In the last few years, the supersymmetry method was generalized to real-symmetric, Hermitean, and Hermitean self-dual random matrices drawn from ensembles invariant under the orthogonal, unitary, and unitary symplectic group, respectively. We extend
this supersymmetry approach to chiral random matrix theory invariant under the three chiral unitary groups in a unifying way. Thereby we generalize a projection formula providing a direct link and, hence, a `short cut between the probability density in ordinary space and the one in superspace. We emphasize that this point was one of the main problems and critiques of the supersymmetry method since only implicit dualities between ordinary and superspace were known before. As examples we apply this approach to the calculation of the supersymmetric analogue of a Lorentzian (Cauchy) ensemble and an ensemble with a quartic potential. Moreover we consider the partially quenched partition function of the three chiral Gaussian ensembles corresponding to four-dimensional continuum QCD. We identify a natural splitting of the chiral Lagrangian in its lowest order into a part of the physical mesons and a part associated to source terms generating the observables, e.g. the level density of the Dirac operator.
In many--body and other systems, the physics situation often allows one to interpret certain, distinct states by means of a simple picture. In this interpretation, the distinct states are not eigenstates of the full Hamiltonian. Hence, there is an in
teraction which makes the distinct states act as doorways into background states which are modeled statistically. The crucial quantities are the overlaps between the eigenstates of the full Hamiltonian and the doorway states, that is, the coupling coefficients occuring in the expansion of true eigenstates in the simple model basis. Recently, the distribution of the maximum coupling coefficients was introduced as a new, highly sensitive statistical observable. In the particularly important regime of weak interactions, this distribution is very well approximated by the fidelity distribution, defined as the distribution of the overlap between the doorway states with interaction and without interaction. Using a random matrix model, we calculate the latter distribution exactly for regular and chaotic background states in the cases of preserved and fully broken time--reversal invariance. We also perform numerical simulations and find excellent agreement with our analytical results.
