Twisted longitudinal index theorem for foliations and wrong way functoriality

For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of Connes-Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai-Melrose-Singer. We construct the pushforward map in twisted K-theory associated to any smooth (generalized) map $f:W\longrightarrow M/F$ and a twisting $\sigma$ on the holonomy groupoid $M/F$, next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes-Skandalis when the twisting is trivial and of Carey-Wang for manifolds.

💡 Research Summary

The paper develops a comprehensive analytic index theory for Lie groupoids equipped with a PU(H)‑principal bundle twisting, and extends the longitudinal index theorem to foliated manifolds with such twistings. The authors begin by recalling Connes’ tangent groupoid construction, which provides a deformation from a Lie groupoid G to its Lie algebroid. By incorporating a PU(H)‑principal bundle σ over G, they obtain a twisted groupoid C*‑algebra C*(G,σ) and a corresponding twisted K‑theory Kσ*. Using the deformation quantization framework, they define an analytic index morphism
Ind_a^σ : Kσ^0(TG) → Kσ^0(C(G,σ)).
When σ is trivial this reduces to the classical Connes‑Skandalis analytic index for groupoids, confirming the consistency of the construction.

The second major part treats a smooth foliated manifold (M,F) with holonomy groupoid G_F = Hol(M,F). A twisting σ on G_F yields a σ‑twisted longitudinal symbol class in Kσ^0(TF). The authors construct a twisted longitudinal index map
i_σ : Kσ^0(TF) → Kσ^0(C*(G_F,σ)),
showing that it coincides with the Connes‑Skandalis longitudinal index theorem when σ is trivial. In the special case where the foliation comes from a fibration π:E→B, the holonomy groupoid is Morita equivalent to the pair groupoid of the fibers, and i_σ matches the twisted families index introduced by Mathai‑Melrose‑Singer. This identification demonstrates that the new index captures both the geometric data of the foliation and the topological information encoded in the twisting.

The third and most innovative contribution is the construction of a “wrong‑way” (push‑forward) map in twisted K‑theory associated to any smooth generalized map f : W → M/F together with a twisting σ on the leaf space. By interpreting f as a groupoid correspondence and employing an adiabatic deformation of the associated tangent groupoids, the authors define a push‑forward
f_! : K_{σ∘f}^(W) → K_σ^{+d}(M/F),
where d is the longitudinal dimension of the foliation. They prove functoriality: for composable maps g∘f, (g∘f)! = g!∘f_!. This extends the wrong‑way functoriality results of Connes‑Skandalis (untwisted case) and Carey‑Wang (twisted manifolds) to the broader setting of foliated leaf spaces with arbitrary twistings.

In the final section the authors discuss several applications. They outline a twisted index pairing between K‑homology and twisted K‑theory, a twisted eta‑invariant associated to longitudinal Dirac operators, and the emergence of higher rho‑invariants and secondary index invariants in the twisted context. The work thus provides a robust toolkit for noncommutative geometry, higher index theory, and twisted field theories, bridging deformation quantization, groupoid C*‑algebras, and foliation theory in a unified framework.