A twisted local index formula for curved noncommutative two tori
AA TWISTED LOCAL INDEX FORMULA FOR CURVEDNONCOMMUTATIVE TWO TORI
FARZAD FATHIZADEH, FRANZ LUEF, JIM TAO Abstract.
We consider the Dirac operator of a general metric in the canonicalconformal class on the noncommutative two torus, twisted by an idempotent(representing the K -theory class of a general noncommutative vector bundle),and derive a local formula for the Fredholm index of the twisted Dirac opera-tor. Our approach is based on the McKean-Singer index formula, and explicitheat expansion calculations by making use of Connes’ pseudodifferential cal-culus. As a technical tool, a new rearrangement lemma is proved to handlechallenges posed by the noncommutativity of the algebra and the presence ofan idempotent in the calculations in addition to a conformal factor. Contents
1. Introduction 22. Preliminaries 32.1. Noncommutative two torus 32.2. Heat kernel expansion 33. Noncommutative geometric spaces and index theory 53.1. Spectral triples 53.2. Twisted spectral triples 63.3. Noncommutative conformal geometry 73.4. Dirac operator twisted by a module (vector bundle) 93.5. Index theorems 124. Calculation of a noncommutative local formula for the index 134.1. Computation of τ ( a ( L − )) 144.2. Computation of τ ( a ( L + )) 184.3. Reduction to the flat metric and the Connes-Chern number 20Appendix A. Proof of Lemma 4.1 and Corollary 4.2 20Acknowledgments 25References 25 Department of Mathematics, Computational Foundry, Swansea University Bay Campus,SA1 8EN, Swansea, United Kingdom; Max Planck Institute for Biological Cybernetics, 72076T¨ubingen, Germany, E-mail: [email protected] Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491Trondheim, Norway, E-mail: [email protected] Department of Mathematics, California Institute of Technology (Caltech), MC 253-37,Pasadena, CA 91125, USA, E-mail: [email protected] a r X i v : . [ m a t h . QA ] A p r . Introduction
The celebrated Atiyah-Singer index theorem provides a local formula that cal-culates the analytical (Fredholm) index of the Dirac operator on a compact spinmanifold, with coefficients in a vector bundle, in terms of topological invariants,namely characteristic classes [AS63, AS68b, AS68a, AS68c]. The theorem has deepapplications in mathematics and theoretical physics. For instance, it implies impor-tant theorems such as the Gauss-Bonnet theorem and the Riemann-Roch theorem,and it is used in physics to count linearly the number of solutions of fundamentalpartial differential equations.In noncommutative differential geometry [Con85a, Con94], an analog of the Diracoperator is used to encode the metric information and to use ideas from spectralgeometry and Riemannian geometry in studying a noncommutative algebra, viewedas the algebra of functions on a space with noncommuting coordinates. That is,the data ( C ∞ ( M ) , L ( S ) , D ) of smooth functions on a spin manifold, the Hilbertspace of L -spinors, and the Dirac operator is extended to the notion of a spectraltriple ( A, H , D ) where A is a noncommutative algebra acting on a Hilbert sapce H , and D plays the role of the Dirac operator while acting in H . The idempotentsand thereby K -theory of the algebra is also used to consider the analog of vectorbundles on an ordinary manifold.The identity of the analytical index and the topological index defined by theConnes-Chern character was established in the noncommutative setting in [Con85a].Moreover, a local formula for the Connes-Chern character, which is suitable forexplicit calculations, was derived in [CM95]. However, as we shall elaborate furtherin § twisted spectral triple :they have shown that a twisted version of spectral triples can incorporate type IIIcases and examples that arise in noncommutative conformal geometry, and thatthe index pairing and the coincidence of the analytic and the topological indexcontinues to hold in the twisted case. However, a general local formula for theindex in the twisted case has not yet been found, except for the special case oftwists afforded by scaling automorphism in [Mos10]. In fact, the difficulty of theproblem of finding a general local index formula for twisted spectral triples raisesthe need for treating more examples.The main result of the present article is a local formula for the index of theDirac operator D + e,σ of a twisted spectral triple on the opposite algebra of the non-commutative two torus T θ , which is twisted by a general noncommutative vectorbundle represented by an idempotent e . The Dirac operator that we consider isassociated with a general metric in the canonical conformal class on T θ . Our ap-proach is based on using the McKean-Singer index formula [MS67] and performingexplicit heat kernel calculations by employing Connes’ pseudodifferential calculusdeveloped in [Con80a] for C ∗ -dynamical systems. It should be noted that, followingthe Gauss-Bonnet theorem proved in [CT11] for the canonical conformal class on T θ and its extension in [FK12] to general conformal classes, study of local geomet-ric invariants of curved metrics on noncommutative tori has received remarkableattention in recent years [CM14, FK13a, LM16, FK15, DS15, Liu18, CF16]. For anoverview of these developments one can refer to [FK19, LM18]. his article is organized as follows. In § § σ -connections and derivethe symbols of the operators necessary for the index calculation. In § Preliminaries
In this section we provide some background material about the noncommutativetwo torus T θ , and explain how one can use a noncommutative pseudodifferentialcalculus to derive the heat kernel expansion for an elliptic operator on T θ .2.1. Noncommutative two torus.
For a fixed irrational number θ , the noncom-mutative two torus T θ is a noncommutative manifold whose algebra of continuous functions C ( T θ ) is the universal (unital) C ∗ -algebra generated by two unitary ele-ments U and V that satisfy the commutation relation V U = e πiθ U V.
The ordinary two torus T = ( R / π Z ) acts on C ( T θ ) by(1) α s ( U m V n ) = e is · ( m,n ) U m V n , s ∈ T , m, n ∈ Z . Since this action, in principle, comes from translation by s ∈ T written in theFourier mode, its infinitesimal generators δ , δ : C ∞ ( T θ ) → C ∞ ( T θ ) are the deriva-tions that are analogs of partial differentiations on the ordinary two torus and aregiven by the defining relations δ ( U ) = U, δ ( V ) = 0 , δ ( U ) = 0 , δ ( V ) = V. The space C ∞ ( T θ ) of the smooth elements consists of all elements in C ( T θ ) thatare smooth with respect to the action α given by (1), which turns out to be a densesubalgebra of C ( T θ ) that can alternatively be described as the space of all elementsof the form (cid:80) m,n ∈ Z a m,n U m V n with rapidly decaying complex coefficients a m,n .The analog of integration is provided by the linear functional τ : C ( T θ ) → C whichis the bounded extension of the linear functional that sends any smooth elementwith the noncommutative Fourier expansion (cid:80) m,n a m,n U m V n to its constant term a , . The functional τ turns out to be a positive trace on C ( T θ ) and we shall viewit as the volume form of a flat canonical metric on T θ . In § T θ by means of a positive invertible element e − h , where h is a self-adjointelement in C ∞ ( T θ ).2.2. Heat kernel expansion.
We will explain in § C ( T θ ). The main tool that will be used is the pseudodifferential calculus developedin [Con80a] for C ∗ -dynamical systems (see also [Tao18, HLP18a]). This calculus, n the case of T θ , associates to a pseudodifferential symbol ρ : R → C ∞ ( T θ ) apseudodifferential operator P ρ : C ∞ ( T θ ) → C ∞ ( T θ ) by the formula P ρ ( a ) = 1(2 π ) (cid:90) R (cid:90) R e − is · ξ ρ ( ξ ) α s ( a ) ds dξ, a ∈ C ∞ ( T θ ) . For example, any differential operator given by a finite sum of the form (cid:80) a j ,j δ j δ j with a j ,j ∈ C ∞ ( T θ ) is associated with the polynomial (cid:80) a j ,j ξ j ξ j . In generala smooth map ρ : R → C ∞ ( T θ ) is a pseudodifferential symbol of order m ∈ Z iffor any non-negative integers i , i , j , j there is a constant C such that || ∂ j ∂ j δ i δ i ρ ( ξ ) || ≤ C (1 + | ξ | ) m − j − j , where ∂ and ∂ are respectively the partial differentiations with respect to thecoordinates ξ and ξ of ξ ∈ R . We denote the space of symbols of order m by S m . A symbol ρ of order m is elliptic if ρ ( ξ ) is invertible for larger enough ξ andthere is a constant c such that || ρ ( ξ ) − || ≤ c (1 + | ξ | ) − m . In the case of differential operators, one can see that the symbol is elliptic if itsleading part is invertible away from the origin.An important feature of an elliptic operator is that it admits a parametrix ,namely an inverse in the algebra of pseudodifferential operators modulo infinitelysmoothing operators. The latter are the operators whose symbols belong to theintersection of all symbols S −∞ = ∩ m ∈ Z S m . This illuminates the importance ofpseudodifferential calculus for solving elliptic differential equations.For our purposes in this article, it is crucial to illustrate that given a positiveelliptic differential operator (cid:52) of order 2 on C ∞ ( T θ ), one can use the pseudodif-ferential calculus to derive a small time asymptotic expansion for the trace of theheat kernel exp( − t (cid:52) ). Note that the pseudodifferential symbol of (cid:52) is of the form σ (cid:52) ( ξ ) = p ( ξ ) + p ( ξ ) + p ( ξ ) , where each p k is homogeneous of order k in ξ . Since the eigenvalues of (cid:52) are realand non-negative, using the Cauchy integral formula and a clockwise contour γ thatgoes around the non-negative real line, one can write(2) exp( − t (cid:52) ) = 12 πi (cid:90) γ e − tλ ( (cid:52) − λ ) − dλ. Since (cid:52) − λ is an elliptic operator, its parametrix R λ will approximate ( (cid:52) − λ ) − appearing above in the integrand. Since (cid:52) − λ is of order 2, one can write for thesymbol of R λ : σ R λ ( ξ, λ, (cid:52) ) = b ( ξ, λ, (cid:52) ) + b ( ξ, λ, (cid:52) ) + b ( ξ, λ, (cid:52) ) + · · · , where each r j is of order − − j .Then, by considering the ellipticity of (cid:52) − λ and the following composition rule,which gives an asymptotic expansion for the symbol of the composition of twopseudodifferential operators, σ P P ∼ (cid:88) i ,i ∈ Z ≥ i ! i ! ∂ i ∂ i σ P δ i δ i σ P , one can find a recursive formula for the r j . That is, it turns out that(3) b ( ξ, λ, (cid:52) ) = ( p ( ξ ) − λ ) − , nd for any n ∈ Z ≥ ,(4) b n ( ξ, λ, (cid:52) ) = − (cid:88) i + i + j +2 − k = n ≤ j Spectral triples. Geometric spaces are defined in terms of spectral data innoncommutative geometry [Con94]. A noncommutative geometric space is a spec-tral triple ( A, H , D ), where A is an involutive algebra represented by bounded op-erators on a Hilbert space H , and D is the analog of the Dirac operator. That is, D is an unbounded self-adjoint operator on its domain which is dense in the Hilbertspace H , and the spectrum of D has similar properties to the spectrum of the Diracoperator on a compact Riemannian spin c manifold, or even more generally, that ofan elliptic self-adjoint differential operator of order 1 on a compact manifold. The spectral dimension of such a triple is the smallest positive real number d such that | D | − d is in the domain of the Dixmier trace Tr ω [Dix66] (or one can say that for any (cid:15) > 0, the operator | D | − d − (cid:15) is a trace-class operator which means that its eigenval-ues are summable). Moreover, an important assumption is that the commutatorof D and the action of any a ∈ A on H , [ D, a ] := Da − aD , extends to a boundedoperator on H . In this paradigm, the algebra A is allowed to be noncommutative,viewed as the algebra of functions on a space with noncommuting coordinates, andthe metric information is encoded in the operator D .The main classical example of this setup is the triple (cid:0) C ∞ ( M ) , L ( S ) , D g (cid:1) , where C ∞ ( M ) is the algebra of smooth complex-valued functions on a compact Riemann-ian manifold with a spin c structure S , L ( S ) is the Hilbert space of the L -spinors,and D g is the Dirac operator associated with the metric, which acts on its do-main in L ( S ) and squares to a Laplace-type operator, see for example [BGV04]for details about the Dirac operator. Since S is a vector bundle over M , the al-gebra C ∞ ( M ) acts naturally by bounded operators on the Hilbert space L ( S ),namely: ( f · s )( x ) := f ( x ) s ( x ) , f ∈ C ∞ ( M ) , s ∈ L ( S ) , x ∈ M . More impor-tantly, the latter action interacts in a bounded manner with the Dirac operator D g in the sense that the commutator of D g with the action of each f ∈ C ∞ ( M ) s a bounded operator as [ D g , f ] = D g f − f D g = c ( df ), where c ( df ) denotes theClifford multiplication on S by the de Rham differential of f . By construction, theDirac operator D g depends heavily on the Riemannian metric g on M . It is knownfrom classical facts in spectral geometry that the spectral dimension of the spectraltriple (cid:0) C ∞ ( M ) , L ( S ) , D g (cid:1) is equal to the dimension of the manifold M , and thatthe important local curvature related information can be detected in the small timeasymptotic expansions of the form(7)Tr (cid:0) f exp( − tD g ) (cid:1) ∼ t → + t − dim M/ ∞ (cid:88) j =0 t j (cid:90) M f ( x ) a j ( x ) dx, f ∈ C ∞ ( M ) , where the densities a j ( x ) dx are uniquely determined by the Riemann curvaturetensor and its contractions and covariant derivatives, cf. [CM08a].The importance of the Dirac operator, for using the tools of Riemannian geome-try in studying noncommutative algebras, is fully illustrated in [Con13] by showingthat the Dirac operator D g contains the full metric information. That is, it isshown that any spectral triple ( A, H , D ) whose algebra A is commutative, and sat-isfies suitable conditions, is equivalent to the spectral triple (cid:0) C ∞ ( M ) , L ( S ) , D g (cid:1) of a Riemannian spin c manifold. In fact, following the Gauss-Bonnet theoremfor the noncommutative two torus proved in [CT11] and its extension in [FK12],the calculation and conceptual understanding of the local curvature terms in non-commutative geometry has attained remarkable attention in recent years [CM14,FK13a, BM12, FK15, Fat15, CF16, LM16, DS15, Liu17, Liu18]. In these studies,the noncommutative local geometric invariants are detected in the analogs of thesmall time asymptotic expansion (7) written for spectral triples.3.2. Twisted spectral triples. It turns out that the notion of a spectral triple( A, H , D ) is suitable for studying algebras that possess a non-trivial trace, and a twisted notion of spectral triples is proposed in [CM08b] to incorporate algebras thatdo not have this property. The reason is that if ( A, H , D ) is a spectral triple withspectral dimension d , then using the Dixmier trace Tr ω , one can define the linearfunction φ : A → C by φ ( a ) = Tr ω ( a | D | − d ), which turns out to be a trace, under theminimal regularity assumption that the commutators [ | D | , a ] are also bounded forany a ∈ A . The main reason for the trace property φ ( ab ) = φ ( ba ) , a, b ∈ A, is thatfor any a ∈ A , the commutator [ | D | − d , a ] = | D | − d a − a | D | − d belongs to the ideal ofcompact operators on which the Dixmier trace Tr ω vanishes. Therefore, in order toincorporate algebras coming from type III examples in the Murray-von Neumannclassification of algebras, which do not possess a non-trivial trace functional, thenotion of a twisted spectral triple was introduced in [CM08b]. The difference is thatthey change the definition of a spectral triple ( A, H , D ) by introducing a twist byan algebra automorphism σ : A → A , and by requiring that instead of the ordinarycommutators, the twisted commutators of the form [ D, a ] σ := Da − σ ( a ) D , a ∈ A ,extend to bounded operators on the Hilbert space H . We note that it is naturalto assume the mild regularity condition that the twisted commutators with | D | arealso bounded operators and to consider a grading: a bounded selfadjoint operator γ on H that squares to identity, commutes with the action of A and anti-commuteswith D . The boundedness of the twisted commutators is then used to observe thatfor any a ∈ A , the operator | D | − d a − σ − d ( a ) | D | − d is in the kernel of the Dixmier race (which is an ideal), hence, the linear functional a (cid:55)→ Tr ω ( a | D | − d ), a ∈ A is atwisted trace. That is, Tr ω ( a b | D | − d ) = Tr ω ( b σ − d ( a ) | D | − d ), for any a, b ∈ A .It is emphasized in [CM08b] on the important issue about twisted spectral triplesthat their Connes-Chern character lands in the ordinary cyclic cohomology whosepairing with the K -theory of the algebra can be realized as the index of a Fredholmoperator. That is, let ( A, H , D, σ ∈ Aut( A )) be a twisted spectral triple of spectraldimension d . Then by passing to the phase F = D/ | D | one arrives at an ordinary Fredholm module ( A, H , F ) with the Connes-Chern character [Con85b],(8) Φ( a , a , . . . , a d ) = Tr ( γF [ F, a ][ F, a ] · · · [ F, a d ]) , a , a , . . . a d ∈ A. This multilinear functional is a cyclic cocycle , it pairs with the K -theory of A asthe Fredholm index of an operator, and the result of the pairing depends only onthe cyclic cohomology class of Φ and the K -theory class of any chosen idempotent[Con85b, CM08b]. However, for the purpose of explicit calculations, one needs tohave a local formula that is cohomologous to Φ in cyclic cohomology or equivalentlyin the ( b, B )-bicomplex. For ordinary spectral triples (when the automorphism σ is the identity), the local formula of Connes and Moscovici provides the desiredformula in the ( b, B )-bicomplex [CM95], see also [Con04]. As a first step towardsa local formula for twisted spectral triples, in [CM08b] they have shown that thereis a Hochschild cocycle associated with twisted spectral triples, which is defined by(9)Ψ( a , a , . . . , a d ) = Tr ω (cid:0) γa [ D, σ − ( a )] σ [ D, σ − ( a )] σ · · · [ D, σ − d ( a d )] σ | D | − d (cid:1) , for a , a , . . . a d ∈ A .Explicit calculations are often possible with this multi-linear map Ψ, as for exam-ple one can use the trace theorem of [Con88] and its extension to noncommutativetori [FK13b] to write it in terms of local formulas. However, the issue is that Ψ isonly a Hochschild coccyle and in general it is not cyclic, therefore it does not pairwith the K -theory (see [GBVF01, FK11] for the relation between the Hochschildcocycle and Connes-Chern character in Hochschild cohomology). Finding a twistedversion of the local index formula of [CM95] has so far proved to be a challengingproblem, and it has only been done for the special examples of scaling automor-phisms in [Mos10]. For the treatment of the problem in twisted cyclic cohomologyone can refer to [Kaa11, KS12, CNNR11, RS14, Mat15]. We elaborate further onindex theoretic aspects of the present paper and its relation with classical resultsand results in noncommutative geometry in § Noncommutative conformal geometry. As a motivating example it isshown in [CM08b] that the twisted notion of a spectral triple arises naturally innoncommutative conformal geometry. They consider the fact that on a Riemannianspin manifold, the Dirac operator of the conformal perturbation g (cid:48) = e − h g ofa metric g is unitarily equivalent to e h De h , where D is the Dirac operator of g . Thus, starting from a spectral triple ( A, H , D ) they use a selfadjoint element h ∈ A to encode the conformal perturbation of the metric in the operator D (cid:48) = e h De h . However, when A is noncommutative, ( A, H , D (cid:48) ) is not a spectral triple anymore, and one needs the twist σ ( a ) = e h ae − h , a ∈ A , to have bounded twistedcommutators [ D (cid:48) , a ] σ . In fact, twisted spectral triples of this nature can be constructed in a moreintrinsic manner as in [CT11] for the noncommutative two torus with a conformally at metric, and as in the extension of this construction in [FG16] to ergodic C ∗ -dynamical systems, see also [PW16b]. Because of its intimate relevance to thepresent work, we review here the construction in [CT11].Viewing the canonical normalized trace τ : C ( T θ ) → C on the noncommutativetwo torus as the volume form of the flat metric, an element h = h ∗ ∈ C ∞ ( T θ ) isfixed and the positive invertible element e − h ∈ C ∞ ( T θ ) is used as a conformal factorto perturb the flat metric conformally. The volume form of the curved metric isgiven by the linear functional ϕ : C ( T θ ) → C defined by ϕ ( a ) = τ ( ae − h ) , a ∈ C ( T θ ).This linear functional turns out to be a KMS state with a 1-parameter group ofautomorphisms. The Dirac operator of the curved metric is constructed by usingthe following analogs of the Dolbeault operators: ∂ = δ + iδ , ¯ ∂ = δ − iδ . For conceptual details of the notion of complex structure in noncommutative ge-ometry we refer the reader to [Con94], see also [KLvS11]. Let H + be the Hilbertspace obtained by the GNS construction from ( C ( T θ ) , ϕ ) and consider ∂ ϕ = ∂ : H + → H − , where H − , the analogue of (1 , (cid:80) a∂ ( b ) , a, b ∈ C ∞ ( T θ ), with respect to the inner product (cid:104) a∂b, c∂d (cid:105) = τ ( c ∗ a ( ∂b )( ∂d ) ∗ ) . In this setting the Dirac operator is defined on the Hilbert space H = H + ⊕ H − as: D = (cid:18) ∂ ∗ ϕ ∂ ϕ (cid:19) : H → H , where the adjoint ∂ ∗ ϕ of ∂ ϕ is seen to be given by [CT11] ∂ ∗ ϕ ( a ) = ¯ ∂ ( a ) k , ∂ ∗ ϕ = R k ¯ ∂, where k = exp( h/ R k denotes the right multiplication by k . The crucialpoint is that the action of C ( T θ ) on H gives rise to an ordinary spectral triple whilethe action of the opposite algebra C ( T θ ) op of C ( T θ ) leads to a twisted spectral tripletriple [CT11]. The main reason for this phenomena is that the action of C ( T θ ) on H + induced by left multiplication is a ∗ -representation of the algebra, however, theaction of C ( T θ ) op on H + induced by right multiplication is not a ∗ -representationand requires a modification which can be provided by an algebra automorphism.That is for a ∈ C ( T θ ) op and ξ + ∈ H + , one has to define a op · ξ + = a + ( ξ + ) = ξ + k − ak, hence the appearance of the automorphism of C ( T θ ) op given by σ ( a op ) = k − ak, a ∈ C ( T θ ) , Note that for any ξ − ∈ H − one stays with a op · ξ − = ξ − a since the inner producton (1 , a ∈ C ( T θ ) op the twistedcommutator Da op − σ ( a op ) D is a bounded operator on H = H + ⊕ H − (while theordinary commutators are not necessarily bounded).We also consider the grading γ : H = H + ⊕ H − → H = H + ⊕ H − given by γ = (cid:18) − (cid:19) , nd for convenience write the action of any a ∈ C ∞ ( T θ ) op on H = H + ⊕ H − as π ( a ) = (cid:18) a + a − (cid:19) , where, clearly, the operators a ± : H ± → H ± are induced by a + ( ξ + ) = ξ + k − ak, a + = R k − ak = R σ ( a ) ,a − ( ξ − ) = ξ − a, a − = R a . We also mention that the automorphism σ satisfies the natural condition σ ( a ) ∗ = σ − ( a ∗ ) , σ − ◦ ∗ = ∗ ◦ σ. Dirac operator twisted by a module (vector bundle). Twisting theDirac operator with a hermitian vector bundle by means of a hermitian connectionis a standard technique in differential geometry. In the noncommutative setting,this can be carried out by compressing the operator F = D/ | D | of the Fredholmmodule of a twisted spectral triple ( A, H , D, σ ∈ Aut(A)) by an idempotent e as in [CM08b]. However, in order to implement the twisting with the operator D , one has to consider a twisted compression [PW16a]. That is, the index ofthe operator σ ( e ) De is important to be calculated. Note that in general e is anidempotent matrix in M q ( A ) for some positive integer q , but for our purposes forthe noncommutative two torus, we can consider e ∈ C ∞ ( T θ ), see [PV80, Rie83].Thus, we need to consider the operator σ ( e ) De = (cid:18) σ ( e + ) ∂ ∗ ϕ e − σ ( e − ) ∂ ϕ e + (cid:19) : e H → σ ( e ) H , and calculate a local formula for the index of the operator(10) D + e,σ := σ ( e − ) ∂ ϕ e + : e + H + → σ ( e − ) H − . We will shortly explain in § D + e,σ ) ∗ D + e,σ and D + e,σ ( D + e,σ ) ∗ . We calculate therelevant terms in the heat expansions by employing the pseudodifferential calculusand the method illustrated in § ι + : e + H + → H + , ι − : σ ( e − ) H − → H − , and the fact that ι ∗ + = e + , σ ( e − ) ∗ = ι − . We have: D + e,σ = σ ( e − ) ∂ ϕ ι + : e + H + ι + −→ H + ∂ ϕ −−→ H − σ ( e − ) −−−−→ σ ( e − ) H − . The operator( D + e,σ ) ∗ D + e,σ = ι ∗ + ∂ ∗ ϕ σ ( e − ) ∗ σ ( e − ) ∂ ϕ ι + = e + ∂ ∗ ϕ ι − σ ( e − ) ∂ ϕ ι + : e + H + → e + H + is the restriction of the operator L +1 = e + ∂ ∗ ϕ σ ( e − ) ∂ ϕ : H + → H + to the subspace e + H + , which means that we can write: L +1 = (cid:18) ( D + e,σ ) ∗ D + e,σ ∗ (cid:19) , hich yields exp( − tL +1 ) = (cid:32) exp (cid:16) − t (cid:0) D + e,σ (cid:1) ∗ D + e,σ (cid:17) ∗ (cid:33) , and Tr (cid:0) exp( − tL +1 ) (cid:1) = Tr (cid:16) exp (cid:16) − t (cid:0) D + e,σ (cid:1) ∗ D + e,σ (cid:17)(cid:17) . We have a unitary map W : H → H + defined by W ( a ) = ak, W = R k , and use it to work with the anti-unitarily equivalent operator L + = JW ∗ L +1 W J : H → H , where the operator J is induced by the involution of C ( T θ ). We have: L +1 = R σ ( e ) R k ¯ ∂ R σ ( e ) ∂,W ∗ L +1 W = R k − R σ ( e ) R k ¯ ∂ R σ ( e ) ∂ R k , and L + = JW ∗ L +1 W J = JR k − R σ ( e ) R k ¯ ∂ R σ ( e ) ∂ R k J = ( JR k − J ) ( JR σ ( e ) J ) ( JR k J ) ( J ¯ ∂J ) ( JR σ ( e ) J ) ( J∂J ) ( JR k J )= k − σ ( e ) k ( − ∂ ) σ ( e ) ( − ¯ ∂ ) k = k − σ ( e ) k ∂ σ ( e ) ¯ ∂ k. We can now calculate the pseudodifferential symbol of the operator L + . Lemma 3.1. We have σ L + ( ξ ) = p +2 ( ξ ) + p +1 ( ξ ) + p +0 ( ξ ) , where p +2 ( ξ ) = k − σ ( e ) k σ ( e ) k (cid:0) ξ + ξ (cid:1) = k − ek ek (cid:0) ξ + ξ (cid:1) ,p +1 ( ξ ) = k − σ ( e ) k (cid:16)(cid:0) σ ( e ) δ ( k ) + δ ( σ ( e )) k + iδ ( σ ( e )) k (cid:1) ξ + (cid:0) σ ( e ) δ ( k ) + δ ( σ ( e )) k − iδ ( σ ( e )) k (cid:1) ξ (cid:17) ,p +0 ( ξ ) = k − σ ( e ) k (cid:16) δ ( σ ( e )) δ ( k ) + σ ( e ) δ ( k ) + δ ( σ ( e )) δ ( k ) + σ ( e ) δ ( k )+ i (cid:0) δ ( σ ( e )) δ ( k ) − δ ( σ ( e )) δ ( k ) (cid:1)(cid:17) . Proof. It follows from the fact that L + = k − σ ( e ) k ∂ σ ( e ) ¯ ∂ k : H → H , and for any a ∈ C ∞ ( T θ ) ⊂ H we have (cid:0) ∂ σ ( e ) ¯ ∂ k (cid:1) ( a )= σ ( e ) k δ ( a ) + σ ( e ) k δ ( a )+ σ ( e ) δ ( k ) δ ( a ) + δ ( σ ( e ) k ) δ ( a ) + σ ( e ) δ ( k ) δ ( a ) + δ ( σ ( e ) k ) δ ( a )+ i (cid:16) σ ( e ) δ ( k ) δ ( a ) − σ ( e ) δ ( k ) δ ( a ) + δ ( σ ( e ) k ) δ ( a ) − δ ( σ ( e ) k ) δ ( a ) (cid:17) + δ ( σ ( e ) δ ( k )) a + δ ( σ ( e ) δ ( k )) a + i (cid:16) δ ( σ ( e ) δ ( k )) a − δ ( σ ( e ) δ ( k )) a (cid:17) . (cid:3) imilarly, the operator D + e,σ ( D + e,σ ) ∗ = σ ( e − ) ∂ ϕ ι + ι ∗ + ∂ ∗ ϕ σ ( e − ) ∗ = σ ( e − ) ∂ ϕ e + ∂ ∗ ϕ ι − : σ ( e − ) H − → σ ( e − ) H − is equal to the restriction of the operator L − = σ ( e − ) ∂ ϕ e + ∂ ∗ ϕ : H − → H − to the subspace σ ( e − ) H − . Therefore L − = (cid:18) D + e,σ ( D + e,σ ) ∗ ∗ (cid:19) , which yields, exp( − tL − ) = (cid:32) exp (cid:16) − tD + e,σ (cid:0) D + e,σ (cid:1) ∗ (cid:17) ∗ (cid:33) , and Tr (cid:0) exp( − tL − ) (cid:1) = Tr (cid:16) exp (cid:16) − tD + e,σ (cid:0) D + e,σ (cid:1) ∗ (cid:17)(cid:17) . The operator L − : H − → H − is anti-unitarily equivalent to the operator L − = JL − J : H → H . We have: L − = R σ ( e ) ∂ R σ ( e ) R k ¯ ∂, and L − = JL − J = J R σ ( e ) ∂ R σ ( e ) R k ¯ ∂ J = ( J R σ ( e ) J ) ( J∂J ) ( JR σ ( e ) J ) ( JR k J ) ( J ¯ ∂ J )= σ ( e ) ( − ¯ ∂ ) σ ( e ) k ( − ∂ )= σ ( e ) ¯ ∂ σ ( e ) k ∂. We can now present the pseudodifferential symbol of L − . Lemma 3.2. We have σ L − ( ξ ) = p − ( ξ ) + p − ( ξ ) where p − ( ξ ) = σ ( e ) k ( ξ + ξ ) = k − ek ( ξ + ξ ) ,p − ( ξ ) = σ ( e ) (cid:16) δ ( σ ( e ) k ) ξ + δ ( σ ( e ) k ) ξ + i (cid:0) δ ( σ ( e ) k ) ξ − δ ( σ ( e ) k ) ξ (cid:1)(cid:17) . Proof. It follows from the fact that L − = σ ( e ) ¯ ∂ σ ( e ) k ∂ : H − → H − , and for any a ∈ C ∞ ( T θ ) we have: (cid:0) ¯ ∂ σ ( e ) k ∂ (cid:1) ( a ) = σ ( e ) k δ ( a ) + σ ( e ) k δ ( a )+ δ ( σ ( e ) k ) δ ( a ) + δ ( σ ( e ) k ) δ ( a ) − iδ ( σ ( e ) k ) δ ( a ) + iδ ( σ ( e ) k ) δ ( a ) . (cid:3) .5. Index theorems. We dedicate this subsection to review some fundamentaltechniques and results about the heat equation proof of the index theorem, whichplay an important role in our approach in the noncommutative setting in the presentpaper. For a complete account of the details one can refer to [LM89, Roe98, Gil84]and references therein.We first explain the McKean-Singer index theorem [MS67]. Assume that M isa compact manifold of even dimension m and that V and W are hermitian vectorbundles over M . Also let P : C ∞ ( V ) → C ∞ ( W ) be an elliptic differential operatorfrom the smooth sections of V to those of W , and for simplicity assume that P isof order 1. By constructing suitable Sobolev spaces out of V and W , it is knownthat P extends to a bounded operator between Sobolev spaces, which is a Fredholmoperator as well. However, in order to deal with the Fredholm index of P , one cansafely only consider the smooth sections, since the regularity stemming from theellipticity of P ensures that all the sections in the kernel of (the extension of) P are smooth, see for example [Gil84]. Then, in order to calculate the index,Ind( P ) = Dim Ker( P ) − Dim Ker( P ∗ ) , where P ∗ is the adjoint of P , one can use the heat expansion as follows. TheMcKean-Singer index theorem states that for any t > P ) = Trace (exp( − tP ∗ P )) − Trace (exp( − tP P ∗ )) . On the other hand there are small time asymptotic expansions of the form (7),which depend on the local symbols. That is, there are densities a j ( x, P ∗ P ) dx and a j ( x, P P ∗ ) dx on M obtained locally from the pseudodifferenital symbols of P ∗ P and P P ∗ such that(12) Trace (exp( − tP ∗ P )) ∼ t → + t − m/ ∞ (cid:88) j =0 t j (cid:90) M a j ( x, P ∗ P ) dx, and(13) Trace (exp( − tP P ∗ )) ∼ t → + t − m/ ∞ (cid:88) j =0 t j (cid:90) M a j ( x, P P ∗ ) dx. Since the index of P in (11) is independent of t , after writing the expansions in theright hand side using (12), (13), the only term that is independent of t turns outto match with the index, hence:(14) Ind( P ) = (cid:90) M ( a m ( x, P ∗ P ) − a m ( x, P P ∗ )) dx This shows that there is a local formula for the index, and the celebrated Atiyah-Sinder index theorem identifies the density that integrates to the index in termsof characteristic classes when P is an important geometric operator. For examplewhen P is the de Rham operator d + d ∗ mapping the even differential forms tothe odd ones, the index is the Euler characteristic of the manifold, which is equalto the integral of the pfaffian of the matrix of curvature 2-forms. In general, theindex theorem states that if D E is the Dirac operator of a spin structure D withcoefficients in a vector bundle E , then the ˆ A -class of the tangent bundle T M andthe Chern character of E give the index by the formula(15) Ind( D E ) = (cid:90) M ˆ A ( T M ) ch ( E ) . or various proofs of the index theorem one can refer to [AS63, AS68b, AS68a,AS68c, Con94, HR00, Get86, Bis84, Bis87], and references therein.The elliptic theory of differential operators and pseudodifferential calculus oper-ate perfectly in noncommutative settings as developed in [Con80b], and analyzedin further detail for noncommutative tori in [Tao18, HLP18a, HLP18b]. Moreover,there is a crucial need for local index formulas for twisted spectral triples as weexplained in § 3. Therefore, in the present paper we take the heat expansion ap-proach to find a local formula for the index of the Dirac operator of the twistedspectral triple described in § Calculation of a noncommutative local formula for the index In this section we apply the method explained in § D + e,σ = σ ( e − ) ∂ ϕ e + : e + H + → σ ( e − ) H − , of a conformally flat metric on noncommutative two torus T θ . Based on the dis-cussion following the McKean-Singer formula (11), our treatment in § D + e,σ ) ∗ D + e,σ and D + e,σ ( D + e,σ ) ∗ to derive their anti-unitarily equivalents L + and L − , and the formula (6) for the terms in the heat expansion, we have:(16) Ind( D + e,σ ) = τ (cid:0) a (cid:0) L + (cid:1) − a (cid:0) L − (cid:1)(cid:1) . Thus, our task is now to perform individually for L + and L − the recursive procedurethat leads to the explicit formula (5) and to derive explicit formulas for a ( L + )and a ( L − ).Using the homogeneous components of the pseudodifferential symbols of L ± presented in Lemmas 3.1 and 3.2, we perform symbolic calculations and use formula(4) to calculate b ( ξ, λ, L ± ). This leads to lengthy expressions which we calculatewith computer assistance. The next task is to use (5) to derive a ( L ± ), namely a ( L ± ) = 12 πi (cid:90) R (cid:90) γ e − λ b ( ξ, λ, L ± ) dλ dξ. Using a homogeneity argument, see [CT11, CM14], one can avoid the contour in-tegration in the latter formula by setting λ = − − 1. More precisely, one has:(17) a ( L ± ) = − (cid:90) R b ( ξ, − , L ± ) dξ. Using the trace property of τ appearing in the formula for the index (16), after calcu-lating b ( ξ, − , L ± ), we will rotate cyclically the multiplicative term b ( ξ, − , L ± )appearing at the very right side of our terms and bring it to the very left side.We then carry out the integration over R by passing to the polar coordinates ξ = r cos θ , ξ = r sin θ . The angular integration with respect to θ , from 0 to 2 π ,can be done in a straight forward manner. However, the radial integration withrespect to r , from 0 to ∞ , poses a challenge coming from the noncommutativity.This challenge appeared in [CT11, BM12, CM14, FK13a] as well and was overcome y the rearrangement lemma and led to the appearance of the modular automor-phism in final formulas. In this article we need an even more elaborate version ofthe rearrangement lemma due to the presence of an idempotent in addition to aconformal factor, as we shall see shortly.4.1. Computation of τ ( a ( L − )) . Considering the symbol of L − written in Lemma3.2 and using the recursive formulas (3) and (4), we have b ( ξ, − , L − ) = ( σ ( e ) k | ξ | + 1) − , and up to a right multiplication by − b ( ξ, − , L − ), b ( ξ, − , L − ) is the sum of 395terms: b ( ξ, − , L − ) = − ξ (cid:0) b σ ( e ) k (cid:1) σ ( e ) kδ ( k ) + 2 iξ (cid:0) b σ ( e ) k (cid:1) σ ( e ) k ( δ δ ( k )) − ξ (cid:0) b σ ( e ) k (cid:1) σ ( e ) δ ( k ) δ ( k ) + 2 iξ (cid:0) b σ ( e ) k (cid:1) σ ( e ) δ ( k ) δ ( k ) − ξ (cid:0) b σ ( e ) k (cid:1) σ ( e ) δ ( k ) k + 2 iξ (cid:0) b σ ( e ) k (cid:1) σ ( e ) δ ( k ) δ ( k ) +2 iξ (cid:0) b σ ( e ) k (cid:1) σ ( e ) ( δ δ ( k )) k − ξ (cid:0) b σ ( e ) k (cid:1) δ ( σ ( e ) k ) − · · · Passing to polar coordinates and integrating the angular variable, we get terms ofthe following form, up to a right multiplication by − πb ( ξ, − , L − ), − (cid:0) σ ( e ) k b (cid:1) δ ( σ ( e ) k ) (cid:0) σ ( e ) k b (cid:1) δ ( σ ( e ) k ) r − (cid:0) σ ( e ) k b (cid:1) δ ( σ ( e ) k ) (cid:0) σ ( e ) k b (cid:1) δ ( σ ( e ) k ) r − (cid:0) ( σ ( e ) k ) b (cid:1) δ ( σ ( e ) k ) b δ ( σ ( e ) k ) r − (cid:0) ( σ ( e ) k ) b (cid:1) δ ( σ ( e ) k ) b δ ( σ ( e ) k ) r + 2 (cid:0) ( σ ( e ) k ) b (cid:1) δ ( σ ( e ) k ) r + · · · , a total of 82 terms of the form. The next step is to perform the integration against r d r over 0 to ∞ .An important integration formula we later prove requires that the leftmost pow-ers of b have a factor of σ ( e ) to their right. Leftmost b ’s raised only to the power1 already have this factor σ ( e ) immediately to their right, and leftmost b ’s raisedto powers 2 or 3 have a power of σ ( e ) k to the left which we can move to the right,again giving us σ ( e ) immediately to the right. As we explained earlier, It is alsouseful to use the trace property of τ to permute the multiplicative factors cyclically.Thus, instead of multiplying the above sum by − πb ( ξ, − , L − ).4.1.1. Terms with all b on the left. As it turns out, it is difficult to evaluate im-proper integrals when nontrivial idempotents are involved. For instance, considerthe integral (cid:82) ∞ e/ (1 + ex ) d x . One would expect that (cid:90) ∞ e d x (1 + ex ) = (cid:20) − 11 + ex (cid:21) ∞ = 1 , but that implies that e = e · e · (cid:90) ∞ e d x (1 + ex ) = (cid:90) ∞ e d x (1 + ex ) = 1 , giving us a contradiction. Since e (1 − ez + ( ez ) − · · · ) = e (1 − z + z − · · · ) for | z | < 1, we have e (1 + ez ) = e (1 + z ) , o we can properly evaluate the aforementioned improper integral as follows: (cid:90) ∞ e d x (1 + ex ) = e (cid:90) ∞ d x (1 + x ) = e (cid:20) − 11 + x (cid:21) ∞ = e. To calculate the contribution of terms with all b on the left to the trace, we needa formula for (cid:90) ∞ ( σ ( e ) k u + 1) m +2 \ u m σ ( e ) d u. Since σ ( e ) and k do not necessarily commute, we cannot do the trick we did above.We need to reduce to the case of terms with b in the middle. The trace propertyallows to write: τ (cid:18)(cid:90) ∞ ( σ ( e ) k u + 1) m +2 \ u m σ ( e ) ρ d u (cid:19) = τ (cid:18)(cid:90) ∞ ( σ ( e ) k u + 1) m +1 \ ( u m σ ( e ) ρ ) / ( σ ( e ) k u + 1) d u (cid:19) . Even though we get what looks like a more difficult integral, we can use techniquesfrom combinatorics, Fourier analysis, and complex analysis to evaluate it, inspiredby a proof given in [CT11] for the case e = 1.4.1.2. Terms with b in the middle. As in [CT11], we use integration by parts sothat we can write terms with b as terms with b in the middle. The main differenceis that this time, b = ( σ ( e ) k u + 1) − , so ∂ r ( b ) = − σ ( e ) k rb and instead of aterm like (cid:90) ∞ r ( k b ) δ ( k ) k ( k b ) δ ( k ) kr d r, we would have something like (cid:90) ∞ r ( σ ( e ) k b ) δ ( k ) k ( σ ( e ) k b ) δ ( k ) kr d r, which we would replace by σ ( e ) k (cid:90) ∞ ∂ r ( r b ) δ ( k ) kb δ ( k ) k d r. Again, we are able to reduce to the case of terms with b in the middle.4.1.3. Terms with b in the middle. For integrals involving elements of C ∞ ( T θ )squeezed between powers of b = b ( ξ, − , L − ), we need to prove a rearrangementlemma. In the most basic case, which suffices for our needs, the idempotent appearswith k in the denominator of the b , and we need the following generalized versionof the rearragement lemma [CT11, Lemma 6.2]: Lemma 4.1. For every element ρ ∈ A ∞ θ and every non-negative integer m we have (cid:90) ∞ ( σ ( ek ) u + 1) m +1 \ ( u m σ ( e ) ρ ) / ( σ ( ek ) u + 1) d u = σ ( e ) D m ( k − (2 m +2) σ ( e ) ρσ ( e )) , where D m = L m (∆) , m ( u ) = (cid:90) ∞ x m ( x + 1) m +1 xu + 1 d x = ( − m ( u − − ( m +1) log u − m (cid:88) j =1 ( − j +1 ( u − j j , ∆( a ) = k − ak , a ∈ C ( T θ ) , is the modular automorphism, and σ ( a ) = k − ak is itssquare root.Proof. The proof is provided in Appendix A. (cid:3) The following corollary will be useful in our calculations: Corollary 4.2. (cid:90) ∞ (∆( ek e ) u + 1) m +1 \ ( u m ∆( e ) ρ ) / (∆( ek e ) u + 1) d u = ∆( e ) D m ( k − (2 m +2) ∆( e ) ρ ∆( e ))∆( e ) Proof. The proof is provided in Appendix A. (cid:3) Applying Lemma 4.1 and using the Leibniz rule to simplify, we find that − π τ ( a ( L − )) = τ (cid:16) σ ( e ) D (cid:0) k − σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) + iσ ( e ) D (cid:0) k − σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) − iσ ( e ) D (cid:0) k − σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) + iσ ( e ) D (cid:0) k − σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) − iσ ( e ) D (cid:0) k − σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) + iσ ( e ) D (cid:0) k − σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) − iσ ( e ) D (cid:0) k − σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) − σ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) − iσ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) + iσ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) − σ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) − σ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) − iσ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) + iσ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) − σ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) − σ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) − iσ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) + iσ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) − σ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) − iσ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) kδ ( k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) + iσ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) − iσ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( k ) kσ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) + iσ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) − iσ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) δ ( σ ( e ) k ) +4 σ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) +4 σ ( e ) k σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) δ ( σ ( e ) k ) + σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) kδ ( k ) − iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) kδ ( k ) + σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) δ ( k ) k − iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) δ ( k ) k + iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) kδ ( k ) + σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) kδ ( k ) + iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) δ ( k ) k + σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) δ ( k ) k + σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) δ ( σ ( e )) k − iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) δ ( σ ( e )) k + iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) δ ( σ ( e )) k + σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) σ ( e ) δ ( σ ( e )) k − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) kδ ( k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) kδ ( k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( k ) δ ( k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( k ) kσ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( k ) δ ( k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( k ) kσ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( σ ( e )) kδ ( k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( σ ( e )) δ ( k ) kσ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( σ ( e )) kδ ( k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( σ ( e )) δ ( k ) kσ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) δ ( σ ( e )) k σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) σ ( e ) kδ ( k ) σ ( e ) (cid:1) + iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) σ ( e ) kδ ( k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) σ ( e ) δ ( k ) kσ ( e ) (cid:1) + iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) σ ( e ) δ ( k ) kσ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) δ ( σ ( e )) k σ ( e ) (cid:1) + iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) δ ( σ ( e )) k σ ( e ) (cid:1) − iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) σ ( e ) kδ ( k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) σ ( e ) kδ ( k ) σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) σ ( e ) δ ( k ) kσ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) σ ( e ) δ ( k ) kσ ( e ) (cid:1) − iσ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) δ ( σ ( e )) k σ ( e ) (cid:1) − σ ( e ) D (cid:0) k − σ ( e ) k δ ( σ ( e )) δ ( σ ( e )) k σ ( e ) (cid:1) +2 σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) +2 σ ( e ) D (cid:0) k − σ ( e ) k σ ( e ) k δ ( σ ( e ) k ) σ ( e ) (cid:1) (cid:17) . Computation of τ ( a ( L + )) . In a very similar manner, starting from thesymbol of the operator L + written in Lemma 3.1, we calculate a local formula for τ ( a ( L + )). We find that b ( ξ, − , L + ), up to right multiplication by − b ( ξ, − , L + ),is the sum of 232 terms of the following form:(( b (cid:48) )∆( e )) k − σ ( e ) k σ ( e ) δ ( k ) + (( b (cid:48) )∆( e )) k − σ ( e ) k σ ( e ) δ ( k ) +(( b (cid:48) )∆( e )) k − σ ( e ) k δ ( σ ( e )) δ ( k ) − i (( b (cid:48) )∆( e )) k − σ ( e ) k δ ( σ ( e )) δ ( k ) + i (( b (cid:48) )∆( e )) k − σ ( e ) k δ ( σ ( e )) δ ( k ) + (( b (cid:48) )∆( e )) k − σ ( e ) k δ ( σ ( e )) δ ( k ) − ξ (cid:0) ( b (cid:48) ) ∆( e ) k (cid:1) k − σ ( e ) k σ ( e ) δ ( k ) − · · · . After passing to the polar coordinates and performing the angular integration, upto write multiplication by − πb ( ξ, − , L + ), these terms add up to the sum of 82terms: − (cid:0) ∆( e ) k ( b (cid:48) ) ∆( e ) (cid:1) δ (∆( e ) k ∆( e )) (cid:0) ∆( e ) k ( b (cid:48) ) ∆( e ) (cid:1) δ (∆( e ) k ∆( e )) r − (cid:0) ∆( e ) k ( b (cid:48) ) ∆( e ) (cid:1) δ (∆( e ) k ∆( e )) (cid:0) ∆( e ) k ( b (cid:48) ) ∆( e ) (cid:1) δ (∆( e ) k ∆( e )) r − (cid:0) (∆( e ) k ) ( b (cid:48) ) ∆( e ) (cid:1) δ (∆( e ) k ∆( e )) (( b (cid:48) )∆( e )) δ (∆( e ) k ∆( e )) r − (cid:0) (∆( e ) k ) ( b (cid:48) ) ∆( e ) (cid:1) δ (∆( e ) k ∆( e )) (( b (cid:48) )∆( e )) δ (∆( e ) k ∆( e )) r + · · · Applying Corollary 4.2, which allows replacing b = b ( ξ, − , L + ) with b ∆( e ), wefind that: − π τ ( a ( L + )) = τ (cid:16) ∆( e ) D (cid:0) k − ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) +∆( e ) D (cid:0) k − ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) + ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − i ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) + i ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) + ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − e ) k ∆( e ) D (cid:0) k − ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − e ) k ∆( e ) D (cid:0) k − ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − ∆( e ) k ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) + i ∆( e ) k ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − i ∆( e ) k ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − ∆( e ) k ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) +2∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) +2∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) +∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) + i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) +∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) +4∆( e ) k ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) + ∆( e ) k ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − e ) D (cid:0) k − ∆( e ) k ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − e ) D (cid:0) k − ∆( e ) k ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) δ (∆( e ) k ∆( e )) − e ) D (cid:0) k − ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) kσ ( e ) δ ( k ) − ∆( e ) D (cid:0) k − ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) kδ ( σ ( e )) k − i ∆( e ) D (cid:0) k − ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) kδ ( σ ( e )) k − e ) D (cid:0) k − ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) kσ ( e ) δ ( k ) + i ∆( e ) D (cid:0) k − ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) kδ ( σ ( e )) k − ∆( e ) D (cid:0) k − ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) ∆( e ) kδ ( σ ( e )) k − ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) kσ ( e ) δ ( k ) + i ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) kσ ( e ) δ ( k ) − i ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) kσ ( e ) δ ( k ) − ∆( e ) D (cid:0) k − ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) ∆( e ) kσ ( e ) δ ( k ) +2∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) kσ ( e ) δ ( k ) +∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) kδ ( σ ( e )) k + i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) kδ ( σ ( e )) k +2∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) kσ ( e ) δ ( k ) − i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) kδ ( σ ( e )) k +∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) ∆( e ) kδ ( σ ( e )) k +∆( e ) D (cid:0) k k ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) + ∆( e ) D (cid:0) k k ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) +∆( e ) D (cid:0) k k ∆( e ) kδ ( σ ( e )) δ ( k )∆( e ) (cid:1) − i ∆( e ) D (cid:0) k k ∆( e ) kδ ( σ ( e )) δ ( k )∆( e ) (cid:1) + i ∆( e ) D (cid:0) k k ∆( e ) kδ ( σ ( e )) δ ( k )∆( e ) (cid:1) + ∆( e ) D (cid:0) k k ∆( e ) kδ ( σ ( e )) δ ( k )∆( e ) (cid:1) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) − e ) D (cid:0) k − ∆( e ) k ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) − e ) D (cid:0) k − ∆( e ) k ∆( e ) kσ ( e ) δ ( k )∆( e ) (cid:1) − e ) D (cid:0) k − ∆( e ) k ∆( e ) kδ ( σ ( e )) δ ( k )∆( e ) (cid:1) + i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) kδ ( σ ( e )) δ ( k )∆( e ) (cid:1) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) − i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) kδ ( σ ( e )) δ ( k )∆( e ) (cid:1) − e ) D (cid:0) k − ∆( e ) k ∆( e ) kδ ( σ ( e )) δ ( k )∆( e ) (cid:1) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) kδ ( σ ( e )) k ∆( e ) (cid:1) − e ) D (cid:0) k − ∆( e ) k ∆( e ) δ ( k ) σ ( e ) δ ( k )∆( e ) (cid:1) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ ( k ) δ ( σ ( e )) k ∆( e ) (cid:1) − i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ ( k ) δ ( σ ( e )) k ∆( e ) (cid:1) − e ) D (cid:0) k − ∆( e ) k ∆( e ) δ ( k ) σ ( e ) δ ( k )∆( e ) (cid:1) + i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ ( k ) δ ( σ ( e )) k ∆( e ) (cid:1) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) δ ( k ) δ ( σ ( e )) k ∆( e ) (cid:1) − e ) D (cid:0) k − ∆( e ) k ∆( e ) k σ ( e ) δ ( k ) kσ ( e ) δ ( k )∆( e ) (cid:1) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k σ ( e ) δ ( k ) kδ ( σ ( e )) k ∆( e ) (cid:1) − i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k σ ( e ) δ ( k ) kδ ( σ ( e )) k ∆( e ) (cid:1) − e ) D (cid:0) k − ∆( e ) k ∆( e ) k σ ( e ) δ ( k ) kσ ( e ) δ ( k )∆( e ) (cid:1) + i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k σ ( e ) δ ( k ) kδ ( σ ( e )) k ∆( e ) (cid:1) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k σ ( e ) δ ( k ) kδ ( σ ( e )) k ∆( e ) (cid:1) − e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( σ ( e )) k σ ( e ) δ ( k )∆( e ) (cid:1) − ( e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( σ ( e )) k δ ( σ ( e )) k ∆( e ) (cid:1) − i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( σ ( e )) k δ ( σ ( e )) k ∆( e ) (cid:1) − e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( σ ( e )) k σ ( e ) δ ( k )∆( e ) (cid:1) + i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( σ ( e )) k δ ( σ ( e )) k ∆( e ) (cid:1) − ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( σ ( e )) k δ ( σ ( e )) k ∆( e ) (cid:1) +2∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( k ) k σ ( e ) k σ ( e ) δ ( k )∆( e ) (cid:1) +∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( k ) k σ ( e ) k δ ( σ ( e )) k ∆( e ) (cid:1) + i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( k ) k σ ( e ) k δ ( σ ( e )) k ∆( e ) (cid:1) +2∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( k ) k σ ( e ) k σ ( e ) δ ( k )∆( e ) (cid:1) − i ∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( k ) k σ ( e ) k δ ( σ ( e )) k ∆( e ) (cid:1) +∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k δ ( k ) k σ ( e ) k δ ( σ ( e )) k ∆( e ) (cid:1) +2∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) +2∆( e ) D (cid:0) k − ∆( e ) k ∆( e ) k ∆( e ) δ (∆( e ) k ∆( e ))∆( e ) (cid:1) (cid:17) . Reduction to the flat metric and the Connes-Chern number. Hav-ing calculated explicit local formulas for τ ( a ( L ± )), based on (16), we have founda local formula for the index of the twisted Dirac operator D + e,σ on T θ given by(10), where the Dirac operator is associated with a conformally flat metric and thetwisting is carried out by an idempotent playing the role of a general vector bun-dle. In forthcoming work, we will present a simplification of the local formula andwill elaborate on the relation between properties of the functions of the modularautomorphism in the simplified form and stability of the index under perturbationsthat leave the Dirac operator in the same connected component of Fredholm oper-ators. We end this article by considering the case of the canonical flat metric on T θ , which corresponds to the trivial conformal factor k = 1 whose correspondingmodular automorphism ∆ is the identity. Therefore, our formula for the index, inthis case, reduces to a much simpler form as one can replace the functions of themodular automorphism with the values of the functions at 1. This yields, for k = 1:Ind D + e,σ = 2 πi τ ( eδ ( e ) δ ( e ) − eδ ( e ) δ ( e )) . Note that c ( e ) := 2 πi τ ( eδ ( e ) δ ( e ) − eδ ( e ) δ ( e ))is the Connes-Chern number of our projection e , and that, as proven in [DJLL18],one can construct e as a self-dual or anti-self dual projection with c ( e ) = n for anyinteger n . Appendix A. Proof of Lemma 4.1 and Corollary 4.2 The Hadamard product of two power series f ( z ) = (cid:80) ∞ n =0 f n z n and g ( z ) = (cid:80) ∞ n =0 g n z n is defined by ( f (cid:12) g )( z ) = ∞ (cid:88) n =0 f n g n z n . Proposition A.1. [FS09, p. 424] Suppose f and g are analytic functions on adomain D ⊂ C , and let γ ⊂ D ∩ ( zD − ) be a closed contour. Then the Hadamardproduct of f and g obeys the following identity: ( f (cid:12) g )( z ) = 12 πi (cid:90) γ f ( w ) g ( z/w ) dw/w. emma A.2. Let N be an integer. For every element ρ ∈ C ∞ ( T θ ) and everynon-negative integer m we have (cid:90) ∞ ( σ N ( ek ) u + 1) m +1 \ ( u m σ N ( e ) ρ ) / ( σ N ( ek ) u + 1) d u = σ N ( e ) D m ( k − (2 m +2) σ N ( e ) ρσ N ( e )) , where the modified logarithm D m = L m (∆) of the modular automorphism ∆( a ) = k − ak , a ∈ C ( T θ ) , with the square root σ ( a ) = k − ak , is defined via the function L m ( u ) = (cid:90) ∞ x m ( x + 1) m +1 xu + 1 d x = ( − m ( u − − ( m +1) log u − m (cid:88) j =1 ( − j +1 ( u − j j . Proof. First we look at the N = 0 case. We express the integrand as two Hadamardproducts [FS09, p. 424] of generating functions multiplied together, and then wemake the change of variables u = exp( s ). We will substitute an improper integralusing exp[( h + s ) / h + s − a − iφ ) = exp[ i ( φ − ia ) / (cid:90) ∞−∞ exp[ it ( h + s − a − iφ )]exp( πt ) + exp( − πt ) d t for the closed form of the Fourier transform of the hyperbolic secant, and willexpress an inner integral as the Fourier transform of another function. Let a be apositive real number. We get (cid:90) ∞ ( ek u + 1) m +1 \ ( u m eρ ) / ( ek u + 1) d u = (cid:90) ∞ ( ek u + 1) m +1 \ ( k m +2 u m ( k − (2 m +2) eρ )) / ( ek u + 1) d u = (cid:90) ∞ ( k u ) m +1 / (cid:20) k u ) m +1 (cid:12) ∆ m +1 / (cid:18)(cid:18) − u ∆ R e (1) (cid:19)(cid:19)(cid:21) ∆ − / ( k − (2 m +2) eρ )( k u ) / (cid:20) 11 + k u (cid:12) − u ∆ R e (1) (cid:21) d uu = (cid:90) ∞−∞ (cid:18)(cid:20) π (cid:90) π exp[( m + 1 / s + h )][1 + exp( h + s − a − iθ )] m +1 ∆ m +1 / (cid:18)(cid:18) − ∆ R e exp( iθ + a ) (1) (cid:19)(cid:19) d θ (cid:21) · ∆ − / ( k − (2 m +2) eρ ) (cid:20) π (cid:90) π exp[( s + h ) / h + s − a − iφ ) 11 − ∆ R e exp( iφ + a ) (1) d φ (cid:21)(cid:19) d s π ) (cid:90) ∞−∞ exp( ith )exp( πt ) + exp( − πt ) (cid:18)(cid:90) ∞−∞ (cid:20)(cid:90) π exp[( m + 1 / s + h )][1 + exp( h + s − a − iθ )] m +1 ∆ m + + it (cid:18)(cid:18) − ∆ R e exp( iθ + a ) (1) (cid:19)(cid:19) d θ (cid:21) ∆ − + it ( k − (2 m +2) eρ ) (cid:34)(cid:90) π exp[ iφ + a + its + t ( φ − ia )]1 − ∆ R e exp( iφ + a ) (1) d φ (cid:35) d s (cid:33) d t = 1(2 π ) (cid:90) ∞−∞ exp( ith )exp( πt ) + exp( − πt ) (cid:20)(cid:90) π (cid:18)(cid:90) ∞−∞ exp[( m + 1 / s + h )] exp( its )[1 + exp( h + s − a − iθ )] m +1 d s (cid:19) ∆ m + + it (cid:18)(cid:18) − ∆ R e exp( iθ + a ) (1) (cid:19)(cid:19) d θ (cid:21) ∆ − + it ( k − (2 m +2) eρ ) (cid:20)(cid:90) π exp[( iφ + a ) / t ( φ − ia )]1 − ∆ R e exp( iφ + a ) (1) d φ (cid:21) d t. For the second equality, agreement of the integrands within the radii of convergenceof the geometric series implies agreement of the integrands throughout their analyticcontinuations: (cid:34) ∞ (cid:88) n =0 ( − ek u ) n (cid:35) m +1 k m +2 u m ( k − (2 m +2) eρ ) (cid:34) ∞ (cid:88) n =0 ( − ek u ) n (cid:35) = ∞ (cid:88) n =0 · · · ∞ (cid:88) n m +1 =0 m +1 (cid:89) q =1 ( − k u ) n q (cid:32) n q (cid:89) p =1 ∆ p e (cid:33) ∗ · k m +2 u m ( k − (2 m +2) eρ ) (cid:34) ∞ (cid:88) n =0 ( − k u ) n (cid:32) n (cid:89) p =1 ∆ p e (cid:33) ∗ (cid:35) = ∞ (cid:88) n =0 · · · ∞ (cid:88) n m +1 =0 ( − k u ) (cid:80) m +1 q =1 n q (cid:80) m +1 q =1 n q (cid:89) p =1 ∆ p e ∗ · k m +2 u m ( k − (2 m +2) eρ ) (cid:34) ∞ (cid:88) n =0 ( − k u ) n (cid:32) n (cid:89) p =1 ∆ p e (cid:33) ∗ (cid:35) . Since (cid:90) ∞−∞ exp[( m + )( s + h )] exp( its )[1 + exp( h + s − a − iθ )] m +1 d s = exp[ i ( m + 12 )( θ − ai )] (cid:90) ∞−∞ exp[( m + )( h + s − a − iθ )] exp( its )[1 + exp( h + s − a − iθ )] m +1 d s = exp[ i ( m + 12 )( θ − ai )] exp[ − it ( h − a − iθ )] ˆ f m ( t ) , where ˆ f m ( t ) is the Fourier transform of the function f m ( s ) = exp[( m + 1 / s ][exp( s ) + 1] m +1 , e have: (cid:90) ∞ ( ek u + 1) m +1 \ ( u m eρ ) / ( ek u + 1) d u = 1(2 π ) (cid:90) ∞−∞ ˆ f m ( t )exp( πt ) + exp( − πt )∆ m + + it (cid:20)(cid:18)(cid:18)(cid:90) π exp[ i ( m + 1 / it )( θ − ai )]1 − ∆ R e exp( iθ + a ) d θ (cid:19) (1) (cid:19)(cid:21) ∆ − + it ( k − (2 m +2) eρ ) (cid:20)(cid:18)(cid:90) π exp[ i (1 / − it )( φ − ia )]1 − ∆ R e exp( iφ + a ) d φ (cid:19) (1) (cid:21) d t = 1(2 π ) (cid:90) ∞−∞ ˆ f m ( t )exp( πt ) + exp( − πt ) (cid:20)(cid:18)(cid:18)(cid:90) π exp[ i ( m + 1 / it )( ∇ + θ − ai )]1 − ∆ R e exp( iθ + a ) d θ (cid:19) (1) (cid:19)(cid:21) ∆ − + it (cid:18) k − (2 m +2) eρ (cid:20)(cid:18)(cid:90) π exp[ i (1 / − it )( ∇ + φ − ia )]1 − ∆ R e exp( iφ + a ) d φ (cid:19) (1) (cid:21)(cid:19) d t. For e = 1, the inner integrals evaluate to 2 π , so what we get agrees with theprevious result. For 0 (cid:54) = w ∈ C \ ( −∞ , 0] and a > − ln | w | , we integrate over aHankel contour H (cid:15),R with branch cut along the negative x -axis and get (cid:90) π exp[ i ( m + 1 / it )( ∇ + θ − ai )]1 − w exp( iθ + a ) d θ = (cid:90) C exp a (∆ z ) m +1 / it − zw i d zz = 1 i lim R →∞ lim (cid:15) → (cid:90) H (cid:15),R (exp[( m − / it )(log z + ∇ )])1 − zw d z ∆= 2 π (exp[( m − / it ) log( w − ∆)])∆= 2 π ((∆ − w ) − m − it )∆ , and (cid:90) π exp[ i (1 / − it )( ∇ + φ − ai )]1 − w exp( iφ + a ) d φ = (cid:90) C exp a (∆ z ) / − it − zw i d zz = 1 i lim R →∞ lim (cid:15) → (cid:90) H (cid:15),R (exp[( − / − it )(log z + ∇ )])1 − zw d z ∆= 2 π (exp[( − / − it ) log( w − ∆)])∆= 2 π (∆ − w ) + it ∆ . ince R e is an idempotent operator, we can reduce to the previous result as follows: (cid:90) ∞ ( ek u + 1) m +1 \ ( u m eρ ) / ( ek u + 1) d u = (cid:90) ∞−∞ ˆ f m ( t )(∆ − ∆ R e ) − m − it ∆(1)∆ − + it ( k − (2 m +2) eρ (∆ − ∆ R e ) + it ∆(1)) d t exp( πt ) + exp( − πt )= (cid:90) ∞−∞ ˆ f m ( t )exp( πt ) + exp( − πt ) R e (1)∆ − + it ( k − (2 m +2) eρR e (1)) d t = (cid:90) ∞−∞ ˆ f m ( t )exp( πt ) + exp( − πt ) e ∆ − + it ( k − (2 m +2) eρe ) d t = e (cid:90) ∞−∞ ˆ f m ( t )exp( πt ) + exp( − πt ) ∆ − + it ( k − (2 m +2) eρe ) d t = eD m ( k − (2 m +2) eρe ) . Now that we have proven the lemma for N = 0, we may replace ρ with σ − N ( ρ ) andapply σ N to both sides, which proves the lemma for any integer N . (cid:3) Corollary A.3. Let N be an integer. Then (cid:90) ∞ ( σ N ( ek e ) u + 1) m +1 \ ( u m σ N ( e ) ρ ) / ( σ N ( ek e ) u + 1) d u = σ N ( e ) D m ( k − (2 m +2) σ N ( e ) ρσ N ( e )) σ N ( e ) Proof. The N = 0 case of lemma we just proved implies that, for any ρ ∈ A ∞ θ andevery non-negative integer m , we have (cid:90) ∞ ( ek u + 1) m +1 \ ( u m eρ ) / ( ek u + 1) d u = eD m ( k − (2 m +2) eρe )and (cid:90) ∞ ( ek u + 1) m +1 \ ( u m eρe ) / ( ek u + 1) d u = eD m ( k − (2 m +2) eρe ) . Since 1 / ( ek u +1) − e/ ( ek u +1) = 1 − e , subtracting the two equations immediatelyabove gives us (cid:90) ∞ ( ek u + 1) m +1 \ ( u m eρ (1 − e )) d u = 0 . Also, (cid:90) ∞ ( ek u + 1) − ( m +1) u m eρ ( ek u + 1) − e d u = eD m ( k − (2 m +2) eρe ) e. Since 1 / ( ek eu + 1) = 1 − e + ( ek u + 1) \ e , adding the two equations immediatelyabove gives us (cid:90) ∞ ( ek u + 1) m +1 \ ( u m eρ ) / ( ek eu + 1) d u = eD m ( k − (2 m +2) eρe ) e. Since ( ek u + 1) m +1 \ e = ( ek eu + 1) m +1 \ e , we get (cid:90) ∞ ( ek eu + 1) m +1 \ ( u m eρ ) / ( ek eu + 1) d u = eD m ( k − (2 m +2) eρe ) e. After replacing ρ with σ − N ( ρ ) and applying σ N to both sides, we are done. (cid:3) e also state a corollary of our rearrangement lemma regarding orthogonal pro-jections. Suppose e , e ∈ A ∞ θ are orthogonal self-adjoint idempotents, i.e. e = e = e ∗ , e = e = e ∗ , and e e = e e = 0. Then ( e + e ) = e + e = e + e =( e + e ) ∗ , so e + e is a self-adjoint idempotent and the following corollary followsfrom our rearrangement lemma: Corollary A.4. 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