NONCOMMUTATIVE GEOMETRY AND CONFORMAL GEOMETRY. I.

LOCAL INDEX FORMULA AND CONFORMAL INVARIANTS

RAPHAEL PONGE AND HANG WANG

Abstract. This paper is the first of a series of papers on noncommutative geometry and con-

formal geometry. In this paper, elaborating on ideas of Connes and Moscovici, we establish alocal index formula in conformal-diffeomorphism invariant geometry. The existence of such a

formula was pointed out by Moscovici [Mo2]. Another main result is the construction of a huge

class of global conformal invariants taking into account the action of the group of conformaldiffeomorphisms (i.e., the conformal gauge group). These invariants are not of the same type as

the conformal invariants considered by Spyros Alexakis in his solution of the Deser-Schwimmer

conjecture. The arguments in this paper rely on various tools from noncommutative geometry,although ultimately the main results are stated in a differential-geometric fashion. In particular,

a crucial use is made of the conformal invariance of the Connes-Chern character of conformal

Dirac spectral triple of Connes-Moscovici [CM3].

1. Introduction

This paper is part of a series of papers whose aim is to use noncommutative geometry to studyconformal geometry and noncommutative versions of conformal geometry. Conformal geometryis the geometry up to angle-preserving transformations. It has interactions with various areas ofmathematical sciences, including geometric nonlinear PDEs, geometric scattering theory, parabolicgeometry, asymptotically hyperbolic geometry, conformal field theory, or even conformal gravity.In particular, it plays a fundamental role in the AdS/CFT correspondance on the conjecturedequivalence between string theory of anti-de Sitter spaces and some conformal field theory ontheir conformal boundaries. An important focus of interest in conformal geometry is the study oflocal and global conformal invariants, especially in the context of Fefferman’s program in parabolicgeometry (see, e.g., [Al, BEG, BØ1, BØ2, FG1, FG2, PR, Po2]). Alternatively, given a conformalstructure C on a manifold M we are interested in the action on M of the group of diffeomorphismspreserving this conformal structure (i.e., the conformal gauge group).

One main result of this paper is the reformulation of the local index formula in conformal-diffeomorphism invariant geometry (Theorem 8.3). Another main result is the construction andcomputation of a huge family of conformal invariants taking into account the action of the groupof conformal-diffeomorphisms (Theorem 10.2). These conformal invariants are not of the sametype as the conformal invariants considered by Alexakis [Al] in his solution of the conjecture ofDeser-Schwimmer [DS]. However, there are closely related to conformal invariants exhibited byBranson-Ørsted [BØ2] (see Remark 10.3).

The main aim of noncommutative geometry is to translate the classical tools of differentialgeometry into the operator theoretic language of quantum mechanics, so as to be able to deal withgeometric situations whose noncommutative nature prevents us from using the classical tools ofdifferential geometry [Co4]. An example of such a noncommutative situation is provided by a groupG acting by diffeomorphisms on a manifoldM . In general the quotientM/G need not be Hausdorff,but in the framework of noncommutative geometry its algebra of smooth functions alway makessense when realized as the (possibly noncommutative) crossed-product algebra C∞(M) o G (seeSection 7 for a definition of this algebra). The trade of spaces for algebras is the main impetus fornoncommutative geometry.

R.P. was partially supported by Research Resettlement Fund and Foreign Faculty Research Fund of SeoulNational University and Basic Research Grant 2013R1A1A2008802 of National Research Foundation of Korea.

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In the framework of noncommutative geometry the role of manifolds is played by spectraltriples. A spectral triple (A,H, D) is given by an algebra A represented on a Hilbert space H andan unbounded selfadjoint operator D satisfying suitable conditions (see Section 2 for the precisedefinition). An example is provided by the spectral triple associated to the Dirac operator ona compact spin Riemannian manifold. In the setup of a diffeomorphism group G acting on amanifold M , we thus seek for a spectral triple over the crossed-product algebra C∞(M) o G. Itis well known that the only differential structure on a manifold that is invariant under the fullgroup of diffeomorphisms is the manifold structure itself. This prevents us from getting a naturalrepresentation of G. A solution to this problem is to pass to the total space of metric bundle ofP → M , which carries a wealth of diffeomorphism-invariant structures (see [Co2, CM2]). Thispassage is a geometric version of the reduction of type III factors to type II factors by takingcrossed-products [Co1].

Although there are Thom isomorphisms between the respective K-theory and cyclic cohomologyof the crossed product algebras C∞(M) oG and C∞(P ) oG, it still is desirable to work directlywith the former. As observed by Connes-Moscovici [CM3], when the group G preserves a givenconformal structure this can be done at the expense of twisting the definition of a spectral triple.More precisely, a twisted spectral (A,H, D)σ is like an ordinary spectral triple at the exceptionthat the boundedness of commutators [D, a], a ∈ A, is replaced by that of twisted commutators[D, a]σ = Da− σ(a)D, where σ is a given automorphism of the algebra A. A natural example isgiven by conformal deformations of ordinary spectral triples,

(A,H, D) −→ (A,H, kDk)σ, σ(a) = k2ak−2,

where k ranges over positive invertible elements of A (see [CM3]). A more refined example is theconformal Dirac spectral triple (C∞(M) o G,L2

g(M,/S), /Dg)σg associated to the Dirac operator

/Dg on a compact Riemannian spin manifold (M, g) and a group G of diffeomorphisms preserving

a given conformal structure C (see [CM3]; a review of this example is given in Section 7). Theconformal invariance of the Dirac operator plays a crucial role in this construction. There arevarious other examples of twisted spectral triples (see [CM3, GMT, CT, IM, Mo2, PW3]).

An important motivation and application of Connes’ noncommutative geometry program is thereformulation and extension of the index formula of Atiyah-Singer [ASi1, ASi2] to various newgeometric settings. Given a Dirac operator /D on a closed Riemannian spin manifold Mn and aHermitian vector bundle E we may twist /D with any Hermitian connection ∇E on E to form aDirac operator /D∇E with coefficients in sections of E. The main geometric 1st order differentialoperators are locally of this form. The operators /D∇E are Fredholm and their Fredholm indicesare computed by the Atiyah-Singer index formula,

ind /D∇E = (2iπ)−n2

∫M

A(RM ) ∧ Ch(FE),

where A(RM ) is the A-form of the Riemann curvature and Ch(FE) is the Chern form of thecurvature FE of the connection ∇E .

Likewise, given a spectral triple (A,H, D) and a Hermitian finitely generated projective moduleE over the algebra A, we can twist the operator D with any Hermitian connection ∇E on Eto get a Fredholm operator D∇E with coefficients in E . The analogues of de Rham homologyand cohomology in noncommutative geometry are provided by the cyclic cohomology and cyclichomology [Co3, Ts]. Furthermore, Connes [Co3] associated to any (p-summable) spectral triple(A,H, D) a cyclic cohomology class Ch(D), called Connes-Chern character, which computes theFredholm indices indD∇E . Namely,

(1.1) indD∇E = 〈Ch(D),Ch(E)〉 ,

where Ch(E) is the Chern character of E in cyclic homology. Under further assumptions, theConnes-Chern character is represented by the CM cocycle [CM2]. The components of the CMcocycle are given by formulas that are local in the sense they involve a version for spectral triplesof the noncommutative residue trace of Guillemin [Gu] and Wodzicki [Wo]. Together with (1.1)

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this provides us with the local index formula in noncommutative geometry. In the case of a Diracspectral triple this enables us to recover the Atiyah-Singer index formula (see [CM2, Po1]).

As for ordinary spectral triples, the datum of a twisted spectral triple (A,H, D)σ gives riseto an index problem by twisting the operator D with σ-connections (see [PW1]). The resultingFredholm indices are computed by a Connes-Chern character Ch(D)σ defined as a class in thecyclic cohomology of A (see [CM3, PW1]). However, although Moscovici [Mo2] produced anAnsatz for a version of the CM cocycle for twisted spectral triple, this Ansatz has been verifiedonly in a special class of examples (see [Mo2]). In particular, to date we still don’t know whetherMoscovici’s Asantz holds for conformal deformations of ordinary spectral triples.

The first main result of this paper states that the Connes-Chern character of the conformalDirac spectral triple is a conformal invariant (Theorem 7.8). In fact, for our purpose it is crucialto define the Connes-Chern character in the cyclic cohomology of continuous cochains, which issmaller than the ordinary cyclic cohomology of general cohains. In fact, we show that for a naturalclass of twisted spectral triples (A,H, D)σ over locally convex algebras (which we call smoothtwisted spectral triples) the Connes-Chern character descends to a class Ch(D)σ ∈ HP0(A),where HP•(A) is the periodic cyclic cohomology of continuous cochains (see Proposition 5.9).We then show that the invariance of this class under conformal perturbations of twisted spectraltriples (Proposition 6.8).

The construction of the conformal Dirac spectral triple (C∞(M)oG,L2g(M,/S), /Dg)σg associated

to a conformal class C on a compact spin manifold M alluded to above a priori depends on achoice of a metric g ∈ C . As it turns out, up to equivalence of twisted spectral triples, changinga metric within C only amounts to performing a conformal deformation in the realm pointed outby Moscovici [Mo2]. Combining this with the invariance under conformal deformations of theConnes-Chern character mentioned above shows that the Connes-Chern character Ch(/Dg)σg ∈HP0(C∞(M) oG) is an invariant of the conformal class C .

The index formula in conformal-diffeomorphism invariant geometry is derived from the compu-tation of the Connes-Chern character Ch(/Dg)σg . As we don’t know whether the conformal Diracspectral triple satisfies Moscovici’s Ansatz alluded to above, we a priori cannot make use of a CMcocycle representative to compute the Connes-Chern character Ch(/Dg)σg . However, its conformalinvariance means that we can choose any metric in the conformal class C to compute it. In partic-ular, we may take a G-invariant metric. It follows from Obata-Ferrand theorem that such a metricalways exists when the conformal structure is not flat (i.e., it is not equivalent to the conformalstructure of the round sphere). In the case the metric g is G-invariant the conformal spectral triplebecomes an ordinary spectral triple. We postpone to the sequel [PW2] (referred to as Part II) theexplicit computation of the Connes-Chern character of a general equivariant Dirac spectral triple(C∞(M)oG,L2

g(M,/S), /Dg), where G is a group of smooth isometries. Note that for our purposeit crucial to compute the Connes-Chern character defined as a class in the continuous cochaincyclic cohomology HP•(C∞(M) o G). Using the results of Part II and given any G-invariantmetric g ∈ C , we then can express the Connes-Chern character Ch(/Dg)σg in terms of explicituniversal polynomials in curvatures and normal curvatures of fixed-point manifolds of the variousdiffeomorphisms in G (see Theorem 8.3 for the explicit formulas). These terms are reminiscentof the local equivariant index theorem for Dirac operators of Atiyah-Segal [AS]. Together withthe index formula (1.1) this provides us with a local index formula in conformal-diffeomorphisminvariant geometry (for non-flat conformal structures).

As cyclic cohomology is dual to cyclic homology, the conformal invariance of the Connes-Cherncharacter Ch(/Dg)σg implies that any pairing with cyclic cycles produces a numerical conformalinvariant. In order to understand these invariants we construct geometric cyclic cycles spanningthe cyclic homology of C∞(M) oG. At this stage it is really important to have a representativeof the Connes-Chern character in HP0(C∞(M) o G), since the geometric cycles live in the dualcyclic homology HP0(C∞(M)oG). This cyclic homology was computed by Brylinski-Nistor [BN]

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(see also [Cr]). Namely,

(1.2) HPi(C∞(M) oG) '

⊕〈φ〉∈〈G〉

⊕0≤a≤na even

⊕q≥0

2q+1≤a

H2q+i(Mφa )Gφ , i = 0, 1,

where 〈G〉 is set of conjugacy classes of G and H•(Mφ) is the Gφ-invariant cohomology of thefixed point submanifold Mφ

a , dimMφa = a, where Gφ is the stabilizer in G of a representative φ

in a given conjugacy class 〈φ〉. The results of [BN] and [Cr] are stated in a much more generalcase of the convolution algebra of an etale groupoid. Furthermore, although it can be exhibitedby explicit geometric Hochschild cycles (see, e.g., [Da] in case G is finite), it is difficult to obtainexplicit cyclic cycles due to some incompatibility properties between the cyclic operator and theaction of the group G.

The above issue is resolved by observing that the cyclic mixed cochain-complex of C∞(M)oGis quasi-isomorphic to a sub-complex consisting of what we call G-normalized cochains. These areG-invariant cochains that are further invariant under the transformations,

(a0, . . . , am) −→ (a0, . . . , ajuψ, u−1ψ aj+1, . . . , am), al ∈ C∞(M) oG,

where ψ ranges over G and ψ → uψ is the representation of elements of G in the crossed productalgebra C∞(M)oG. It is fairly natural to consider G-normalized cochains since, for instance, thetranverse fundamental class cocycle of Connes [Co2] is G-normalized. Furthermore, the formulasfor the Connes-Chern character Ch(/Dg)σg mentioned above produce G-normalized representatives.

At the level of chains, the G-normalization condition essentially amounts to replacing the pro-jective tensor product ⊗ by a coarser topological tensor product ⊗G which takes into accountthe action of G and the CG-bimodule structure of the crossed-product algebra C∞(M) oG. Theresults of [BN, Cr] imply that the projection of the cyclic mixed complex onto the G-normalizedcyclic mixed complex is a quasi-isomorphism (see Proposition 9.12). As the Connes-Chern char-acter Ch(/Dg)σg is explicitly represented by G-normalized cochains, it then follows that we onlyneed to exhibit geometric G-normalized cyclic chains. This is done along the line of argumentsof [BN, Cr]. Given any φ ∈ G and a fixed-point submanifold Mφ

a , we then have a natural mixedcomplex morphism from the Gφ-invariant de Rham complex of Mφ

a to the G-normalized cyclicmixed complex of C∞(M) oG arising from the map,

ω = f0df0 ∧ · · · ∧ dfm −→ ηω :=∑σ∈Sm

f0⊗Gfσ(1)⊗G · · · ⊗Gfσ(m−1)⊗Gfσ(m)uφ,

where Sm is the m’th symmetric group and f j is a suitable extension of f j ∈ C∞(Mφa ) into a

smooth function on M . As it turns out, gathering all these maps yields the isomorphism (1.2). Thisalso shows that, any Gφ-invariant closed even form ω on Mφ

a gives rise to a G-normalized cycliccycle ηω on C∞(M) oG. Pulling back its homology class to a class π∗[ηω] in HP0(C∞(M) oG)and pairing it with the Connes-Connes character Ch(/Dg)σg we then obtain a conformal invariant,

(1.3) Ig(ω) :=⟨Ch(/Dg)σg , π

∗[ηω]⟩.

Furthermore, using the geometric expression of the Connes-Chern character Ch(/Dg)σg that weobtained in case of a G-invariant metric, we see that, for any G-invariant metric in the conformalclass at stake,

(1.4) Ig(ω) = (−i)n2 (2π)−a2

∫Mφa

A(RTM

φ)∧ νφ

(RN

φ)∧ ω,

where νφ

(RN

φ)

is an explicit polynomial in the normal curvature RNφ

.

The above conformal invariants are not of the same type of the conformal invariants consideredby Alexakis [Al] in his solution of the conjecture of Deser-Schwimmer [DS] on the characterizationof global conformal invariants. In [Al, DS] the conformal invariants appear as integrals overthe whole manifold M of local Riemannian invariants, whereas the invariants (1.3)–(1.4) involveintegrals over some fixed-point submanifolds. As explained in Remark 10.3, our invariants arerelated to conformal invariants exhibited by Branson-Ørsted [BØ2]. It would be interesting to

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find, even conjecturally, a characterization of all conformal invariants encompassing the invariantsof [Al, DS] and the invariants (1.3)–(1.4) and those in [BØ2].

In this paper, we focus on conformal structures of even dimension that are no equivalent tothe conformal structure of the round sphere. We would like to stress out that our results can beextended to the odd dimension case. Although, the index theory in the odd-dimensional case issomewhat different, as it involves spectral flows, there is a Connes-Chern character which is almostidentical to that in the even case. This enables us to obtain a local index formula and exhibitconformal invariants as in the even case. We defer the details to a sequel of this paper. In the caseof round spheres, the conformal group is essential, and so the computation of the Connes-Cherncharacter requires a different type of analysis than those carried in Part II (see also [Mo2] for adiscussion on this issue). We hope to deal with round spheres in a future article.

The paper is organized as follows. In Section 2, we review the main definitions and examplesregarding twisted spectral triples and the construction of their index maps. In Section 3, wereview the main facts about cyclic cohomology, cyclic homology and the Chern character in cyclichomology. In Section 4, we review the construction of the Connes-Chern character of a twistedspectral triple. In Section 5, we show that for smooth twisted spectral triples the Connes-Cherncharacter descends to the cyclic cohomology of continuous cochains. In Section 6, we establishthe invariance of the Connes-Chern character under conformal deformations. In Section 7, afterreviewing the construction of the conformal Dirac spectral triple, we prove that its Connes-ChernCharacter is a conformal invariant. In Section 8, we compute this Connes-Chern character by usingthe results of Part II. This provides us with a local index formula in conformal-diffeomorphisminvariant geometry. In Section 9, we construct geometric cyclic cycles spanning the cyclic homologyof the crossed-product algebra C∞(M)oG. In Section 10, we combine the results of the previoussections to construct our conformal invariants.

Acknowledgements

The authors would like to thank Sasha Gorokhovsky, Xiaonan Ma, Victor Nistor, HesselPosthuma, Xiang Tang and Bai-Ling Wang for helpful discussions related to the subject mat-ter of this paper. They also would like to thank the following institutions for their hospitalityduring the preparation of this manuscript: Seoul National University (HW); University of Ade-laide, University of California at Berkeley, Kyoto University (Research Institute of MathematicalSciences and Department of Mathematics), University Paris 7, and Mathematical Science Centerof Tsinghua University (RP); Australian National University, Chern Institute of Mathematics ofNankai University, and Fudan University (RP+HW).

2. Index Theory on Twisted Spectral Triples.

In this section, we recall how the datum of a twisted spectral triple naturally gives rise to anindex problem ([CM3, PW1]). The exposition closely follows that of [PW1] (see also [Mo1] for thecase of ordinary spectral triples).

In the setting of noncommutative geometry, the role of manifolds is played by spectral triples.

Definition 2.1. A spectral triple (A,H, D) is given by

(1) A Z2-graded Hilbert space H = H+ ⊕H−.(2) A unital ∗-algebra A represented by bounded operators on H preserving its Z2-grading.(3) A selfadjoint unbounded operator D on H such that

(a) D maps dom(D) ∩H± to H∓.(b) The resolvent (D + i)−1 is a compact operator.(c) a dom(D) ⊂ dom(D) and [D, a] is bounded for all a ∈ A.

Remark 2.2. The condition (3)(a) implies that with respect to the splitting H = H+ ⊕ H− theoperator D takes the form,

D =

(0 D−

D+ 0

), D± : dom(D) ∩H± → H∓.

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Example 2.3. The paradigm of a spectral triple is given by a Dirac spectral triple,

(C∞(M), L2g(M,/S), /Dg),

where (Mn, g) is a compact spin Riemannian manifold of even dimension n and /Dg is its Dirac

operator acting on the spinor bundle /S = /S+ ⊕ /S−.

The definition of a twisted spectral triple is similar to that of an ordinary spectral triple, exceptfor some “twist” given by the conditions (3) and (4)(b) below.

Definition 2.4 ([CM3]). A twisted spectral triple (A,H, D)σ is given by

(1) A Z2-graded Hilbert space H = H+ ⊕H−.(2) A unital ∗-algebra A represented by even bounded operators on H.(3) An automorphism σ : A → A such that σ(a)∗ = σ−1(a∗) for all a ∈ A.(4) An odd selfadjoint unbounded operator D on H such that

(a) The resolvent (D + i)−1 is compact.(b) a(domD) ⊂ domD and [D, a]σ := Da− σ(a)D is bounded for all a ∈ A.

The relevance of the notion of twisted spectral triples in the setting of conformal geometrystems from the following observation. Let (C∞(M), L2

g(M,/S), /Dg) be a Dirac spectral triple as inExample 2.3, and consider a conformal change of metric,

g = k−2g, k ∈ C∞(M), k > 0.

We then can form a Dirac spectral triple (C∞(M), L2g(M,/S), /Dg) associated to the new metric g.

As it turns out (see [PW1]) this spectral triple is equivalent to the following spectral triple,(C∞(M), L2

g(M,/S),√k /Dg

√k).

We note that the above spectral triple continues to make sense if we only assume k to be a positiveLipschitz function on M .

More generally, let (A,H, D) be an ordinary spectral triple and k a positive element of A.If we replace D by its conformal deformation kDk then, when A is noncommutative, the triple(A,H, kDk) need not be an ordinary spectral triple. However, as the following result shows, italways gives rise to a twisted spectral triple.

Proposition 2.5 ([CM3]). Let σ : A → A be the automorphism defined by

(2.1) σ(a) := k2ak−2 ∀a ∈ A.Then (A,H, kDk)σ is a twisted spectral triple.

Remark 2.6. A more elaborate version of the above example, and the main focus of this paper, isthe conformal Dirac spectral triple of Connes-Moscovici [CM3]. This is a twisted spectral tripletaking into account of the action of the group of diffeomorphisms preserving a given conformalstructure. We refer to Section 7.1 for a review of this example.

Remark 2.7. We refer to [CM3, GMT, CT, IM, Mo2, PW3] for the constructions of various otherexamples of twisted spectral triples.

From now on, we let (A,H, D)σ be a twisted spectral triple. In addition, we let E be a finitelygenerated projective right module over A, i.e., E is a direct summand of a free module E0 ' AN .

Definition 2.8 ([PW1]). A σ-translate for E is a finitely generated projective right module Eσequipped with the following data:

(i) A linear isomorphism σE : E → Eσ.(ii) An idempotent e ∈MN (A), N ∈ N.(iii) Right module isomorphisms φ : E → eAN and φσ : Eσ → σ(e)AN such that

(2.2) φσ σE = σ φ.

Remark 2.9. The condition (2.2) implies that

σE(ξa) = σE(ξ)σ(a) for all ξ ∈ E and a ∈ A.6

Remark 2.10. When E = eAN with e = e2 ∈ MN (A) we always may take Eσ = σ(e)AN asσ-translate of eAN . In this case σE agrees on eAN with the lift of σ to AN . In particular, Eσ = Ewhen σ = id.

Throughout the rest of the section we let Eσ be a σ-translate of E . In addition, we consider the(A,A)-bimodule of twisted 1-forms,

Ω1D,σ(A) =

Σai[D, bi]σ : ai, bi ∈ A

.

The “twisted” differential dσ : A → Ω1D,σ(A) is given by

(2.3) dσa := [D, a]σ ∀a ∈ A.

This is a σ-derivation, in the sense that

(2.4) dσ(ab) = (dσa)b+ σ(a)dσb ∀a, b ∈ A.

Definition 2.11. A σ-connection on E is a C-linear map ∇ : E → Eσ ⊗A Ω1D,σ(A) such that

(2.5) ∇(ξa) = (∇ξ)a+ σE(ξ)⊗ dσa ∀ξ ∈ E ∀a ∈ A.

Example 2.12. Suppose that E = eAN with e = e2 ∈ MN (A). Then a natural σ-connection on Eis the Grassmannian σ-connection ∇E0 defined by

(2.6) ∇E0 ξ = σ(e)(dσξj) for all ξ = (ξj) in E .

Definition 2.13. A Hermitian metric on E is a map (·, ·) : E × E → A such that

(1) (·, ·) is A-sesquilinear, i.e., it is A-antilinear with respect to the first variable and A-linearwith respect to the second variable.

(2) (·, ·) is positive, i.e., (ξ, ξ) ≥ 0 for all ξ ∈ E.(3) (·, ·) is nondegenerate, i.e., ξ → (ξ, ·) is an A-antilinear isomorphism from E onto itsA-dual HomA(E ,A).

Example 2.14. The canonical Hermitian structure on the free module AN is given by

(2.7) (ξ, η)0 = ξ∗1η1 + · · ·+ ξ∗qηq for all ξ = (ξj) and η = (ηj) in AN .

It gives a Hermitian metric on any direct summand E = eAN , e = e2 ∈MN (A), (see, e.g., [PW1]).

From now on we assume that E and its σ-translate carry a Hermitian metric. We denote byH(E) the pre-Hilbert space consisting of E ⊗A H equipped with the Hermitian inner product,

(2.8) 〈ξ1 ⊗ ζ1, ξ2 ⊗ ζ2〉 := 〈ζ1, (ξ1, ξ2)ζ2〉 , ξj ∈ E , ζj ∈ H,

where (·, ·) is the Hermitian metric of E . It can be shown H(E) actually is a Hilbert space and itstopology is independent of the choice of the Hermitian inner product of E (see, e.g., [PW1]). Wealso note there is a natural Z2-grading on H(E) given by

(2.9) H(E) = H+(E)⊕H−(E), H±(E) := E ⊗A H±.

We denote by H(Eσ) the similar Z2-graded Hilbert space associated to Eσ and its Hermitianmetric.

Let ∇E be a σ-connection on E . Regarding Ω1D,σ(A) as a subalgebra of L(H) we have a natural

left-action c : Ω1D,σ(A)⊗A H → H given by

c(ω ⊗ ζ) = ω(ζ) for all ω ∈ Ω1D,σ(A) and ζ ∈ H.

We then denote by c(∇E)

the composition (1Eσ ⊗ c) (∇E ⊗ 1H) : E ⊗ H → Eσ ⊗H. Thus, for

ξ ∈ E and ζ ∈ H, and upon writing ∇Eξ =∑ξα ⊗ ωα with ξα ∈ Eσ and ωα ∈ Ω1

D,σ(A), we have

(2.10) c(∇E)

(ξ ⊗ ζ) =∑

ξα ⊗ ωα(ζ).

In what follows we regard the domain of D as a left A-module, which is possible since the actionof A on H preserves domD.

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Definition 2.15. The operator D∇E : E ⊗A dom(D)→ H(Eσ) is defined by

(2.11) D∇E (ξ ⊗ ζ) := σE(ξ)⊗Dζ + c(∇E)(ξ ⊗ ζ) for all ξ ∈ E and ζ ∈ domD.

Remark 2.16. Although the operators σE , D and ∇E are not module maps, the operator is welldefined as a linear map with domain E ⊗A dom(D) (see [PW1]).

Remark 2.17. With respect to the Z2-gradings (2.9) for H(E) and H(Eσ) the operator D∇E takesthe form,

(2.12) D∇E =

(0 D−∇E

D+∇E 0

), D±∇E : E ⊗A domD± −→ H∓(Eσ).

That is, D∇E is an odd operator.

Example 2.18 (See [PW1]). Suppose that E = eAN with e = e2 ∈ MN (A) and let ∇E0 be theGrassmanian σ-connection of E . Then up to the canonical unitary identifications H(E) ' eHNand H(Eσ) ' σ(e)HN the operators D∇E0 agrees with

σ(e)(D ⊗ 1N ) : e(domD)N −→ σ(e)HN .Example 2.19. In the case of a Dirac spectral triple (C∞(M), L2

g(M,/S), /Dg), we may take E to

the module C∞(M,E) of smooth sections of a vector bundle E over M . Any Hermitian metricand connection on E give rise to a Hermitian metric and a connection ∇E on E . Furthermore, ifwe set H = L2

g(M,E), then, under the natural identification H(E) ' L2(M,/S ⊗ E), the operator(/Dg)∇E agrees with the usual twisted Dirac operator /D∇E as defined, e.g., in [BGV].

Proposition 2.20 ([PW1]). The operator D∇E is closed and Fredholm.

Note that the above result implies that the operators D±∇E in (2.12) are Fredholm. This leadsus to the following definition.

Definition 2.21. The index of the operator D∇E is

indD∇E =1

2

(indD+

∇E − indD−∇E),

where indD±∇E is the usual Fredholm index of D±∇E (i.e., indD±∇E = dim kerD±∇E−dim cokerD±∇E ).

Remark 2.22. In general the indices ± indD±∇E do not agree, so that it is natural take their meanto define the index of D∇E . However, as shown in [PW1], the index indD,σ is an integer whenthe automorphism σ is ribbon, in the sense there is another automorphism τ : A → A such thatσ = τ τ and τ(a)∗ = τ−1(a∗) for all a ∈ A. The ribbon condition is satisfied in all main examplesof twisted spectral (see [PW1]). When this condition holds, it is shown in [PW3] that we alwayscan endow E with a “σ-Hermitian structure” (see [PW3] for the precise definition). In that case,for any connection ∇E compatible with the σ-Hermitian structure,

indD∇E = dim kerD+∇E − dim kerD−∇E ∈ Z.

The above formula is the generalization of the usual formula for the index of a Dirac operatortwisted by a Hermitian connection on a Hermitian vector bundle.

As it turns out the index indD∇E only depends on the K-theory class of E (see [PW1]). Moreprecisely, we have the following result.

Proposition 2.23 ([CM3, PW1]). There is a unique additive map indD,σ : K0(A) → 12Z such

that, for any finitely generated projective module E over A and any σ-connection on E, we have

indD,σ[E ] = indD∇E .

3. Cyclic Cohomology and the Chern Character

Cyclic cohomology, and its dual version cyclic homology, were discovered by Connes [Co3] andTsygan [Ts] independently. In this section, we review the main facts about cyclic cochomologyand cyclic homology, and how this enables us to define a Chern character in cyclic homology. Werefer to [Co3, Co4, Lo] for more complete accounts on these topics. Throughout this section welet A be a unital algebra over C.

8

3.1. Cyclic cohomology. The Hochschild cochain-complex of A is defined as follows. The spaceof m-cochains Cm(A), m ∈ N0, consists of (m + 1)-linear maps ϕ : Am+1 → C. The Hochschildcoboundary b : Cm(A)→ Cm+1(A), b2 = 0, is given by

bϕ(a0, . . . , am+1) =

m∑j=0

(−1)jϕ(a0, . . . , ajaj+1, . . . , am+1)(3.1)

+ (−1)m+1ϕ(am+1a0, . . . , am), aj ∈ A.(3.2)

A cochain ϕ ∈ Cm(A) is called cyclic when Tϕ = ϕ, where the operator T : Cm(A)→ Cm(A)is defined by

(3.3) Tϕ(a0, . . . , am) = (−1)mϕ(am, a0, . . . , am−1), aj ∈ A.We denote by Cmλ (A) the space of cyclic m-cochains. As b(C•λ(A)) ⊂ C•+1

λ (A), we obtain a sub-complex (C•λ(A), b), the cohomology of which is denoted HC•(A) and called the cyclic cohomologyof A.

The operator B : Cm(A)→ Cm−1(A) is given by

(3.4) B = AB0(1− T ), where A = 1 + T + · · ·+ Tm,

and the operator B0 : Cm(A)→ Cm−1(A) is defined by

(3.5) B0ϕ(a0, . . . , am−1) = ϕ(1, a0, . . . , am−1), aj ∈ A.Note that B is annihilated by cyclic cochains. Moreover, it can be checked that B2 = 0 andbB + Bb = 0. Therefore, in the terminology of [Lo, §2.5.13], we obtain a mixed cochain-complex(C•(A), b, B), which is called the cyclic mixed cochain-complex of A. Associated to this mixedcomplex is the periodic cyclic complex (C [•](A), b+B), where

C [i](A) =

∞⊕q=0

C2q+i(A), i = 0, 1,

and we regard b and B as operators between C [0](A) and C [1](A). The corresponding cohomologyis called the periodic cyclic cohomology of A and is denoted by HP•(A). Note that a periodiccyclic cocycle is a finitely supported sequence ϕ = (ϕ2q+i) with ϕ2q+i ∈ C2q+i(A), q ≥ 0, suchthat

bϕ2q+i +Bϕ2q+2+i = 0 for all q ≥ 0.

As the operator B is annihilated by cyclic cochains, any cyclic m-cocycle ϕ is naturally identifiedwith the periodic cyclic cocycle (0, . . . , 0, ϕ, 0, . . .) ∈ C [i](A), where i is the parity of m. This givesrise to natural morphisms,

HC2q+•(A) −→ HP•(A), q ≥ 0.

Connes’ periodicity operator S : Cmλ (A) → Cm+2λ (A) is obtained from the cup product with the

unique cyclic 2-cocycle on C taking the value 1 at (1, 1, 1) (see [Co3, Co4]). Equivalently,

S =1

(m+ 1)(m+ 2)

m+1∑j=1

(−1)jSj ,

where the operator Sj : Cmλ (A)→ Cm+2λ (A) is given by

Sjϕ(a0, . . . , am+2) =∑

0≤l≤j−2

(−1)lϕ(a0, . . . , alal+1, . . . , ajaj+1, . . . , am+2)

+ (−1)j+1ϕ(a0, . . . , aj−1ajaj+1, . . . , am+2).(3.6)

Here (cf. [Co4]) the operator S is normalized so that the induced map on HC•(A) satisfies

(3.7) S = −bB−1 on HC•(A).

In particular, if ϕ is any cyclic cocycle, then Sϕ is a cyclic cocycle whose class in HP•(A) agreeswith that of ϕ. Furthermore, Connes [Co3, Theorem II.40] proved that

(3.8) lim−→(HC2q+•(A), S

)= HP•(A),

9

where the left-hand side is the inductive limit of the direct system (HC2q+•(A), S).It is sometimes convenient to “normalize” the cyclic mixed complex. More precisely, we say

that a cochain ϕ ∈ Cm(A) is normalized when

(3.9) ϕ(a0, . . . , am) = 0 whenever aj = 1 for some j ≥ 1.

We denote by Cm0 (A) the space of normalized m-cochains. As the operators b and B preserve thespace C•0 (A), we obtain a mixed subcomplex (C•0 (A), b, B) of the cyclic mixed complex. Note thatB = B0(1−T ) on C•0 (A). We denote by HP•0(A) the cohomology of the normalized periodic com-

plex (C[•]0 (A), b+B), where C

[•]0 (A) =

⊕q≥0 C

2q+•0 (A). Furthermore (see [Lo, Corollary 2.1.10]),

the inclusion of C•0 (A) in C•(A) gives rise to an isomorphism,

(3.10) HP•0(A) ' HP•(A).

Example 3.1. Let A = C∞(M), where M is a closed manifold. For m = 0, 1, . . . , n, let Ωm(M)be space of m-dimensional currents. Any current C ∈ Ωm(M) defines a cochain ϕC ∈ Cm(A) by

ϕC(f0, . . . , fm) =1

m!

⟨C, f0df1 ∧ · · · ∧ dfm

⟩, fj ∈ C∞(M).

Note that ϕC is a normalized cochain. Moreover it can be checked that bϕC = 0 and BϕC = ϕdtC ,where dt is the de Rham boundary for currents. Therefore, we obtain a morphism from the mixedcomplex (Ω•(M), 0, dt) to the cyclic mixed complex of A = C∞(M). In particular, we have anatural linear map,

(3.11) αM : H[•](M,C) −→ HP•(C∞(M)), H[i](M,C) :=⊕q≥0

H2q+i(M,C), i = 0, 1,

where H2q+i(M,C) is the de Rham homology of M of degree 2q + i.

3.2. Cyclic homology. The cyclic mixed cochain-complex defined above is the dual of a mixedchain-complex defined as follows. The space of m-chains, m ≥ 0, is Cm(A) = A⊗(m+1). TheHochschild boundary b : Cm(A)→ Cm−1(A), b2 = 0, is given by

b(a0 ⊗ · · · ⊗ am) =

m−1∑j=0

(−1)ja0 ⊗ · · · ⊗ ajaj+1 ⊗ · · · ⊗ am(3.12)

+ (−1)mama0 ⊗ · · · ⊗ am−1, aj ∈ A.(3.13)

A chain η ∈ Cm(A) is cyclic when Tη = η, where the cyclic operator T : Cm(A) → Cm(A) isgiven by

T (a0 ⊗ · · · ⊗ am) = (−1)mam ⊗ a0 ⊗ · · · ⊗ am−1, aj ∈ A.We denote by Cλm(A) the space of cyclic m-chains. This provides us with a sub-complex (Cλ• (A), b)of the Hochschild chain-complex. Its homology is called the cyclic homology of A and is denotedby HC•(A).

The operator B : Cm(A)→ Cm+1(A) is given by

(3.14) B := (1− T )B0A, where A := 1 + T + · · ·+ Tm,

and the operator B0 : Cm(A)→ Cm+1(A) is defined by

(3.15) B0(a0 ⊗ · · · ⊗ am) = 1⊗ a0 ⊗ · · · ⊗ am, aj ∈ A.

It can be checked that B2 = 0 and bB + Bb = 0, so that we obtain a mixed chain-complex(C•(A), b, B), called the cyclic mixed chain-complex of A. The associated periodic chain-complexis (C[•](A), b+B), where

C[i](A) =

∞∏q=0

C2q+i(A), i = 0, 1.

The homology of this complex is the periodic cyclic homology of A and is denoted by HP•(A).10

The periodicity operator S : Cλm(A)→ Cλm−2(A) is given by

(3.16) S =1

m(m− 1)

m−1∑j=1

(−1)jSj ,

where the operator Sj : Cλm(A)→ Cλm(A) is defined by

Sj(a0 ⊗ · · · ⊗ am) =

∑0≤l≤j−2

(−1)la0 ⊗ · · · ⊗ alal+1 ⊗ · · · ⊗ ajaj+1 ⊗ · · · ⊗ am

+ (−1)j+1a0 ⊗ · · · ⊗ aj−1ajaj+1 ⊗ · · · ⊗ am+2.

The operator S commutes with the Hochschild boundary b and any cyclic cycle η is cohomologousto Sη in HP•(A). Furthermore, the dual version of (3.8) holds, namely,

lim←− (HC2q+•(A), S) = HP•(A),

where the left-hand side is the projective limit of the system (HC2q+•(A), S).The mixed chain-complex (C•(A), b, B) is normalized as follows. For m ∈ N and l = 1, . . . ,m,

the operator Nm,l : Cm−1(A)→ Cm(A) is defined by

Nm,l(a0 ⊗ · · · ⊗ am−1) = a0 ⊗ · · · ⊗ al ⊗ 1⊗ al+1 ⊗ · · · ⊗ am−1, aj ∈ A.

We then set Nm(A) :=∑ml=1Nm,l(Cm(A)). The spaces of normalized chains C0

m(A), m ≥ 0, arethen defined by

C00 (A) = C0(A) and C0

m(A) = Cm(A)/Nm(A) for m ≥ 1.

Note that C0m(A) ' A ⊗ (A/C)⊗m for m ≥ 1. The operators b, B, and S descend to operators

on normalized chains. In particular, we obtain another mixed chain-complex (C0•(A), b, B). The

associated periodic complex is (C0[•](A), b + B), where C0

[i](A) =∏∞q=0 C

02q+i(A). Its homology

is denoted HP0•(A). Moreover, the canonical projection C•(A) → C0

•(A) gives rise to an isomor-phism,

(3.17) HP•(A) ' HP0•(A).

The duality pairing between C•(A) and C•(A) is given by

(3.18)⟨ϕ, a0 ⊗ · · · ⊗ am

⟩= ϕ(a0, . . . , am), ϕ ∈ Cm(A), aj ∈ A.

This induces a duality pairing between cyclic cochains and cyclic chains. This also extends to a

duality pairing between C [•](A) (resp., C[•]0 (A)) and C[•](A) (resp., C0

[•](A)). For instance, for

ϕ = (ϕ2q+i) ∈ C [i](A) and η = (η2q+i) ∈ C[i](A), we have

(3.19) 〈ϕ, η〉 =∑q≥0

〈ϕ2q+i, η2q+i〉 .

Moreover, the operators b, T , A, B0, B, S on cochains are the transposes of the correspondingoperators on chains. In particular, the aformentioned pairings descend to duality pairings betweenthe cohomology space HC•(A) (resp., HP•(A), HP•0(A)) and the homology space HC•(A), (resp.,HP•(A), HP0

•(A)).

Example 3.2. Let A = C∞(M), where M is a closed manifold. Let Ωm(M) be the space ofdifferential forms on M of degree m. There is a natural linear map αM : Cm(A)→ Ωm(M) givenby

(3.20) αM (f0 ⊗ · · · ⊗ fm) =1

m!f0df1 ∧ . . . ∧ dfm, f j ∈ C∞(M).

It is a mixed-complex morphism from the cyclic mixed chain-complex of A to (Ω•(M), 0, d),where d is de Rham’s differential. This thus gives rise to a morphism of chain-complexes from the

11

periodic complex (C[•](C∞(M)), b + B) to the de Rham complex (Ω[•](M), d), where Ω[i](M) =⊕∞

q=0 Ω2q+i(M), i = 0, 1. In particular, we get a linear map,

αM : HP•(C∞(M)) −→ H [•](M,C), H [i](M,C) :=

⊕q≥0

H2q+i(M,C), i = 0, 1,

where H2q+i(M,C) is the de Rham cohomology of M in degree 2q+ i. This map is the transposeof the map αM defined by (3.11). Thus, for any de Rham homology class ω ∈ H [i](M,C) and anyperiodic cyclic homology class η ∈ HPi(C

∞(M)), we have

(3.21)⟨αM (ω), η

⟩= 〈ω, αM (η)〉 ,

where the pairing on the r.h.s. is the pairing between de Rham homology and de Rham cohomology.

3.3. Chern character in cyclic homology. Given a positive integer N , the trace tr on thealgebra MN (A) = A⊗MN (C) gives rise to a linear map tr : Cm(MN (A))→ Cm(A) defined by

tr[(a0 ⊗ µ0)⊗ · · · ⊗ (am ⊗ µm)

]= tr

[µ0 · · ·µm

]a0 ⊗ · · · ⊗ am, aj ∈ A, µj ∈MN (C).

This map is compatible with the operators b, T , and B and yields isomorphisms at the level ofcyclic homology and periodic cyclic homology (see [Co3, Lo]).

The Chern character in cyclic homology [Co3, GS, Lo] is defined as follows. Let e be anidempotent in MN (A) and define the even normalized chain Ch(e) = (Ch2q(e))q≥0 ∈ C

0[0](A) by

(3.22) Ch0(e) = tr[e], Ch2q(e) = (−1)q(2q)!

q!tr

(e− 1

2

)⊗

2q times︷ ︸︸ ︷e⊗ · · · ⊗ e

, q ≥ 1.

It can be checked that Ch(e) = 0 in a cocycle in C0[•](A) whose class in HP0

0(A) ' HP0(A) depends

only on the K-theory class of e. In fact, this gives rise to an additive map Chern character map,

(3.23) Ch : K0(A) −→ HP0(A).

Incidentally, given any finite generated projective module E over A, we define its Chern characterby

(3.24) Ch(E) = class of Ch(e) in HP0(A),

where e is any idempotent in some algebra MN (A), N ≥ 1, such that E ' eAN .Composing the Chern character map Ch : K0(A)→ HP0(A) with the duality pairing between

HP0(A) and HP0(A) provides us with a pairing between HP0(A) and K0(A). Furthermore, givenany cyclic 2q-cocycle ϕ it can be shown (see, e.g., [PW1, Remark 6.4]) that

(3.25) 〈ϕ,Ch(e)〉 = (−1)q(2q)!

q!

⟨ϕ, tr

[e⊗(2q+1)

]⟩∀e ∈MN (A), e2 = e.

Therefore, the pairing between HP0(A) and K0(A) given by the above Chern character map agreeswith the original pairing defined by Connes [Co3, Co4].

Example 3.3. Suppose now thatA = C∞(M), whereM is closed manifold, and let e ∈Mm(C∞(M)),e2 = e. Consider the vector bundle E = ran e, which we regard as a subbundle of the trivial vectorbundle E0 = M×Cm. By Serre-Swan theorem any vector bundle over M is isomorphic to a vectorbundle of this form. We then equip E with the Grassmannian connection ∇E defined by e, i.e.,

∇EXξ(x) = e(x) · ((Xξj)(x))) for all X ∈ C∞(M,TM) and ξ = (ξj) ∈ C∞(M,E).

As the curvature of ∇E is FE = e(de)2 = e(de)2e, it can be checked that

(3.26) αM (Ch(E)) = αM (Ch(e)) = Ch(FE) = Ch(E) in H [0](M).

12

3.4. Locally convex algebras. In various geometric situations we often work with algebras thatcarry natural locally convex algebra topologies (e.g., A = C∞(M), where M is a smooth manifold).In such cases it is often more convenient to work with continuous versions of cyclic cohomology(see [Co3]). We shall now briefly recall how the corresponding cyclic cohomologies and homologiesare defined in such a context.

We shall now assume that A is a (unital) locally convex algebra, i.e., it carries a locally convexspace topology with respect to which its product is continuous. Given m ∈ N0, we denote byCm(A) the space of continuous cochains on A, i.e., continuous (m + 1)-linear forms on Am+1.All the operators b, T , A, B0, B, S introduced earlier preserve C•(A). Therefore, all the above-mentioned results for arbitrary cochains hold verbatim for continuous cochains. In particular,we get cyclic mixed complexes (C•(A), b, B) and (C•0(A), b, B), where C•0(A) is the space ofnormalized continuous cochains. This then gives rise to the following cochain-complexes:

- The cyclic complex (C•λ(A), b), where C•λ(A) is the space of cyclic continuous cochains.

- The periodic cyclic complex (C[•](A), b+B), where C[i](A) =⊕

q≥0 C2q+i(A), i = 0, 1.

- The normalized periodic cyclic complex (C[•]0 (A), b+B), where C

[i]0 (A) =

⊕q≥0 C2q+i

0 (A).

The respective cohomologies of these complexes are denoted by HC•(A), HP•(A), and HP•0(A).Furthermore, the analogues of (3.8) and (3.10) hold. Namely,

lim−→(HC2q+•(A), S

)= HP•(A) and HP•0(A) ' HP•(A).

The corresponding chains are defined by replacing the algebraic tensor product ⊗ (over C) bythe projective topological tensor product ⊗ (see [Gr, Tr]). Thus, the space of m-chains is

Cm(A) = A⊗(m+1) =

m+ 1 times︷ ︸︸ ︷A⊗ · · · ⊗A .

All the operators b, T , A, B0, B, S on algebraic chains extends continuously to operators ontopological chains. Therefore, all the results on algebraic chains mentioned continue to hold fortopological chains. In particular, we obtain a mixed chain-complex (C•(A), b, B). This gives rise

to a cyclic complex (Cλ•(A), b) and a periodic cyclic complex (C[•](A), b + B). Here Cλ

•(A) isthe space of cyclic topological chains and C[i](A) =

∏q≥0 C2q+i(A), i = 0, 1. The respective

homologies of these chain-complexes are denoted by HC•(A) and HP•(A).The mixed complex (C•(A), b, B) is normalized as follows. The space of normalized m-chains

is given by the topological quotient,

C0•(A) = C•(A)/N•(A).

This provides us with a normalized mixed complex (C0•(A), b, B) and a normalized periodic cyclic

chain-complex (C0[•](A), b + B), where C0

[i](A) =∏q≥0 C0

2q+i(A), i = 0, 1. The homology of the

normalized periodic complex is denoted by HP0•(A).

Finally, the inclusion of C[•](A) into C[•](A) gives rise to an embedding of HP•(A) into HP•(A).Composing it with the Chern character map (3.23) yields a map,

Ch : K0(A) −→ HP0(A).

Example 3.4. Suppose that A = C∞(M), where M is a closed manifold. Then A carries a naturalFrechet algebra topology. The map C → ϕC in Example 3.1 maps even/odd currents to continuousperiodic (normalized) cochains. As proved by Connes [Co3], it descends to an isomorphism,

HP• (C∞(M)) ' H[•](M,C),

where H[•](M,C) is the even/odd de Rham homology. Likewise, the map αM given by (3.20) is

continuous and extends to C0• (C∞(M)) and gives rise to an isomorphism,

HP• (C∞(M)) ' H [•](M,C),

where H [•](M,C) is the even/odd de Rham cohomology. Furthemore, under the above isomor-phisms the pairing (3.19) is the usual duality pairing between de Rham homology and de Rhamcohomology.

13

4. The Connes-Chern Character of a Twisted Spectral Triple

In this section, we shall recall the construction the Connes-Chern character of a twisted spectraltriple and how this enables us to compute the associated index map [CM3, PW1]. This extendsto twisted spectral triples the construction of the Connes-Chern character of an ordinary spectraltriple by Connes [Co3]. The exposition follows closely that of [PW1].

Let (A,H, D)σ be a twisted spectral triple. We assume that (A,H, D)σ is p-summable for somep ≥ 1, that is,

(4.1) Tr |D|−p <∞.In what follows, letting L1(H) be the ideal of trace-class operators on H, we denote by Str itssupertrace, i.e., Str[T ] = Tr[γT ], where γ := idH+ − idH− is the Z2-grading of H.

Definition 4.1. Assume D is invertible and let q be an integer ≥ 12 (p − 1). Then τD,σ2q is the

2q-cochain on A defined by

(4.2) τD,σ2q (a0, . . . , a2q) = cq Str(D−1[D, a0]σ · · ·D−1[D, a2q]σ

)∀aj ∈ A,

where we have set cq = 12 (−1)q q!

(2q)! .

Remark 4.2. The right-hand side of (4.2) is well defined since the p-summability condition (4.1)and the fact that q ≥ 1

2 (p− 1) imply that

D−1[D, a0]σ · · ·D−1[D, aq]σ ∈ L1(H) ∀aj ∈ A.

Proposition 4.3 ([CM3, PW1]). Assume D is invertible and let q be an integer ≥ 12 (p−1). Then

(1) The cochain τD,σ2q is a normalized cyclic cocycle whose class in HP0(A) is independent ofthe value of q.

(2) For any finitely generated projective module E over A and σ-connection on E,

(4.3) indD∇E =⟨τD,σ2q ,Ch(E)

⟩,

where Ch(E) is the Chern character of E in cyclic homology.

When D is not invertible, we can reduce to the invertible case by passing to the unital invertibledouble of (A,H, D)σ as follows.

Consider the Hilbert space H = H⊕H, which we equip with the Z2-grading given by

γ =

(γ 00 −γ

),

where γ is the grading operator of H. On H consider the selfadjoint operator,

D =

(D 11 −D

), dom(D) := dom(D)⊕ dom(D).

Noting that

D2 =

(D2 + 1 0

0 D2 + 1

),

we see that D is invertible and |D|−p is trace-class. Let A = A ⊕ C be the unitalization of Awhose product and involution are given by

(a, λ)(b, µ) = (ab+ λb+ µa, λµ), (a, λ)∗ = (a∗, λ), a, b ∈ A, λ, µ ∈ C.

The unit of A is 1A = (0, 1). Thus, identifying any element a ∈ A with (a, 0), any element

a = (a, λ) ∈ A can be uniquely written as (a, λ) = a + λ1A. We represent A in H using the

representation π : A → L(H) given by

π(1A) = 1, π(a) =

(a 00 0

)∀a ∈ A.

In addition, we extend the automorphism σ into the automorphism σ : A → A given by

σ(a+ λ1A) = σ(a) + λ1A ∀(a, λ) ∈ A× C.14

It can be verified that any twisted commutator [D, π(a)]σ, a ∈ A, is bounded. We then deduce

that (A, H, D)σ is a p-summable twisted spectral triple. Moreover, as D is invertible we may

define the normalized cyclic cocycles τ D,σ2q , q ≥ 12 (p− 1).

Definition 4.4. Let q ≥ 12 (p− 1). Then τD,σ2q is the 2q-cochain on A defined by

τ D,σ2q (a0, . . . , a2q) = τ D,σ2q (a0, . . . , a2q)

= cq Str(D−1[D, π(a0)]σ · · · D−1[D, π(a2q)]σ

)∀aj ∈ A.

Remark 4.5. The cochain τ D2q is the restriction to A2q+1 of the cochain τ D,σ2q . Note that, as therestriction to A of the representation π is not unital, unlike in the invertible case, we don’t obtaina normalized cochain.

Proposition 4.6 ([PW1]). Let q be an integer ≥ 12 (p− 1).

(1) The cochain τD,σ2q is a cyclic cocycle whose class in HP0(A) is independent of q.

(2) If D is invertible, then the cocycles τD,σ2q and τD,σ2q are cohomologous in HP0(A).

(3) For any finitely generated projective module E over A and σ-connection on E,

(4.4) indD∇E =⟨τD,σ2q ,Ch(E)

⟩.

All this leads us to the following definition.

Definition 4.7. The Connes-Chern character of (A,H, D)σ, denoted by Ch(D)σ, is defined asfollows:

• If D is invertible, then Ch(D)σ is the common class in HP0(A) of the cyclic cocycles τD,σ2q

and τD,σ2q , with q ≥ 12 (p− 1).

• If D is not invertible, then Ch(D)σ is the common class in HP0(A) of the cyclic cocycles

τD,σ2q , q ≥ 12 (p− 1).

Remark 4.8. When σ = id the Connes-Chern character is simply denoted Ch(D); this is the usualConnes-Chern Character of an ordinary spectral triple constructed by Connes [Co3].

With this definition in hand, the index formulas (4.3) and (4.4) can be merged onto the followingresult.

Proposition 4.9 ([CM3, PW1]). For any Hermitian finitely generated projective module E overA and any σ-connection on E,

(4.5) indD∇E = 〈Ch(D)σ,Ch(E)〉 ,where Ch(E) is the Chern character of E in cyclic homology.

Example 4.10 ([Co3, BF, Po1]). Let (Mn, g) be a compact Riemannian manifold. The Connes-Chern character Ch(/Dg) of the Dirac spectral triple (C∞(M), L2(M,/S), /Dg) is cohomologous to

the even periodic cocycle ϕ = (ϕ2q)q≥0 given by

(4.6) ϕ2q(f0, . . . , f2q) =

(2iπ)−n2

(2q)!

∫M

A(RM ) ∧ f0df1 ∧ . . . ∧ df2q, f j ∈ C∞(M).

Together with (3.20), (3.22) and (3.26) this enables us to recover the index theorem of Atiyah-Singer [ASi1, ASi2] for Dirac operators.

Remark 4.11. The definitions of the cocycles τD,σ2q and τD,σ2q involve the usual (super)trace on trace-class operators, but this is not a local functional since it does vanish on finite rank operators. Asa result this cocycle is difficult to compute in practice (see, e.g., [BF]). Therefore, it stands forreason to seek for a representatative of the Connes-Chern character which is easier to compute. Forordinary spectral triples, and under further assumptions, such a representative is provided by theCM cocycle of Connes-Moscovici [CM2]. This cocycle is an even periodic cycle whose componentsare given by formulas which are local in the sense of noncommuative geometry. More precisely,

15

they involve a version for spectral triples of the noncommutative residue trace of Guillemin [Gu]and Wodzicki [Wo]. This provides us with the local index formula in noncommutative geometry.In Example 4.10 the cocycle ϕ = (ϕ2q) given by (4.6) is precisely the CM cocycle of the Diracspectral triple (C∞(M), L2(M,/S), /Dg) (see [CM2, Remark II.1] and [Po1]).

Remark 4.12. In the case of twisted spectral triples, Moscovici [Mo2] attempted to extend thelocal index formula to the setting of twisted spectral triples. He devised an Ansatz for such alocal index formula and verified it in the special case of a ordinary spectral triples twisted byscaling automorphisms (see [Mo2] for the precise definition). Whether Moscovici’s Ansatz holdsfor a larger class of twisted spectral triples still remains an open question to date. For instance,it is not known if Moscovici Ansatz holds for conformal deformations of ordinary spectral triplessatisfying the local index formula in noncommutative geometry.

5. Twisted Spectral Triples over Locally Convex Algebras

In this section, we shall explain how to refine the construction of the Connes-Chern characterfor twisted spectral triples over locally convex algebras.

In what follows by locally convex ∗-algebra we shall mean a ∗-algebra equipped with a locallyconvex tolopolgy with respect to which the product and involution are continuous.

Definition 5.1. A twisted spectral triple (A,H, D)σ over a locally convex ∗-algebra A is calledsmooth when the following conditions hold:

(1) The representation of A in L(H) is continuous.(2) The map a→ [D, a]σ is continuous from A to L(H).(3) The automorphism σ : A → A is a homeomorphism.

Example 5.2. Let (Mn, g) be a closed spin Riemannian manifold of even dimension. Then theassociated Dirac spectral triple (C∞(M), L2

g(M,/S), /Dg) is smooth.

Example 5.3. Any conformal deformation of a smooth ordinary spectral triple yields a smoothtwisted spectral.

Remark 5.4. As we shall see in Section 7 the conformal Dirac spectral triple of [CM3] is smooth.

Throughout the rest of this subsection we let (A,H, D)σ be a smooth twisted spectral triplewhich is p-summable for some p ≥ 1. We shall now show that the Connes-Chern characterof (A,H, D)σ, which is originally defined as a class in HP0(A), actually descends to a class inHP0(A).

Lemma 5.5. Let q be any integer ≥ 12 (p− 1).

(1) If D is invertible, then the cyclic cocycle τD,σ2q is continuous and its class in HP0(A) isindependent of q.

(2) The cyclic cocycle τD,σ2q is continuous and its class in HP0(A) is independent of q.

Proof. Assume that D is invertible and let q be an integer ≥ 12 (p − 1). By assumption the map

a→ [D, a]σ is continuous from A to L(H). Combining this with Holder’s inequality for Schattenideals we deduce that the map (a0, . . . , a2q) → γD−1[D, a0] · · ·D−1[D, a2q] is continuous from

A2q+1 to L1(H). As τD,σ2q is (up to a multiple constant) the composition of this map with the

operator trace, we deduce that τD,σ2q is a continuous cochain.

Moreover, by Lemma 7.4 and Lemma 7.5 of [PW1, ] we may write

(5.1) τD,σ2q+2 = b(ϕ2q+1 − ψ2q+1), τD,σ2q = −B(ϕ2q+1 − ψ2q+1),

where, up to normalization constants, the cochains ϕ2q+1 and ψ2q+1 are given by

ϕ2q+1(a0, . . . , a2q+1) = Str(a0D−1[D, a1]σ · · ·D−1[D, a2q+1]σ),

ψ2q+1(a0, . . . , a2q+1) = Str(σ(a0)[D, a1]σD−1 · · · [D, a2q+1]σD

−1), aj ∈ A.Note that ϕ2q+1 and ψ2q+1 are normalized cochains. Moreover, in the same way as with the

cocycle τD,σ2q we can show that these cochains are continuous. As (5.1) implies that τD,σ2q+2−τD,σ2q =

16

(B + b)(ϕ2q+1 − ψ2q+1), we then deduce that τD,σ2q and τD,σ2q+2 define the same class in HP0(A).

Alternatively, using (3.7) we get

(5.2) SτD,σ2q = −SB(ϕ2q+1 − ψ2q+1) = b(ϕ2q+1 − ψ2q+1) = τD,σ2q+2 in HC2q+2(A).

In case D is not invertible, we observe that the unitalization A = A⊕ C is a locally convex ∗-algebra with respect to the direct sum topology. It then can be checked that the invertible double(A, H, D)σ considered in Section 4 is a smooth twisted spectral triple. Therefore, by part (1) the

cocycle τ D,σ2q is continuous. As the inclusion of A into A is continuous and τD,σ2q is the restriction

to A of τ D,σ2q , we then see that τD,σ2q is a continuous cocycle on A. Note also that SτD,σ2q is the

restriction to A of Sτ D,σ2q (cf. Eq. (3.6)). Therefore using (5.2) we deduce that SτD,σ2q = τD,σ2q+2 in

HC2q+2(A), which implies that SτD,σ2q and τD,σ2q+2 define the same class in HP0(A). The proof iscomplete.

In addition, we will also need the following version for smooth twisted spectral triples of [PW1,Propositon C.1] on the homotopy invariance of the Connes-Chern character of a twisted spectraltriple.

Lemma 5.6. Assume D is invertible and consider an operator homotopy of the form,

Dt = D + Vt, 0 ≤ t ≤ 1,

where (Vt)0≤t≤1 is a C1-family of selfadjoint operators in L(H) such that Dt is invertible for all

t ∈ [0, 1] and (D−1t )0≤t≤1 is a bounded family in Lp(H). Then

(1) (A,H, Dt)σ is a smooth p-summable twisted spectral triple for all t ∈ [0, 1].

(2) For any q ≥ 12 (p + 1), the cocycles τD0,σ

2q and τD1,σ2q are cohomologous in HC2q(A), and

hence define the same class in HP0(A).

Proof. We know from [PW1, Propositon C.1] that (A,H, Dt)σ is a p-summable twisted spectral

triple for all t ∈ [0, 1] and, for any q ≥ 12 (p+ 1), the cocycles τD0,σ

2q and τD1,σ2q are cohomologous in

HC2q(A). Therefore, we only need to show that the twisted spectral triples (A,H, Dt)σ, t ∈ [0, 1],

are smooth and the cocycles τD0,σ2q and τD1,σ

2q differs by the coboundary of a continuous cycliccochain.

By assumption (A,H, D)σ is a smooth twisted spectral triple. In particular, the representationof A in L(H) is continuous and the automorphism σ is a homeomorphism. Moreover, for allt ∈ [0, 1] and a ∈ A,

[Dt, a]σ = [D, a]σ + Vta− σ(a)Vt.

We then see that the map (t, a)→ [Dt, a]σ is continuous from [0, 1]×A to L(H). It then followsthat (A,H, Dt)σ is a smooth twisted spectral triple for all t ∈ [0, 1].

Let q be an integer ≥ 12 (p − 1). In [PW1] the explicit homotopy between τD0,σ

2q and τD1,σ2q is

realized as follows. For t ∈ [0, 1] and a ∈ A set

δt(a) = D−1t [VtD−1t , σ(a)]Dt.

In addition, for j = 0, . . . , 2q + 1 we set

αtj(a) = a if j is even and αtj(a) = D−1t σ(a)Dt if j is odd.

We note that

δt(a) = D−1t (Vtαt1(a)− σ(a)Vt) and αt1(a) = a−D−1t [Dt, a]σ

In particular we see that the maps (t, a)→ αtj(a) are continuous from [0, 1]×A to L(H). Moreover,

it is shown in [PW1, Appendix C] that (D−1t )0≤t≤t is a C1-family in Lp(H). Therefore, we alsosee that the map (t, a)→ δt(a) is continuous from [0, 1]×A to Lp(H).

Bearing this mind it is shown in [PW1] that

(5.3) τD1,σ2q − τD0,σ

2q = Bη,17

where η is the Hochschild (2q + 1)-cocycle given by

η(a0, . . . , a2q+1) = cq

2q+1∑j=0

∫ 1

0

Str(αj(a

0)D−1t [Dt, a1]σ · · · δt(aj) · · ·D−1t [Dt, a

2q+1]σ)dt,

where cq is some normalization constant. It follows from all the previous observations and the factthat q ≥ 1

2 (p− 1) that all the maps

(t, a0, . . . , a2q+1)→ αj(a0)D−1t [Dt, a

1]σ · · · δt(aj) · · ·D−1t [Dt, a2q+1]σ

are continuous from [0, 1]×A2q+2 to L1(H). Therefore the above formula for η defines a contiuousHochschild cocycle on A2q+2. Moreover, using (5.2) and applying the operator S to both sidesof (5.3) gives

τD1,σ2q+2 − τ

D0,σ2q+2 = SτD1,σ

2q − SτD0,σ2q = SBη = −bη = 0 in HC2q+2(A).

This proves the 2nd part of the lemma and completes the proof.

When D is invertible we know that, for q ≥ 12 (p− 1), the cyclic cocycles τD,σ2q and τD,σ2q define

the same class in HP0(A). If we argue along the same lines as that of the proof in [PW1] and useLemma 5.6, then we obtain the following result.

Lemma 5.7. Assume D is invertible and let q be a an integer ≥ 12 (p−1). Then the cyclic cocycles

τD,σ2q and τD,σ2q define the same class in HP0(A).

Granted Lemma 5.5 and Lemma 5.7 the Connes-Chern character descends to a class in HP0(A)as follows.

Definition 5.8. Ch(D)σ is the common class in HP0(A) of the cocycles τD,σ2q , q ≥ 12 (p− 1), and

of the cocycles τD,σ2q , q ≥ 12 (p− 1), when D is invertible.

Proposition 5.9. Let (A,H, D)σ be a p-summable smooth twisted spectral triple. Then

(1) The class Ch(D)σ agrees with the Connes-Chern character Ch(D)σ under the linear mapHP0(A)→ HP0(A) induced by the inclusion of C•(A) into C•(A).

(2) For any Hermitian finitely generated projective module E over A and σ-connection on E,

indD∇E = 〈Ch(D)σ,Ch(E)〉 ,

where Ch(E) is the Chern character of E seen as a class in HP0(A).

Proof. The first part is immediate since by their very definitions of Ch(D)σ and Ch(D)σ arerepresented by the same cyclic cocycles. As for the 2nd part, consider an idempotent e in someMN (A), N ≥ 1 such that E ' eAN . Then, for any q ≥ 1

2 (p− 1),

〈Ch(D)σ,Ch(E)〉 =⟨τD,σ2q ,Ch(e)

⟩= 〈Ch(D)σ,Ch(E)〉 .

Combining this with Proposition 4.9 gives the result.

Remark 5.10. In what follows by Connes-Chern character of a smooth (p-summable) twistedspectral triple (A,H, D)σ we shall mean the cohomology class Ch(D)σ ∈ HP0(A).

6. Invariance of the Connes-Chern Character

In preparation for the next section, we prove in this section the invariance of the Connes-Cherncharacter under equivalences and conformal deformations of twisted spectral triples.

18

6.1. Equivalence of twisted spectral triples. The equivalence of two twisted spectral triplesover the same algebra is defined as follows.

Definition 6.1. Let (A,H1, D1)σ1and (A,H2, D2)σ2

be twisted spectral triples over the samealgebra. For i = 1, 2 let us denote by πi the representation of A into Hi. Then we say that(A,H1, D1)σ1

and (A,H2, D2)σ2are equivalent when there is a unitary operator U : H1 → H2

such that

D1 = U∗D2U,(6.1)

π1(a) = U∗π2(a)U and π1(σ1(a)) = U∗π2(σ2(a))U for all a ∈ A.(6.2)

Remark 6.2. We may also define the equivalence of a pair of spectral triples (A1,H1, D1)σ1and

(A2,H2, D2)σ2with A1 6= A2 by requiring the existence of a ∗-algebra isomorphism ψ : A1 → A2

and replacing the condition (6.2) by

π1(a) = U∗π2 (ψ(a))U and π1(σ1(a)) = U∗π2 (σ2 (ψ(a)))U for all a ∈ A1.

Proposition 6.3. Let (A,H1, D1)σ1 and (A,H2, D2)σ2 be equivalent twisted spectral triples thatare p-summable for some p ≥ 1. In addition, let q be an integer ≥ 1

2 (p− 1). Then

(1) The cyclic cocycles τD1,σ1

2q and τD2,σ2

2q agree.

(2) The same result holds for the cocycles τD1,σ1

2q and τD2,σ2

2q when D1 and D2 are invertible.

(3) The twisted spectral triples (A,H1, D1)σ1and (A,H2, D2)σ2

have same Connes-Cherncharacter.

Proof. The last part is an immediate consequence of the first two parts, so we only need to provethese parts. In addition, we note that the invertible doubles of (A,H1, D1)σ1

and (A,H2, D2)σ2

are equivalent, where the equivalence is implemented by the unitary operator U ⊕ U acting onH = H⊕H. Therefore, it is enough to assume that D1 and D2 are invertible and prove the 2ndpart.

Under the aforementioned assumption and using (6.1)–(6.2) we see that, for all a ∈ A,

D−11 [D1, a]σ1 = U∗D−12 [D2, a]σ2U.

Therefore, for all aj ∈ A, we have

τD1,σ1

2q (a0, . . . , a2q) =1

2(−1)q

q!

(2q)!StrU∗D−12 [D2, a

0]σ2U · · ·U∗D−12 [D2, a2q]σ2U

= τD2,σ2

2q (a0, . . . , a2q).

This proves the result.

6.2. Conformal deformations of twisted spectral triples. For future purpose it will beuseful to extend to the setting of twisted spectral triples the conformal deformations of ordinaryspectral triples. Let (A,H, D)σ be a twisted spectral triple and k a positive element of A. We letσ : A → A be the automorphism of A given by

σ(a) = kσ(kak−1)k−1 ∀a ∈ A.

Proposition 6.4. (A,H, kDk)σ is a twisted spectral triple.

Proof. We only need to check the boundedness of twisted commutators [kDk, a]σ, a ∈ A. To seethis we note that

(6.3) [kDk, a]σ = kDka− σ(a)kDk = k(D(kak−1)− (k−1σ(a)k)D

)k = k[D, a]σk.

As the twisted commutator [D, a]σ is bounded, it follows that [kDk, a]σ is bounded as well. Theproof is complete.

The following shows that the Connes-Chern character is invariant under conformal deforma-tions.

19

Proposition 6.5. Assume that (A,H, D)σ is p-summable for some p ≥ 1. Then, for any positiveelement k ∈ A,

Ch(kDk)σ = Ch(D)σ ∈ HP0(A).

Remark 6.6. The above result is proved in [CM3] in the special case σ = id and D is invertible.

Proof of Proposition 6.5. Set Dk = kDk. We shall first prove the result when D is invertible,as the proof is simpler in that case. Given an integer q > 1

2 (p − 1) let aj ∈ A, j = 0, . . . , 2q.Using (4.2) and (6.3) we get

τDk,σ2q (a0, . . . , a2q) =cq StrD−1k [Dk, a

0]σ · · ·D−1k [Dk, a2q]σ

(6.4)

=cq Str

(k−1D−1[D, ka0k−1]σk) · · · (k−1D−1[D, ka2qk−1]σk)

(6.5)

=cq StrD−1[D, ka0k−1]σ · · ·D−1[D, ka2qk−1]σ

(6.6)

=τD,σ2q

(ka0k−1, · · · , ka2qk−1

).(6.7)

As cyclic cohomology is invariant under the action of inner automorphisms (see [Co4, Prop. III.1.8]

and [Lo, Prop. 4.1.3]), we deduce that the cyclic cocycles τD,σ2q and τDk,σ2q are cohomologous in

HC2q(A). Therefore, they define the same class in HP0(A), and so the the twisted spectral triples(A,H, D)σ and (A,H, Dk)σ have same Connes-Chern character.

Let us now prove the result when D is not invertible. To this end consider the respectiveunital invertible doubles (A, H, D)σ and (A, H, Dk)σ of (A,H, D)σ and its conformal deformation(A,H, Dk)σ, where by a slight abuse of notation we have denoted by σ and σ the extensions to

A of the automorphisms σ and σ. As it turns out, (A, H, Dk)σ is not a conformal deformation

of (A, H, D)σ so that the proof in the invertible case does not extend to the invertible doubles.

Nevertheless, as we shall see, (A, H, Dk)σ is a pseudo-inner twisting in the sense of [PW3] of a

twisted spectral triple which is homotopy equivalent to (A, H, D)σ.

To wit consider the selfadjoint unbounded operator on H = H⊕H given by

D1 =

(D k−2

k−2 −D

), dom(D1) = dom(D)⊕ dom(D).

As D1 agrees with D up to a bounded operator, we see that (A, H, D1)σ is a p-summable twistedspectral triple. Note also that

(6.8) Dk =

(kDk 1

1 −kDk

)= ωD1ω, where ω :=

(k 00 k

).

In particular, we see that D1 is invertible. An explicit homotopy between (A, H, D)σ and (A, H, D1)σis given by the family of twisted spectral triples (A, H, Dt)σ, t ∈ [0, 1], with

(6.9) Dt =

(D k−2t

k−2t −D

)= D + Vt, where Vt =

(0 k−2t − 1

k−2t − 1 0

).

We observe that (Vt)t∈[0,1] is a C1-family in L(H) and, in the same way as for D1, it can be

shown that the operator Dt is invertible for all t ∈ [0, 1]. Therefore, we may use the homotopyinvariance of the Connes-Chern character in the form of [PW1, Appendix C] to deduce that, for

all q ≥ 12 (p+ 1), the cyclic cocycles τ D,σ2q and τ D1,σ

2q are cohomologous in HC2q(A). Incidentally, if

denote by τ D,σ2q and τ D1,σ2q their respective restrictions to A, then we see that τ D,σ2q and τ D1,σ

2q are

cohomologous cyclic cocycles in HC2q(A).Bearing this in mind, let a ∈ A. Using (6.8) and noting that ωπ(a)ω−1 = π(kak−1), we see

that Dkπ(a) = ωD1(ωπ(a)ω−1)ω = ωD1π(kak−1)ω. Likewise,

π(σ(a))Dk = ω(ω−1π(σ(a))ω)D1ω = ωπ(k−1σ(a)k)D1ω = ωπ(σ(kak−1))D1ω.

Thus,

(6.10) D−1k [Dk, π(a)]σ =(ω−1D−11 ω−1

)(ω[D1, π(kak−1)]σω

)= ω−1D−11 [D1, π(kak−1)]σω.

20

Given an integer q > 12 (p − 1) let aj ∈ A, j = 0, . . . , 2q. Using (4.2) and (6.10) and arguing as

in (6.7) we obtain

τ Dk,σ2q (a0, . . . , a2q) =cq StrDk[Dk, π(a0)]σ · · · Dk[Dk, π(a2q)]σ

(6.11)

=cq Str

(ω−1D1[D1, π(ka0k−1)]σω) · · · (ω−1D1[D1, π(ka2qk−1)]σω)

(6.12)

=cq StrD1[D1, π(ka0k−1)]σ · · · D1[D1, π(ka2qk−1)]σ

(6.13)

=τ D1,σ2q (ka0k−1, . . . , ka2qk−1).(6.14)

Therefore, in the same way as in the invertible case, we deduce that the cocycles τ Dk,σ2q and τ D1,σ2q

are cohomologous in HC2q(A). It then follows that, for q ≥ 12 (p+ 1), the cocycles τ D,σ2q and τ Dk,σ2q

are cohomologous in HC2q(A), and hence define the same class in HP0(A). This shows that thetwisted spectral triples (A,H, D)σ and (A,H, Dk)σ have same Connes-Chern character. The proofis complete.

6.3. Smooth twisted spectral triples. We shall now explain how to extend to smooth twistedspectral triples the previous results of this section. First, we have the following result.

Proposition 6.7. Let (A,H1, D1)σ1and (A,H2, D2)σ2

be equivalent smooth twisted spectraltriples that are p-summable for some p ≥ 1. Then Ch(D1)σ1

= Ch(D2)σ2in HP0(A).

Proof. This is immediate consequence of the first two parts of Proposition 6.3, since they implythat Ch(D1)σ1

and Ch(D2)σ2are represented by the same cocycles.

Proposition 6.8. Assume that (A,H, D)σ is p-summable for some p ≥ 1. Then, for any positiveelement k ∈ A,

Ch(kDk)σ = Ch(D)σ ∈ HP0(A).

Proof. We shall continue using the notation of the proof of Proposition 6.5. This proof shows that,for any q ≥ 1

2 (p+1), the cocycles τD,σ and τDk,σ define the same class in HC2q(A) by establishing

they both are cohomologous to the cocycle τ D1,σ. In order to completes the proof we only needto show that τD,σ and τDk,σ define the same class in HC2q(A).

The equality between the classes of τD,σ and τ D1,σ is a consequence of the homotopy invarianceof the Connes-Chern character and the fact that the operators D and D1 can be connected by a

operator homotopy of the form (6.9). Therefore, using Lemma 5.6 we see that the cocycles τ D,σ

and τ D1,σ are cohomologous in HC2q(A), and so are their restrictions to A, that is, the cocycles

τD,σ and τ D1,σ, define the same class in HC2q(A).

In addition, Eq. (6.14) shows that the cocycle τDk,σ is obtain from τ D1,σ by composing with theinner automorphism defined by k. The proof of the invariance of cyclic cohomology by the actionof inner automorphisms in [Lo] holds verbatim for the cyclic cohomology of continuous cochains.

It then follows that the cocycles τDk,σ and τ D1,σ define the same class HC2q(A), and so τD,σ andτDk,σ are cohomologous in HC2q(A). The proof is complete.

7. The Conformal Connes-Chern Character

In this section, after recalling the construction of the conformal Dirac spectral triple of [CM3]associated to any given conformal structure, we show that its Connes-Chern character (defined incontinuous cyclic cohomology) is actually a conformal invariant.

7.1. The conformal Dirac spectral triple. Throughout this section and the rest of the paperwe let M be a compact (closed) spin oriented manifold of even dimension n. We also let C be aconformal structure on M , i.e., a conformal class of Riemannian metrics on M . We then denote byG the identity component of the group of (smooth) orientation-preserving diffeomorphisms of Mpreserving the conformal structure C and the spin structure of M . Let g be a representative metricin the conformal class C , and consider the associated Dirac operator /Dg : C∞(M,/S)→ C∞(M,/S)

21

on the sections of the spinor bundle /S = /S+ ⊕ /S

−. In addition, we denote by L2

g(M,/S) the

corresponding Hilbert space of L2-spinors.In the setup of noncommutative geometry, the role of the quotient space M/G is played by the

(discrete) crossed-product algebra C∞(M) o G. The underlying vector space of C∞(M) o G isC∞c (M ×G), where G is equipped with the discrete topology and smooth structure. The productand involution of C∞(M) oG are given by

F1 ∗ F2(x, φ) =∑

φ1φ2=φ

F1(x, φ1)F2(φ−11 (x), φ2), F ∗(x, φ) = F (x, φ−1).

Alternatively, if we denote by uφ the characteristic function of M × φ, then uφ ∈ C∞c (M ×G)and any F ∈ C∞c (M ×G) is uniquely written as a finitely supported sum,

F =∑φ∈G

fφuφ,

where fφ(x) := F (x, φ) ∈ C∞(M). Moreover, we the following relations hold:

uφ1uφ2 = uφ1φ2 , u∗φ1= uφ−1

1= u−1φ1

, φj ∈ G,(7.1)

uφf = (f φ−1)uφ, f ∈ C∞(M), φ ∈ G.(7.2)

Let φ ∈ G. As φ is a diffeomorphism preserving the conformal class C , there is a uniquefunction kφ ∈ C∞(M), kφ > 0, such that

(7.3) φ∗g = k2φg.

Moreover, φ uniquely lifts to a unitary vector bundle isomorphism φ/S : /S → φ∗/S, i.e., a unitarysection of Hom(/S, φ∗/S) (see [BG]). We then let Vφ : L2

g(M,/S) → L2g(M,/S) be the bounded

operator given by

(7.4) Vφu(x) = φ/S(u φ−1(x)

)∀u ∈ L2

g(M,/S) ∀x ∈M.

The map φ→ Vφ is a representation of G in L2g(M,/S), but this is not a unitary representation. In

order to get a unitary representation we need to take into account the Jacobian |φ′(x)| = kφ(x)n

of φ ∈ G. This is achieved by using the unitary operator Uφ : L2g(M,/S)→ L2

g(M,/S) given by

(7.5) Uφ = kn2

φ Vφ, φ ∈ G.

Then φ → Uφ is a unitary representation of G in L2g(M,/S). Combining this with the action

of C∞(M) by multiplication operators provides with a ∗-representation of the crossed-productalgebra C∞(M) oG. In addition, we let σg be the automorphism of C∞(M) oG given by

(7.6) σg(fuφ) := kφfuφ ∀f ∈ C∞(M) ∀φ ∈ G.

Proposition 7.1 ([CM3]). The triple (C∞(M) oG,L2g(M,/S), /Dg)σg is a twisted spectral triple.

Remark 7.2. The bulk of the proof is showing the boundedness of the twisted commutators[/Dg, Uφ]σg , φ ∈ G. We remark that

Uφ/DgU∗φ = k

n2

φ (Vφ/DgV−1φ )k

−n2φ = k

n2

φ /Dφ∗gk−n2φ = k

n2

φ /Dk2φgk−n2φ .

Combining this with the conformal invariance of the Dirac operator (see, e.g., [Hi]) we obtain

Uφ/DgU∗φ = k

n2

φ

(k−(n+1

2 )φ /Dgk

n−12

φ

)k−n2φ = k

− 12

φ /Dgk− 1

2

φ

Using this we see that the twisted commutator [/Dg, Uφ]σg = /DgUφ − kφUφ/Dg is equal to(/Dgk

12

φ − kφ(Uφ/DgU∗φ)k

12

φ

)k− 1

2

φ Uφ =(/Dgk

12

φ − k12

φ /Dg

)k− 1

2

φ Uφ = [/Dg, k12

φ ]k− 1

2

φ Uφ.

This shows that [/Dg, Uφ]σg is bounded.

Remark 7.3. We shall refer to (C∞(M)oG,L2g(M,/S), /Dg)σg as the conformal Dirac spectral triple

associated to the representative metric g.

22

Remark 7.4. The automorphism σg is ribbon in the sense mentioned in Remark 2.22. This is seenby using the automorphism τg of C∞(M) oG defined by

τg(fuφ) =√kφfuφ ∀f ∈ C∞(M) ∀φ ∈ G,

where kφ is the conformal factor in the sense (7.3).

In addition, recall that C∞(M) oG has a natural locally convex ∗-algebra topology defined asfollows. For any finite set F ⊂ G set

C∞(M) o F :=

∑φ∈F

fφuφ; fφ ∈ C∞(M)

.

There is a natural linear isomorphism from C∞(M) o F onto C∞(M)F , which enables us topullback to C∞(M) o F the locally convex space topology of C∞(M)F . The locally convexspace topology of C∞(M) oG is then obtained as the inductive limit of the locally convex spacetopologies of the spaces C∞(M) o F . In particular, the following result holds.

Lemma 7.5. Given any topological vector space X, a linear map Φ : C∞(M) o G → X iscontinuous if and only if, for all φ ∈ G, the map f → Φ(fuφ) is a continuous from C∞(M) to X.

Using this we see that the product and involution of C∞(M)oG are continuous. Furthermore,we have the following.

Lemma 7.6. For any metric g ∈ C , the twisted spectral triple (C∞(M) o G,L2g(M,/S), /Dg)σg is

smooth in the sense of Definition 5.1.

Proof. Let φ ∈ G. The map f → fUφ is continuous from C∞(M) to L(L2g(M,/S)

). As σg(fuφ) =

kφfuφ and σ−1g (fuφ) = k−1φ fuφ, the maps φ → σ±1g (fuφ) are continuous from C∞(M) to

C∞(M) oG. In addition, note that, for any f ∈ C∞(M),

[/Dg, fUφ]σg = [/Dg, f ]Uφ + f [/Dg, Uφ]σg = ic(df)Uφ + f [/Dg, Uφ]σg ,

where c(df) is the Clifford representation of the differential df . Thus, the map f → [/Dg, fUφ]σgis continuous from C∞(M) to L

(L2g(M,/S)

). Combining all this with Lemma 7.5 shows that the

conditions (1)–(3) of Definition 5.1 are satisfied, and so (C∞(M)oG,L2g(M,/S), /Dg)σg is a smooth

twisted spectral triple . The lemma is thus proved.

7.2. Conformal invariance of the Connes-Chern character. The construction of the confor-mal Dirac spectral triple (C∞(M)oG,L2

g(M,/S), /Dg)σg , depends on the choice of a representativemetric g in the conformal class C . The following proposition describes the dependence on thischoice.

Proposition 7.7. Let g be another metric in the conformal class C , i.e., g = k2g for somefunction k ∈ C∞(M), k > 0. Let σ be the automorphism of C∞(M) oG defined by

(7.7) σ(a) = k−12σg(k

− 12 ak

12 )k

12 = k−1σg(a)k, a ∈ C∞(M) oG.

Then the conformal Dirac spectral (C∞(M) o G,L2g(M,/S), /Dg)σg associated to g is equivalent to

the conformal deformation (C∞(M) oG,L2g(M,/S), k−

12 /Dgk

− 12 )σ.

Proof. Let U : L2g(M,/S) → L2

g(M,/S) be the operator given by the multiplication by kn2 . Let

u ∈ L2g(M,/S). As |dx|g = k(x)n|dx|g, we have

‖Uu‖2L2g(M,/S) =

∫M

(k(x)

n2 u(x), k(x)

n2 u(x)

)|dx|g =

∫M

(u(x), u(x)) |dx|g = ‖u‖2L2g(M,/S).

This shows that U is a unitary operator. Moreover, using the conformal invariance of the Diracoperator [Hit] we see that

(7.8) /Dg = k−12 (n+1)/Dgk

12 (n−1) = U∗k−

12 /Dgk

− 12U.

23

Let φ ∈ G. We have two representations of φ. One is the unitary operator Vφ of L2g(M,/S)

given by (7.4) using the representative metric g. We have another representation of φ as a unitary

operator Vφ of L2g(M,/S) given by the same formula using the metric g. That is,

Vφu = enhφφ∗u, u ∈ L2g(M,/S),

where hφ ∈ C∞(M,R) is so that φ∗g = e2hφ g. Set h = log k. Then

φ∗g = φ∗(k2g) = (k φ−1)2φ∗g = e2he2hφg = e2(hφ

−1+hφ−h)g,

which shows that hφ = h φ−1 + hφ − h. Let u ∈ L2g(M,/S). Then

Vφu = e−nhenhφe−nhφφ∗u = k−n2 enhφφ∗(k

n2 u) = U∗VφUu.

Likewise,

σg(Vφ)u = e−(n+1)hφφ∗u = k−(n+1)e(n+1)hφφ∗(kn+1u) = Uk−1σg(Vφ)kUu = U−1σ(Vφ)Uu.

We then deduce that, for all f ∈ C∞(M) and φ ∈ G,

fVφ = U∗(fVφ)U and σg(fVφ) = U∗σ(fVφ)U.

Combining this with (7.8) shows that the twisted spectral triples (C∞(M) o G,L2g(M,/S), /Dg)σg

and (C∞(M) oG,L2g(M,/S), k−

12 /Dgk

− 12 )σ are equivalent. The proof is complete.

Given any metric g ∈ C , the j-th eingenvalue of |/Dg| grows like j1n as j → ∞. Therefore,

the associated twisted spectral triple (C∞(M) o G,L2g(M,/S), /Dg)σg is p-summable for all p >

n. Combining this with Proposition 6.3 shows that the Connes-Chern character of (C∞(M) oG,L2

g(M,/S), /Dg)σg is well defined as a class in HP0(C∞(M) o G) for every metric g in theconformal class C .

We are now in a position to state the main result of this section.

Theorem 7.8. The Connes-Chern character Ch(/Dg)σg ∈ HP0 (C∞(M) oG) is independent ofthe choice of the metric g ∈ C , i.e, it is an invariant of the conformal structure C .

Proof. Let g be another metric in the conformal class C , so that g = k2g with k ∈ C∞(M), k > 0.Combining Proposition 7.7 with Proposition 6.3 and Proposition 6.8 we get

Ch(/Dg)σg = Ch(k−

12 /Dgk

− 12

)σ

= Ch(/Dg)σg in HP0 (C∞(M) oG),

where σ is defined as in (7.7). This proves the result.

This leads us to the following definition.

Definition 7.9. The Connes-Chern character of the conformal class C , denoted by Ch(C ), isthe common class in HP0 (C∞(M) oG) of the Connes-Chern characters of the conformal Diracspectral triples (C∞(M) oG,L2

g(M,/S), /Dg)σg as the metric g ranges over the conformal class C .

Combining this with Proposition 5.9 we obtain the following index formula.

Proposition 7.10. Let E be a finitely generated projective over C∞(M) o G. Then, for anymetric g ∈ C and any σg-connection on E, it holds that

indD∇E = 〈Ch(C ),Ch(E)〉 .24

8. Local Index Formula in Conformal Geometry

In this section, we shall compute the conformal Connes-Chern character Ch(C ) when theconformal structure C is not flat. Together with Proposition 7.10 this will provide us with thelocal index formula in conformal-diffeomorphism invariant geometry. We recall that the conformalstructure C is flat when it is equivalent to the conformal structure of the round sphere Sn. Incase M is simply connected and has dimension ≥ 4 this is equivalent to the vanishing of the Weylcurvature tensor of M (see [Ku]).

As pointed out in Remark 4.12, the Ansatz of Moscovici [Mo2] is not known to hold for confor-mal deformations of ordinary spectral triples sastisfying the local index formula in noncommutativegeometry of [CM2]. As a result, unless the conformal structure C is flat and G is a maximal para-bolic subgroup of PO(n+ 1, 1), we cannot claim that for a general metric in C the correspondingconformal Dirac spectral triple (C∞(M) oG,L2

g(M,/S), /Dg)σg satisfies Moscovici’s Ansatz.

Having said this, the conformal invariance of the Connes-Chern character Ch(/Dg)σg provided

by Theorem 7.8 allows us to choose any metric in the conformal class C to compute Ch(C ). Inparticular, as we shall see below (and as also observed by Moscovici [Mo2]), the computation isgreatly simplified by choosing a G-invariant metric in the conformal class C . When the conformalstructure C is non-flat, the existence of such a metric is ensured by the following result.

Proposition 8.1 (Ferrand-Obata [Ba, Fe, Sc]). If the conformal structure C is not flat, then thegroup of smooth diffeomorphisms of M preserving C is a compact Lie group and there is a metricin C that is invariant by this group.

The relevance of using a G-invariant metric g ∈ C stems from the observation that in thiscase φ∗g = g for all φ ∈ G, and so, for every φ ∈ G, the conformal factor kφ is always theconstant function 1. This implies that, for any φ ∈ G, the unitary operator Uφ agrees with thepushforward by φ. This also implies that the automorphism σg given by (7.6) is trivial, so thatthe conformal Dirac spectral triple (C∞(M)oG,L2

g(M,/S), /Dg)σg is actually an ordinary spectraltriple. Therefore, we arrive at the following statement.

Proposition 8.2. The conformal Connes-Chern character Ch(C ) agrees with the ordinary Connes-Chern character of the equivariant Dirac spectral triple (C∞(M) oG,L2

g(M,/S), /Dg) associated toany G-invariant metric g in the conformal class C .

We postpone the computation of the Connes-Chern character of an equivariant Dirac spectraltriple (C∞(M)oG,L2

g(M,/S), /Dg) associated to any G-invariant metric g ∈ C to the sequel [PW2]

(which we shall refer to as Part II). In order to use the results of Part II a bit of notation need tobe introduced.

In what follows we let g be a G-invariant metric in the conformal class C . Let φ ∈ G and denoteby Mφ its fixed-point set. As φ preserves the orientation and the metric g, the fixed-point set Mφ

is a disconnected union Mφa of submanifolds of even dimension a = 0, 2, · · · , n. Therefore, we can

merely treat Mφ as if it were a manifold. Let N φ = (TMφ)⊥ be the normal bundle of Mφ, whichwe regard as a vector bundle over each of the submanifold components of Mφ. We denote by φN

the isometric vector bundle isomorphism induced on N φ by φ′. We note that the eigenvalues ofφN are either −1 (which has even multiplicity) or complex conjugates e±iθ, θ ∈ (0, π), with samemultiplicity. In addition, we shall orient Mφ

a like in [BGV, Prop. 6.14], so that the vector bundleisomorphism φ/S : /S → φ∗/S gives rise to a section of Λb(N φ)∗ which is positive with respect to theorientation of N φ defined by the orientations of M and Mφ. Here b is the dimension of the fibersof N φ.

Let RTM be the curvature of (M, g), seen as a section of Λ2T ∗M ⊗ End(TM). As the Levi-Civita connection ∇TM is preserved by φ, it preserves the splitting TM|Mφ = TMφ ⊕ N φ over

Mφ, and so it induces connections ∇TMφ

and ∇Nφ on TMφ and N φ respectively, in such a waythat

∇TM|TMφ = ∇TMφ

⊕∇Nφ

on Mφ.

25

We note that ∇TMφ

is the Levi-Civita connection of TMφ. Let RTMφ

and RNφ

be the respective

curvatures of ∇TMφ

and ∇Nφ . We define

(8.1) A(RTMφ

) := det12

(RTM

φ

/2

sinh(RTMφ/2)

)and νφ

(RN

φ)

:= det−12

(1− φN e−R

Nφ),

where det−12

(1− φN e−RN

φ)is defined in the same way as in [BGV, Section 6.3]. In addition,

given a differential form ω on M we shall denote by∫Mφaω the integral over Mφ

a of the top-degree

component of ι∗ω, where ι is the inclusion of Mφa in M .

Combining the results of Part II with Proposition 8.2 we obtain the following index formula inconformal-diffeomorphism invariant geometry.

Theorem 8.3. Assume that the conformal structure C is non-flat.

(1) For any G-invariant metric g ∈ C , the conformal Connes-Chern character Ch(C ) isrepresented in HP0(C∞(M) oG) by the cocycle ϕ = (ϕ2q)q≥0 given by

(8.2) ϕ2q(f0uφ0

, · · · , f2quφ2q) =

(−i)n2(2q)!

∑0≤a≤na even

(2π)−a2

∫Mφa

A(RTM

φa

)∧ νφ

(RN

φ)∧ f0df1 ∧ · · · ∧ df2q,

where we have set φ := φ0 · · · φ2q and f j := f j φ−1j−1 · · · φ−10 , j = 1, . . . , 2q.

(2) Let E be a finitely generated projective module over C∞(M) o G. Then, for any metricg ∈ C and any σg-connection on E, it holds that

indD∇E = 〈ϕ,Ch(E)〉 ,

where ϕ is the cocycle (8.2) associated to any G-invariant metric in C and Ch(E) is theChern character of E.

Remark 8.4 (See also [Mo2]). For q = 12n the right-hand side of (8.2) reduces to an integral over

Mφn and this submanifold is empty unless φ = id. Thus,

(8.3) ϕn(f0uφ0, · · · , fnuφn) =

(2iπ)−

n2

n!

∫M

f0df1 ∧ · · · ∧ dfn if φ0 · · · φn = id,

0 otherwise.

That is, ϕn agrees with the transverse fundamental class cocycle of Connes [Co2].

Remark 8.5. The computation in Part II of the Connes-Chern character in HP(C∞(M)oG) of anequivariant Dirac spectral triple (C∞(M)oG,L2

g(M,/S), /Dg) has two main steps. The first step is

showing that the Connes-Chern is represented in HP0(C∞(M)oG) (not just in HP0(C∞(M)oG))by the CM cocycle of [CM2]. This requires extending the local index formula to the framework ofsmooth spectral triples. The second step is the explicit computation of that CM cocycle. Relatedcomputations were carried out independently by Azmi [Az] in the case of a finite group and Chern-Hu [CH] in the equivariant setting by elaborating on the approach of Lafferty-Yu-Zhang [LYZ]to the proof of the equivariant local index theorem of Patodi [Pa], Donnelly-Patodi [DP] andGilkey [Gi] (see also [Bi, BV, BGV, LM]). In Part II this computation is obtained as a simpleconsequence of a new proof of the equivariant local index theorem.

9. The Cyclic Homology of C∞(M) oG

In this section, we present a geometric construction of cycles in HP0(C∞(M)oG). This will beused in the next section to construct conformal invariants. In what follows we continue to assumethat the conformal structure C is non-flat.

26

9.1. The Cyclic Homology of C∞(M)oG. The periodic cyclic homology HP• (C∞(M) oG)is known [BN, Cr] and can be described as follows.

Let 〈G〉 be the set of conjugacy classes of G. Given φ ∈ G we denote by 〈φ〉 its conjugacy classand let Gφ = ψ ∈ G; ψ φ = φ ψ be its stabilizer. Note that the action of Gφ preserves eachfixed-point submanifold Mφ

a , a = 0, 2, . . . , n. For i = 0, 1 we then set

H [i](Mφa )Gφ =

⊕q≥0

2q+i≤a

H2q+i(Mφa )Gφ ,

where H2q+i(Mφa )Gφ is the Gφ-invariant de Rham cohomology of degree 2q + i of Mφ

a . We thenhave the following description of HP• (C∞(M) oG).

Proposition 9.1 (Brylinski-Nistor [BN]; see also [Cr]). It holds that

(9.1) HP• (C∞(M) oG) '⊕〈φ〉∈〈G〉

⊕0≤a≤na even

H [•](Mφa )Gφ .

Remark 9.2. Brylinski-Nistor [BN] actually computed the cyclic homology of the convolution alge-bras of Hausdorff etale groupoids. Crainic [Cr] extended their results to non-Hausdorff groupoids.In the case of the discrete transformation groupoid associated to the action of G on M we obtainthe isomorphism (9.1). We refer to [BC, BGJ, Co4, Da, FT, GJ, KKL, Ni1, Ni2, NPPT] forvarious results related to the cyclic homology of crossed-product algebras.

In what follows, we will construct explicit maps from each cohomology space H [•](Mφa )Gφ to

HP• (C∞(M) oG). This construction is carried out in several steps. It essentially follows theapproach of [BN, Cr] except for the specialization of the arguments to group actions and the useof the mixed complexes of G-normalized chains, which does not appear explicitly in [BN, Cr].

We refer to Definition 9.3 and Definition 9.9 below for the precise definitions of G-normalizedcochains and G-normalized chains. It will be shown in the next section that, for any G-invariantmetric g ∈ C , the cocycle (8.2) representing the conformal Connes-Chern character Ch(E) is G-normalized. Therefore it is a crucial step in our approach to obtain cycles arising from explicitG-normalized chains.

9.2. G-normalized cochains. In what follows we set A = C∞(M) and AG = C∞(M) oG. Wealso note that the group G acts on C•(AG) as follows: given ϕ ∈ Cm(AG) and ψ ∈ G, the actionof ψ on ϕ is the cochain ψ∗ϕ ∈ Cm(AG) defined by

(9.2) ψ∗ϕ(a0, . . . , am) = ϕ(uψa0u−1ψ , . . . , uψa

mu−1ψ ) ∀aj ∈ AG.

Definition 9.3. A continuous cochain ϕ ∈ Cm(AG) is G-normalized when, for all a0, . . . , am inAG and ψ in G, the following holds

ϕ(a0, . . . , aju−1ψ , uψaj+1, . . . , am) = ϕ(a0, . . . , am) for j = 0, . . . ,m− 1,(9.3)

ϕ(uψa0, a1, . . . , am−1, amu−1ψ ) = ϕ(a0, . . . , am).(9.4)

We denote by CmG (AG) the space of G-normalized m-cochains on AG.

The following lemma is useful for checking the G-normalization conditions (9.3) and (9.4).

Lemma 9.4. Let ϕ ∈ Cm(AG). Then the following are equivalent:

(i) The cochain ϕ is G-normalized.(ii) The cochain ϕ is satisfy (9.3) and is G-invariant with respect to the action (9.2).(iii) For all f0, . . . , fm in C∞(M) and φ0, . . . , φm, φ, ψ in G, we have

ϕ(f0uφ0, . . . , fmuφm) = ϕ

(f0, f1 φ−10 , . . . , (fm φ−1m−1 · · · φ

−10 )uφ0···φm

),(9.5)

ϕ(f0 ψ−1, . . . , fm−1 ψ−1, (fm ψ−1)uψφψ−1) = ϕ(f0, . . . , fm−1, fmuφ).(9.6)

27

Proof. If (9.3) holds, then, for all a0, . . . , am in AG and ψ ∈ G, we have

ψ∗ϕ(a0, . . . , am) = ϕ(uψa0u−1ψ , uψa

1u−1ψ . . . , uψamu−1ψ ) = ϕ(uψa

0, a1, . . . , amu−1ψ ).

Therefore, in this case (9.4) is equivalent to the G-invariance of ϕ. This gives the equivalence of(i) and (ii).

Assume that (iii) holds. Let f0, . . . , fm be in C∞(M) and let φ0, . . . , φm and ψ be in G.Then (7.2) implies that, for j = 1, . . . ,m− 1,

ϕ(f0uφ0 , . . . , f

muφm)

= ϕ(f0uφ0 , . . . , f

juφjψ, (fj+1 ψ−1)uψ−1φj+1

, . . . , fmuφm)

(9.7)

= ϕ(f0uφ0 , . . . , (f

juφj )uψ, u−1ψ (f j+1uφj+1), . . . , fmuφm

).(9.8)

Note also that

ϕ(uψ(f0uφ0)u−1ψ , . . . , uψ(fmuφm)u−1ψ

)= ϕ

((f0 ψ−1)uψφ0ψ−1 , . . . , (fm ψ−1)uψφmψ−1

).

Therefore, using (9.5) and (9.6) we see that ϕ(uψ(f0uφ0

)u−1ψ , . . . , uψ(fmuφm)u−1ψ

)is equal to

ϕ(f0 ψ−1, (f1 φ−10 ) ψ−1, . . . , (fm φ−1m−1 · · · φ

−10 ) ψ−1uψ(φ0···φm)ψ−1

)= ϕ

(f0, f1 φ−10 , . . . , (fm φ−1m−1 · · · φ

−10 )uφ0···φm

)= ϕ

(f0uφ0

, . . . , fmuφm).

Together with (9.8) this shows that ϕ satisfies (ii).Conversely, assume that ϕ satisfies (ii) and let us show that (iii) then holds. As (9.6) is only a

special case of (9.2) (cf. Eq. (7.2)), we only have to check (9.5). To see this let f0, . . . , fm be inC∞(M) and φ0, . . . , φm in G. We observe that the relation f = u−1φ (f φ−1)uφ for all f ∈ C∞(M)

and φ ∈ G implies that ϕ(f0uφ0

, . . . , fmuφm)

is equal to

ϕ(f0uφ0 , u

−1φ0

(f φ−10 )uφ0φ1 , . . . , u−1φ0···φm(fm (φ0 · · ·φm−1)−1)uφ0···φm

).

Combining this with (9.3) yields (9.5) and shows that ϕ satisfies (ii). The proof is complete.

Remark 9.5. As we shall see (cf. Proposition 10.1), for any given G-invariant metric g ∈ C ,the cocycle given by (8.2) is G-normalized. In particular, this implies that Connes’ transversefundamental class cocycle (8.3) is G-normalized.

Remark 9.6. When the metric g is G-invariant and the Dirac operator /Dg is invertible, it also can

be shown that the cocycles τ/Dg2q , q ≥ n

2 , are G-normalized and differ from the cocycle ϕ in (8.2)by G-normalized coboundaries.

Lemma 9.7. The operators b and B preserve C•G(AG).

Proof. Let ϕ ∈ CmG (AG). The operator b is equivariant with respect to the action of G. Therefore,

in view of Lemma 9.4, in order to check that bϕ is G-normalized we only have to check it satisfiesthe condition (9.3). Let a0, . . . , am+1 be in AG and ψ in G. Using (3.1) it is straightforwardto check that bϕ(a0, . . . , aju−1ψ , uψa

j+1, . . . , am+1) agrees with bϕ(a0, . . . , am) for j = 0, . . . ,m.

Likewise, it is immediate to check that bϕ(a0, . . . , amu−1ψ , uψam+1)− bϕ(a0, . . . , am) is equal to

(9.9) ϕ(uψam+1a0, a1, . . . , amu−1ψ )− ϕ(am+1a0, a1, . . . , am),

which is seen to be zero by using (9.4). Therefore, the cochain bϕ satisfies (9.3)-(9.4), and henceis G-normalized. This shows that the operator b preserves the G-normalization condition.

As B = (1− T )B0T , in order to check that the G-normalization condition is preserved by theoperator B, it is sufficient to check this property for the operators T and B0. Let ϕ ∈ Cm

G (AG).As with the coboundary bϕ above, in order to check that Tϕ and B0ϕ are G-normalized weonly have to check they satisfy (9.3). Let ψ be in G and let a0, . . . , am be in AG. Using (3.3)

28

it is straighforward to check that (Tϕ)(a0, . . . , aju−1ψ , uψaj+1, . . . , am) = (Tϕ)(a0, . . . , am) for

j = 0, . . . ,m− 2. In addition, (Tϕ)(a0, a1, . . . , am−1u−1ψ , uψam)− (Tϕ)(a0, . . . , am) is equal to

(−1)mϕ(uψam, a0, a1, . . . , am−1u−1ψ )− (−1)mϕ(am, a0, . . . , am−1),

which like (9.9) is seen to be zero by using (9.4). This implies that Tϕ satisfies (9.3)-(9.4), andhence is G-normalized.

It is also immediate to check that B0ϕ(a0, . . . , aju−1ψ , uψaj+1, . . . , am−1) = B0ϕ(a0, . . . , am−1)

for j = 0, . . . ,m− 2. Moreover, by using (9.3)-(9.4) we obtain

B0(uψa0, a1, . . . , am−2, amu−1ψ ) = ϕ(uψ1u−1ψ , uψa

0, a1, . . . , am−2, am−1u−1ψ ) = B0ϕ(a0, . . . , am).

Thus, the cochain B0ϕ satisfies (9.3)-(9.4), and hence is G-normalized. All this shows that theG-normalization condition is preserved by the operators T and B0. The proof is complete.

It follows from Lemma 9.7 that we obtain a mixed sub-complex (C•G(AG), b, B) of the cyclic

mixed complex (C•(AG), b, B). In particular we obtain a periodic complex (C[•]G (AG), b + B),

where C[i]G (AG) =

⊕q≥0 C2q+i

G (AG), i = 0, 1. We denote by HP•G(AG) the cohomology of this

complex. We note that the inclusion of C•G(AG) into C•(AG) gives rise to an injective linear map,

(9.10) ι∗ : HP•G(AG) −→ HP•(AG).

Remark 9.8. As we shall see (cf. Proposition 9.12), the results of [BN, Cr] imply that this map isactually an isomorphism. There are similar isomorphisms at level of Hochschild cohomology and(non-periodic) cyclic cohomology.

9.3. G-normalized chains. The G-normalization of chains is defined as follows. There is anatural action of G on C•(AG) so that the action of ψ ∈ G on a chain a0 ⊗ · · · ⊗ am, aj ∈ AG, isgiven by

(9.11) ψ∗(a0 ⊗ · · · ⊗ am) := (uψa

0u−1ψ )⊗ · · · ⊗ (uψamu−1ψ ).

For j = 0, . . . ,m we denote by ψ∗;m,j the endomorphism of C•(AG) defined by

ψ∗;m,j(a0 ⊗ · · · ⊗ am) =

a0 ⊗ · · · ⊗ aju−1ψ ⊗ uψaj+1 ⊗ · · · ⊗ am for j = 0, . . . ,m− 1,

uψa0 ⊗ a1 ⊗ · · · ⊗ am−1 ⊗ amu−1ψ for j = m.

In addition, we have an inhomogeneization linear map θ : C•(AG) → C•(AG) such that, for allf0, . . . , fm in A and φ0, . . . , φm in G,

(9.12) θ(f0uφ0 ⊗ · · · ⊗ fmuφm

)= f0 ⊗ f1 φ−10 ⊗ · · · ⊗ (fm φ−1m−1 · · · φ

−10 )uφ0···φm .

We note that θ is an idempotent, i.e., θ2 = θ. We also observe that

θ (Cm(AG)) = Spanf0 ⊗ · · · ⊗ fm−1 ⊗ fmuφ; f j ∈ A, φ ∈ G

.

For m ∈ N0, set

NGm (AG) =

m∑j=0

∑ψ∈G

(ψ∗;m,j − id) (Cm(AG)) .

Definition 9.9. For m ∈ N0, the space of G-normalized m-chains is the topological quotient,

CGm(AG) = Cm(AG)/NG

m (AG).

Remark 9.10. CmG (AG) is naturally identified with the topological dual of CG

m(AG).

Lemma 9.11. Let m ∈ N0. Then

NGm(AG) =

m−1∑j=0

∑ψ∈G

(ψ∗;m,j − id) (Cm(AG)) +∑ψ∈G

(ψ∗ − id) (Cm(AG))(9.13)

= ker θ ∩ Cm(AG) +∑ψ∈G

(ψ∗ − id) θ (Cm(AG)) .(9.14)

29

Proof. A (pre-)dualization of the arguments of the proof of Lemma 9.4 together with (9.12) gives

NGm(AG) =

m−1∑j=0

∑ψ∈G

(ψ∗;m,j − id) (Cm(AG)) +∑ψ∈G

(ψ∗ − id) (Cm(AG))

= (θ − id) (Cm(AG)) +∑ψ∈G

(ψ∗ − id) θ (Cm(AG)) .

The proof is completed by noting that the idempotency of θ implies that ran(θ − id) = ker θ.

In what follows, given a0, . . . , am in AG, we shall denote by a0⊗G · · · ⊗Gam the class of the

chain a1 ⊗ · · · ⊗ am modulo NGm (AG). As an immediate consequence of the very definition of

NGm(AG) and Lemma 9.11 we obtain that, for all a0, . . . , am in AG and ψ ∈ G, we have

a0⊗G · · · ⊗Gaju−1ψ ⊗Guψaj+1⊗G · · · ⊗Gam = a0⊗G · · · ⊗Gam for j = 1, . . . ,m− 1,(9.15)

uψa0u−1ψ ⊗G · · · ⊗Guψa

mu−1ψ = uψa0⊗Ga1⊗G · · · am−1⊗Gamu−1ψ = a0⊗G · · · ⊗Gam.(9.16)

It also follows from (9.14) that, for all f0, . . . , fm in A and all φ0, . . . , φm, φ, ψ in G, we have thefollowing identities:

(f0 ψ−1)⊗G · · · ⊗G(fm−1 ψ−1)⊗G(fm ψ−1)uψφψ−1 = f0⊗G · · · ⊗Gfm−1⊗Gfmuφ,(9.17)

f0uφ0⊗G · · · ⊗Gfmuφm = f0⊗G(f1 φ−10

)⊗G · · · ⊗G

(fm φ−1m−1 · · · φ

−10

)uφ0···φm .(9.18)

The operators b and B descend to continuous operators on G-normalized chains. Therefore, weobtain a mixed chain-complex (CG

• (AG), b, B) and a periodic complex (CG[•](AG), b + B), where

CG[i](AG) =

∏q≥0 CG

2q+i(AG), i = 0, 1. We shall denote by HPG• (AG) the homology of this

periodic complex.The canonical projection π : C•(AG) → CG

[•](AG) is a morphism of mixed complexes, and

hence gives rise to linear map,

π∗ : HP•(AG) −→ HPG• (AG).

In addition, the duality between C•G(AG) and CG• (AG) yields a duality pairing,

〈·, ·〉 : HP•G(AG)×HPG• (AG) −→ C.

In particular, the linear map ι∗ given by (9.10) is related to the linear map π∗ by

(9.19) 〈ι∗ϕ, η〉 = 〈ϕ, π∗η〉 ∀ϕ ∈ HP•G(AG) ∀η ∈ HP•(AG).

Proposition 9.12. The linear map π∗ is an isomorphism.

Proof. As π is onto, we only have to establish that π∗ is one-to-one. That result is a consequenceof Proposition 4.1 of [Cr]. To see this, it is convenient to identify each chain space CG

m(AG) withthe function space C∞c (Mm+1×Gm+1) in such way that any m-cochain f0uφ0

⊗· · ·⊗fmuφm withf j ∈ A and φj ∈ G is identified with the function,

(x0, . . . , xm, ψ0, . . . , ψm) −→ f0(x0) · · · fm(xm)δφ0(ψ0) · · · δφm(ψm),

where δφj (ψ) is the characteristic function of φj on G. Under this identification an m cochainappears as a finite sum,

(9.20) η =∑

ηφ0,...,φm(x0, . . . , xm)δφ0(ψ0) · · · δφm(ψm), ηφ0,...,φm ∈ C∞(Mm+1).

In addition, the Burghelea space B(m) is the subspace of Mm+1 ×Gm+1 given by

B(m) :=

(x0, . . . , xm, φ0, . . . , φm); xj+1 = φ−1j (xj), j = 0, . . . ,m− 1, x0 = φ−1m (xm).

In the terminology of [Cr] this the (m + 1)-th Burghelea space of the transformation groupoidassociated to the right action (x, φ)→ φ−1(x) of G on M . We also note that

B(m) =⋃φ∈G

(x, φ−10 (x), . . . , φ−1m−1 · · · φ

−10 (x), φ0, . . . , φm); x ∈Mφ, φ0 · · · φm = φ

.

30

Bearing this in mind, by [Cr, Proposition 4.1] a periodic cycle η = (η2q+i)q≥0 in CG[i](AG),

i = 0, 1, is a boundary if and only if it is cohomologous to a cycle η = (η2q+i)q≥0 such that

(9.21)(η2q+i

)|B(2q+i) = 0 for all q ≥ 0.

Thus in order to prove that π∗ is one-to-one we only have to show that any periodic cycle containedin∏NG

2q+i(AG) is cohomologous to a cycle of the form (9.21). We observe that the action of G onHP•(AG) is trivial, since cyclic homology is invariant under inner automorphisms (cf. [Co4, Lo]).Combining this with (9.14) we deduce that in order to check that any periodic cycle contained in∏NG

2q+i(AG) is cohomologous to a cycle of the form (9.21) it is sufficient to show that any chain

in ker θ ∩CGm(AG) vanishes on the Burghelea space B(m). To see this we observe that if η is an

m-chain of the form (9.20), then (θη)(x0, . . . , xm, ψ0, . . . , ψm) is equal to∑ηφ0,...,φm

(x1, φ

−10 (x1), . . . , (φ−1m−1 · · · φ

−10 )(xm)

)δ1(ψ0) · · · δ1(ψm−1)δφ0···φm(ψm).

Therefore, we see that θη = 0 if and only if∑φ0···φm=φ

ηφ0,...,φm

(x0, φ

−10 (x1), . . . , (φ−1m−1 · · · φ

−10 )(xm)

)= 0 ∀φ ∈ G.

This condition clearly implies that η vanishes on B(m). Therefore, we see that the kernel of θconsists of chains vanishing on Burghelea spaces. This completes the proof.

Remark 9.13. A dual version of Proposition 9.12 shows that the map ι∗ : HP•G(AG) −→ HP•(AG)is an isomorphism as well.

9.4. Splitting along conjugacy classes. The same way as in [BN, Co4, Cr, FT, Ni1, Ni2], the

cyclic homologies HP•(AG) and HPG• (AG) split along the set 〈G〉 of conjugacy classes of G as

follows. Let φ ∈ G and denote by 〈φ〉 its conjugacy class. For m ∈ N0, define

(9.22) Cm(AG)φ = Cm(AG)φ and CGm(AG)φ = CGm(AG)φ

where

Cm(AG)φ = Spanf0uφ0

⊗ · · · ⊗ fmuφm ; f j ∈ C∞(M), φj ∈ G, φ0 · · · φm ∈ 〈φ〉,

CGm(AG)φ = Spanf0uφ0

⊗G · · · ⊗Gfmuφm ; f j ∈ C∞(M), φj ∈ G, φ0 · · · φm ∈ 〈φ〉.

We also set

NGm(AG)φ =

m∑j=0

∑ψ∈G

(ψ∗;m,j − id) (Cm(AG)φ) ,

= ker θ ∩ Cm(AG)φ +∑ψ∈G

(ψ∗ − id) θ (Cm(AG)φ) .

We observe that

θ (Cm(AG)φ) = Spanf0 ⊗ · · · ⊗ fm−1 ⊗ fmuκ−1φκ; f j ∈ A, κ ∈ G

.

Moreover, the canonical projection Cm(AG) → CGm(AG) induces a topological vector space iso-

morphism,

(9.23) CGm(AG)φ ' Cm(AG)φ/NG

m(AG)φ.

Bearing this in mind, we have the obvious splitting,

(9.24) CGm(AG) =

⊕〈φ〉∈〈G〉

CGm(AG)φ,

with a similar decomposition for each chain space Cm(AG). We also observe that the conditionφ0 · · · φm ∈ 〈φ〉 is preserved by any cyclic permutation of the diffeomorphisms φ0, . . . , φm. Itthen follows that the space CG

• (AG)φ is preserved by the action of the cyclic homology opera-tors b, T , B0, B, S. Therefore, we obtain a mixed sub-complex (CG

• (AG)φ, b, B) of the mixedcomplex (CG

• (AG), b, B). This yields a periodic complex (CG[•](AG)φ, b + B), where CG

[i](AG)φ =

31

∏∞q=0 CG

2q+i(AG)φ, i = 0, 1. The direct sum over conjugacy classes of these complexes yields a

complex (⊕〈φ〉∈〈G〉C

G[•](AG)φ, b+B). The splitting (9.24) then yields an inclusion of complexes,

(9.25)⊕〈φ〉∈〈G〉

CG[•](AG)φ → CG

[•](AG).

9.5. Twisted cyclic mixed complexes. Let φ ∈ G. The twisted cyclic mixed complex (C•(A), bφ, Bφ)(cf. [FT, Ni1, Cr]) is defined as follows. The boundary bφ : Cm(A)→ Cm−1(A) is given by

bφ(f0 ⊗ · · · ⊗ fm) =

m−1∑j=0

(−1)jf0 ⊗ · · · ⊗ f jf j+1 ⊗ · · · ⊗ fm

+ (−1)m(fm φ)f0 ⊗ · · · ⊗ fm−1, f j ∈ A.

The operator Bφ : Cm(A) → Cm+1(A) is (1 − Tφ)B0Aφ, where the operator B0 is defined as

in (3.15) and Aφ = 1 + Tφ + · · ·Tm−1φ , where Tφ : Cm(A)→ Cm(A) is given by

Tφ(f0 ⊗ · · · ⊗ fm) = (−1)m(fm φ)⊗ f0 ⊗ · · · ⊗ fm−1, f j ∈ A.In addition, the group G acts on Cm(A) by

ψ∗(f0 ⊗ · · · ⊗ fm) = f0 ψ−1 ⊗ · · · ⊗ fm ψ−1, f j ∈ A, ψ ∈ G.

Note this action agrees with the action (9.11) on chains of the form f0 ⊗ · · · ⊗ fm, f j ∈ A.Furthermore, this induces an action on Cm(A)φ by the stabilizer Gφ := ψ ∈ G; φ ψ = ψ φ.We then observe that the operators bφ and Bφ are equivariant with respect to the action ofGφ. Therefore, we obtain a mixed complex

(C•(A)Gφ , bφ, Bφ

), where C•(A)Gφ is the space

of Gφ-invariant chains in C•(A). This yields a periodic complex(C[•](A)Gφ , bφ +Bφ

), where

C[i](A)Gφ =⊕

q≥0 C2q+i(A)Gφ , i = 0, 1. The homology of this periodic complex is denoted by

HPGφ• (A)φ.

Bearing this in mind, the mixed complexes(C•(A)Gφ , bφ, Bφ

)and (C•(AG)φ, b, B) are related

as follows. Let χφ : C•(A)→ C•(AG)φ be the linear map defined by

(9.26) χφ(f0⊗ · · · ⊗fm) = f0⊗G · · · ⊗Gfm−1⊗Gfmuφ ∀f j ∈ A.

Lemma 9.14. The map χφ is a morphism of mixed complexes.

Proof. Let f1, . . . , fm be in A and set η = f0 ⊗ · · · ⊗ fm. Then

bχφ(η) =m−1∑j=0

(−1)jf0⊗G · · · ⊗Gf jf j+1⊗G · · · ⊗Gfmuφ

+ (−1)m−1f0⊗G · · · ⊗Gfm−1fmuφ + (−1)mfmuφf0⊗G · · · ⊗Gfm−1.

Using (9.16) we see that fmuφf0⊗G · · · ⊗Gfm−1 is equal to

(9.27) uφ(fm φ)f0⊗G · · · ⊗Gfm−1 = (fm φ)f0⊗G · · · ⊗Gfm−1uφ.It then follows that bχφ(η) = χφ(bφη).

We also have B0χφ(η) = 1⊗Gf0⊗G · · · ⊗Gfm−1 ⊗G fmuφ = χφ(B0η). Moreover, in the sameway as in (9.27) we see that T χφ(η) is equal to

(−1)mfmuφ⊗Gf0⊗G · · · ⊗Gfm−1 = (−1)m(fm φ)⊗Gf0⊗G · · · ⊗Gfm−1uφ = χφ(Tφη).

We then deduce that Bχφ(η) = χφ(Bφη). The proof is complete.

We shall denote by χφ the restriction of χφ to Gφ-invariant chains. Thanks to Lemma 9.14 thisprovides us with a mixed complex morphism from

(C•(A)Gφ , bφ, Bφ

)to (C•(AG)φ, b, B).

We also observe that the linear maps χφ and χφ are related by means of the average mapΛφ : C•(A)→ C•(A)Gφ defined by

Λφ(η) =

∫Gφ

ψ∗ηdλ(ψ) ∀η ∈ C•(A).

32

where λ is the Haar probability measure of Gφ. More precisely, we have the following result.

Lemma 9.15. It holds that χφ = χφ Λφ.

Proof. Let f0, . . . , fm be in A and set η = f0 ⊗ · · · ⊗ fm. As χφ is a continuous linear map,

(9.28) χφ Λφ(η) = χφ

(∫Gφ

ψ∗ηdλ(ψ)

)=

∫Gφ

χφ (ψ∗η) dλ(ψ).

Let ψ ∈ Gφ. Then

χφ ψ∗(η) = χφ(f0 ψ−1 ⊗ · · · ⊗ fm ψ−1

)= f0 ψ−1⊗G · · · ⊗Gfm−1 ψ−1⊗G(fm ψ−1)uφ.

As φ = ψ φ ψ−1 since ψ ∈ Gφ, using (9.17) we see that χφ ψ∗(η) is equal to

f0 ψ−1⊗G · · · ⊗Gfm−1 ψ−1⊗G(fm ψ−1)uψφψ−1 = f0⊗G · · · ⊗Gfm−1⊗Gfmuφ = χφ(η).

Combining this with (9.28) then gives

χφ Λφ(η) =

∫Gφ

χφ (η) dλ(ψ) = χφ

(∫Gφ

ηdλ(ψ)

)= χφ(η).

This proves the lemma.

We shall now construct an explicit inverse for χφ. To this end let f0, . . . , fm be in A andφ0, . . . , φm in G such that φ0 · · · φm ∈ 〈φ〉, i.e., φ0 · · · φm = κ−1 φ κ for some κ ∈ G.Using (9.17)–(9.18) and Lemma 9.15 we observe that

f0uφ0⊗G · · · ⊗Gfmuφm = f0⊗Gf1 φ−10 ⊗G · · · ⊗G(fm φ−1m−1 · · · φ−10 )uκ−1φκ

= f0 κ−1⊗Gf1 φ−10 κ−1⊗G · · · ⊗G(fm φ−1m−1 · · · φ−10 κ−1)uφ

= χφ κ∗(f0 ⊗ f1 φ−10 ⊗ · · · ⊗ fm φ−1m−1 · · · φ

−10 ).

Combining this with Lemma 9.15 then gives

(9.29) f0uφ0⊗G · · · ⊗Gfmuφm = χφ Λφ κ∗(f0 ⊗ f1 φ−10 ⊗ · · · ⊗ fm φ−1m−1 · · · φ

−10 ).

This leads us to define

(9.30) µφ(f0uφ0⊗G · · · ⊗Gfmuφm) = Λφ κ∗(f0 ⊗ f1 φ−10 ⊗ · · · ⊗

(fm φ−1m−1 · · · φ

−10

)).

We observe that κ is unique up to the left-composition with a diffeomorphism in Gφ. Moreover,as Λφ ψ∗ = Λφ for all ψ ∈ Gφ, we deduce that the r.h.s. of (9.30) is independent of the choice ofκ. We thus obtain a continuous linear map µφ : C•(AG)φ → C•(A)Gφ .

Lemma 9.16. The linear map µφ is Gφ-equivariant and annihilated by NG• (AG)φ.

Proof. In view of (9.14) we only have to show that µφ vanishes on ker θ∩Cm(AG)φ and, for everyψ ∈ G, the map µφ (ψ∗ − id) vanishes θ (Cm(AG)φ). It is immediate from the definition of µφthat µφ θ = µφ. As ker θ ∩ Cm(AG)φ = (θ − id) (Cm(AG)φ), it follows that ker θ ∩ Cm(AG)φ iscontained in the kernel of µφ.

Let f0, . . . , fm be in A, and κ and ψ in G. Then(9.31)ψ∗(f0 ⊗ · · · ⊗ fm−1 ⊗ fmuκ−1φκ

)=(f0 ψ−1 ⊗ · · · ⊗ fm−1 ψ−1 ⊗ fm ψ−1u(κψ−1)−1φ(κψ−1)

).

Therefore, by the very definition (9.30) of µφ we have

µφ ψ∗(f0 ⊗ · · · ⊗ fm−1 ⊗ fmuκ−1φκ

)= Λφ (κ ψ−1)∗(f

0 ψ−1 ⊗ · · · ⊗ fm ψ−1)

= Λφ κ∗(f0 ⊗ · · · ⊗ fm)

= µφ(f0 ⊗ · · · ⊗ fm−1 ⊗ fmuκ−1φκ.

Combining this with (9.31) shows that µφ (ψ∗− id) vanishes on θ (Cm(AG)φ). The proof is thuscomplete.

33

Combining Lemma 9.16 with (9.23) shows that µφ descends to a continuous linear map,

µφ : CG• (AG)φ −→ C•(A)Gφ .

Proposition 9.17. The linear maps χφ and µφ are inverses of each other.

Proof. It immediately follows from (9.29) that χφ µφ = id. Moreover, we have

µφ χφ(f0 ⊗ · · · ⊗ fm

)= µφ

(f0⊗G · · · ⊗Gfmuφ

)= f0 ⊗ · · · ⊗ fm.

Thus µφ χφ = id. The proof is complete.

9.6. Concentration on fixed-point sets. Let φ ∈ G. For a = 0, 2, . . . , n set Aφa = C∞(Mφa ).

As the action of Gφ on M preserves each fixed-point component Mφa , we see that the restriction

map f → f|Mφa

gives rise to a Gφ-equivariant continuous linear map,

ρφa : C•(A)→ C•(Aφa).

As φ acts likes identity on Mφa this actually provides with a morphism of mixed complexes from

the twisted cyclic mixed complex (C•(A), bφ, Bφ) to the cyclic mixed complex(C•(Aφa), b, B

). A

right-inverse of ρφa is obtained as follows.Let g be a G-invariant metric in the conformal class C . Given r > 0 we denote by Br(TM) the

ball bundle in TM of radius r about the zero-section. For r small enough the exponential map is adiffeomorphism from Br(TM) onto its image. Let N φ

a be the restriction of the normal bundle N φ

to Mφa . This is a smooth vector bundle over Mφ

a such that the fiber at x ∈Mφa is (TxM

φa )⊥ ⊂ TaM .

Let pa : N φa → Mφ

a be the corresponding fibration map. In additon, we denote by Br(N φa ) the

ball bundle in N φa of radius r about the zero-section. Then the map Φa : X 7→ exppa(X)(X) is a

diffeomorphism from Br(N φa ) onto a tubular neighborhood Va,r of Mφ

a . Composing pa with Φ−1awe then obtain a smooth fibration pa : Va,r →Mφ

a such that pa|Mφa

= idMφa

.

As G is contained in the isometry group of the metric g, its action preserves the ball bundleBr(TM) and the exponential map is a G-equivariant map. If we restrict this action to Gφ, then theaction further preserves Mφ

a and the ball bundle Br(N φa ) and the fibration pa is a Gφ-equivariant

map. Therefore, the diffeomorphism Φa is a Gφ-equivariant map and the action of Gφ preservesthe tubular neighborhood Va,r. It then follows that pa is a Gφ-equivariant fibration.

In addition, let r1 and r2 be real numbers such that 0 < r1 < r2 < r. For j = 1, 2 setVa,rj = Φa

(Brj (N φ

a )), where Brj (N φ

a ) is the ball the ball bundle in N φa of radius rj about the

zero-section. As Va,r both Va,r1 and Va,r2 are preserved by the action of Gφ. Therefore, we canfind a Gφ-invariant function ψa ∈ C∞(Va,r) such that ψa = 1 on Va,r1 and ψa = 0 outside Va,r2 .Extending ψa to be zero outside Va,r we obtain a Gφ-invariant smooth function on M which isequal to 1 near Mφ

a .

For any given f ∈ C∞(Mφa ) we denote by f the smooth function on M defined by

f(x) = ψa(x)(f pa)(x) ∀x ∈M.

This provides us with a continuous linear map from Aφa to A. We note that f = f on Mφa ,

so that we have obtain a right-inverse of the restriction map f → f|Mφa

. Furthermore, given

b ∈ 0, 2, . . . , n \ a the fact that Mφb and N φ

a are disjoint implies that

Mφb ∩ Va,r = Φ(Mφ

b ) ∩ Φ(Br(M

φa ))

= ∅.

Therefore f vanishes on Mφb .

The map f → f gives rise to a continuous linear map εφa : C•(Aφa)→ C•(A), so that

εφa(f0 ⊗ · · · ⊗ fm) = f0 ⊗ · · · ⊗ fm ∀f j ∈ Aφa .

This is a right-inverse of ρφ,a. Furthermore, the Gφ-equivariance of pa and the Gφ-invariance ofψa imply that εφa is a Gφ-equivariant linear map. In addition, as φ is contained in Gφ and actslike the identity on Mφ

a , we further see that εφa is a mixed complex morphism from(C•(Aφa), b, B

)to (C•(A), bφ, Bφ).

34

As pointed out in Example 3.3, the map αMφa

given by (3.20) is a quasi-isomorphism from the

periodic cyclic complex (C[•](Aφa), b+B) to the de Rham complex (Ω[•](Mφa ), d). A quasi-inverse

(cf. [Co3]) is provided by the linear map βMφa

: Ω[•](Mφa )→ C[•](Aφa) defined by

βMφa

(f0df1 ∧ · · · ∧ fm) =∑σ∈Sm

ε(σ)f0 ⊗ fσ(1) ⊗ · · · ⊗ fσ(m), f j ∈ Aφa ,

where Sm is the m-th symmetric group. Note that βMφa

is actually a right-inverse of αMφa

.Moroever, both αMφ

aand βMφ

aare Gφ-equivariant maps.

Setting ρφa = αMφa ρφa and εφa = εφa βMφ

awe obtain Gφ-equivariant morphisms of complexes,

ρφa : C[•](A) −→ Ω[•](Mφa ) and εφa : Ω[•](Mφ

a ) −→ C[•](A),

such that εφa is a right-inverse of ρφa . Note also that

(9.32) εφa(f0df1 ∧ · · · ∧ dfm) =1

m!

∑σ∈Sm

ε(σ)f0 ⊗ fσ(1) ⊗ · · · ⊗ fσ(m), f j ∈ Aφa .

In addition both ρφa and εφa are Gφ-equivariant maps, and hence descends to morphisms of com-plexes,

ρφa : C[•](A)Gφ −→ Ω[•](Mφa )Gφ and εφa : Ω[•](Mφ

a )Gφ −→ C[•](A)Gφ ,

such that the latter is a right-inverse of the former.

9.7. Geometric classes in HP•(AG). We are now in a position to construct geometric cyclichomology classes in HP•(AG). Let φ ∈ G and a ∈ 0, 2, . . . , n. For any ω ∈ Ωm(Mφ

a )Gφ we set

ηω = m!χφ εφa(ω) ∈ CGm(AG).

If we write ω in the form ω =∑l f

0l df

1l ∧ · · · ∧ dfml , f jl ∈ Aφa , then (9.32) and (9.26) give

(9.33) ηω =∑l

∑σ∈Sm

ε(σ)f0l ⊗Gfσ(1)l ⊗G · · · ⊗Gfσ(m−1)l ⊗Gfσ(m)

l uφ.

The map ω → ηω is a morphism of mixed complexes from (Ω•(Mφa )Gφ , 0, d) to (CG

• (AG), b, B).

Therefore, if ω ∈ Ωm(Mφa )Gφ is closed, then ηω is a cycle in HPG

i (AG), where i is the parity of

m. We shall then denote by [ηω] its class in HPGi (AG).

By Proposition 9.12, the canonical projection π∗ : HP•(AG)→ HPG• (AG) is an isomorphism.

Denoting by π∗ its inverse we arrive at the following result.

Proposition 9.18. Let φ ∈ G and a ∈ 0, 2, . . . , n. Then any closed form ω ∈ Ω[•](Mφa )Gφ

defines a cyclic homology class π∗[ηω] ∈ HP•(AG) which depends only on the class of ω inH [•](Mφ

a )Gφ .

9.8. Proof of Proposition 9.1. Although this is not our main purpose, let us briefly disgressand explain how the previous constructions enable us to compute the cyclic homology of AG.Given φ ∈ G and using the splitting Ω•(Mφ) =

⊕a Ω•(Mφ

a ) we may combine together all themaps ρφa (resp., εφa), a = 0, 2, . . . , n, so as to get linear maps,

(9.34) ρφ : C•(A)Gφ −→ Ω•(Mφ)Gφ and εφ : Ω•(Mφ)Gφ −→ C•(A)Gφ .

such that εa is a right-inverse of ρa. These maps are mixed complex morphisms between the deRham mixed complex (Ω•(Mφ)Gφ , 0, d) and the twisted cyclic mixed complex (C•(A)Gφ , bφ, Bφ).A result of Brylinski-Nistor [BN, Lemma 5.2] ensures us that ρφ gives rise to a quasi-isomorphismfrom the twisted Hochschild complex (C•(A)Gφ , bφ) to (Ω•(Mφ)Gφ , 0). As this is a morphismof mixed complexes, a routine exercise in cyclic homology shows that ρφ gives rise to a quasi-

isomorphism from the periodic complex (C[•](A)Gφ , bφ+Bφ) to the de Rham complex (Ω[•](Mφ)Gφ , d).Using the Gφ-equivariance of ρφ and Proposition 9.17 we then obtain the isomorphisms,

HPG• (AG)φ ' HP

Gφ• (A) ' H [•](Mφ) =

⊕a

H [•](Mφa )Gφ .

35

This implies that the inclusion ω → ηω in Proposition 9.18 is actually a quasi-isomorphism.Combining all this with Proposition 9.12 gives the isomorphisms,

HP•(AG) ' HPG• (AG) '

⊕〈φ〉∈〈G〉

HPG• (AG)φ '

⊕〈φ〉∈〈G〉

⊕a

H [•](Mφa )Gφ .

This proves Proposition 9.1. In particular, this shows that HP•(AG) is spanned by the cyclichomology classes provided by Proposition 9.18.

10. Conformal Invariants

In this section, we shall combine together the results from the previous sections to constructand compute a wealth of new conformal invariants. We shall continue using the notation andassumptions of the previous sections.

To a large extent the introduction of G-normalized cochains in the previous section was moti-vated by the following result.

Proposition 10.1. Let g be a G-invariant metric in C . Then the cocycle ϕ = (ϕ2q) definedby (8.2) is G-normalized.

Proof. Let q be a nonnegative integer ≤ 12n. For j = 0, . . . , 2q let f j ∈ C∞(M) and φj ∈ G. It

immediately follows from the formula (8.2) that

ϕ2q(f0uφ0 , . . . , f

2quφ2q ) = ϕ2q

(f0, f1 φ−10 , . . . , (f2q φ−12q−1 · · · φ

−10 )uφ0···φ2q

).

In addition, let φ and ψ be in G, and set φ = ψ φ ψ−1. Then, for a = 0, 2, . . . , n, we have∫M φa

A(RTM

φ)∧ νφ

(RN

φ)∧ (f0 ψ−1)d

(f1 ψ−1

)∧ · · · ∧ d

(f2q ψ−1

)=

∫Mφa

A(RTM

φ)∧ νφ

(RN

φ)∧ f0df1 ∧ · · · ∧ df2q.

Thus,

ϕ2q

(f0 ψ−1, . . . , f2q−1 ψ−1, (f2q ψ−1)uφ

)= ϕ2q

(f0, . . . , f2q−1, f2quφ

).

Combining all this with Lemma 9.4 shows that ϕ2q is G-normalized, proving the lemma.

We are now in a position to extract the geometric contents of Theorem 9.1. More precisely, wehave the following result.

Theorem 10.2. Given φ ∈ G and a ∈ 0, 2, . . . , n, let ω be a Gφ-invariant closed even form onMφa . For any metric in the conformal class g ∈ C define

(10.1) Ig(ω) =⟨Ch(/Dg)σg , π

∗[ηω]⟩,

where Ch(/Dg)σg is the Connes-Chern character of the conformal Dirac spectral triple associated

to the metric g and the class π∗[ηω] ∈ HP0(AG) is defined as in Proposition 9.18. Then

(1) The number Ig(ω) is an invariant of the conformal structure C which depends only on the

classs of ω in H [0](Mφa )Gφ .

(2) For any G-invariant metric g ∈ C , we have

(10.2) Ig(ω) = (−i)n2 (2π)−a2

∫Mφa

A(RTM

φ)∧ νφ

(RN

φ)∧ ω.

Proof. The first part is an immediate consequence of Theorem 7.8 and Proposition 9.18. To provethe second part, let us assume that g is a G-invariant metric in C . Then by Theorem 8.3 theConnes-Chern character Ch(/Dg)σg is represented by the cocycle ϕ = (ϕ2q) given by (8.2). By

Proposition 10.1 this cocycle is G-normalized, so that Ch(/Dg)σg = ι∗[ϕ]. Therefore, using (9.19)we get

(10.3) Ig(ω) = 〈ι∗[ϕ], π∗[ηω]〉 = 〈ϕ, ηω〉 =∑q≥0

⟨ϕ2q, ηω2q

⟩,

36

where ω2q is the component of degree 2q of ω. Let us write ω2q =∑l f

0l df

1l ∧ · · · ∧ df

2ql , with

f jl ∈ Aφa . Then (9.33) gives

〈ϕ2q, ω2q〉 =∑l

∑σ∈S2q

ε(σ)⟨ϕ2q, f

0l ⊗Gf

σ(1)l ⊗G · · · ⊗Gfσ(2q−1)l ⊗Gfσ(2q)l uφ

⟩(10.4)

=∑l

∑σ∈S2q

ε(σ)ϕ2q

(f0l , f

σ(1)l , · · · , fσ(2q−1)l , f

σ(2q)l uφ

).(10.5)

Recall that if f ∈ Aφa , then f = f on Mφa and f = 0 on Mφ

b with b 6= 0. Therefore, theformula (10.5) for ϕ2q gives

ϕ2q

(f0i , f

σ(1)i , · · · , fσ(m−1)i , f

σ(m)i uφ

)=

∫Mφa

Υφa ∧ f0l df

σ(1)l ∧ · · · ∧ dfσ(m)

l ,

where we have set Υφa = (−i)n2 (2π)−

a2 A(RTM

φ)∧ νφ

(RN

φ)

. Using this together with (10.5)

shows that the pairing 〈ϕ2q, ω2q〉 is equal to

1

(2q)!

∑l

∑σ∈S2q

ε(σ)

∫Mφa

Υφa ∧ f0l df

σ(1)l ∧ · · · ∧ dfσ(2q)l =

∑l

∫Mφa

Υφa ∧ f0l df1l ∧ · · · ∧ df

2ql

=

∫Mφa

Υφa ∧ ω2q.

Combining this with (10.3) yields the formula (10.2) for Ig(ω). The proof is complete.

Remark 10.3. The conformal invariants Ig(ω) are closely related to conformal invariants con-structed by Branson-Ørsted [BØ2, §1]. It can be shown (see, e.g., [Gi] or Part II) that, given aG-invariant metric g ∈ C and φ ∈ G, we have

Str[e−t/D

2gUφ

]∼

∑0≤a≤na even

∑j≥0

t−a2+j

∫Mφa

I(j)φ,a(g)(x)

√g(x)dax as t→ 0+,

where the I(j)φ,a(g) are smooth functions on Mφ

a . Using variational formulas for equivariant heat

kernel asymptotics Branson-Ørsted [BØ2] proved that each constant coefficient,∫Mφa

I(a/2)φ,a (g)(x)

√g(x)dax, a = 0, 2, . . . , n,

remains constant when g ranges over all φ-invariant metrics in C . Thus, they give rise to invariantsof the conformal class C . The results of Branson-Ørsted actually hold for any even power of aconformally invariant elliptic selfadjoint differential operator. Bearing this in mind, given φ ∈ Gand any closed Gφ-invariant form ω ∈ Ω2q(Mφ

a ), the computation in Part II [PW2] of the Connes-Chern character Ch(/Dg) for a G-invariant metric in C shows that

Ig(ω) = limt→0+

tq Str[c(ω)e−t/D

2gUφ

],

where c(ω) is the Clifford action of ω on spinors. In particular, the r.h.s. above agrees with (10.1)by taking ω to be a scalar multiple of the volume form of Mφ

a . Therefore, Theorem 10.2 providesus with a cohomological interpretation and explicit computations of Branson-Ørsted’s conformalinvariants in the case of Dirac operators.

Remark 10.4. In general, the invariants of the form (10.2) are not of the same type of theglobal conformal invariants considered by Alexakis [Al] in his solution of the conjecture of Deser-Schwimmer [DS] on the characterization of global conformal invariants. His invariants are of theform,

Ig =

∫M

Pg(R)(x)√g(x)dnx,

where Pg(R)(x) is a linear combination of complete metric contractions of tensor products of theRiemann curvature tensor and its covariant derivatives in such way that Ig remains constant on

37

each conformal class of metrics. Such a conformal invariant is of the form (10.2) when φ = id andω is a constant function. However, in this case Theorem 10.2 reduces to the conformal invarianceof the total A-class of M , which is obvious since this is a topological invariant.

Remark 10.5. As mentioned in the Introduction, it would be interesting to find, at least conjec-turally, a characterization of conformal invariants encompassing the invariants of [Al, DS] and theinvariants from [BØ2] and Theorem 10.2.

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Department of Mathematical Sciences, Seoul National University, Seoul, South Korea

E-mail address: ponge.snu@gmail.com

School of Mathematical Sciences, University of Adelaide, Adelaide, Australia

E-mail address: hang.wang01@adelaide.edu.au

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