  Given a gerbe $L$, on the holonomy groupoid $\mathcal G$ of the foliation
$(M, \mathcal F)$, whose pull-back to $M$ is torsion, we construct a Connes
$\Phi$-map from the twisted Dupont-Sullivan bicomplex of $\mathcal G$ to the
cyclic complex of the $L$-projective leafwise smoothing operators on $(M,
\mathcal F)$. Our construction allows to couple the $K$-theory analytic indices
of $L$-projective leafwise elliptic operators with the twisted cohomology of
$B\mathcal G$ producing scalar higher invariants. Finally by adapting the
Bismut-Quillen superconnection approach, we compute these higher twisted
indices as integrals over the ambiant manifold of the expected twisted
characteristic classes.
