Microscopic foundations of nuclear supersymmetry

We discuss a microscopic framework for phenomenological boson-fermion models of nuclear structure based on the U($n/m$) type of superalgebras. The generalized Dyson mapping of fermion collective superalgebras provides a basis to do so and to understand how collectivity selects the required preservation of boson plus fermion number as a good quantum number. We also consider the difference between dynamical and invariant supersymmetries based on possible supermultiplets of spectra of neighboring odd and even nuclei. We point out that different criteria exist for choosing the appropriate single particle transfer operators in the two cases and discuss a microscopically based method to construct these operators in the case of dynamical supersymmetry.