Quantization via real polarization of the moduli space of flat connections and Chern-Simons gauge theory in genus one

Abstract

We study the quantization of the moduli space of flat connections on a surface of genus one, using the real polarization of this space described in [10]. The quantum wave functions in this formalism are exponential functions supported along the integral fibres of the polarization. The space of wave functions obtained in this way is isomorphic to a space of theta functions. We use our construction to construct part of what may be a topological field theory in genus one, and to compute the associated invariants of some three manifolds. These computations agree with those of Witten [12], but the invariants are expressed as sums of quantities computed at a discrete set of connections with curvature concentrated on a link in the three manifold. A similar prescription is used to produce knot invariants.