N ) Super - Current Algebras for Disordered Dirac Fermions in Two Dimensions

We consider the non-hermitian 2D Dirac Hamiltonian with (A): real random mass, imaginary scalar potential and imaginary gauge field potentials, and (B): arbitrary complex random potentials of all three kinds. In both cases this Hamiltonian gives rise to a delocalization transition at zero energy with particle-hole symmetry in every realization of disorder. Case (A) is in addition time-reversal invariant, and can also be interpreted as the random-field XY Statistical Mechanics model in two dimensions. The supersymmet-ric approach to disorder averaging results in current-current perturbations of gl(N |N) super-current algebras. Special properties of the gl(N |N) algebra allow the exact computation of the βeta-functions, and of the correlation functions of all currents. One of them is the Edwards-Anderson order parameter. The theory is 'nearly conformal' and possesses a scale-invariant subsector which is not a current algebra. For N = 1, in addition , we obtain an exact solution of all correlation functions. We also study the delocalization transition of case (B), with broken time reversal symmetry, in the Gade-Wegner (Random-Flux) universality class, using a GL(N |N ; C)/U (N |N) sigma model, as well as its P SL(N |N) variant, and a corresponding generalized random XY model. For N = 1 the sigma model is shown to be identical to the current-current perturbation. For the delocal-ization transitions (case (A) and (B)) a density of states, diverging at zero energy, is found.