CHERN-SIMONS THEORY, MODULAR FUNCTIONS AND QUANTUM MECHANICS IN AN ALCOVE

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Abstract
We show how the vacuum expectation value of the Wilson loop of the trivial knot in the left-regular representation in a Chern-Simons theory is basically the partition function for a quantum particle confined to a certain bounded region (namely, an alcove of the gauge group Lie algebra). For example, for su(3) the particle is confined to an equilateral triangle. The result follows from mathematical work on the category-theoretic rank of quantum groups obtained in a previous paper. In the present paper we give the details of the physical interpretation and discuss the implications. In particular, both these physical systems are connected with number theory.

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