Deligne–Beilinson Cohomology and Abelian Link Invariants

For the Abelian Chern–Simons field theory, we consider the quantum functional integration over the Deligne–Beilinson cohomology classes and we derive the main properties of the observables in a generic closed orientable 3-manifold. We present an explicit path-integral non-perturbative computation of the Chern–Simons link invariants in the case of the torsion-free 3-manifolds $S^3$, $S^1\times S^2$ and $S^1\times\Sigma_g$.