  We lay down a general framework for how to construct a Topological Quantum
Field Theory $Z_A$ defined on shaped triangulations of orientable 3-manifolds
from any Pontryagin self-dual locally compact abelian group $A$. The partition
function for a triangulated manifold is given by a state integral over the LCA
$A$ of a certain combinations of functions which satisfy Faddeev's operator
five term relation. In the cases where all elements of the LCA $A$ are
divisible by 2 and it has a subgroup $B$ whose Pontryagin dual is isomorphic to
$A/B$, this TQFT has an alternative formulation in terms of the space of
sections of a line bundle over $(A/B)^{2}$. We apply this to the LCA
$\mathbb{R}\times \mathbb{Z}/N\mathbb{Z}$ and obtain a TQFT, which we show is
Quantum Chern-Simons theory at level $N$ for the complex gauge group
$SL(2,\mathbb{C})$ by the use of geometric quantization.
