion in the nilpotent multiplet.

Supertrace formulae for nonlinearly realized supersymmetry