  It is built a map between an Abelian Topological Quantum Field Theory, $2+1D$
compact U(1) gauge Maxwell Chern-Simons Theory and the nonrelativistic quantum
mechanics Azbel-Hofstadter model of Bloch electrons. The $U_q(sl_2)$ quantum
group and the magnetic translations group of the Azbel-Hofstadter model
correspond to discretized subgroups of U(1) with linear gauge parameters. The
magnetic monopole confining and condensate phases in the Topological Quantum
Field Theory are identified with the extended (energy bands) and localized
(gaps) phases of the Bloch electron. The magnetic monopole condensate is
associated, at the nonrelativistic level, with gravitational white holes due to
deformed classical gauge fields. These gravitational solutions render the
existence of finite energy pure magnetic monopoles possible. This mechanism
constitutes a dynamical symmetry breaking which regularizes the solutions on
those localized phases allowing physical solutions of the Shr\"odinger equation
which are chains of electron filaments connecting several monopole-white
holes.To test these results would be necessary a strong external magnetic field
$B\sim 5 T$ at low temperature $T<1 K$. To be accomplished, it would test the
existence of magnetic monopoles and classical gravity to a scale of $\sim
10^{-15}$ \textit{meters}, the dimension of the monopole-white hole. A proper
discussion of such experiment is out of the scope of this theoretical work.
