The higher twisted index theorem for foliations

Abstract

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.