Quantum corrections to the conductivity of fermion - gauge field models: Application to half filled Landau level and high-T_c superconductors

We calculate the Altshuler-Aronov type quantum correction to the conductivity of $2d$ charge carriers in a random potential (or random magnetic field) coupled to a transverse gauge field. The gauge fields considered simulate the effect of the Coulomb interaction for the fractional quantum Hall state at half filling and for the $t-J$ model of high-$T_c$ superconducting compounds. We find an unusually large quantum correction varying linearly or quadratically with the logarithm of temperature, in different temperature regimes.