  We present a microscopic model for calculating the AC conductivity of a
finite length line junction made up of two counter or co-propagating single
mode quantum Hall edges with possibly different filling fractions. The effect
of density-density interactions and the tunneling conductance (\sigma), between
the two edges, is considered, and exact expressions are derived for the AC
conductivity as a function of the frequency (\omega), the length of the line
junction and other parameters of the system. We reproduce the results of Phys.
Rev. B 78, 085430 (2008) in the DC limit (\omega \to 0), and generalize those
results for an interacting system. As a function of \omega, the AC conductivity
shows significant oscillations if \sigma is small; the oscillations become less
prominent as \sigma increases. A renormalization group analysis shows that the
system may be in a metallic or an insulating phase depending on the strength of
the interactions. We discuss the experimental implications of this for the
behavior of the AC conductivity at low temperatures.
