In the theory of superalgebras, if A is a commutative superalgebra, V is a free right A-supermodule and T is an endomorphism from V to itself, then the supertrace of T, str(T) is defined by the following trace diagram: More concretely, if we write out T in block matrix form after the decomposition into even and odd subspaces as follows, then the supertrace Let us show that the supertrace does not depend on a basis.
[1] Such a supertrace is not uniquely defined; it can always at least be modified by multiplication by an element of A.
In supersymmetric quantum field theories, in which the action integral is invariant under a set of symmetry transformations (known as supersymmetry transformations) whose algebras are superalgebras, the supertrace has a variety of applications.
In such a context, the supertrace of the mass matrix for the theory can be written as a sum over spins of the traces of the mass matrices for particles of different spin:[2] In anomaly-free theories where only renormalizable terms appear in the superpotential, the above supertrace can be shown to vanish, even when supersymmetry is spontaneously broken.
are the respective tree-level mass matrices for the separate bosonic and fermionic degrees of freedom in the theory and