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Non-abelian theta functions a la Andre Weil
by
Razvan Gelca
Texas Tech University
Coauthors: Alejandro Uribe, University of Michigan
In 1964 Andre Weil pointed out to the existence of an action of a Heisenberg group on classical theta functions. This action induces, via a Stone-von Neumann theorem, the previously known Hermite-Jacobi action of the modular group. From the quantum mechanical point of view, theta functions, the finite Heisenberg group, and the Hermite-Jacobi action (given by discrete Fourier transforms) are the analogues, for a particle with periodic position and momenta, of what the square integrable functions, the Heisenberg with real entries, and the metaplectic representation (given by Fourier transforms) are for a free particle.
Non-abelian theta functions are defined as holomorphic sections of the Chern-Simons line bundle over the moduli space of connections on a surface. In the case where the gauge group is U(1) one recovers the classical theta functions. There is a combinatorial description of theta functions, as graphs colored by irreducible representations of quantum groups. The functions on the moduli space which are traces of holonomies along simple closed curves, the so called Wilson lines, admit a quantization as operators acting on theta functions, which can again be described using quantum groups. In the present talk we will explain that the algebra of quantum group quantizations of Wilson lines and the Reshetikhin-Turaev representation of the mapping class group of the surface are non-abelian analogues of the group algebra of the finite Heisenberg group and of the Hermite-Jacobi action.
Date received: March 11, 2010
Copyright © 2010 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbaf-03.