The higher twisted index theorem for foliations

Abstract

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.

arXiv admin note: text overlap with arXiv:1007.3667