Twisted Spectral Triples and Connes Character Formula

Twisted Sp ectral T riples and Connes’ Character F orm ula F arzad F athizadeh Departmen t of Mathematics and Statistics Y ork Univ e rsit y T oron to, On tario, Canada, M3J 1P3 ffathiza@mathstat.y orku.ca Masoud Khal khali Departmen t of Mathematics The Univ ers ity of W estern Ontario London, On tario, Canada, N6A 5B7 masoud@u w o.ca Abstract W e giv e a p roof of an analogue of Connes’ Ho c hschild c haracter theo- rem for twi sted sp ectral triples obtained from twisti ng a spectral triple by scaling automorphisms, un der s ome suitable conditions. W e also surv ey some of th e p roperties of tw isted sp ectral t riples t hat are known so far. 1 In tro duction In [8] Connes and Moscovici intro duced a refinement of the notion o f sp ectral triple called twiste d sp e ct r al triple . As is well known by now, the metric asp ects of a geometr ic space in noncomm utative geometry is enco ded b y a sp e ctr al triple [5]. This concept how ever puts ra ther severe re strictions o n the underlying noncommutativ e spa c e , as it for ces the existence of a tracial sta te. In particular it is inapplica ble in typ e III situations according to Murr a y-von Neumann’s classification of von Neumann algebras into types. It w as to deal with these t yp e I II ex a mples that twisted sp e c tral triples were introduced. In a twisted sp ectral triple, the t wist is afforded by an automorphism . More precisely , there is an automorphism σ of the underly ing noncommutativ e space A such tha t all twisted commutators [ D , a ] σ = D a − σ ( a ) D are b ounded oper ators ( D is a s elfadjoin t op erator playing the role of the Dira c op erator). While there are many natural examples of twisted sp ectral triples, pr o ving general results ab out them, parallel to the unt wisted case, hav e prov ed to b e a challenging task. The mo s t celebra ted among such questions are p erhaps the extensio n of Connes’ 1 character fo rm ula for the Hochsc hild class of the Connes- Chern character of a sp ectral triple [5 ], as well as an extension of Co nnes-Moscovici’s lo cal index formula, [7] to twisted sp ectral triples. In this pap er we give a pro of of a character formula for twisted sp ectral triples obtained from twisting a sp ectral triple by s caling automorphisms , under suitable conditions. W e shall also br iefly outline Moscovici’s lo cal index formula for twisted sp ectral triples obtained by conformal p erturbation of a sp ectral triple [17]. This pap er is organized as follows. In Section 2 we rec a ll the basic notions of sp ectral triples, noncommutativ e in tegration a nd residue functionals for a sp ectral triple. W e also recall Connes and Moscovici’s lo cal index form ula [7]. W e then reca ll the notion of t wisted sp ectral triples [8]. In Sec tion 3 we rec a ll several genera l metho ds to construct twisted sp ectral triples from [8], [17] and [11]. An imp ortant idea here is t wisting a spectr al triple by scaling automor- phisms and the cor r esponding algebra o f twisted pseudo differential op erators int ro duced b y Moscovici [1 7]. In Section 4 we r ecall some non-obvious prop er- ties of twisted sp ectral triples from [8], a mong them the fact that o ne can define a Connes- Chern c haracter for t wisted sp ectral triples with v alues in ordinary non-twisted cyclic cohomology . E quiv alently o ne can define a pairing b et ween a finitely summable twisted sp ectral triple and K -theory . In s ection 5 we give a pro of of an ana logue o f Connes ’ Ho chsc hild character theorem for t wisted sp ec- tral triples o btained from twisting a sp ectral triple by scaling automorphisms. Finally in the las t section we recall Moscovici’s Ansatz and pro o f of the local index for m ula for t wisted spec tral triples obtained by sca ling automorphisms [17]. 2 F rom sp ectral triples to t wisted sp ectral triples The no tion of a geometr ic s pa ce in noncommutativ e g eometry is enco ded by a sp e ctr al triple . This concept how ev er puts rather severe r estrictions on the un- derlying noncommutativ e s pace that renders it inapplica ble in typ e III situations according to Mur r a y-von Neumann classification of von Neumann alg ebras into t yp es. W e sha ll first try to explain this p oint. W e s ta rt with a quick review of the D ix mier t r ac e a nd the nonc ommut ative inte gr al , following closely [5]. Let H be a Hilber t spa ce and let K ( H ) denote the t wo sided ideal of compac t o perato rs on H . F or a compact o p erator T : H → H , let µ 1 ( T ) ≥ µ 2 ( T ) ≥ · · · ≥ 0 denote the s equence of eigenv a lues of its absolute v alue | T | := ( T ∗ T ) 1 2 , written in decrea sing order . Thu s, by the minimax principle, µ 1 ( T ) = || T || , and in general µ n ( T ) = inf || T | V || , n ≥ 1 , where the infimum is over the set of subspaces of co dimension n − 1, and T | V denotes the restriction of T to the subspace V . A compact op erator T ∈ K ( H ) 2 is called trace class if P µ n ( T ) < ∞ , and in that case the trace o f T is defined by T ra ce ( T ) = X n ( T e n , e n ) where { e n } ∞ n =1 is an orthonor mal basis for H . It is known that this is the unique no rmal tra ce on the C ∗ -algebra L ( H ) o f b ounded o perator s on H . The question o f existence of non-normal traces on L ( H ) was left op en until it w as settled affirmativ ely b y Jacques Dixmier. In [9] Dixmier shows that there a re uncountably many non-no rmal traces o n L ( H ). Man y y ears later , Alain Connes discov ered that these no n-normal traces ca n in fa ct b e used to define a pro cess of noncommutativ e integration in no ncomm utative geometry as we describ e next. The natura l do main of a Dixmier trace is the set of op erators L 1 , ∞ ( H ) = { T ∈ K ( H ); N X 1 µ n ( T ) = O (log N ) } . Notice that tr ace class oper ators ar e automatically in L 1 , ∞ ( H ). The Dixmier trace o f an op erator T ∈ L 1 , ∞ ( H ) mea sures the lo garithmic diver genc e of its ordinary trace. More precisely , we are in terested in the limit of the bounded sequence σ N ( T ) = P N 1 µ n ( T ) log N , N = 1 , 2 , . . . as N → ∞ . The first pr o blem of course is that, while by our assumption the sequence is bounded, the us ual limit may not exist and must b e replaced by a gener alize d limit , similar to Banach limits of non-convergen t b ounded sequences. A mor e challenging task is to make sure that our gener a lized limit still defines a trace. T o this end, let T race Λ ( T ) , Λ ∈ [1 , ∞ ), b e the piecewise affine interpo lation of the partial trace function T race N ( T ) = P N 1 µ n ( T ). Reca ll tha t a state on a C ∗ -algebra is a non-zero positive linear functional on the algebra . Let ω : C b [ e, ∞ ) → C b e a normaliz ed state on the algebra of b ounded contin uous functions on [ e, ∞ ) such that ω ( f ) = 0 for all f v a nishing at ∞ . Now, using ω , the Dixmier trace o f a p ositive op erator T ∈ L 1 , ∞ ( H ) is defined as T r ω ( T ) := ω ( τ Λ ( T )) , where τ Λ ( T ) = 1 log Λ Z Λ e T ra ce r ( T ) log r dr r is the Cezar o mean of the function T race r ( T ) log r ov er the multiplicativ e g roup R ∗ + . One then extends T r ω to all of L 1 , ∞ ( H ) by linea rit y . The resulting linear functional T r ω is a p ositive trace on L 1 , ∞ ( H ). It is easy to see from its definition that if T actua lly happ ens to be a trace class op erator then T r ω ( T ) = 0 for all ω , i.e., the Dixmier tra ce is inv ariant under 3 per turbations by trace clas s op erator s. T his is a very useful prop erty and makes T r ω a flexible to ol in computatio ns . The Dixmier trace, T r ω , in gener a l dep ends on the limiting pro cedure ω , how ever, fo r the class of op erators T for which Lim Λ →∞ τ Λ ( T ) e xits, it is in- depe ndent of the c hoice of ω and is equal to Lim Λ →∞ τ Λ ( T ). One of the main results prov ed in [4] is that if M is a closed n -dimensional manifold, E is a smo oth vector bundle on M , P is a pseudo different ial op e rator of order − n act- ing b et ween L 2 -sections of E , a nd H = L 2 ( M , E ) , then P ∈ L 1 , ∞ ( H ) a nd, for any choice of ω , T r ω ( P ) = n − 1 Res( P ). Here Res denotes W odz icki’s no ncom- m utative residue [19, 15]. F or example, if D is a n elliptic first orde r differential op erator, | D | − n is a pseudodifferential op erator of or der − n , and the Dixmier trace T r ω ( | D | − n ) is indep enden t o f the choice of ω . Next, we would like to explain the no tion of sp ectral triple. This co ncept has its ro ots in K -homology a nd Riemannian g e o metry simultaneously . W e sta rt by explaining the notion o f a F redholm mo dule which is the co nfo r mal counterpart of a sp ectral triple. An o dd F r e dholm mo du le over a unital algebra A is a pair ( H , F ) where H is a Hilber t spa ce on which the algebr a A a cts by b ounded op erators and F ∈ L ( H ) is a selfadjoin t op erator such that F 2 = id , and such that the commutators [ F, π ( a )] a re compa ct op erators for all a ∈ A . Here π : A → L ( H ) denotes a unital action of A on H . A F redholm mo dule is ca lled p - summable (1 ≤ p < ∞ ), if [ F , π ( a )] ∈ L p ( H ) for all a ∈ A , wher e L p ( H ) is the Schatten ide al of p - summable compact op erators [3]. F r e dholm mo dules should b e thought o f as representing K -homolo gy clas ses defined by a bs tract elliptic partial differ en tial op erators on the noncommutativ e s pa ce A . Spec tr al triples provide a refinement of F redholm mo dules. Going from F redholm mo dules to sp ectral triples is akin to going from the c onformal class of a Riemannian metr ic to the metric itself. Spec tr al triples s imultaneously provide a notion of Dir ac op er ator in nonco mm utativ e geometr y , as well as a Riemannian type distanc e fun ct io n for no ncomm utative spaces as w e shall explain next. T o motiv ate the definition o f a sp ectral tr iple, we recall that the Dira c ope ra- tor D on a compact Riemannian spin c manifold acts as an unbounded selfadjoint op erator on the Hilb ert spa ce L 2 ( M , S ) of L 2 -spinors on the manifold M . If we let C ∞ ( M ) a ct on L 2 ( M , S ) b y mu ltiplication op erators, then o ne can chec k that for any smo oth function f , the c o mm utator [ D , f ] = D f − f D extends to a bo unded op erator on L 2 ( M , S ). Now the ge o desic distanc e d on M can b e recov ered fro m the following b eautiful distanc e formula of Connes [5]: d ( p, q ) = Sup {| f ( p ) − f ( q ) | ; k [ D , f ] k≤ 1 } , ∀ p, q ∈ M . The triple ( C ∞ ( M ) , L 2 ( M , S ) , D ) is a comm utative example of a s pectral triple. Its general definition, in the o dd c ase, is as fo llows. Definition 2. 1. L et A b e a unital algebr a. An o dd sp e ctr al triple on A is a triple ( A , H , D ) c onsisting of a Hilb ert sp ac e H , a selfadjo int unb ounde d op er ator D : Dom ( D ) ⊂ H → H with c omp act r esolvent, i.e., ( D + λ ) − 1 ∈ K ( H ) for al l λ / ∈ R , 4 and a unital r epr esentation π : A → L ( H ) of A such t ha t for al l a ∈ A , the c ommu t ator [ D , π ( a )] is define d on Dom ( D ) and extends to a b ounde d op er ator on H . The finite summability as sumption fo r F redholm mo dules has a finer ana - logue for s pectral triples. F or simplicity we shall assume that D is inv ertible (in general, since Ke r( D ) is finite dimensional, by restricting to its orthogo nal co m- plement we can alw ays r educe to this case). A spec tr al triple is called fin itely summable if for s ome n ≥ 1, | D | − n ∈ L 1 , ∞ ( H ) . (2.1) A simple example of an o dd spectra l tr iple is ( C ∞ ( S 1 ) , L 2 ( S 1 ) , D ), wher e S 1 = R / 2 π Z is the circle and D is the unique s elfadjoin t extens io n of the op erator 1 i d dx . The eigenv alues of | D | a re | n | ; n ∈ Z which s hows that if we restrict D to the orthogo nal compliment of co nstan t functions then | D | − 1 ∈ L 1 , ∞ ( L 2 ( S 1 )) . Given a sp ectral tr iple ( A , H , D ), one can obtain a F redholm mo dule ( A , H , F ) by c ho osing F = Sign ( D ) = D | D | − 1 . Connes ’ Ho c hschild c haracter for m ula gives a lo cal expression for the Ho c hschild clas s of the Connes-Chern character of ( A , H , F ) in terms o f D itself. F or this o ne ha s to ass ume that the sp ectral triple ( A , H , D ) is r e gular in the sense that fo r all a ∈ A , a and [ D , a ] ∈ ∩ Dom( δ k ) where the deriv ation δ is given by δ ( x ) = [ | D | , x ] . Now, assuming (2 .1) holds, Co nnes defines an ( n + 1)-linear functional ϕ on A by ϕ ( a 0 , a 1 , . . . , a n ) = T r ω ( a 0 [ D , a 1 ] · · · [ D, a n ] | D | − n ) . It ca n b e shown tha t ϕ is a Ho c hschild n -co cycle on A . W e reca ll that a Ho c hschild n -cycle c ∈ Z n ( A, A ) is an element c = P a 0 ⊗ a 1 ⊗ · · · ⊗ a n ∈ A ⊗ ( n +1) such that its Hochsc hild boundar y b ( c ) = 0, where b is defined by (5.4). The following res ult, known as Connes’ Ho chschil d char acter formula , computes the Ho c hschild class of the Che r n character by a lo cal formula, i.e., in terms of ϕ : Theorem 2.1. [5] Le t ( A , H , D ) b e a r e gu lar sp e ctra l t rip le. L et F = Sign ( D ) denote the sign of D and τ n ∈ H C n ( A ) denote the Connes-Chern char acter of ( H , F ) . F or every n - di mensional Ho chschild cycle c = P a 0 ⊗ a 1 ⊗ · · · ⊗ a n ∈ Z n ( A , A ) , one has h τ n , c i = X ϕ ( a 0 , a 1 , . . . , a n ) . Ident ifying the full cyclic coho mology class of the Connes-C he r n character of ( A , H , D ) by a lo cal formula is the conten t of Connes- Mo sco vici’s lo cal index formula. F or this we hav e to assume the sp ectral tr iple satisfies a no ther technical condition. Let B denote the subalgebra of L ( H ) generated by op erators δ k ( a ) and δ k ([ D , a ]) , k ≥ 1 . A sp ectral triple is sa id to hav e a discrete dimension sp e ctrum Σ if Σ ⊂ C is discrete and fo r any b ∈ B the function ζ b ( z ) = T ra ce ( b | D | − z ) , Re z > n, 5 extends to a holomorphic function on C \ Σ. It is further assumed that Σ is simple in the sense that ζ b ( z ) has o nly simple p oles in Σ. The lo cal index formula of Connes and Moscovici [7] is given by the following theorem (we hav e used the for m ulation in [6]): Theorem 2. 2. [6] 1. The e quality Z − P = Res z =0 T ra ce ( P | D | − z ) defines a tr ac e on the algebr a gener ate d by A , [ D, A ] , and | D | z , z ∈ C . 2. Ther e ar e only a finite nu mb er of n on-zer o t erms in the fo l lowing formula which defines the o dd c omp onents ( ϕ n ) n =1 , 3 ,... of an o dd cyc lic c o cycle in the ( b, B ) bic omplex of A : F or e ach o dd inte ger n let ϕ n ( a 0 , · · · , a n ) := X k c n,k Z − a 0 [ D , a 1 ] ( k 1 ) . . . [ D, a n ] ( k n ) | D | − n − 2 | k | wher e T ( k ) := ∇ k and ∇ ( T ) = D 2 T − T D 2 , k is a multi-index, | k | = k 1 + · · · + k n and c n,k := ( − 1 ) | k | √ 2 i ( k 1 ! . . . k n !) − 1 (( k 1 + 1 ) · · · ( k 1 + k 2 + · · · k n )) − 1 Γ( | k | + n 2 ) . 3. The p airing of t he cyclic c ohomolo gy class ( ϕ n ) ∈ H C ∗ ( A ) with K 1 ( A ) gives the F re dholm index of D with c o efficients in K 1 ( A ) . Given an n -summable regular spec tr al triple ( A , H , D ), the linear functiona l a 7→ T r ω ( a | D | − n ) defines a tr ace on the algebra A ( cf . P ropo sition 4.1 ). Th us to deal with type II I alg ebras which carry no non-trivial trac e s , the no tion o f sp ectral triple must be mo dified. In [8] Co nnes and Mo sco vici define a notion of twiste d sp e ctr al triple , wher e the twist is affor de d b y an algebr a automorphism (related to the mo dular automor phism gro up). More pr ecisely , one po s tulates that there exists an auto morphism σ of A such that the twisted commutators D a − σ ( a ) D are bo unded o perator s for a ll a ∈ A . Here is the full definition: Definition 2.1. L et A b e an algebr a which is r epr esente d by b ounde d op er ators in a Hilb ert sp ac e H , and D b e an unb oun de d selfadjoint op er ator in H with c omp act re solvent. With σ b eing an automorphism of A , the triple ( A , H , D ) is c al le d a twiste d sp e ctr al triple or a σ -sp e ctra l triple if for any a ∈ A , the twiste d c ommu t ator [ D , a ] σ := D a − σ ( a ) D is define d on the domain of D , and extends to a b ounde d op er ator in H . A twiste d sp e ctr al triple is said t o b e Lipschitz-r e gular if the twiste d c ommu tators | D | a − σ ( a ) | D | ar e b ounde d as wel l for al l a ∈ A . A gr ade d twiste d sp e ctr al triple is one that is endowe d with a gr ading op er ator γ = γ ∗ ∈ L ( H ) such that γ 2 = id , γ c ommut es with the action of A , and antic ommut es with D . 6 When the alg ebra A is in volutiv e, the re presen tation is assumed to b e in vo- lutive a s well. In this case, it is natural to impose the following compatibility condition b et ween the a utomorphism and the inv olution: σ ( a ) ∗ = σ − 1 ( a ∗ ) , ∀ a ∈ A . It is shown in [8 ] tha t in the t wisted case, the Dixmier tra ce induces a twisted trace o n the algebr a A , but surprisingly , under some regularity co nditions, the Connes-Chern character of the phas e space lands in ordinar y cyclic coho mology . Thu s its pa iring with o rdinary K -theor y makes sense, and it can be recov ered a s the index o f F redholm op erator s . This sug gests the s ig nificance of developing a lo cal index formula for twisted sp ectral triples, i.e. finding a formula for a cocy - cle, cohomolo gous to the Connes- Che r n character in the ( b, B )-bico mplex, which is giv en in terms o f twisted commutators and res idue functionals. W e b eliev e that this new theme o f twisted sp ectral triples, and type I II noncommutativ e geometry in gener al, shall dominate the sub ject in near future. F or example, very recently a lo cal index for m ula has b een proved for a cla ss of t wisted sp ectral triples b y Henri Moscovici [17]. W e will discuss this result in detail in Section 6 of this pap er. This c la ss is obtained by twisting an or dinary sp ectral triple ( A , H , D ) b y a subgro up G of sc aling aut omorp hisms of the tr iple, i.e. the s et of all unitary op erators U ∈ U ( H ) such tha t U A U ∗ = A , and U D U ∗ = µ ( U ) D , with µ ( U ) > 0. It is shown that the cro ssed pro duct algebra A ⋊ G admits an automorphis m σ , given b y the formula σ ( aU ) = µ ( U ) − 1 aU , for all a ∈ A , U ∈ G ; a nd ( A ⋊ G, H , D ) is a twisted s p ectral triple. 3 Examples of t wisted sp ectral triples In this section, w e reca ll g eneral metho ds to construct twisted sp ectral triples. 3.1 P erturbing sp ectral triples In [8 ], it is s ho wn tha t starting from an ordina ry sp ectral triple ( A , H , D ) a nd a selfadjoint element h = h ∗ ∈ A , the pertur bed triple ( A , H , D ′ ) , D ′ = e h D e h is a σ -spe c tr al triple, where the a uto morphism σ is given b y σ ( a ) = e 2 h ae − 2 h , a ∈ A . In fact for any a ∈ A , one has D ′ a − σ ( a ) D ′ = e h D e h a − e 2 h ae − 2 h e h D e h = e h [ D , e h ae − h ] e h . Therefore the twisted comm utators [ D ′ , a ] σ are bounded since the commu ta- tors [ D , b ] are bounded op erators for all b ∈ A . A concrete example of this construction is obtained when o ne c ompares the Dirac op erators of conformally equiv ale nt Riema nnian metrics as follows: 7 Example 3.1. Let ( M , g ) be a compact Riema nnian s pin manifold and D = D g be the ass ociated Dirac o perator acting o n the Hilb ert space of L 2 -spinors H = L 2 ( M , S g ). Let g ′ = e − 4 h g be a c onformally equiv a len t metric wher e h ∈ C ∞ ( M ) is a selfadjoint elemen t. It can be s hown that the gauge tr a nsform op erator g D g ′ = β g ′ g ◦ D g ′ ◦ β g g ′ has the form g D g ′ = e ( n +1) h ◦ D g ◦ e − ( n − 1) h . After the canonic a l identification o f the s pace o f g -spinor s with g ′ -spinors, it ca n be seen [8] that the g a uge transfor m sp ectral tr iple is obtained from the o riginal one by replacing D with D ′ = e h D e h . More generally , one can start from a σ -spectr al triple ( A , H , D ) and a self- adjoint elemen t h = h ∗ ∈ A , and investigate the prope r ties of the p erturb ed triple ( A , H , D ′ ) , D ′ = e h D e h . Lemma 3 .2. L et ( A , H , D ) b e a σ -sp e ctr al triple, and h = h ∗ ∈ A . Then the p ertu rb e d triple ( A , H , D ′ ) , D ′ = e h D e h is a σ ′ -sp e ctr al t rip le wher e σ ′ ∈ Au t ( A ) is given by σ ′ ( a ) = e h σ ( e h ae − h ) e − h , a ∈ A . Pr o of. F o r any a ∈ A , o ne has D ′ a − σ ′ ( a ) D ′ = e h D e h a − e h σ ( e h ae − h ) e − h e h D e h = e h [ D , e h ae − h ] σ e h . Therefore the twisted commutators [ D ′ , a ] σ ′ are b ounded since the t wisted c o m- m utator s [ D , b ] σ are b ounded op erators for all b ∈ A . 3.2 Twisted connections on crossed pro duct algebras In [11], a metho d for constructing twisted spectra l triples o n cr ossed pr oduct algebras is s uggested. This metho d uses 1 -cocy cles to construct automorphisms for cr ossed pro duct algebras, and explains how one can obtain t wisted traces, t wisted deriv atives, and t wisted c onnections on these algebr as. W e shall explain this metho d and show that it can recons truct a n ex ample of twisted spectral triples, first given by Connes a nd Moscovici in [8]. First we reca ll some defini- tions: Definition 3.3. L et σ : A → A b e an automorphism of an algebr a A . 1. A σ -derivation on A is a line ar map δ : A → A s u ch that δ ( ab ) = δ ( a ) b + σ ( a ) δ ( b ) , ∀ a, b ∈ A . 8 2. A σ -t ra c e on A is a line ar map τ : A → C such that τ ( ab ) = τ ( σ ( b ) a ) , ∀ a, b ∈ A . Now, le t A be a n alg ebra with a right a ction o f a gro up Γ by automo rphisms: A × Γ → A, ( a, γ ) 7→ a · γ . W e consider the a lgebraic cross ed pro duct A ⋊ Γ with the standard multiplica- tion: ( a ⊗ γ )( b ⊗ µ ) = ( a · µ ) b ⊗ γ µ, a, b ∈ A, γ , µ ∈ Γ . Let Z ( A ) denote the center of the algebra A , and A ∗ its group o f invertible elements. Definition 3.4. 1. A map j : Γ → Z ( A ) ∩ A ∗ is a 1-c o cycle if j ( γ µ ) =  ( j γ ) · µ  j µ, ∀ γ , µ ∈ Γ . 2. Given a map j : Γ → A , a line ar functional τ : A → C is said t o have the change of variable pr op erty with r esp e ct to j , if τ (( a · γ ) j γ ) = τ ( a ) , ∀ a ∈ A, γ ∈ Γ . Notice that (3.2) amounts to saying that j is a (m ultiplicative) gr oup 1 − co cycle for H 1 (Γ , Z ( A ) ∩ A ∗ ) . Prop osition 3.5 . [1 1] L et A b e an algebr a with a right action of a gr oup Γ by automorphisms, and j : Γ → Z ( A ) ∩ A ∗ b e a 1-c o cycle. 1. The m ap σ : A ⋊ Γ → A ⋊ Γ given by σ ( a ⊗ γ ) =  ( j γ − 1 ) · γ  a ⊗ γ is an automorphism. 2. L et δ : A → A b e a derivation such that δ ( a · γ ) = ( δ ( a ) · γ ) j γ for al l a ∈ A, γ ∈ Γ . Then for any s = 1 , 2 , . . . , the map δ ′ s : A ⋊ Γ → A ⋊ Γ define d by δ ′ s ( a ⊗ γ ) =  δ  ( a · γ − 1 )( j γ − 1 ) s  ( j γ − 1 ) − s ⊗ 1  1 ⊗ γ  is a σ -derivation on A ⋊ Γ . Also δ ′ s ◦ σ = σ ◦ δ ′ s if δ ◦ j = 0 . 3. If τ : A → C is a tr ac e su ch that τ ◦ δ = 0 , then τ ′ ◦ δ ′ s = 0 wher e t he line ar functional τ ′ : A ⋊ Γ → C is define d by τ ′ ( a ⊗ γ ) = 0 if γ 6 = 1 , and τ ′ ( a ⊗ 1) = τ ( a ) . Also , if τ has t he change of variable pr op erty with r esp e ct to j , then τ ′ is a σ -tr ac e on A ⋊ Γ . 9 Example 3 .6. 1. L et M b e a smo oth oriente d manifold and ω a volume form on M . L et Γ = D if f ( M ) b e the gr oup of diffe omorphisms of M . The m ap j : Γ → C ∞ ( M ) define d by γ ∗ ( ω ) = j ( γ ) ω is e asily se en to b e a 1-c o cycle. 2. L et χ : Γ → C ∗ b e a 1-dimensional char acter of a gr oup Γ which acts on an algeb r a A by automorph isms, and let j ( γ ) = χ ( γ )1 A . Then j is a 1-c o cycle, and a derivation δ : A → A is c omp atible with j if and only if δ ( a · γ ) = χ ( γ ) δ ( a ) · γ for any a ∈ A , and γ ∈ Γ . Definition 3.7. L et A b e an algebr a, δ : A → A a deriv ation, and E a left A -mo dule. A line ar map ∇ : E → E is said t o b e a c onne ction if it satisfies t he L eibniz rule, i.e. ∇ ( aξ ) = δ ( a ) ξ + a ∇ ( ξ ) , ∀ a ∈ A , ξ ∈ E . If σ : A → A is an automorph ism and δ : A → A a σ -derivation, then a line ar map ∇ : E → E is said to b e a twiste d c onne ction if it satisfies the twiste d L eibniz rule: ∇ ( aξ ) = δ ( a ) ξ + σ ( a ) ∇ ( ξ ) , ∀ a ∈ A , ξ ∈ E . This notion of t wisted connection comes from [18]. Given a twisted connectio n ∇ , one can try to define a twisted sp ectral tr iple by letting D = ∇ . Then ∇ a − σ ( a ) ∇ = δ ( a ) s ho ws the b oundedness co ndition is satisfied provided δ ( a ) acts by a bounded op erator. In the following propo sition, A is an alg ebra endo wed with a r igh t action of a g roup Γ by automorphisms, with represe n tations π : A → End ( E ), and ρ : Γ → GL( E ), defining a cov ar ian t sys tem, i.e. π ( a · γ ) = ρ ( γ − 1 ) π ( a ) ρ ( γ ) , ∀ a ∈ A, γ ∈ Γ . Then we obtain a r epresent ation π ′ : A ⋊ Γ → End( E ) given by π ′ ( a ⊗ γ ) = ρ ( γ ) π ( a ) , ∀ a ∈ A, γ ∈ Γ . Also let j : Γ → Z ( A ) ∩ A ∗ be a 1 -cocycle and δ : A → A a deriv ation as in Prop osition 3.5. Therefor e we have an a utomorphism σ : A ⋊ Γ → A ⋊ Γ and we fix a σ -deriv ation δ ′ s : A ⋊ Γ → A ⋊ Γ for so me s ∈ N . Prop osition 3.8. [1 1] A c onne ction ∇ : E → E for A is a twiste d c onn e ction for A ⋊ Γ if and only if ∇ ρ ( γ ) = ρ ( γ )  ( s − 1) π  δ ( j γ − 1 · γ )  + ∇ π ( j γ − 1 · γ )  , ∀ γ ∈ Γ . (3.2) 10 Remark 3. 9. Let s > 0 b e a real num b e r and assume  j ( γ )  s ∈ A is defined for all γ ∈ Γ. Prop ositions 3.5 and 3.8 cont inue to hold for these v alue s o f s as well. F or this w e need the extra co ndition δ ( x s ) = sx s − 1 δ ( x ) to hold for a ll x = j ( γ ), γ ∈ Γ. Example 3.10. Let C ∞ ( S 1 ) b e the algebra of smo oth functions on the circle S 1 = R / Z , and Γ ⊂ D if f + ( S 1 ) a group of orientation preserving diffeomor - phisms of the circle as in [8]. W e represent the algebra C ∞ ( S 1 ) by bo unded op erators in the Hilber t spa ce L 2 ( S 1 ) by ( π ( g ) ξ )( x ) = g ( x ) ξ ( x ) , ∀ g ∈ C ∞ ( S 1 ) , ξ ∈ L 2 ( S 1 ) , x ∈ R / Z . Define a repr e s en tation of Γ b y b ounded op erators in L 2 ( S 1 ) by ( ρ ( φ − 1 ) ξ )( x ) = φ ′ ( x ) 1 2 ξ ( φ ( x )) , ∀ φ ∈ Γ , ξ ∈ L 2 ( S 1 ) , x ∈ R / Z . The group Γ ac ts on C ∞ ( S 1 ) from rig h t b y comp osition and one ca n ea sily chec k that the ab ov e r e presen tations g ive a cov a r ian t system whic h yields the representation o f C ∞ ( S 1 ) ⋊ Γ as in [8]. The map j : Γ → C ∞ ( S 1 ) defined b y j ( φ ) = φ ′ is a 1 -cocy cle and the deriv ation δ : C ∞ ( S 1 ) → C ∞ ( S 1 ) , δ ( f ) = 1 i f ′ is compatible with j . Now by using Prop osition 3.5, we obtain a n automorphism σ of C ∞ ( S 1 ) ⋊ Γ whic h agrees with the automorphism in [8], and a twisted deriv a tion δ ′ 1 2 . Note that s ince it is po ssible to take the square ro ot of the elements in the image of j in this example, we can let s = 1 2 to obtain a twisted deriv a tion. Now if w e let ∇ = 1 i d dx , one can s ee that the equality (3.2) holds, therefore ∇ is a t wisted co nnection for C ∞ ( S 1 ) ⋊ Γ by Pr opositio n 3.8. Also the line a r map τ : C ∞ ( S 1 ) → C defined by τ ( g ) = Z R / Z g ( x ) dx, ∀ g ∈ C ∞ ( S 1 ) , is a trace which has the change o f v ar ia ble pr operty with resp ect to j , and τ ◦ δ = 0. Therefore by Prop osition 3 .5, one obtains a twisted tra c e τ ′ : C ∞ ( S 1 ) ⋊ Γ → C such that τ ′ ◦ δ ′ 1 2 = 0 . 3.3 Twisting sp ectral triples by scaling automorphism s The last example of the preceding subsection g iv es a sp e c ial case of a cla ss of t wisted spe ctral triples that arise naturally in conformal g eometry [17]. Le t ( M , g ) b e a connected compact Riemannian s pin manifold o f dimension n and D = D g be the as sociated Dirac oper ator acting on the Hilbert space of L 2 - spinors H = L 2 ( M , S g ). Let S C O ( M , [ g ]) denote the Lie group of diffeomor- phisms of M that pr eserve the conformal structure [ g ] consisting of a ll Rieman- nian metrics that are confor ma lly equiv alent to g , the orientation, and the spin structure, a nd let G = S C O ( M , [ g ]) 0 denote the connected comp onen t of the ident ity . In [1 7], using a suitable a utomorphism o f the crossed pro duct algebra C ∞ ( M ) ⋊ G , a t wisted spectra l triple of the form ( C ∞ ( M ) ⋊ G, L 2 ( M , S g ) , D ) is 11 constructed. Similarly , by endowin g R n with the Euclidean metric, and cons id- ering the g roup G of conformal transfor mations of R n , a twisted sp ectral triple is constructed over the cro ssed pro duct algebra C ∞ c ( R n ) ⋊ G . An abstract for- m ulation of this cla ss of twisted spe c tral triples lea ds in [17] to the idea of t wisting an ordinar y s pectral triple by its sc aling automorph isms o r c onformal similarities which we explain in this subsection. Using scaling automorphisms of a spec tr al triple ( A , H , D ), one can construct a twisted spectr al triple [17]. The set o f s caling automorphisms, a lso called conformal similarities, of a sp ectral tr iple ( A , H , D ), deno ted by Sim( A , H , D ), consists of all unitary op erators U o n H such that U A U ∗ = A , a nd U D U ∗ = µ ( U ) D , for some µ ( U ) > 0 . It is easy to s e e that Sim( A , H , D ) is a group and the ma p µ : Sim( A , H , D ) → (0 , ∞ ) is a character. W e fix a subgroup G ⊂ Sim( A , H , D ) and let A G be the crossed pro duct a lgebra A ⋊ G . It is shown in [17] that the for m ula σ ( aU ) = µ ( U ) − 1 aU, ∀ a ∈ A , U ∈ G, defines an automor phism o f A G , and ( A G , H , D ) is a twisted sp ectral triple. In fact the twisted co mm utators [ D , aU ] σ := D aU − σ ( aU ) D = [ D , a ] U are bo unded o perator s for all a ∈ A , U ∈ G . F or this cla ss of twisted sp ectral tr iple s , one can form the cros sed pr oduct algebra Ψ( A ⋊ G, H , D ) := Ψ( A , H , D ) ⋊ G , wher e Ψ ( A , H , D ) is the a lgebra of pseudo differen tial op erators asso ciated to the base sp ectral triple [7, 13, 14], and under the extende d simple dimension sp e ctrum hyp othesis , the residue functiona l R D − : Ψ( A G , H , D ) → C given b y Z D − P := Res z =0 T ra ce ( P | D | − 2 z ) is a trace [17]. This s e e ms to b e in agreement with the twisted a nalogue o f the Adler-Manin tra ce [1, 16, 11]: F or a triple ( A, σ , δ ) consisting of an alg ebra A , an algebr a automorphism σ : A → A , a nd a σ -deriv a tion δ : A → A , we define an algebra of formal t wisted pseudo differential symbols Ψ( A, σ, δ ) whose elements are for ma l series of the form N X n = −∞ a n ξ n , N ∈ Z , a n ∈ A, ∀ n ≤ N . The m ultiplication in this alg ebra is essentially derived from these r elations ξ a = σ ( a ) ξ + δ ( a ) , ∀ a ∈ A, ξ ξ − 1 = ξ − 1 ξ = 1 . In [11], we prove that starting from a δ -inv ariant twisted trace on A , the non- commutativ e r esidue is a tr ace on Ψ ( A, σ , δ ): 12 Theorem 3.11. [11] L et σ : A → A b e an automorphi sm of an algebr a A . If τ : A → C is a σ - tr ac e and δ : A → A is a σ -derivatio n such t ha t τ ◦ δ = 0 , then t he line ar functional Res : Ψ( A, σ, δ ) → C define d by Res  n X i = −∞ a i ξ i  = τ ( a − 1 ) is a tr ac e on Ψ( A, σ, δ ) . 4 Prop erties of t wisted sp ectral triples In this section, w e reca ll the basic prop erties o f twisted sp ectral triples [8]. 4.1 Twisted trace Given a σ -sp ectral tr iple ( A , H , D ) with D − 1 ∈ L n, ∞ , it is observed in [8] that D − n a − σ − n ( a ) D − n ∈ L 1 , ∞ 0 ( H ) , ∀ a ∈ A . Here L 1 , ∞ 0 ( H ) is the ideal { T ∈ K ( H ); N X i =0 µ i ( T ) = o (log N ) } , on whic h the Dixmier trace T r ω v anishes . Then it follows that the Dixmier trace induces a twisted hyper trace o n A : Prop osition 4.1. [8 ] L et ( A , H , D ) b e a σ -sp e ctr al triple with D − 1 ∈ L n, ∞ . Then the line ar functional ϕ : A → C define d by ϕ ( a ) = T r ω ( aD − n ) , a ∈ A is a σ n -hyp ertr ac e, i.e. T r ω ( T aD − n ) = T r ω ( σ n ( a ) T D − n ) for any a ∈ A , T ∈ L ( H ) . In p articular ϕ is a twiste d tr ac e on A . Note t ha t for Lipschitz-r e gular twiste d sp e ctr al triples, t he same ho lds whe n D − n is r eplac e d by | D | − n . 4.2 Connes-Chern c haracter Let ( A , H , D ) b e a Lipschitz-regular σ -sp ectral triple suc h that D − 1 ∈ L n, ∞ ( H ) for some n ∈ N . Here L n, ∞ ( H ) = { T ∈ K ( H ); N X i =0 µ i ( T ) = O ( N 1 − 1 n ) } , if n > 1 , 13 L 1 , ∞ ( H ) = { T ∈ K ( H ); N X i =0 µ i ( T ) = O (log N ) } , where for any compa ct o perator T ∈ K ( H ), its singular v alues are written in decreasing order : µ 1 ( T ) ≥ µ 2 ( T ) ≥ · · · ≥ 0 . W e recall from [8] that pas sage to the phase F = D | D | − 1 of s uc h a twisted sp ectral triple giv es a finitely summable F redholm mo dule whic h has a well- defined Connes-Che r n character in cyclic cohomolog y g iv en by Φ F ( a 0 , a 1 , . . . , a n ) = T race( γ F [ F , a 0 ][ F, a 1 ] · · · [ F, a n ]) , a 0 , a 1 , . . . , a n ∈ A , where γ = id in the ungra de d ca se. Since for any a ∈ A [ F, a ] = | D | − 1 ([ D , a ] σ − [ | D | , a ] σ F ) , the commutators [ F , a i ] a r e differentials of the sa me or der as D − 1 and the op erator γ F [ F, a 0 ][ F, a 1 ] · · · [ F, a n ] is a trace clas s op erator. One can see tha t Φ F is a cyclic c o cycle, i.e . Φ F ( a n , a 0 , . . . , a n − 1 ) = ( − 1) n Φ F ( a 0 , a 1 , . . . , a n ) , ∀ a 0 , a 1 , . . . , a n ∈ A , b Φ F = 0 , where b is the Ho c hschild cob oundary o perator : b Φ F ( a 0 , a 1 , . . . , a n +1 ) = n X i =0 ( − 1) i Φ F ( a 0 , . . . , a i a i +1 , . . . , a n +1 ) + ( − 1) n +1 Φ F ( a n +1 a 0 , a 1 , . . . , a n ) , for all a 0 , a 1 , . . . , a n +1 ∈ A . Moreov er, if D − 1 ∈ L n, ∞ ( H ) for a n even n ∈ N , then the Co nnes-Chern character can b e defined without the Lipschit z-re gularit y assumption: Prop osition 4.2. [8 ] L et ( A , H , D ) b e a gr ade d σ -sp e ctr al triple with D − 1 ∈ L n, ∞ ( H ) for some even n ∈ N . Then the multiline ar fun ctional Φ D,σ define d by Φ D,σ ( a 0 , a 1 , . . . , a n ) = T race( γ D − 1 [ D , a 0 ] σ D − 1 [ D , a 1 ] σ · · · D − 1 [ D , a n ] σ ) (4.3 ) for a 0 , . . . , a n ∈ A , is a cyclic c o cycle. 4.3 Index pairing and K -theory In [8], it is shown that in the t wisted case, the index pairing with ordinar y K - theory makes sense and it is given by the pairing of the Connes-Chern character with K -theory . 14 Given a graded σ - spectral triple ( A , H , D ) with D − 1 ∈ L n, ∞ ( H ) for some even n ∈ N , one ca n write an o rthogonal de c o mposition for the Hilb ert space H using the gr ading γ : H = H + ⊕ H − , where H + = { x ∈ H ; γ ( x ) = x } , H − = { x ∈ H ; γ ( x ) = − x } . Now a close look at the Connes-Chern c hara cter Φ D,σ defined b y (4.3) shows the existence of a pa ir of F redholm mo dules ov er A and t wo co cycles as follows. With resp ect to the decomp osition H = H + ⊕ H − , w e can write D =  0 D − D + 0  , a =  a + 0 0 a −  , ∀ a ∈ A , since we have D γ = − γ D , and aγ = γ a , for all a ∈ A . Then define tw o Hilb ert spaces e H + = H + ⊕ H + , e H − = H − ⊕ H − . There a re repres en tations π + and π − of the a lgebra A on L ( e H + ) a nd L ( e H − ) resp ectiv ely given by π ± ( a ) =  a ± 0 0 D − 1 ± σ ( a ) ∓ D ±  . In [8], letting F =  0 I ± I ± 0  , where I + and I − are identit y op erators on H + and H − resp ectiv ely , it is shown that the co mm utators [ F ± , π ± ( a )] are co mpact op erators, hence they obta in a pair of F redholm mo dules over the algebra A . It is also sho wn that for a n y idempo ten t e ∈ A , the bo unded c losure o f op erators D − 1 ± σ ( e ) ∓ D ± denoted b y f ± are idemp oten ts, and f ± e ± : e ± H ± → f ± H ± are F redholm op erators. Since the index dep e nds only on the K -theory class of the idemp otent , they define a pair of index ma ps by Index ± [ e ] = Index( f ± e ± ) , for all idemp oten ts e ∈ M N ( A ). On the other hand, the cyclic co cycle Φ D,σ defined by (4.3) consists of a pair of co cycles Φ ± D,σ given by Φ ± D,σ ( a 0 , . . . , a n ) = T race( D − 1 ± ( D ± a 0 ± − σ ( a 0 ) ∓ D ± ) · · · D − 1 ± ( D ± a n ± − σ ( a n ) ∓ D ± )) , for a 0 , . . . , a n ∈ A . The following prop osition states that the index pairing c an b e expr e ssed as the pairing of these c o cycles with K -theory: 15 Prop osition 4 .3. [8] Given a gr ade d σ -sp e ctr al triple ( A , H , D ) with D − 1 ∈ L n, ∞ ( H ) for some even n ∈ N , and an idemp otent e ∈ M N ( A ) , one has Index ± [ e ] = Φ ± D,σ ( e, . . . , e ) . If e ∗ = σ ( e ) , then Index + [ e ] = − Index − [ e ] . If there exists a stro ngly con tinuous 1-parameter gro up of isometric auto- morphisms { σ t } t ∈ R with an ana lytic extens io n coinc iding with σ at t = − i , then the a bov e index maps co incide. This as sumption is denoted by (1PG). Such an analytic ex tension defined on a dense subalgebr a O of A is a ssured to exist by a theorem of B ost in [2]. Prop osition 4.4 . [8 ] If A satisfies (1PG), then Index + [ e ] = − Index − [ e ] , ∀ e ∈ M N ( A ) . Accordingly , in [8], the rela tion be t w een the cyclic cocycles Φ ± D,σ has b een studied under the (1P G) a ssumption. Theorem 4.5. [8] L et ( A , H , D ) b e a gr ade d σ -sp e ctr al triple as ab ove and assume that A satisfies (1PG). Then [Φ − D,σ ] = − [(Φ + D,σ ) ∗ ] ∈ H P ev ( O ) , wher e (Φ + D,σ ) ∗ ( a 0 , . . . , a n ) := Φ + D,σ ( a ∗ n , . . . , a ∗ 0 ) , ∀ a 0 , . . . , a n ∈ O . W e note that in the pro of of the latter [8], the homotop y inv a r iance of the Connes-Chern c hara c ter of a finitely s ummable F redholm mo dule, established in Lemma 1, in Part I, section 5 o f [3] plays a crucia l r ole. 4.4 Lo cal Ho c hsc hild co cycle In [8], a s a first step to extend the lo cal index for m ula [7 , 13, 14] to twisted sp ec- tral tr iples, using the Dixmier trace, a lo cal Ho ch schild co cycle is constructed for twisted sp ectral triples: Prop osition 4. 6. [8] L et ( A , H , D ) b e a gr ade d σ -sp e ct r al triple such that D − 1 ∈ L n, ∞ ( H ) for some even n ∈ N . Then t he n + 1 -line ar form on A define d by Ψ D,σ ( a 0 , a 1 , . . . , a n ) = T r ω ( γ a 0 [ D , σ − 1 ( a 1 )] σ · · · [ D, σ − n ( a n )] σ | D | − n ) for a 0 , . . . , a n ∈ A , is a Ho chschild c o cycle. In the ungr ade d c ase, for a Lipschitz-r e gular σ -sp e ctr al triple of o dd summa- bility de gr e e, t he Ho chschild c o cycle is given by Ψ D,σ ( a 0 , a 1 , . . . , a n ) = T r ω ( a 0 [ D , σ − 1 ( a 1 )] σ · · · [ D , σ − n ( a n )] σ | D | − n ) , for any a 0 , . . . , a n ∈ A . 16 The ab o ve co cycle identities a re prov ed in [8 ] using Pr opositio n 4 .1 and the fact that for any a, b ∈ A : [ D , ab ] σ = [ D , a ] σ b + σ ( a )[ D, b ] σ . In [8, 1 7], the above lo cal Ho c hschild co cycle is o bta ined in a heuristic ma n- ner as follows. F or an ordinary spe ctral triple ( A , H , D ), consider the lo cal Ho c hschild co cycle Ψ D ( a 0 , . . . , a n ) = T r ω ( γ a 0 [ D , a 1 ] · · · [ D, a n ] D − n ) , for any a 0 , . . . , a n ∈ A . One ca n move D − n to the left and distribute it among the factors to w r ite this co cycle in the form: Ψ D ( a 0 , . . . , a n ) = T r ω ( γ a 0 ( D a 1 D − 1 − a 1 ) · · · ( D n a n D − n − D n − 1 a n D − n +1 ) D − n ) . In or de r to construct a lo cal Ho c hschild co cycle fo r twisted sp e ctral triples, they replace each D k aD − k in the latter by D k σ − k ( a ) D − k and re verse the pro cess of moving the D − n to the le ft whic h leads to the ab ov e lo cal Hochschild co cycle for t wisted spectral triples. W e note that the Connes-Cher n character in tro duced in Pro position 4.2 can b e obtained in a similar manner. 5 Ho c hsc h ild class of the Connes-Chern c harac- ter F or any algebra A , there is an obvious pairing b et ween the s pace of Ho chsc hild n -co c hains C n ( A , A ∗ ), i .e. the space of ( n + 1)-linear functionals on A , and the space of Ho c hschild n -chains A ⊗ ( n +1) , given by h ϕ, a 0 ⊗ a 1 ⊗ · · · ⊗ a n i := ϕ ( a 0 , . . . , a n ) . This pairing sa tis fies h bϕ, c i = h ϕ, b ( c ) i where the Ho chsc hild o p erator s for the co chains and chains ar e given by: bϕ ( a 0 , a 1 , . . . , a n +1 ) = n X i =0 ( − 1) i ϕ ( a 0 , . . . , a i a i +1 , . . . , a n +1 ) + ( − 1) n +1 ϕ ( a n +1 a 0 , a 1 , . . . , a n ) , b ( a 0 ⊗ a 1 ⊗ · · · ⊗ a n +1 ) = n X i =0 ( − 1) i a 0 ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n +1 + ( − 1) n +1 a n +1 a 0 ⊗ a 1 ⊗ · · · ⊗ a n , (5.4) 17 for a 0 , . . . , a n +1 ∈ A . It follows that, if t wo co cycles ϕ 1 and ϕ 2 are cohomo logous with ϕ 1 − ϕ 2 = bψ , then they yield the same v alue on a n y Ho c hschild cycle since we hav e h ϕ 1 , c i − h ϕ 2 , c i = h bψ , c i = h ψ , b ( c ) i = 0 if b ( c ) = 0 . As we s a w in Sec tion 4, to any twisted spectr al triple, in particular to an ordinary sp ectral tr iple , with certa in conditio ns one can a ssocia te the Co nnes- Chern character and a lo cal Ho chsc hild co cycle. Connes’ character for m ula (or Co nnes’ Ho c hschild c hara cter theorem) states that in the case of ordinary sp ectral triples, these tw o co cycles yield the same v alue on any Ho c hschild cycle [5, 12, 14]. In this section the analo gue of Connes’ character form ula for the class of t wisted spectra l triples intro duced in Subsection 3.3 is in vestigated which w e explain next following [1 0]. 5.1 Connes’ c haracter form ula and twis ting b y scaling au- tomorphisms In Subsection 3 .3, we saw that using confor mal s imila rities of a spectral triple ( A , H , D ), one can construct a t wisted sp ectral triple ( cf . [1 7]) . W e recall that the set of conformal similarities of a sp ectral tr iple ( A , H , D ), denoted by Sim( A , H , D ), consists of all unitary op erators U o n H , such that U A U ∗ = A , a nd U D U ∗ = µ ( U ) D , for some µ ( U ) > 0 . Recall that Sim( A , H , D ) is a group and the map µ : Sim( A , H , D ) → (0 , ∞ ) is a character. W e fix a subgroup G ⊂ Sim( A , H , D ) and let A G be the crossed pro duct alg ebra A ⋊ G . The formula σ ( aU ) = µ ( U ) − 1 aU, ∀ a ∈ A , U ∈ G, defines an automorphis m o f A G , and ( A G , H , D ) is a σ -spectra l triple [1 7]. W e will assume tha t the base sp ectral triple ( A , H , D ) is r egular, i.e. the op erators A , [ D , A ] are in the doma in o f all p o wers of the deriv atio n δ ( · ) = [ | D | , · ]. W e also assume that D − 1 ∈ L n, ∞ ( H ) for so me fixed even num b er n ∈ 2 N , a nd that the twisted sp ectral triple ( A G , H , D ) is g raded. F rom the reg ularit y o f the ba se sp ectral tr iple, it easily follows that the t wisted sp ectral triple ( A G , H , D ) is Lipschitz-regular. Therefor e, by passage to the phase, o ne can as sociate the Connes-Chern character to the F redholm mo dule ( A G , H , F = D | D | − 1 ). T o rec a ll, this is a cyclic n -co cycle given by Φ F ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) = T race( γ F [ F , a 0 U 0 ][ F, a 1 U 1 ] · · · [ F, a n U n ]) for all a i ∈ A , U i ∈ G, i = 0 , . . . , n . Here γ deno tes the grading . Also there is a Ho c hschild n -co cycle given by Ψ D,σ ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) = T r ω ( γ a 0 U 0 [ D , σ − 1 ( a 1 U 1 )] σ · · · [ D, σ − n ( a n U n )] σ | D | − n ) 18 for all a i ∈ A , U i ∈ G, i = 0 , . . . , n . W e r ecall fro m [12] how Φ F ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) = T race( γ F [ F , a 0 U 0 ][ F, a 1 U 1 ] · · · [ F, a n U n ]) can b e approximated by the trace o f finite rank o perato r s using a cutoff. Let g : R → R b e a smo oth function which ta kes the v a lue 1 on [0 , 1 2 ], decr eases to 0 on [ 1 2 , 1], is 0 fo r t > 1, g ( − t ) = g ( t ) for t < 0, and R ∞ 0 g ′ ( u )d u = − 1. Define the op erators A t = g ( t | D | ) for t > 0. The op erators A t , t > 0, a r e positive, finite rank, and sa tisfy P 1 / 2 t ≤ A t ≤ P 1 /t , where P N is the sp ectral pro jector of | D | o n the interv al [0 , N ]. Given a Hochschild n -cycle c = k X j =1 a 0 j U 0 j ⊗ a 1 j U 1 j ⊗ · · · ⊗ a nj U nj , it is proved in [12] that Φ F ( c ) = 2 lim t ↓ 0 Ψ t ( c ) , (5.5) where Ψ t ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) := − T ra ce ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ] F [ A t , a n U n ]) . The proo f of this is essentially based on tw o facts ( cf . [1 2]) . The first is that the op erator trace is nor mal, and the s econd is that, since b ( c ) = 0, one has k X j =1 a 0 j U 0 j [ F, a 1 j U 1 j ] · · · [ F, a n − 1 ,j U n − 1 ,j ] a nj U nj = k X j =1 a nj U nj a 0 j U 0 j [ F, a 1 j U 1 j ] · · · [ F, a n − 1 ,j U n − 1 ,j ] . F or conv enience, we drop the index j and the summation in the formula for the Ho c hschild cycle c , and simply write c = a 0 U 0 ⊗ a 1 U 1 ⊗ · · · ⊗ a n U n . Lemma 5.1. [10] The op er ator a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ] F | D | n − 1 is b ounde d. Pr o of. Using the identit y U [ F , a ] = [ F , U aU ∗ ] U , for all a ∈ A, U ∈ G , one can see that a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ] F | D | n − 1 = µ ( U 0 U 1 · · · U n − 1 ) n − 1 a 0 [ F, U 0 a 1 U ∗ 0 ] · · · [ F, U 0 U 1 · · · U n − 2 a n − 1 U ∗ n − 2 · · · U ∗ 1 U ∗ 0 ] F | D | n − 1 U 0 U 1 · · · U n − 1 . The bo undedness of the latter follows from the fac t that the o perato r a ′ 0 [ F, a ′ 1 ] · · · [ F, a ′ n − 1 ] F | D | n − 1 is bo unded for any a ′ 0 , . . . , a ′ n − 1 ∈ A , which is prov ed in [12]. 19 F or co n venience let R = − γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ] F | D | n − 1 ∈ L ( H ) . Then we hav e: Ψ t ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) = T race( R | D | − ( n − 1) [ A t , a n U n ]) . In the sequel we will imp ose the following extra condition on the Hochschild cycle c = a 0 U 0 ⊗ a 1 U 1 ⊗ · · · ⊗ a n U n (note that we have dropp ed the summation): Condition 5. 2. We shal l assume that lim t ↓ 0 T ra ce ( R | D | − ( n − 1) [ A t , a n U n ]) = lim t ↓ 0 T ra ce ( R | D | − ( n − 1) [ A t , a n ] U n ) . F or ex amp le, if µ ( U n ) = 1 t hen this c ondition is satisfie d. The function Ψ ′ t ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) := T race( R | D | − ( n − 1) [ A t , a n ] U n ) is contin uous on the interv al 1 t ≥ e , and ha s a limit as 1 t → ∞ , therefore it is bo unded. Hence the ev aluation o f the s tate ω on the cor respo nding element in the C ∗ -algebra C b ([ e, ∞ )) /C 0 ([ e, ∞ )) will yield this limit which will b e denoted by lim t − 1 → ω Ψ ′ t ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) . T o compute this limit, one c a n use the following prop osition of Co nnes. F o r a detailed discussion, we r efer the r eader to [12]. Prop osition 5.3. L et f : [0 , ∞ ) → [0 , ∞ ) b e a c ont inu ous function, p > 1 , and P k m k ( f ) e pk < ∞ , wher e for e ach k m k ( f ) = sup { f ( u ); k ≤ log u ≤ k + 1 } . Then M p = R ∞ 0 f ( u ) u p − 1 d u is finite and lim t − 1 → ω t p T ra ce ( f ( t | D | ) S ) = M p T r ω ( S | D | − p ) , pr ovide d S ∈ L ( H ) , and D − 1 ∈ L p, ∞ ( H ) . Also we use Le mma 10.29 of [12]. Lemma 5. 4 . If g ( t ) = h ( t ) 2 wher e h ∈ D ( R ) is also a cutoff, and if a ∈ A , then k [ g ( t | D | ) , a ] − 1 2 { g ′ ( t | D | ) , tδ a } k n − = o ( t ) a s t ↓ 0 . W e no te that k · k n − is defined in [1 2]. The argument following the abov e lemma in [1 2] a pplies verbatim to our case b y simply r eplacing R by U n R , a nd it follows that lim t − 1 → ω Ψ ′ t ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) = n T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ] D − 1 δ ( a n ) U n ) = n T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ] δ σ σ − 1 ( a n U n ) D − 1 ) , (5.6) 20 where δ ( x ) = [ | D | , x ] and δ σ ( x ) = [ | D | , x ] σ . The last step is to prov e that the lo cal Ho c hschild co cycle Ψ D,σ is coho mol- ogous to ζ σ n defined by ζ σ n ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) = n T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ] δ σ σ − 1 ( a n U n ) D − 1 ) for any a i ∈ A , and U i ∈ G , i = 0 , . . . , n. This will b e achieved by defining the co c hains ζ σ 1 , . . . , ζ σ n − 1 by defining ζ σ k ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) as n T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a k − 1 U k − 1 ] D − 1 δ σ ( a k U k )[ F, a k +1 U k +1 ] · · · [ F, a n U n ]) for a n y a i ∈ A , U i ∈ G , i = 0 , . . . , n . Then using the following tw o lemmas, one can se e that Ψ D,σ is co ho mologous to ζ σ n , hence they yield the sa me v a lue on any Ho c hschild cycle. Lemma 5.5. [10] The c o chains ζ σ 1 , . . . , ζ σ n ar e mutu al ly c ohomol o gous H o chschild c o cycles. Pr o of. First we show that bζ σ n = 0 . F or any a i ∈ A , U i ∈ G, w e hav e bζ σ n ( a 0 U 0 , . . . , a n +1 U n +1 ) = T r ω ( γ a 0 U 0 a 1 U 1 [ F, a 2 U 2 ] · · · [ F, a n U n ][ | D | , σ − 1 ( a n +1 U n +1 )] σ D − 1 ) − T r ω ( γ a 0 U 0 [ F, a 1 U 1 a 2 U 2 ] · · · [ F, a n U n ][ | D | , σ − 1 ( a n +1 U n +1 )] σ D − 1 ) − · · · + T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ][ | D | , σ − 1 ( a n U n a n +1 U n +1 )] σ D − 1 ) − T r ω ( γ a n +1 U n +1 a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ][ | D | , σ − 1 ( a n U n )] σ D − 1 ) = T r ω ( γ a 0 U 0 a 1 U 1 [ F, a 2 U 2 ] · · · [ F, a n U n ][ | D | , σ − 1 ( a n +1 U n +1 )] σ D − 1 ) − T r ω ( γ a 0 U 0 a 1 U 1 [ F, a 2 U 2 ] · · · [ F, a n U n ][ | D | , σ − 1 ( a n +1 U n +1 )] σ D − 1 ) − T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] a 2 U 2 · · · [ F, a n U n ][ | D | , σ − 1 ( a n +1 U n +1 )] σ D − 1 ) · · · +T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ] a n U n [ | D | , σ − 1 ( a n +1 U n +1 )] σ D − 1 ) +T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ][ | D | , σ − 1 ( a n U n )] σ σ − 1 ( a n +1 U n +1 ) D − 1 ) − T r ω ( γ a n +1 U n +1 a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n − 1 U n − 1 ][ | D | , σ − 1 ( a n U n )] σ D − 1 ) . The latter v anishes b ecause o f s uccessiv e cancellations , where the last tw o ter ms cancel each other since using the identit y D − 1 a n +1 U n +1 − σ − 1 ( a n +1 U n +1 ) D − 1 = − D − 1 [ D , σ − 1 ( a n +1 U n +1 )] σ D − 1 , one obtains a tra ce class oper ator which v anishes under the Dixmier trace. Now we in tro duce cochains η σ 1 , . . . , η σ n − 1 such that ζ σ k − ζ σ k +1 = bη σ k for a ll k = 1 , . . . , n − 1, and this will finish the pro of. 21 Using the identities D − 1 δ σ ( a k U k ) − δ σ σ − 1 ( a k U k ) D − 1 = − D − 1 δ ([ D, a k ]) D − 1 U k , | D | − 1 [ F, a j U j ] − [ F , σ − 1 ( a j U j )] | D | − 1 = − | D | − 1 [ F, δ σ σ − 1 ( a j U j )] | D | − 1 , F [ F, a j U j ] = − [ F, a j U j ] F, we can mov e D − 1 in the form ula for ζ σ k to the right under the Dixmier trace and obtain the following expressio n. ζ σ k ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) = n T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a k − 1 U k − 1 ] D − 1 δ σ ( a k U k )[ F, a k +1 U k +1 ] · · · [ F, a n U n ]) = ( − 1) n − k n T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a k − 1 U k − 1 ] δ σ ( σ − 1 ( a k U k ))[ F, σ − 1 ( a k +1 U k +1 )] · · · [ F, σ − 1 ( a n U n )] D − 1 ) . Therefore ( ζ σ k − ζ σ k +1 )( a 0 U 0 , a 1 U 1 , . . . , a n U n ) is eq ua l to ( − 1) n − k n T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a k − 1 U k − 1 ] R σ k [ F, σ − 1 ( a k +2 U k +2 )] · · · [ F , σ − 1 ( a n U n )] D − 1 ) , where R σ k = δ σ σ − 1 ( a k U k )[ F, σ − 1 ( a k +1 U k +1 )] + [ F , a k U k ] δ σ σ − 1 ( a k +1 U k +1 ) . Now we define η σ k ( a 0 U 0 , . . . , a n − 1 U n − 1 ) = ( − 1) k n T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a k − 1 U k − 1 ] [ F, δ σ σ − 1 ( a k U k )][ F, σ − 1 ( a k +1 U k +1 )] · · · [ F, σ − 1 ( a n − 1 U n − 1 )] D − 1 ) . Finally , using the identit y [ F, δ σ σ − 1 ( a k U k a k +1 U k +1 )] = R σ k + [ F , δ σ σ − 1 ( a k U k )] σ − 1 ( a k +1 U k +1 ) + a k U k [ F, δ σ σ − 1 ( a k +1 U k +1 )] , one can see that ζ σ k − ζ σ k +1 = bη σ k . Lemma 5. 6. [1 0] The c o chain Ψ D,σ − 1 n ( ζ σ 1 + · · · + ζ σ n ) is a Ho chschild c ob ou n d- ary. Pr o of. Let a 0 U 0 , a 1 U 1 , . . . , a n U n ∈ A G . Since | D | − 1 [ D , a j U j ] σ − [ D , σ − 1 ( a j U j )] σ | D | − 1 = | D | − 1 [ D , a j ] U j − µ ( U j )[ D , a j ] U j | D | − 1 = | D | − 1 [ D , a j ] U j − [ D , a j ] | D | − 1 U j = −| D | − 1 δ ([ D, a ]) | D | − 1 U, 22 in the expr ession for Ψ D,σ we can replace ea c h [ D , σ − 1 ( a j U j )] σ | D | − 1 by | D | − 1 [ D , a j U j ] σ . Therefore Ψ D,σ ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) = T r ω ( γ a 0 U 0 [ D , σ − 1 ( a 1 U 1 )] σ · · · [ D, σ − n ( a n U n )] σ | D | − n ) = T r ω ( γ a 0 U 0 [ D , σ − 1 ( a 1 U 1 )] σ | D | − 1 · · · [ D, σ − 1 ( a n U n )] σ | D | − 1 ) . (5.7) Also we hav e [ D , σ − 1 ( a j U j )] σ | D | − 1 = [ F , a j U j ] + δ σ σ − 1 ( a j U j ) D − 1 + [ F , δ σ σ − 1 ( a j U j )] | D | − 1 . Since [ F, δ σ σ − 1 ( a j U j )] = µ ( U j )[ F, δ ( a j ) U j ] = µ ( U j )[ F, δ ( a j )] U j = µ ( U j ) | D | − 1 ( δ ([ D, a j ]) − δ 2 ( a j ) F ) U j , the terms containing [ F, δ σ σ − 1 ( a j U j )] | D | − 1 yield trace clas s op erator s which v anish under the Dixmier trac e . So w e can repla ce each [ D , σ − 1 ( a j U j )] σ | D | − 1 by [ F, a j U j ] + δ σ σ − 1 ( a j U j ) D − 1 . Therefore (5.7) is the sum of 2 n terms. The term T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n U n ]) is zero b ecause one can write a 0 U 0 = F [ F, a 0 U 0 ] + F a 0 U 0 F , and γ F [ F, a 0 U 0 ][ F, a 1 U 1 ] · · · [ F, a n U n ] is trace class . Therefo r e the term is equal to T r ω ( γ F a 0 U 0 F [ F, a 1 U 1 ] · · · [ F, a n U n ]) = ( − 1) n T r ω ( γ F a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n U n ] F ) = − T r ω ( F γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n U n ] F ) = − T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a n U n ]) . Hence it has to b e zero . The terms having e xactly one facto r of the form δ σ σ − 1 ( a j U j ) D − 1 add up to 1 n ( ζ σ 1 + · · · + ζ σ n ), and to finish the pro of we show that the terms with mor e than one fa c tors of the fo rm δ σ σ − 1 ( a j U j ) D − 1 are Ho chsc hild cob oundaries. F or example let us consider the case when tw o consecutive factor s of the a bov e form 23 yield the term T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a j − 1 U j − 1 ] δ σ σ − 1 ( a j U j ) D − 1 δ σ σ − 1 ( a j +1 U j +1 ) D − 1 [ F, a j +2 U j +2 ] · · · [ F, a n U n ]) = T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a j − 1 U j − 1 ] δ σ σ − 1 ( a j U j ) δ σ σ − 2 ( a j +1 U j +1 ) D − 2 [ F, a j +2 U j +2 ] · · · [ F, a n U n ]) = T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a j − 1 U j − 1 ] δ σ σ − 1 ( a j U j ) δ σ σ − 2 ( a j +1 U j +1 )[ F, σ − 2 ( a j +2 U j +2 )] · · · [ F, σ − 2 ( a n U n )] D − 2 ) . (5.8 ) Now using the identit y δ 2 σ σ − 2 ( a j a j +1 ) = δ 2 σ σ − 2 ( a j ) σ − 2 ( a j +1 )+2 δ σ σ − 1 ( a j ) δ σ σ − 2 ( a j +1 )+ a j δ 2 σ σ − 2 ( a j +1 ) , one can see that (5 .8) is equal to bϕ σ j ( a 0 U 0 , a 1 U 1 , . . . , a n U n ) where ϕ σ j ( a 0 U 0 , a 1 U 1 , . . . , a n − 1 U n − 1 ) = ( − 1) j 2 T r ω ( γ a 0 U 0 [ F, a 1 U 1 ] · · · [ F, a j − 1 U j − 1 ] δ 2 σ σ − 2 ( a j U j )[ F, σ − 2 ( a j +1 U j +1 )] · · · [ F, σ − 2 ( a n − 1 U n − 1 )] D − 2 ) . Hence, co nsidering (5.5), Condition 5.2, and (5 .6) w e ha ve prov ed the fol- lowing: Theorem 5.7. [10] The cyclic c o cycle Φ F and the lo c al Ho chschild c o cycle 2Ψ D,σ yield the same value on any Ho chschild n -cycle c = k X j =1 a 0 j U 0 j ⊗ a 1 j U 1 j ⊗ · · · ⊗ a nj U nj that satisfies Co ndition 5.2. 6 A lo cal index form ula for t wisted sp ectral triple s In [17], an Ansatz for a lo cal index formula for twisted sp ectral tr iples is given and its v alidity for t wisted sp ectral tr iples obtained fro m scaling automor phisms of spectral triple, has b een v erified. This form ula is g iv en in terms of residue functionals a nd twisted commutators. In this section we sketc h very briefly some of the ideas in [17]. 6.1 Mosco vici’s Ansatz In Subsection 4.4, we ex plained the heuristic manner for obtaining the lo cal Ho c hschild co cycle for twisted sp ectral triples from the one for ordina ry sp ectral 24 triples. In a rather similar manner one can obtain Moscovici’s Ansatz for a lo c a l index fo rm ula for twisted sp ectral triples from the or iginal lo cal index formula of Connes and Moscovici discuss ed in Sectio n 2. The Ansatz sugg ests that for a σ -sp ectral triple ( A , H , D ), the twisted version of the loc a l character should be given by ϕ σ n ( a 0 , . . . , a n ) := X k c n,k Z − a 0 [ D , σ − 2 k 1 − 1 ( a 1 )] ( k 1 ) σ · · · [ D, σ − 2( k 1 + ··· + k n ) − n ( a n )] ( k n ) σ | D | − n − 2 | k | , where the iter ate d twiste d c ommutators [ D , a ] ( k ) σ are defined in [17]. Here ag ain Z − P = Res z =0 T ra ce ( P | D | − 2 z ) . Therefore a n ana lo gue of the s imple dimension sp ectrum hypo thesis is a ssumed. Namely , one needs to ass ume that ther e e xists a discrete subset of the complex plane such that for a ny op erator P in an algebr a of twisted pseudo differential op- erators asso ciated to the twisted spectr al triple ( A , H , D , σ ), P | D | − 2 z is a trace class oper ator provided Re( z ) is larg e enough, and ζ P ( z ) := T r ace( P | D | − 2 z ) has a meromorphic extension to the plane with a t mo st simple p oles in this discrete set. W e note that, as it is emphasized in [17], there is no canonical w ay of c o nstructing an algebra o f twisted pseudo differen tial op erators for a general t wisted sp ectral triple. Also, in the t wisted case, for formulating analogue of the reg ularit y condition, one needs to p ostulate that the automorphism σ has an extension to a larg er algebr a which contains the t wiste d differ ent ia l forms . Another necessa r y condition for the v alidity of the Ansatz is a σ -inv ar iance of the res idue functionals which is related to the Selb erg principle for reductive Lie groups [17]. 6.2 A lo cal index form ula for sp ectral t riples twisted by scaling automorphisms Let ( A G , H , D ) denote a t wisted sp ectral tr iple obtained by scaling automor- phisms of an ordinary finitely s ummable reg ula r sp ectral triple, in tro duced in Subsection 3.3 and let Ψ( A G , H , D ) := Ψ ( A , H , D ) ⋊ G denote the asso ciated algebra of twisted pseudo differen tial opera tors. The automorphism σ extends to an auto mo rphism of Ψ( A G , H , D ) and it is req uired that for an y P in this algebra [17]: Z − P = Z − σ ( P ) . (6.9) Definition 6.1. [17] The twiste d JLO br acket of or der q as a q + 1 - line ar form on Ψ( A G , H , D ) is define d by h a 0 U ∗ 0 , . . . , a q U ∗ q i D = Z ∆ q T ra ce  γ a 0 U ∗ 0 e − s 0 µ ( U 0 ) 2 D 2 a 1 U ∗ 1 e − s 1 µ ( U 0 U 1 ) 2 D 2 · · · · · · a q U ∗ q e − s q µ ( U 0 ··· U q ) 2 D 2  25 for al l a 0 , . . . , a q ∈ A and U 0 , . . . , U q ∈ G , wher e ∆ q := { s = ( s 0 , . . . , s q ) ∈ R q +1 ; s j ≥ 0 , s 0 + · · · + s q = 1 } . In [17], it is assumed that a 0 , . . . , a q are polyno mials in D and elements of A and [ D , A ] which are homogeneous in λ as D is r eplaced by λD . The expre s sion obtained fro m re placing ea c h D occurr ing in a 0 , . . . , a q by ǫ 1 / 2 D is denoted by h a 0 U ∗ 0 , . . . , a q U ∗ q i D ( ǫ ). Equiv a len tly , one has h a 0 U ∗ 0 , . . . , a q U ∗ q i D ( ǫ ) = ǫ m 2 h a 0 U ∗ 0 , . . . , a q U ∗ q i ǫ 1 / 2 D , where m is the total degree of λ in a 0 · · · a q after replacing e a c h D b y λD . Prop osition 6.2. [17] If a 0 ∈ A and a 1 , . . . , a q ∈ [ D , A ] , then ther e is an asymptotic exp ans ion h a 0 U ∗ 0 , . . . , a q U ∗ q i D ( ǫ ) ∼ X j ∈ J ( c j + c ′ j log ǫ ) ǫ q 2 − ρ j + O (1) as ǫ ց 0 wher e ρ 0 , . . . , ρ m ar e p oints in the half plane Re ( z ) ≥ q 2 . The twisted version of the JLO coc ycles is defined b y J q ( D )( A 0 , . . . , A q ) = h A 0 , [ D , σ − 1 ( A 1 )] σ , . . . , [ D , σ − q ( A q )] σ i D for A 0 , . . . , A q ∈ A G . Since in the twisted case this doe s not define a co cycle, in [17], by co nsidering J q ( ǫ 1 / 2 D )( A 0 , . . . , A q ) = ǫ q 2 h A 0 , [ D , σ − 1 ( A 1 )] σ , . . . , [ D , σ − q ( A q )] σ i ǫ 1 / 2 D for ǫ > 0 , a nd passing to the constant term using the ab o ve prop osition, ϕ σ q is defined by ϕ σ q ( A 0 , . . . , A q ) := h A 0 , [ D , σ − 1 ( A 1 )] σ , . . . , [ D, σ − q ( A q )] σ i D | 0 . It follows from the following tw o results [17] that for an y twisted spectra l triple obta ined b y conformal p erturbation of an ordinary finitely summable reg- ular sp ectral triple and s atisfying a Selb erg t yp e in v a riance condition (6.9), o ne can asso ciate a lo c al cyclic co cycle for its Co nnes-Chern ch ar a cter. Theorem 6. 3. [17] The c o chain { ϕ σ q } s at isfies the c o cycle identity in the (b, B)-bic omplex: bϕ σ q − 1 ( a 0 U ∗ 0 , . . . , a q U ∗ q ) + B ϕ σ q +1 ( a 0 U ∗ 0 , . . . , a q U ∗ q ) = 0 , for al l a 0 , . . . , a q ∈ A and U 0 , . . . , U q ∈ G . Theorem 6.4 . [17] The c o cycle { ϕ σ q } is c ohomolo gous to the Connes-Chern char acter asso ciate d to the twiste d s p e ctra l triple ( A G , H , D ) in the p erio dic cyclic c ohomolo gy H P ∗ ( A G ) . 26 References [1] M. Adler, On a t r ac e functional for formal pseudo differ ential op er ators and the symple ctic struct ur e of the Kortewe g-de V ries typ e e qu atio ns. Inv ent. 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