The higher twisted index theorem for foliations

TL;DR

This paper develops a higher twisted index theorem for foliations, linking twisted K-theory indices with cohomological invariants via a Connes $\

Contribution

It introduces a new Connes $\

Findings

Constructs a Connes $\

couples K-theory indices with twisted cohomology

Computes higher twisted indices as integrals of twisted characteristic classes

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.