The Quantization of the Moduli Space of Flat Connections on a Surface

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The present research lies on the boundary between low dimensional topology and quantum physics. It aims to construct a geometric model for the quantization of the moduli space of flat SU(2)-connections on a surface, with the Lagrangian given by the Chern-Simons functional. This would yield a new method for studying the Jones polynomial of a knot, a fundamental topological invariant, with the tools of quantum mechanics. The main problem is to understand the observables of the quantum mechanical system associated with the Jones polynomial. The quantum physical model in discussion is currently applied to the study of the behaviour of electrons in a strong magnetic field at low temperatures. Physicists use the Chern-Simons functional to couple a magnetic charge to the electric charge of the electron and transform it formally into a particle whose physics is easier to describe. It is believed that the behaviour of electrons is sophisticated enough so that it can be used for writing the code of a quantum computer. We are concerned with the mathematical aspects of this theory.