Theta functions, root systems and 3-manifold invariants (Q1913689)

The author presents an extension of the Witten theory for the quantization of the Chern-Simons action in the case of a general gauge group \(G\). He considers a Chern-Simons type action on the higher dimensional Jacobian tori \(\text{Jac}(\Sigma_g) \times \mathbb{R}\), where \(\Sigma_g\) is a Riemann surface of genus \(g\). The space \(N_{\Sigma_g} = \text{Hom}(\pi_1 (\text{Jac} (\Sigma_g)), G)/G\) of the representations of \(\pi_1(\text{Jac} (\Sigma_g))\) has a simple description and the associated bundle may be extended to a projectively flat bundle over the moduli space of principally polarized abelian varieties. As a consequence, it is obtained a family of invariants for closed oriented 3-manifolds which do coincide with those defined by Witten for lens spaces and torus bundles.