  It is shown that the Jain mapping between states of integer and fractional
quantum Hall systems can be described dynamically as a perturbative
renormalization of an effective Chern-Simons field theory. The effects of
mirror duality symmetries of toroidally compactified string theory on this
system are studied and it is shown that, when the gauge group is compact, the
mirror map has the same effect as the Jain map. The extrinsic ingredients of
the Jain construction appear naturally as topologically non-trivial field
configurations of the compact gauge theory giving a dynamical origin for the
Jain hierarchy of fractional quantum Hall states.
