Geometric quantization of Chern-Simons gauge theory

Abstract

We present a new construction of the quantum Hubert space of Chern-Simons gauge theory using methods which are natural from the threedimensional point of view.To show that the quantum Hubert space associated to a Riemann surface Σ is independent of the choice of complex structure on Σ, we construct a natural projectively flat connection on the quantum Hubert bundle over Teichmuller space.This connection has been previously constructed in the context of two-dimensional conformal field theory where it is interpreted as the stress energy tensor.Our construction thus gives a (2 + 1 )-dimensional derivation of the basic properties of (1 + 1)-dimensional current algebra.To construct the connection we show generally that for affine symplectic quotients the natural projectively flat connection on the quantum Hubert bundle may be expressed purely in terms of the intrinsic Kahler geometry of the quotient and the Quillen connection on a certain determinant line bundle.The proof of most of the properties of the connection we construct follows surprisingly simply from the index theorem identities for the curvature of the Quillen connection.As an example, we treat the case when Σ has genus one explicitly.We also make some preliminary comments concerning the Hubert space structure.