From Compact U ( 1 ) Maxwell Chern - Simons to Azbel - Hofstadter : Testing Magnetic Monopoles and Gravity to ∼ 10 − 17 m on a Nearby Laboratory

It is built a map between the Topological Quantum Field Theory (TQFT) 3D compact U(1) gauge Maxwell Chern-Simons 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 dis-cretized U(1) gauge transformations, they correspond to subgroups with linear gauge parameter. The monopole confining and monopole condensate phases in the TQFT are identified with the extended (energy bands) and localized (gaps) phases of the bloch electron. The monopole condensate induces at the nonrelativistic level gravi-tational white holes due to deformed classical gauge fields, furthermore these grav-itational solutions render the existence of finite energy pure magnetic monopoles possible. This mechanism constitutes a dynamical symmetry breaking of the theory which regularizes the solutions on those localized phases allowing physical solutions of the Shrödinger equation which are chains of electron strings connecting several monopole-white holes. An experiment to test these results is suggested which would test the existence of magnetic monopoles and classical gravity to a scale of ∼ 10 −17 meters, the dimension of the monopole-white hole.