The higher twisted index theorem for foliations

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