  Let $L$ be a closed, orientable, monotone Lagrangian 3-manifold of a
symplectic manifold $(M, \omega)$, for which there exists a local system such
that the corresponding Lagrangian quantum homology vanishes. We show that its
cohomology ring satisfies a certain dichotomy, which depends only on the parity
of the first Betti number of $L$. Essentially, the triple cup product on the
first cohomology group is shown to be either of maximal rank or identically
zero. This in turn influences the Landau-Ginzburg superpotential of $L$: either
one of its partial derivatives do not vanish on the corresponding local system,
or it is globally constant. We use this to prove that quantum Reidemeister
torsion is invariant and can be expressed in terms of open Gromov-Witten
invariants of $L$.
