Chern-Simons Theory on Seifert 3-Manifolds

TL;DR

This work provides an exact, non-perturbative evaluation of the Chern-Simons partition function on Seifert 3-manifolds by applying Abelianisation to reduce the problem to a 2D Abelian theory on the orbifold base Σ. The authors derive a precise finite-dimensional integral over the Cartan subalgebra modulo the affine Weyl group, with the integrand given by the square root of the Ray-Singer torsion and a calculable phase from the eta invariant, incorporating orbifold data via Kawasaki index-type reasoning. The construction reproduces and unifies the Lawrence-Rozansky and Mariño results and extends them to orbifold bases, while also allowing inclusion of Wilson loops along the fibre. These results offer a tractable exact framework for non-Abelian CS theory on a broad class of 3-manifolds and connect to broader geometric structures such as torsion, orbifold index theory, and line-bundle data. The approach is poised to impact related theories (e.g., BF theory, gravity) and provides a bridge between path-integral methods and geometric/topological invariants on orbifold bases.

Abstract

We study Chern-Simons theory on 3-manifolds M that are circle-bundles over 2-dimensional orbifolds S by the method of Abelianisation. This method, which completely sidesteps the issue of having to integrate over the moduli space of non-Abelian flat connections, reduces the complete partition function of the non-Abelian theory on M to a 2-dimensional Abelian theory on the orbifold S which is easily evaluated.