---
abstract: 'We explain how a simple twisting of the notion of spectral triple allows to incorporate type III examples, such as those arising from the transverse geometry of codimension one foliations.  We show that the classical cyclic cohomology valued Chern character of finitely summable spectral triples extends to the twisted case and lands in ordinary (untwisted) cyclic cohomology.  The index pairing with ordinary (untwisted) K-theory continues to make sense and the index formula is given by the pairing of the corresponding Chern characters.  This opens the road to extending the local index formula to the type III case.' 
address:
- |
    A. Connes: Collège de France\
    3, rue d’Ulm\
    Paris, F-75005 France\
    I.H.E.S. and Vanderbilt University
- 'H. Moscovici: Department of mathematics, The Ohio State University, Columbus, OH 43210, USA'
author:
- Alain Connes
- Henri Moscovici
title: Type III and spectral triples
---

[^1]

Introduction
============

The basic paradigm of noncommutative geometry is that of a spectral triple $(\cA , \fH , D) $ ([[*cf.*]{}  ]{}[@book], [@C-M2]), where the algebra $\cA$ encodes the space and the operator $D$ encodes the metric.  In the finite dimensional situation, [[*i.e.*]{} ]{}when there is an $\, \alpha > 0$ such that the $n$-th characteristic value of the resolvent of $D$ decays as $n^{- \alpha}$ for $n\to \infty$, the Dixmier trace [@book] induces a nontrivial trace on the algebra $\cA$. The existence of a trace is a characteristic of the type II situation in the Murray-von Neumann classification of rings of operators.  Thus, in essence the theory is, in its finite dimensional form, restricted to the type II case. 

We shall explain in this note how a simple twisting of the notion of spectral triple allows to incorporate type III examples, such as those arising from the transverse geometry of codimension one foliations.  Since the twisting of the commutators turns the usual hypertrace constructed out of the Dixmier trace into a twisted trace on the coordinate algebra, one would be tempted to interpret that as a manifestation of twisting at the level of cyclic cohomology, akin to that introduced by the authors in the context of Hopf cyclic cohomology [@C-M].  The main point of this note, besides giving simple natural examples of the general notion and developing the first basic steps of the theory, is to show that contrary to the initial expectations no cohomological twisting is in fact required.  The Chern character of finitely summable spectral triples extends to the twisted case, and lands in fact in ordinary (untwisted) cyclic cohomology.  The same holds true for the local Hochschild character.  The index pairing with ordinary (untwisted) K-theory continues to make sense and the index formula is still given by the pairing of the corresponding Chern characters.  This opens the road to extending the local index formula, as well as the analogue of the hypoelliptic construction on the dual system together with the corresponding Thom isomorphism, to the context of twisted spectral triples of type III. 

Two motivating examples
=======================

Dirac operator
--------------

We start by recalling the classical comparison formula for the Dirac operators associated to conformally equivalent metrics, [[*cf.*]{}  ]{}[@hitchin], [@bourg].  Given a compact spin manifold $M^n$, to each Riemannian metric $g$ on $M$ one can canonically associate a Dirac operator $\Dirac = \Dirac^g$ acting on the Hilbert space $\fH = \fH^g : = L^2(M,S^g)$ of $L^2$-sections of the spin bundle $S= S^g$, and thus a corresponding spectral triple (cf. [ @book]) $(\cA, \fH,\,  \Dirac)$ over the algebra $\cA: = C^\infty(M)$. Let $h\in C^\infty(M)$ be a self-adjoint element and replace $g$ by the rescaled metric $g' = e^{-4h}\,g$. After identifying the corresponding spin bundles via the $Spin_n$-equivariant transformation $\beta^g_{g'}$ from $g$-spinorial frames to $g^{\,\prime}$-spinorial frames defined in [@bourg], the gauge transformed operator $\,^g \Dirac^{\,g'}\,  := \, \beta^{g'}_g \circ \Dirac^{\,g'} \circ
\beta^g_{g'}$ has the expression, [[*cf.*]{}  ]{}[@bourg (26)], $$^g \Dirac^{\,g'} \,  = \, e^{(n+1) h}\circ {\Dirac}^{\,g} \circ
e^{-(n-1)h} \, .$$ In order to account for the change of the Riemannian volume form, $ \vol_{g'} = e^{-2nh}\,\vol_g$, at the level of $L^2$-sections one needs to further rescale the canonical identification by setting $$\tilde{\beta}^g_{g'} \, := \, e^{nh} \circ \beta^g_{g'} \, = \,
  \beta^g_{g'} \circ e^{nh}: \fH^g \rightarrow \fH^{g'} .$$ This shows that the gauge transformed spectral triple is simply obtained by replacing $\Dirac$ with $${\Dirac}^{\,\prime} \, =\, e^{h}\,\Dirac \,e^{h} \, .$$

Perturbed spectral triple {#example1}
-------------------------

In the general case, when one starts from an arbitrary spectral triple $(\cA , \fH , D) $ ([[*cf.*]{}  ]{}[@book], [@C-M2]) and a self-adjoint element $h = h^* \in \cA$, it is natural to wonder what are the properties of the ‘perturbed’ triple $$(\cA , \fH , D') \,,\quad D'=\,e^{h}\,D\,e^{h} .\label{spectrip}$$ The operator $D'$ is still self-adjoint but the basic boundedness condition $$[D,a] \quad \hbox{is bounded} \quad \forall \, a \in \cA \, ,
\label{boundedcom}$$ will not necessarily hold, unless $h$ is in the center of $\cA$.

However, the following simple fact holds true. 

\[conjugate\] For any self-adjoint element $h \in \cA$, letting $$\sigma(a)=\,e^{2h}\,a\,e^{-2h}, \quad  a \in \cA , \label{dfesigma}$$ one has $$d^{\,\prime}_\sigma a \, := \, D'\,a-\,\sigma(a)\,D' \quad \hbox{is
bounded} \quad \forall \, a \in \cA \, .  \label{boundedcom1}$$

The fact that the usual commutators $[D,b]$, $b \in \cA$, are bounded implies the boundedness of $$\label{bound'}
d^{\,\prime}_\sigma a \, = \, e^{h}\,D\,e^{h}\,a\,
-\,e^{2h}\,a\,e^{-h}\,D\, e^{h} \, = \, e^{h} \, [D, b] \, e^{h} \,
, \qquad b=e^{h}\,a\,e^{-h} .$$

Transverse spectral triple  {#trans1}
---------------------------

As a next example we take a codimension $1$ foliation and consider the corresponding noncommutative algebra $\cA$ of ‘transverse coordinates’.  In fact, as a further simplification we restrict to a complete transversal and take for $\cA$ the algebraic crossed product of the algebra $C^\infty(S^1)$ of smooth functions on $S^1$ by a group $\Gamma$ of orientation preserving diffeomorphisms.  Any element of $\cA=\,C^\infty(S^1)\rtimes \Gamma$ is represented as a finite sum of the form $$a=\,\sum_\Gamma
a_\phi\, U^*_\phi \,,$$ the product rule is determined by $$U^*_\phi\,f=\,(f \circ\phi)\, U^*_\phi \,,\quad
U^*_\phi\,U^*_\psi=U^*_{\psi\,\phi}\, , \label{product}$$ and the involution $$a=\,\sum_\Gamma
a_\phi\, U^*_\phi \quad \longmapsto \quad
a^* =  \,\sum_\Gamma U_\phi \, \bar{a}_\phi \, .  \label{involution}$$

One represents $\cA=\,C^\infty(S^1)\rtimes \Gamma$ in the Hilbert space $\fH=\,L^2(S^1)$ by the $\ast$-representation $$(\pi
(g\,U^*_\phi)\,\xi)(x)=\,g(x)\,\phi'(x)^{\frac{1}{2}}\,\xi(\phi(x))\,
, \quad \forall \xi \in \fH \, , \quad x\in \R/\Z .  \label{repa}$$ In the role of $D$ we take the operator $\, \Dirac =\,\frac{1}{i}\,\frac{d}{dx}$, while the automorphism $\sigma \in \Aut \cA$ is defined on the monomials generating $\cA$ by $$\label{dfesigma1}
 \sigma(g\,U^*_\phi)=\,\frac{d\phi(x)}{dx}\,g\,U^*_\phi \, .$$ One then has the following boundedness property. 

\[foliation\] For any $\, a \in \cA$, the twisted commutators $$\begin{aligned}
\Dirac \circ \pi (a) \,&-&\,\pi (\sigma(a))\circ \Dirac
  \, , \label{boundedcom2} \\
\text{\rm and} \qquad |\Dirac| \circ \pi (a) \,&-&\,\pi (\sigma(a))\circ |\Dirac|
 \label{boundedcom3}\end{aligned}$$ are bounded. 

For any $ a = g\,U^*_\phi$ one has: $$\begin{aligned}
\Dirac (\pi (a) \xi) (x)&= \frac{1}{i}\,\frac{d}{dx} \left(g(x)
\phi'(x)^{\frac{1}{2} }\right)\,\xi(\phi(x)) +
g (x) \phi'(x)^{\frac{1}{2} } \, \Dirac  (\xi \circ \phi)(x)\\
 &= \frac{1}{i}\,\frac{d}{dx} \left(g(x) \phi'(x)^{\frac{1}{2} }\right)\,\xi (\phi(x)) +
 g (x) \phi'(x)^{\frac{1}{2}} \, \left( \phi' (x) \, (\Dirac  (\xi) )(\phi(x)) \right) \, ;\end{aligned}$$ equivalently, $$\label{bdcom}
\Dirac  ((\pi (a)) (\xi) ) (x) -  \pi (\sigma (a)) (\Dirac
(\xi) ) (x) =  \frac{1}{i}\,\frac{d}{dx} \left(g(x) \phi'(x)^{\frac{1}{2}}\right)\,\phi'(x)^{-\frac{1}{2} }\, 
(\pi (U^*_\phi) \xi)(x),$$ which proves the boundedness of the twisted commutator . 

To prove the same for we shall switch from direct calculation to an equally elementary symbolic argument.  Letting $\, V_\phi$ denote the translation operator by $\, \phi \in \Gamma$, $$(V_\phi\,\xi)(x)\, =\,\xi(\phi^{-1}(x))\,
, \quad \forall \xi \in \fH \, , \quad x\in \R/\Z \, ,$$ one has for any $ a = g\,U^*_\phi \in \cA$, $$\big( |\Dirac| \circ \pi(a)\, -\,\pi(\sigma(a)) \circ |\Dirac| \big)\circ V_\phi \, = \,
|\Dirac|  \circ g \,\phi'^{\frac{1}{2}}  \, - \, g \,\phi'^{\frac{1}{2}} \, \phi' \, V^{-1}_\phi \circ |\Dirac|  \circ V_\phi \, .$$ Now $V^{-1}_\phi \circ |\Dirac|  \circ V_\phi $ is a $1$-st order (classsical) pseudodifferential operator whose principal symbol is $\displaystyle \frac{1}{\phi'}$ times the principal symbol of $|\Dirac| $.  It follows that the right hand is a pseudodifferential operator of order $0$, hence bounded. 

\[orderdecrease\] [The symbolic argument given above applies more generally to any pseudodifferential operator of arbitrary order $m \in \R$. Thus, if $\, P \in \Psi DO^m (S^1)$, then for all $\, a \in \cA$]{} $$\label{lowerorder}
\big( P \circ \pi(a)\, -\,\pi(\sigma^m(a)) \circ P \big)\circ V_\phi \ \in \, \Psi DO^{m-1} (S^1) \, .$$

[The canonical state $\varphi$ on $\cA$, $$\varphi(f \, U^*_\phi)=0 \;\textit{ if }\; \phi\neq 1\,,\quad \text{and} \quad
\varphi(f)=\int_{\R/\Z} \,f (x) \,dx \label{state}$$ is a $\sigma^{-1}$-*trace*, [[*i.e.*]{} ]{}satisfies $$\label{sigmatrace}
\varphi(a\, b)\, =\, \varphi(b \, \sigma^{-1} (a))\, , \qquad \forall \, a, b \in \cA \, ,$$ and its modular automorphism group is precisely the one-parameter group of automorphisms $$\label{dfesigma1t}
 \sigma_t(g\,U^*_\phi)=\,\left(\frac{d\phi(x)}{dx}\right)^{it}\,g\,U^*_\phi\, , \qquad t \in \R \, ,$$ whose value (after analytic continuation) at $\, t = -i$ coincides with $\sigma$.]{}

$\sigma$-spectral triples and their basic properties
====================================================

Elementary properties
---------------------

The usual definition of a spectral triple extends to the context illustrated by the preceding examples as follows. 

With $\sigma$ being an automorphism of $\cA$, an [ungraded]{} $\sigma$-spectral triple $(\cA , \fH , D)$ is given by an action of $\cA$ in the Hilbert space $\fH$, while $D$ is a self-adjoint operator with compact resolvent and such that $$D\,a\, -\,\sigma(a)\,D \quad \hbox{is bounded} \quad \forall \, a \in \cA \, . 
\label{boundedcom4}$$ A [graded]{} $\sigma$-spectral triple is similarly defined, with the additional datum of a grading operator $$\gamma = \gamma^* \in \cL (\fH) \, , \quad \gamma^2 = I$$ which commutes with the action of $\cA$, while $$D\,\gamma \, = \, -\,\gamma\,D \, . 
\label{anticomm}$$ A [Lipschitz-regular]{} $\sigma$-spectral triple is one that satisfies the additional condition $$\forall \, a \in \cA \, , \quad |D|\,a-\,\sigma(a)\,|D| \quad  \hbox{is bounded}. 
\label{boundedcom5}$$

In the case when $\cA $ is an involutive algebra and the representation is involutive, to ensure the compatibility between the automorphism $\sigma$ and the $\ast$-involution, we impose the additional *unitarity condition*: $$\sigma(a^*)=\,(\sigma^{-1}(a))^* {{\,,\quad \forall\,}}\, a \in \cA \, . 
\label{sigmainv}$$

Lemma \[foliation\] shows that the transverse Dirac operator on a codimension $1$ foliation gives rise to a Lipschitz-regular $\sigma$-spectral triple with $\sigma$ given by the Jacobian of the holonomy.  In particular, since this spectral triple is $1$-summable, one gets examples of finitely summable $\sigma$-spectral triples for which the representation of $\cA$ in $\fH$ generates a type III factor. 

Let us spell out the extensions to the twisted case of some basic properties of spectral triples.  For background on spectral triples, including notational conventions used below, we refer the reader to [@book IV.2] and [@C-M2 Appendix A], while for the notion of $\sigma$-trace see [@C-M]. 

First of all, we note that any twisted spectral triple which is Lipschitz-regular can be canonically ‘untwisted’ by passage to its ‘phase’.  This is quite clear in the case of the second example ([[*cf.*]{}  ]{}§\[trans1\]), since the phase of $\, \Dirac
=\,\frac{1}{i}\,\frac{d}{dx}$ is the Hilbert transform, and is actually easy to prove in full generality. 

\[untwist\] If the $\sigma$-spectral triple $(\cA , \fH , D)$ is Lipschitz-regular and $\, F =  D \, |D|^{-1} $, then $( \fH , F)$ is a Fredholm module over $\cA$. If moreover $(\cA , \fH , D)$ is finitely summable, so is $( \fH , F)$.

Indeed, for any $\, a \in \cA$, $$D\,a\, -\,\sigma(a)\,D \, = \, |D|\, (F\,a\, -\, a\, F ) \, + \, (|D|\,a-\,\sigma(a)\,|D| )\, F \, ,$$ therefore $$[F, a] \, = \, |D|^{-1} \, \big(  (D\,a\, -\,\sigma(a)\,D) \, -
\, (|D|\,a-\,\sigma(a)\,|D| )\, F \big) \, .$$ Thus, all these commutators are compact operators, and in fact they are quantized differentials of the same order as $\, D^{-1}$.

\[resandsig\] Let $(\cA , \fH , D)$ be a $\sigma$-spectral triple with $D^{-1}\in \cL^{n,\infty}$.

-   The linear functional $$a \in \cA \mapsto \varphi(a)=\,\cutint \,a\,D^{-n} \, := \Tr_\omega (a\,D^{-n})$$ is a $\sigma^{-n}$-trace on $\cA$: $\quad \varphi(a\, b)\, =\, \varphi(b \, \sigma^{-n} (a))$, $\quad \forall \, a, b \in \cA$.

-   More generally, for any bounded operator $T \in \cL (\fH)$, $$\label{dixsig2}
    Tr_\omega (T \, \sigma^{-n} (a) \,D^{-n}) \, = \, Tr_\omega (a\, T \, D^{-n} ) \, ,
    \qquad \forall \, a \in \cA \, .$$

-   When the $\sigma$-spectral triple $(\cA , \fH , D)$ is Lipschitz-regular, the same hold true when $\,D^{-n}$ is replaced by $\,|D|^{-n}$.

Let us show by induction that for any $\, 1 \leq k \leq n$ one has $$\label{ord-k}
D^{-k}\,a-\,\sigma^{-k}(a)\,D^{-k}\in \cL_0^{\frac{n}{k} \, ,\infty}{{\,,\quad \forall\,}}a \in \cA .$$ Clearly, $$D^{-1}\,a-\,\sigma^{-1}(a)\,D^{-1}=\,D^{-1}\,(a\,D-\,D\,\sigma^{-1}(a))\,D^{-1}  \, \in  \, \cL_0^{n ,\infty} .$$ To verify the inductive step we write $$\begin{aligned}
D^{-k}\,a-\,\sigma^{-k}(a)\,D^{-k} &=& D^{-1} \big(D^{-(k-1)}\,a-\,\sigma^{-(k-1)}(a)\,D^{-(k-1)}\big)\\
&+& \big(D^{-1}\,\sigma^{-(k-1)}(a)\, -\,\sigma^{-k}(a)\,D^{-1}\big) D^{-(k-1)}\end{aligned}$$ and observe that, by Hölder’s inequality and the induction hypothesis, each of the two summands in the right hand side belongs to $\,\cL_0^{\frac{n}{k} \, ,\infty}$.

Applying now for $k=n$, one obtains $$\varphi(T\,\sigma^{-n}(a))=\,\cutint \,T\,\sigma^{-n}(a)\,D^{-n}=\,
\cutint \,T\,D^{-n}\,a=\, \varphi(a\,T)  \, , \quad \forall \, T \in \cL (\fH) \, .$$ To prove the third statement, one replaces $\,D$ by $\,|D| $ throughout the above argument. 

We now consider the analogue of the bimodule of gauge potentials given in the usual case by the $\cA$-bimodule $\Omega_D^1 \subset
\cL (\fH)$ of operators of the form $$\label{pot}
A = \Sigma  a_i [D,b_i], \ \ \  a_i , b_i \in \cA \, .$$ Let $(\cA , \fH , D)$ be a $\sigma$-spectral triple, then we let $\Omega_D^1 \subset \cL (\fH)$ be the linear space of operators of the form $$\label{pot1}
A = \Sigma  a_i (D\,b_i-\sigma(b_i)\,D)\, , \ \ \  a_i , b_i \in \cA
\, .$$

\[bim\] Let $(\cA , \fH , D)$ be a $\sigma$-spectral triple, then $\Omega_D^1 $ is an $\cA$-bimodule for the action $$\label{bim1}
a \cdot\omega\cdot b=\,\sigma(a)\,\omega\,b {{\,,\quad \forall\,}}a,\,b \in \cA {{\,,\quad \forall\,}}\omega \in \Omega_D^1$$ and the map $$\label{der}
a \mapsto d_\sigma (a)=\,D\,a-\sigma(a)\,D$$ is a derivation of $\cA$ in $\Omega_D^1 $. 

One has $$d_\sigma (ab)=\,D\,ab-\sigma(ab)\,D=\,(D\,a-\sigma(a)\,D)\,b+\sigma(a)\,(D\,b-\sigma(b)\,D)$$ which shows that $$\label{bim2}
d_\sigma (ab)=\,d_\sigma (a)\cdot b+ a\cdot d_\sigma (b) {{\,,\quad \forall\,}}a, b \in \cA$$ Since $\sigma$ is an automorphism of $\cA$, the linear space $\Omega_D^1$ is the linear span of the $a\cdot d_\sigma (b)$ for $a, b \in
\cA$. By this is stable under right multiplication by elements of $\cA$. Thus $\Omega_D^1$ is an $\cA$-bimodule.  Finally shows that $d_\sigma $ is a derivation. 

Chern character
---------------

By Proposition \[untwist\], any $\sigma$-spectral triple $(\cA , \fH , D)$ of finite summability degree, [[*i.e.*]{} ]{}such that with $D^{-1}\in \cL^{n,\infty}$ for some $n
\in \bN$, which in addition is [*Lipschitz-regular*]{} has a well-defined Chern character in cyclic cohomology, namely the Chern character of its ‘phase’ Fredholm module $( \fH , F)$ over $\, \cA$, $$\label{CCh}
\Phi_F (a^0 , a^1, \ldots , a^n)\, :=  \,
\Tr \, (\gamma\, F\,  [F, a^0] \, [F, a^1] \cdots \, [F, a^n]) \, ,
\qquad  \forall \, a^0 , a^1, \ldots a^n \in \cA ,$$ with $\gamma$ omitted in the ungraded case ([[*cf.*]{}  ]{} [@ncdg Part I]). 

On the other hand let us assume for a moment that $(\cA , \fH , D')$, $D' = e^h \, D \, e^h$, is a graded twisted spectral triple as in §\[example1\], with the property that $D^{-1}\in \cL^{n,\infty}$ for some even $n \in \bN$. Applying one sees that $$D'^{-1} \, d'_\sigma a \, = \, e^{-h} \, D^{-1} \, [D, b] \, e^h \, ,  \quad \text{where} \quad
b = e^h \, a \, e^{-h} .$$ Therefore, for any $a^0 ,  a^1, \ldots ,  a^n \in \cA$, $$\label{compchern}
\Tr \, (\gamma\, D'^{-1}  d'_\sigma a^0 \, D'^{-1} d'_\sigma a^1 \cdots \, D'^{-1} d'_\sigma a^n)
=  \Tr \, (\gamma\, D^{-1} [D, b^0] \, D^{-1} [D, b^1] \cdots \,
D^{-1} [D, b^n]) \, ,$$ with $\, b^i \, =\, e^h \, a^i \, e^{-h}$, $ \forall \, i = 0, \ldots , n$. The right hand side of the above identity is a cyclic cocycle on $\cA$ that represents, up to normalization, the Chern character of $(\cA , \fH , D)$, [[*cf.*]{}  ]{} [@ncdg Part I, §6].  It follows that the left hand side is also a cyclic cocycle, obtained via conjugation by an inner automorphism, and thus determining the same periodic cyclic cohomological class $$Ch^* (\cA , \fH , D') \, = \,  Ch^* (\cA , \fH , D) \in HP^* (\cA) .$$

This suggests that it should be possible to define a ‘straight’ Chern character for any finitely sumable twisted spectral triple, not just for those that are Lipschitz-regular.  The proposition below confirms that this is indeed the case. 

\[straightchern\] Let $(\cA , \fH , D)$ be a graded $\sigma$-spectral triple such that $D^{-1}\in \cL^{n,\infty}$ for some even $n \in \bN$. Then the following multilinear form $$\label{Cycocycle}
\Phi_{D, \sigma} (a^0 , a^1, \ldots , a^n)\, :=  \,
\Tr \, (\gamma\, D^{-1}  d_\sigma a^0 \, D^{-1} d_\sigma a^1 \cdots \, D^{-1} d_\sigma a^n),
\quad  a^0, \ldots a^n \in \cA$$ is a cyclic cocycle in $Z_\lambda^n (\cA)$.

The proof of Proposition 1 in [@ncdg Part I, §6] ([[*cf.*]{}  ]{}also [*infra*]{}) applies verbatim to the twisted case if one simply replaces the representation $\, a \mapsto D^{-1} \, a \, D\, $ by $\, a \mapsto D^{-1} \, \sigma(a) \, D$, $\, \forall \, a \in \cA$.

Pairing with $K$-theory
-----------------------

Implicit in the above proof is the existence ‘behind the scene’ of a pair of graded Fredholm module over $\cA$, $\, (\tilde\fH^{\pm}, F^{\pm})\,$, canonically associated to the $\sigma$-spectral triple $(\cA , \fH , D)$. To wit, we decompose $(\cA , \fH , D)$ according to the grading by $\gamma$ into $$\fH =  \fH_{+} \oplus \fH_{-} \, , \qquad
D  = \begin{pmatrix} 0  &D_{-}\\ \, D_{+}   &0  \end{pmatrix} \, , \qquad
a  =  \begin{pmatrix}  \, a_{+}  &0\\  0   &a_{-}  \end{pmatrix} \, ,
\quad \forall a \in \cA ,$$ then define $$\tilde\fH^{\pm}  :=  \fH_{\pm}  \oplus \fH_{\pm} , \quad
\pi^{\pm} (a)  :=
\begin{pmatrix}   a_{\pm}   &0\\ 0 &D_{\pm}^{-1}  \sigma(a_{\mp})  D_{\pm}  \end{pmatrix} \, \,
\text{on}\, \, \Dom D_{\pm} , \quad
F^{\pm} := \, \begin{pmatrix} 0  &I_{\pm}\\ \, I_{\pm}   &0   \end{pmatrix},$$ and note that for all $\,   a \in \cA \,$, firstly, $$\label{Fred1}
D_{\pm}^{-1}  \sigma(a_{\mp})  D_{\pm}  \,= \, a_{\pm} \,
- \, D_{\pm}^{-1} (D_{\pm}\, a_{\pm} \, - \, \sigma (a_{\mp}) \, D_{\pm})  \quad
\text{is bounded} ,$$ and secondly, $$\label{Fred2}
[F^{\pm} , \pi^{\pm} (a)] \, =
\begin{pmatrix}   0  &- D_{\pm}^{-1} \left(D_{\pm} \, a_{\pm} - \sigma(a_{\mp}) \, D_{\pm}\right) \\
D_{\pm}^{-1} \left(D_{\pm} \, a_{\pm} - \sigma(a_{\mp}) \, D_{\pm}\right) &0\end{pmatrix}
\in \cL^{n,\infty}.$$

\[pairing\] Let $(\cA , \fH , D)$ be a graded $\sigma$-spectral triple such that $D^{-1}\in \cL^{n,\infty}$ for some even $n \in \bN$, and let $e \in  \cA$ be an idempotent.  Denote by $f_{\pm}$ the bounded closure of of $D_{\pm}^{-1}  \sigma(e_{\mp})  D_{\pm}$. Then $\, f_{\pm}^2 = f_{\pm}\,$ and $\, f_{\pm} e_{\pm}  : e_{\pm} \fH_{\pm} \longrightarrow f_{\pm} \fH_{\pm}\, $ are Fredholm operators. 

The first claim is obvious and the second follows from the fact that $\, f_{\pm}\, - \, e_{\pm}\,$ is compact. 

The integer $\,\Index (f_{\pm} e_{\pm} )$ depends only on the $K$-theory class of the idempotent, and thus one can define a pair of index maps $\, \Index^{\pm}_{D, \sigma} : K_0 (\cA) \rightarrow \Z $, by setting $$\label{indmap}
\Index^{\pm}_{D, \sigma} [e] \, =  \, \Index (f_{\pm} e_{\pm} ) \, , \qquad
\forall \, e^2 = e \in M_N (\cA) ;$$ taken together, they give rise to a [*double index map*]{} $$\label{doublemap}
 \Index_{D, \sigma} = \big(\Index^{+}_{D, \sigma} ,  \, \Index^{-}_{D, \sigma} \big)
: K_0 (\cA) \rightarrow \Z \times \Z .$$ On the other hand, the cyclic cocycle is itself made of two cocycles in $Z_\lambda^n (\cA)$, $$\label{pmcocycle}
\Phi^{\pm}_{D, \sigma} (a^0, \ldots , a^n) := 
\Tr \,\big( D_{\pm}^{-1} (D_{\pm} \, a^0_{\pm} - \sigma(a^0_{\mp}) \, D_{\pm})\,
 \cdots \, D_{\pm}^{-1} (D_{\pm} \, a^n_{\pm} - \sigma(a^n_{\mp}) \, D_{\pm})\big).$$

\[doubleindex\] Let $(\cA , \fH , D)$ be a graded $\sigma$-spectral triple such that $D^{-1}\in \cL^{n,\infty}$ for some even $n \in \bN$. For any $\, e^2 = e \in M_N (\cA) $, one has $$\label{doubleindexformula}
\Index^{\pm}_{D, \sigma} [e] \, =  \, \Phi^{\pm}_{D, \sigma} (e , \ldots , e) \, .$$ If in addition $\, e^* = \sigma (e) $, then $$\label{sigidemp}
\Index^{+}_{D, \sigma} [e] \, =  \, - \, \Index^{-}_{D, \sigma} [e] \, .$$

Indeed, $\, \Index^{\pm}_{D, \sigma} $ is precisely the index map associated to the Fredholm module $\, (\tilde\fH_{\pm} , F_{\pm} )\,$, and therefore is given by the corresponding index formula, [[*cf.*]{}  ]{} [@ncdg Part I, §3, Theorem 1]. 

The second claim follows from the fact that, if $\, e^* = \sigma (e)$, then $$\label{sigadj1}
\big(D  \, e - \sigma(e) \, D\big)^* \, = \, - \, (D  \, e - \sigma(e) \, D) \, ,$$ which in turn implies $$\overline{\Phi^{+}_{D, \sigma} (e , \ldots , e)} \, =  \, - \Phi^{-}_{D, \sigma} (e , \ldots , e)  \, .$$

One-parameter group of automorphisms
------------------------------------

We now make the the additional assumption that the involutive Banach algebra $\cA$ is equipped with a strongly continuous $1$-parameter group of isometric automorphisms $\,  \{ \sigma_t \}_{t \in \R} $ such that

-   [*$\sigma$ coincides with the value at $t = -i$ of the analytic extension of $ \{ \sigma_t \}_{t \in \R}$*]{}. 

The existence of such an analytic extension defined on a [*dense subalgebra*]{} $\cO$ of $ \cA$, which is moreover [*stable under holomorphic functional calculus*]{}, is ensured by a theorem of Bost [@bost Thm.  1.1.1]. 

To begin with, let us note that in the presence of the above hypothesis, which by the way is automatically satisfied by the transverse spectral triple, [[*cf.*]{}  ]{}, the double index map reduces to a single index pairing map. 

Assume that $\cA$ satisfies [(1PG)]{}.  Then the two signed index maps coincide, [[*i.e.*]{} ]{}$$\Index^{+}_{D, \sigma} \, =  \, - \, \Index^{-}_{D, \sigma} : \, K_0 (\cA) \rightarrow \Z  \, .$$

Let $\, e^2 = e = e^* \in M_N (\cO) $ be a projection.  Instead of we now use the identity $$\label{sigadj2}
\big(D  \, \sigma^{-1}(e)\, - \, e \, D\big)^* \, = \, - \big( D  \, e \,  - \,\sigma (e) \, D \big) \, ,$$ together with the fact that the idempotents $e$ and $\sigma^{-1} (e)$ are homotopic via the path $ t \in [0, 1] \mapsto \sigma_{it} (e)$, to obtain $$\begin{aligned}
\Index^{+}_{D, \sigma} [e] &=&
\Index^{+}_{D, \sigma} [\sigma^{-1}(e)] =
\overline{\Phi^{+}_{D, \sigma} (\sigma^{-1}(e) , \ldots , \sigma^{-1}(e))} =
- \Phi^{-}_{D, \sigma} (e , \ldots , e) \\
 &=& - \Index^{-}_{D, \sigma} [e]  \, .\end{aligned}$$

As a matter of fact, an elaboration of the above argument gives a more comprehensive result that elucidates the relationship between the two ‘half-character’ cocycles .  Note however that unlike the classical case, where this relationship manifests itself already at the level of cocycles ([[*cf.*]{}  ]{} [@ncdg Part I, §6]), in the twisted case it only occurs at the cohomological level. 

Let $(\cA , \fH , D)$ be a graded $\sigma$-spectral triple such that $D^{-1}\in \cL^{n,\infty}$ for some even $n \in \bN$, Under the assumption [(1PG)]{}, the two Chern characters $[\Phi^{\pm}_{D, \sigma}]  \in HP^{\rm ev} (\cO)$ are related by the identity $$\label{adjointchern}
[\Phi^{-}_{D, \sigma}]\,  =\, - [(\Phi^{+}_{D, \sigma})^*]  \, ,$$ where $$(\Phi^{+}_{D, \sigma})^* (a_0 ,  \ldots , a_n) \, := \,
\overline{\Phi^{+}_{D, \sigma} (a_n^* , \ldots , a_0^*)} \, , \qquad \forall \,
a^0 , a^1, \ldots , a^n \in \cO \, .$$

For any $\, a \in \cO$ and $\, t \in [0, 1]$, let $$\label{homotop}
\pi_t  (a) \, := \, \big((\pi^{+} \circ \sigma_{it}) (a^*)\big)^* \,= \,
\begin{pmatrix}   \sigma_{-it} (a_{+})   &0\\ 0 &D_{-} \sigma_{i-it} (a_{-})  D_{-}^{-1}   \end{pmatrix} ,$$ and define the family of Fredholm modules $\{ (\fH_t , F_t )\}_{t \in [0, 1]}$, by taking $$\fH_t  =  \fH_{+}  \oplus \fH_{+}  \quad \text{acted upon by} \,\,  \cO \, \text{via} \,\,
\pi_t \, , \, \, \text{and} \quad
F_t := \, \begin{pmatrix} 0  &I_{+}\\ \, I_{+}   &0   \end{pmatrix}.$$ As in one has $\, [F_t , \pi_t (a)] \, \in \cL^{n,\infty}$, because $$D \, \sigma_{i-it} (a) \, D^{-1} \, - \,  \sigma_{-it} (a)  \, = \,
\big( D \, \sigma_{i-it} (a) \, - \,  \sigma(\sigma_{i-it} (a) ) \, D\big)\, D^{-1}
\, \in \cL^{n,\infty}.$$ Note also that $$\label{pi0}
\pi_0 (a) \,= \,
\begin{pmatrix}  a_{+}   &0\\ 0 &D_{-} \sigma^{-1}(a_{-})  D_{-}^{-1}   \end{pmatrix}
  \,= \, \big(\pi^{+} (a^*)\big)^* \, ,$$ and $$\label{pi1}
\pi_1 (a) \,= \,
\begin{pmatrix}   \sigma (a_{+})   &0\\ 0 &D_{-} a_{-}  D_{-}^{-1}   \end{pmatrix}
  \,= \,  \begin{pmatrix} 0  &D_{-} \\ \, D_{-}   &0  \end{pmatrix} \, \pi^{-} (a)
  \,  \begin{pmatrix} 0  &D_{-}^{-1} \\ \, D_{-}^{-1}   &0  \end{pmatrix} .$$ By Lemma 1 in [@ncdg Part I, §5], the periodic cyclic cohomology class $\, Ch^* (\fH_t , F_t ) \in HP^{\rm ev} (\cO)$, is independent of $t \in [0, 1]$. We also recall that this class can be represented by the cyclic cocycle $$\Phi_t (a^0 ,  \ldots , a^n) := 
\frac{1}{2} \, \Tr \left(\begin{pmatrix} I  &\, 0\\ \, 0   &-I   \end{pmatrix}
F_t \, [F_t , \pi_t (a^0)] \, \cdots \, [F_t, \pi_t (a^n)] \right), \quad
a^0 , \ldots , a^n \in \cO .$$ In particular, using and , one has $$\begin{aligned}
\Phi_0 (a^0 ,  \ldots , a^n )&=& \,
 \Tr \left( (a^0_{+} - D_{-} \sigma^{-1}(a^0_{-})  D_{-}^{-1} ) \, \cdots \,
(a^n_{+} - D_{-} \sigma^{-1}(a^n_{-})  D_{-}^{-1} ) \right) \\
&=&  \,  (\Phi^{+}_{D, \sigma})^*  (a^0 ,  \ldots , a^n ) \, ,\end{aligned}$$ while by $$\begin{aligned}
\Phi_1 (a^0 ,  \ldots , a^n ) \, &=& \,
 \Tr \left( (\sigma (a^0_{+} )- D_{-} a^0_{-}  D_{-}^{-1} ) \, \cdots \,
(\sigma (a^n_{+} )- D_{-} a^n_{-}  D_{-}^{-1} ) \right) \\
&=&\,   - \Phi^{-}_{D, \sigma}  (a^0 ,  \ldots , a^n ) .\end{aligned}$$

Local Hochschild cocycles {#hoch}
-------------------------

As a first step in the direction of extending the local index formula of [@C-M2] to twisted spectral triple, we shall construct an analogue of the local Hochschild cocycle that gives the Hochschild class of the Chern character in the untwisted case, [[*cf.*]{}  ]{} [@book IV.2.$\gamma$].  We begin by revisiting the latter and then proceed in a heuristic manner. 

Given a graded spectral triple $(\cA , \fH , D)$ such that $D^{-1}\in \cL^{n,\infty}$ for some $n \in 2 \bN$, the Hochschild class of its Chern character is represented in local form by the following cocycle: $$\label{Hcocycle1}
\Psi_D (a^0 , a^1, \ldots , a^n)\, :=
\,\cutint \, \gamma \,a^0 \, [D, a^1]\, \cdots \, [D, a^n] \,D^{-n} \, ,
\qquad  \forall \, a^0 , a^1, \ldots , a^n \in \cA \, .$$ Noting that $$\label{switch1}
[D, a] \,D^{-k} \, = \, D^{-k+1} \, \big(D^k \, a \, D^{-k} \, - \, D^{k-1} \, a \, D^{-k +1}\big) \, ,
\qquad  \forall \, a \in \cA \, ,$$ we can successively move $\,D^{-n} \,$ to the left with loss of a power at each step, and thus rewrite the cocycle in the form $$\label{Hcocycle2}
\Psi_D (a^0 , a^1, \ldots , a^n) \, =  \,
 \cutint \gamma \, a^0 \, (D \, a^1\, D^{-1} \, - \,  a^1) \, \cdots \,
(D^n \, a^n \, D^{-n} \, - \, D^{n-1} \, a^n\, D^{-n +1}) .$$ In the twisted case, inspired by the formulas , , we make the formal substitution $$\label{subst}
D^k \, a \, D^{-k}  \quad \longmapsto \quad
D^k \, \sigma^{-k} (a) \, D^{-k}  \, , \qquad  \forall \, a \in \cA \, ,$$ and obtain the following candidate for a Hochschild character cocycle: $$\begin{split}
\Psi_{D, \sigma}& (a^0 , a^1, \ldots , a^n) \, =  \\
 &\cutint \gamma \, a^0 \, \big(D \, \sigma^{-1} (a^1)\, D^{-1} \, - \,  a^1\big) \, \cdots \,
\big(D^n \, \sigma^{-n} (a^n) \, D^{-n} \, - \, D^{n-1} \, \sigma^{-n+1} (a^n)\, D^{-n +1}\big) . 
\end{split}$$ The counterpart of being $$\label{switch2}
d_\sigma (\sigma^{-k} (a)) \,D^{-k} \, =
\, D^{-k+1} \, \big(D^k \, \sigma^{-k} (a) \, D^{-k} \, - \, D^{k-1} \, \sigma^{-k+1} (a) \, D^{-k +1}\big) \, ,$$ one can reverse the process of distributing $\,D^{-n} \,$ among the factors, which leads to the expression stated below. 

\[tHclass\] Let $(\cA , \fH , D)$ be a graded $\sigma$-spectral triple such that $D^{-1}\in \cL^{n,\infty}$ for some even $n \in \bN$. Then the $n+1$-linear form on $\cA$ $$\label{tHcocycle}
\Psi_{D, \sigma} (a^0 , a^1, \ldots , a^n)\, :=
\,\cutint \, \gamma \,a^0 \, d_\sigma (\sigma^{-1}(a^1))\, \cdots \,
d_\sigma (\sigma^{-n} (a^n)) \,D^{-n}$$ is a Hochschild cocycle in $Z^n (\cA, \cA^*)$.

In the ungraded case, for a $\sigma$-spectral triple of odd summability degree which is Lipschitz-regular, the corresponding Hochschild cocycle is defined by the expression $$\label{oddtHcocycle}
\Psi_{D, \sigma} (a^0 , a^1, \ldots , a^n)\, :=
\,\cutint \,a^0 \, d_\sigma (\sigma^{-1}(a^1))\, \cdots \,
d_\sigma (\sigma^{-n} (a^n)) \,|D|^{-n} \, .$$

We check that $\Psi_{D, \sigma}$ is a Hochschild cocycle by computing its coboundary, using the derivation rule , as follows: $$\begin{split}
&b\Psi (a^0 , a^1, ..., a^{n+1})= \sum_{i=0}^n (-1)^i\, \Psi (a^0,..., a^i a^{i+1},..., a^{n+1})
+ (-1)^{n+1} \Psi (a^{n+1} a^0 , a^1, ..., a^n) \\
 &=\, \cutint \, \gamma\, a^0 \, a^1\, d_\sigma (\sigma^{-1} (a^2)) \cdots \,
 d_\sigma (\sigma^{-n}  (a^{n+1}))\,D^{-n} \,\\
 &\quad - \,  \cutint \, \gamma \, a^0 a^1 \,
 d_\sigma (\sigma^{-1} (a^2)) \cdots \,   d_\sigma  (\sigma^{-n}  (a^{n+1}))\,D^{-n} \\
&\quad  - \, \cutint \, \gamma \, a^0\, d_\sigma (\sigma^{-1} (a^1)) \, \sigma^{-1} (a^2) \cdots \,
 d_\sigma (\sigma^{-n}  (a^{n+1}))\,D^{-n} \, + \ldots \\
&\quad \ldots  + \,   (-1)^n \cutint \, \gamma\, a^0\, d_\sigma (\sigma^{-1} (a^1)) \cdots
 \, \sigma^{-n+1} (a^{n-1})\, \sigma^{-n+1}  (a^n) \, \sigma^{-n} (a^{n+1}) \,D^{-n} \\
&\qquad \qquad + \,   (-1)^n \cutint \, \gamma \,a^0\, d_\sigma (\sigma^{-1} (a^1)) \cdots
 \, \sigma^{-n+1} (a^{n-1})\, d_\sigma ( \sigma^{-n}(a^n) )\,  \sigma^{-n}(a^{n+1}) \,D^{-n} \\
&\qquad  \qquad +\, (-1)^{n+1} \cutint  \, \gamma \,  a^{n+1} \, a^0\,
d_\sigma (\sigma^{-1} (a^1))  \cdots \, d_\sigma ( \sigma^{-n}(a^n) )\,D^{-n} \, . 
\end{split}$$ The resulting expression vanishes because of successive cancelations, with the last two terms canceling each other in view of . 

In the ungraded Lipschitz-regular case, the very same calculation holds true provided $\gamma$ is replaced by the phase operator $\, F
= \, D \, |D|^{-1}$.

The cyclic group generated by $\sigma \in \Aut (\cA)$ acts in a natural way on the set of such Hochschild cocycles.  For each integer $m \in \Z$, the corresponding ‘gauge transformed’ cocycle via the action of $\sigma^m \in \Aut (\cA)$, has the expression: $\quad  \forall \, a^0 , a^1, \ldots a^n \in \cA$, $$\label{mtHcocycle}
\Psi^{(m)}_{D, \sigma} (a^0 , a^1, \ldots , a^n)\, :=
\,\cutint \, \gamma \,\sigma^{m}(a^0) \, d_\sigma (\sigma^{m-1}(a^1))\, \cdots \,
d_\sigma (\sigma^{m-n} (a^n)) \,D^{-n} \, .$$

As an illustration, let us specialize formula to the twisted spectral triple $(\cA, \Dirac, \fH)$ associated to a codimension 1 foliation ([[*cf.*]{}  ]{}§\[trans1\]).  We shall assume throughout that the group $\Gamma$ consists of orientation preserving diffeomorphisms with *nondegenerate isolated fixed points*.  Denoting the local Hochschild cocycle by $\Psi_1$, and using , one has $$\begin{aligned}
&&\Psi_1 (f\,U^*_\phi , \, g\, U^*_\psi )  \, = \,
\cutint \,\pi (f\, U^*_\phi)
 \, \big( \Dirac \, \pi(\sigma^{-1}(g\, U^*_\psi)) \, - 
 \, \pi (g\, U^*_\psi) \,  \Dirac \big) \,|\Dirac|^{-1} \\ 
 &&= \,\cutint \, f  \, (\phi')^{\frac{1}{2}}\,  V^{-1}_\phi \, \frac{1}{i} \frac{d}{dx}
 \big(g\, (\psi')^{-\frac{1}{2}} \big) \,  V^{-1}_\psi \,|\Dirac|^{-1} 
  = \frac{1}{i} \,\cutint \, f  \, (\phi')^{\frac{1}{2}}\,
 \big(g\, (\psi')^{-\frac{1}{2}} \big)^\prime \circ \phi\, \, V^{-1}_{\psi\circ \phi} \,|\Dirac|^{-1} . 
  \end{aligned}$$ Here the functional $\displaystyle \cutint$ is extended to the algebra $\Psi DO^{\infty} (S^1) \rtimes \Gamma$ by the formula $$\label{res}
\cutint \, V^{-1}_\chi\, P \, = \, {\rm Res}_{s=0} \, \Tr (V^{-1}_\chi\, P \, |\Dirac|^{-s} )  \, ,
\qquad P \in \Psi DO^{\infty} (S^1) \, , \quad \chi \in \Gamma.$$ By [@book IV.  2.$\beta$], when $\chi = I$ this coincides with the Wodzicki residue of $P$. On the other hand, when $P = F \, |\Dirac|^{-1} $, with $ F \in C^\infty (S^1)$, and $\chi$ is a diffeomorphism with nondegenerate isolated fixed points the above expression vanishes.  Indeed, using the Mellin transform $$\Tr (V^{-1}_\chi\, F \, |\Dirac|^{-s-1} )  \, = \, \Tr \left(V^{-1}_\chi\, F \, 
  \left(\Dirac^2\right)^{-\frac{s+1}{2}} \right)   \, = \,
  \frac{1}{\Gamma (\frac{s+1}{2} )} \, \int_0^\infty \, t^{\frac{s-1}{2}} \, 
  \Tr \left(V^{-1}_\chi\, F \, e ^{-t \, \Dirac^2} \right) \, dt .$$ By [@gilkey §1.8], given that the fixed point set is $0$-dimensional and that the differential operator $F$ is of degree $0$, the asymptotic expansion of $\, \Tr \left(V^{-1}_\chi\, F \, e ^{-t \, \Dirac^2} \right) $ as $\,  t \rightarrow 0$ involves only positive half-integer powers $\, t^{\frac{n}{2}} $, with $n \geq 0$. Thus, the residue at $s=0$ of the analytic continuation of the above expression is equal to $0$.

It follows, firstly, that the cocycle $\Psi_1$ is localized to the identity.  Secondly, for $\, \psi = \phi^{-1}$, one obtains the following explicit formula: $$\label{psi1}
\Psi_1 (f\,U^*_\phi , \, g\, U^*_\psi )  \, = \,
 \frac{2}{i}  \int_{\R/\Z} \, f (x)  \, \phi'(x)^{\frac{1}{2}}\,
 \big(g\, (\psi')^{-\frac{1}{2}} \big)^\prime (\phi (x)) \, dx \, .$$ An elementary manipulation of this expression gives $$\label{psi2}
 \Psi_1 (f\,U^*_\phi , \, g\, U_\phi ) \,  =\,
  -2i \,  \int_{\R/\Z}  \, f \, \phi^\ast (dg) \,  -\, i \,  \int_{\R/\Z}  \, 
f  \, \phi^\ast (g) \, d \log \phi'   \, .$$ Both terms in the right hand side are recognizable cyclic cocycles, [[*cf.*]{}  ]{}[@transverse] and [@book III.  6.$\beta$].  The first equals $\, -2i$ times the cyclic cocycle $$\label{tfc1}
\tau (f\,U^*_\phi , \, g\, U^*_\psi )\, = \, \varphi (f\,U^*_\phi \cdot dg\, U^*_\psi ) \, ,$$ with $\varphi$ the canonical state , which represents the *transverse fundamental class* of the ‘quotient space’ $[S^1/\Gamma]$. The second coincides with the *Lie derivative* of $\, \tau$ with respect to the generator $\delta$ of the modular automorphism group $\{\sigma_t \}_{t \in \R}$ (see ), $\delta (f\,U^*_\phi) \, = \, i \, \log \phi' \cdot f\,U^*_\phi$. Indeed, for $\, \psi = \phi^{-1}$, $$\begin{aligned}
 \label{Lder}
\cL_\delta \, \tau (f\,U^*_\phi , \, g\, U^*_\psi )&:=& \, 
\tau (\delta(f\,U^*_\phi) , \, g\, U^*_\psi ) \, + \, \tau (f\,U^*_\phi , \, \delta (g\, U^*_\psi) ) \\ 
&=& \varphi (\delta(f\,U^*_\phi) \cdot dg\, U^*_\psi ) \, + \, 
\varphi (f\,U^*_\phi \cdot d(\delta (g\, U^*_\psi )) \\
&=& i  \int_{\R/\Z}  \, f \, \log \phi'   \, \phi^\ast (dg)  \, +
 \, i  \int_{\R/\Z}  \, f  \, \phi^\ast (g\, d \log \psi') \\
&=&- i \,  \int_{\R/\Z}  \, 
f  \, \phi^\ast (g) \, d \log \phi'   \, . 
 \end{aligned}$$ By the cyclic analogue of the Cartan homotopy formula $\, \cL_\delta \, = \, [e_\delta  + E_\delta , b + B]$, and since $\tau$ is a cyclic cocycle, one sees that $ \cL_\delta \, \tau$ is a coboundary: $$\begin{aligned}
 \label{Lder2}
 \cL_\delta \, \tau  \, &=& \, B (e_\delta \, \tau ) + b (E_\delta \, \tau)  \, , \\ \nonumber
\text{where}  \quad
e_\delta \tau (a^0 , a^1, a^2) &:=& - \tau (\delta (a^2) a^0, a^1) \quad
\text{and }  \quad E_\delta  \tau (a^0):=  \tau (1, a^0)  =  B  \tau (a^0) = 0 . 
 \end{aligned}$$ We have thus proved the following statement, showing that the local Hochschild cocycle does represent the ‘correct’ class. 

\[tHclass1\] The local Hochschild cocycle associated to the Dirac spectral triple over $\cA=\,C^\infty(S^1)\rtimes \Gamma$, with $\Gamma$ acting by orientation preserving diffeomorphisms with nondegenerate isolated fixed points, has the explicit expression $$\label{psi3}
 \Psi_1  \,  =\, -2i \, \tau  \, + \,  \cL_\delta \, \tau  .$$ In particular, it is a cyclic cocycle and its periodic cyclic cohomology class is proportional to the transverse fundamental class $[S^1/\Gamma]$.

The facility of the above computation stands in sharp contrast to the tremendous amount of calculations required to explicitly compute the local index cocycle for the type II – but $3$-dimensional – lift of the above spectral triple to the frame bundle.  Those calculations were kept unpublished, due to their excessive length, but we recorded their end result in [@C-M3 Appendix, eq. ( 9)]; the expression of the corresponding $3$-dimensional cocycle is quite similar to equation . 

Future developments
===================

We conclude by listing, roughly in their increasing order of complexity, a few themes for future research in this direction. 

Symbolic calculus and local index formula
-----------------------------------------

The symbolic calculus developed for spectral triples ([[*cf.*]{}  ]{} [@C-M2 Appendix B]) needs to be adapted to allow, in particular, establishing that if $(\cA , \fH , D)$ is a $\sigma$-spectral triple with $D^{-1}\in \cL^{n,\infty}$ that satisfies a stronger regularity assumption, then $$\label{extrareg}
 |D|^{-t} \left(|D|^t \, a \, - \, \sigma^t (a) \, |D|^t \right)
\in \cL^{n,\infty} \, , \qquad \forall \, t \in \R \, .$$ By Remark \[orderdecrease\], the transverse spectral triple example fulfills this property.  Note also that the extra regularity assumption immediately reconciles the two definitions given above to the Chern character, [*viz.*]{}  and .  Indeed, one can then produce the following homotopy between the cyclic cocycles $\Phi_F$ and $\Phi_{D, \sigma}$: $$\Phi_t (a^0 , a^1, \ldots , a^n):= 
 \Tr \,\big(\gamma\,  D_t^{-1} (D_t \, a^0 - \sigma^{1-t}(a^0) \, D_t)\,
 \cdots \, D_t^{-1} (D_t \, a^n - \sigma^{1-t}(a^n) \, D_t)\big),$$ where $\, D_t = D\, |D|^{-t}$ and $\, t \in [0, 1]$.

The full expression of the local formula for the Chern character of a finitely summable $\sigma$-spectral triple, based on $\sigma$-twisted commutators and extending the noncommutative local index formula in [@C-M2 Part II], remains to be worked out. 

Relation to type II and Thom isomorphism
----------------------------------------

One should expect that the constructions in the foliation context and in the context of modular forms of hypoelliptic spectral triples on frame bundles extend to the general context of twisted spectral triples satisfying (1PG).  The noncommutative space associated to the total space of the frame bundle corresponds to the cross product algebra by the one-parameter group $ \{ \sigma_t \}_{t \in \R} $.  The $K$-homology classes on the base and on the total space as well as their local index cyclic cocycles should be related by a Thom isomorphism. 

Higher dimensions
-----------------

The most challenging task ahead consists in extending the above considerations to the case of foliations of higher codimension, which has been put in the framework of a higher form of Tomita’s theory in [@transverse], Section 3.  The above notion of twisting only allows to handle the determinant part of the cocycle given by the Jacobian.  One expects the general case to involve dual actions of Lie groups such as $\GL(n)$ and more generally of quantum groups. 

Relation with quantum groups
----------------------------

The domain of quantum groups is a natural arena where twisting frequently occurs (see [@krahmer]) and where the above extension of the notion of spectral triple could be useful.  One would expect that the higher dimensional generalizations alluded to above would also extend to the braided context that arises from quantum groups. 

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[^1]: The work of the second named author was partially supported by the National Science Foundation award no.  DMS-0245481
