  We present a detailed theory for finite-frequency conductivities
Re$[\sigma_{\alpha\beta}(\omega)]$ of quantum Hall stripes, which form at
Landau level $N\geq 2$ close to half filling, in the presence of weak Gaussian
disorder. We use an effective elastic theory to describe the low-energy
dynamics of the stripes with the dynamical matrix being determined through
matching the density-density correlation function obtained in the microscopic
time-dependent Hartree-Fock approximation. We then apply replicas and the
Gaussian variational method to deal with the disorder. Within this method, a
set of saddle point equations for the retarded self energies are obtained,
which are solved numerically to get Re$[\sigma_{\alpha\beta}(\omega)]$. We find
a quantum depinning transition as $\Delta\nu$, the fractional part of the
filling factor, approaches a critical value $\Delta\nu_c$ from below. For
$\Delta\nu<\Delta\nu_c$, the pinned state is realized in a replica symmetry
breaking (RSB) solution, and the frequency-dependent conductivities in both the
directions perpendicular and parallel to the stripes show resonant peaks. These
peaks shift to zero frequency as $\Delta\nu\to \Delta\nu_c$. For
$\Delta\nu\ge\Delta \nu_c$, we find a partial RSB (PRSB) solution in which
there is RSB perpendicular to the stripes, but replica symmetry along the
stripes, leading to free sliding along the stripe direction. The quantum
depinning transition is in the Kosterlitz-Thouless universality class. The
result is consistent with a previous renormalization group analysis.
