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 non-relativistic quantum mechanics (semi-classical) Azbel-Hofstadter model of Bloch-electrons. The U q (sl 2) quantum group of magnetic translations in the Azbel-Hofstadter model is interpreted as discretized gauge transformations on the TQFT. The monopole confining and monopole condensate phases in the TQFT are identified with the extended (energy bands) and localised (gaps) phases of the bloch electron. The monopole condensate induces at the semi-classical level gravitational white holes due to deformed classical gauge fields. This mechanism regularises the theory on those phases allowing physical solutions of the Shrödinger equation which are chains of electron strings and 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 which is the dimension of the monopole-white hole).