  In this paper we construct a topological model for the
Witten-Reshetikhin-Turaev invariants for $3$-manifolds as graded intersection
pairings of homology classes in configuration spaces. More precisely, for a
fixed level $\cN \in \N$ we show that the level $\cN$ WRT invariant for a
$3-$manifold is a state sum of Lagrangian intersections in a covering of a
fixed configuration space in the punctured disk. This model brings a new
perspective on the structure of the level $\cN$ Witten-Reshetikhin-Turaev
invariant, showing that it is completely encoded by the intersection points
between certain Lagrangian submanifolds in a fixed configuration space, with
additional gradings which come from a particular choice of a local system.
