Index of elliptic operators for a diffeomorphism

Index of elliptic op erators for a diffeomorphism An ton Sa vin and Bo ris Sternin No v ember 12, 2 018 Abstract W e dev elop elli ptic theory of op erators associated with a diffeo morphism of a closed smo oth ma nifold. T h e aim of the presen t pap er is to obtain an index form ula for suc h op erators in terms of top olo gical inv ariants of the manifold and of the sym b ol of the op erator. The symb ol in this situation is an elemen t of a certain crossed p rod uct. W e express the index as the pairing of the class in K-theory defined b y the symbol and the T o dd class in p erio dic cyclic cohomology of the crossed pro duct. Con ten ts In tro duction 2 1 Ellipticit y and fi niteness theorem 3 2 Chern c haracters for crossed pro ducts 5 2.1 Chern character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Twisted Chern c haracter (Chern c haracter with co efficien ts in a v ector bundle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Chern character of a n elliptic sym b ol . . . . . . . . . . . . . . . . . . . . . 10 3 The equa tion ch y = Td x 11 4 Index theorem 14 5 Reduction of a noncomm utat ive op erator t o a ψ DO on a closed manifold 15 5.1 Reduction to a sp ecial t wo-term op erators . . . . . . . . . . . . . . . . . . 15 5.2 Reduction to a b oundary v alue problem . . . . . . . . . . . . . . . . . . . . 17 5.3 Homotop y of the b oundary conditio n . . . . . . . . . . . . . . . . . . . . . 17 5.4 Reduction to a ψ DO on the to r us . . . . . . . . . . . . . . . . . . . . . . . 19 6 Comparison of top ological indices 20 6.1 Computation o f the index of ψ DO on the torus using Atiy ah–Singer formula 20 6.2 Comparison of the top o logical indices of the ψ D O on the torus and of the original op erator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1 7 Index form ula in cyclic cohomology 25 7.1 Equiv arian t Che rn c haracter . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.2 Index fo rm ula in cyclic cohomology . . . . . . . . . . . . . . . . . . . . . . 27 8 Examples. Remarks 29 8.1 Example. Op erators on the torus T 3 . . . . . . . . . . . . . . . . . . . . . 29 8.2 Remark. Sp ecial tw o-term op erators as opera t ors in subspaces . . . . . . . 33 8.3 Remark. A gene ralization of the notion of ellipticity . . . . . . . . . . . . . 34 References 34 In t ro duction Let M b e a smo oth manifold and g : M → M b e a diffeomor phism. W e dev elop elliptic theory for op erators of the form D = X k D k T k : C ∞ ( M ) − → C ∞ ( M ) . (0.1) Here T is the shift op erator T u ( x ) = u ( g ( x )) along the orbits of g , D k are pseudo differ- en tial operat o rs ( ψ DO) on M , and the sum is assumed to be finite. The aim of the presen t pap er is to obtain an index f o rm ula for the op erator (0.1) in terms o f top ological in v arian ts of the ma nif o ld and of the sym b ol of the op erator. Precise definitions of all the ob jects will b e given b elow but no w let us note an imp ortan t c harac- teristic prop ert y of this theory . Namely , the algebra of sym b ols of op erators (0.1) is not c ommutative . More precisely , a n explicit computation shows that the algebra of sym b ols is the crossed pro duct C ∞ ( S ∗ M ) ⋊ Z of the alg ebra of functions o n the cosphere bundle b y the action of the group Z . This essen tially means that we consider noncomm utativ e elliptic theory . Sp ecial cases of opera t o rs (0.1) w ere considered by a num b er of a uthors (e.g., see [1 – 9]). In these pap ers, as a rule, certain conditions w ere imp osed on the ma nif o ld M and on the diffeomorphism g . F or example , in the b o ok [2 ] it is ass umed that the diffeomorphism is an isometry , in [7] the diffeomorphism is arbitrary but the manifold is one-dimensional, and so on. F urther, we w o uld lik e to men tio n in teresting pap ers [10] and [11], where the authors study elliptic op erators of the form (0.1) that are asso ciated with the Dirac op erator and conformal diffeomorphisms. Let us stress that in the presen t pa p er w e consider an arbit r a ry compact smo oth manif o ld and a n arbitrary diffeomorphism without any r estrictions. The main result of the pap er is an explicit index form ula for the elliptic op erator (0 .1). More explicitly , the a nsw er is giv en b y the formu la ind D = (2 π i ) − n h [ σ ( D )] , T o dd( T ∗ C M ) i , dim M = n, (0.2) in terms o f cyclic cohomolo gy , where [ σ ( D )] ∈ K 0 ( C ∞ ( S ∗ M × S 1 ) ⋊ Z ) is the class of sym b ol in K -theory , T o dd( T ∗ C M ) ∈ H P ev ( C ∞ ( S ∗ M × S 1 ) ⋊ Z ) is the T o dd class in cyclic cohomology and the bra ck ets h , i denote the pairing of K -theory and cyclic cohomology . 2 Let us briefly describe the metho ds used in the presen t pap er. It is clear more or less that a noncomm utativ e elliptic theory requires a noncomm utativ e apparatus: noncomm u- tativ e differen tial forms, noncommutativ e trace, etc. Moreov er, since the diffeomorphism generates an action of the group Z , the relev an t top ological in v arian ts are naturally ele- men t s of the Haefliger cohomology gro up H ∗ ( S ∗ M / Z ) (see [12]). In this framew ork, w e define the Chern character and establish an imp ortan t in termediate index form ula (inter- esting in its o wn righ t) as an in tegral of a Haefliger form ov er S ∗ M × S 1 . After this, w e reduce the obta ined formula to the nat ural and elegan t form ula (0 .2). The latter index form ula can b e considered as an analog ue of the A tiyah–Singer fo rm ula in our situation. W e now describe the conten ts of the pap er. In the first section, w e in tro duce a notion of ellipticit y and prov e finiteness theorem. In the second section, w e define Chern c haracters for crossed pro ducts, including t wisted Chern c haracter, and define the Chern c haracter of an elliptic sym b ol. Section three is de v oted to the solution of t he equation c h y = Td x, (0.3) where Td x is the T o dd class of a complex v ector bundle x and c h is the Chern c haracter. The p oint here is that in our situation the T o dd class is generally sp eaking undefined. Ho w eve r, it can b e replaced by t he Chern c haracter of a bundle satisfying Eq. (0.3). Finally , in the fourth section w e formu late an index theorem (in Haefliger cohomology). The pro of of the index theorem is give n in Sections 5 and 6. Namely , in Sec. 5 w e reduce our initial op erator to a sp ecial b oundary v alue pro blem o n the cylinder M × [0 , 1 ] (see [13, 14]), whic h is then reduced to a certain pseudodifferential op erator on the torus of the original manifold t wisted b y the diffeomorphism g (cf. [15, 16]). The index of the latter op erator can b e computed by the A tiy ah–Singer fo rm ula. How ev er, to give an index form ula in terms of the original op erator, w e need to compare the index form ula for the pseudo differen tial op erator on the to rus and the form ula a nno unced in Sec. 4. So, the pro of of the index form ula f o r the o r iginal o perato r is complete, at least in the fra mew ork of Haefliger cohomology . In Sec. 7 we in terpret the index formula in Haefliger cohomolo g y as an A tiy ah–Singer form ula in cyclic cohomology . Here w e use equiv arian t c ha r a cteristic classes in cyclic cohomology (see [17 ]) . In the eigh th section w e giv e some remarks and consider a n example. This work w as done at t he Institute o f Analysis of Leibniz Univ ersity o f Ha nnov er during our stay in the summer 20 11. W e are grateful to Prof. Elmar Schrohe, in whose w orking group this res earc h w as done, for a t t en tion and excellen t w orking conditions. 1 Elliptici t y and fini teness theorem Let M b e a smo o th closed manifold and g : M → M b e a diffeomorphism. Consider an op erator of the form D = X k D k T k : C ∞ ( M , C N ) − → C ∞ ( M , C N ) , (1.1) where T : C ∞ ( M ) → C ∞ ( M ) , T u ( x ) = u ( g ( x )) , 3 is the shift op erator corr esponding to g and the co efficie n ts D k : C ∞ ( M , C N ) − → C ∞ ( M , C N ) are pseudodifferential op erators ( ψ D O) of order zero and the sum in (1.1) is finite. Denote the principal sym b ols of the co efficie n ts b y σ ( D k ) ∈ C ∞ ( S ∗ M , Mat N ( C )) . Here S ∗ M = T ∗ 0 M / R + stands f o r the cosphere bundle with the pro jection π : S ∗ M → M , where T ∗ 0 M = T ∗ M \ 0 is the cotangen t bundle with the zero s ection deleted. Definition 1.1. The symb ol of the op erator D is a collection σ ( D ) = { σ ( D k ) } of sym b ols of its co efficien ts. If B = X B k T k : C ∞ ( M , C N ) → C ∞ ( M , C N ) (1.2) is another op erator of the fo r m (1.1), then the sym b ol of t he comp osition of (1.1) and (1.2) is determined by the fo r m ula σ ( D B )( k ) = X l + m = k σ ( D l )  ( ∂ g ) l ∗ σ ( B m )  . (1.3) The pro duct of sym b ols in the r ig h t- ha nd side of Eq. (1.3) is called the crossed pro duct of sym b ols. Here ∂ g = ( dg t ) − 1 : T ∗ M → T ∗ M is the co differen tial of g . Definition 1.2. An o p erato r D is el l i p tic if there exists a sym b ol σ ( D ) − 1 with a finite n um b er of nonzero comp onen t s suc h that σ ( D ) σ ( D ) − 1 = 1 , σ ( D ) − 1 σ ( D ) = 1 , where the pro duct of symbols is defined b y Eq. (1.3), while the sym b ol of the iden tit y op erator I d = T 0 is denoted b y 1 . The comp osition form ula (1.3) readily implies the follo wing finiteness theorem. Theorem 1.1. An el liptic op er ator (1.1) is F r e dholm in Sob ole v sp ac es D : H s ( M , C N ) − → H s ( M , C N ) for al l s , and its kernel and c o kernel c onsist of smo oth functions. Pr o of. Indeed, since D is elliptic, t he inv erse sym b ol σ ( D ) − 1 has finitely man y nonzero comp onen ts. Denote b y D − 1 an arbitrary op erator with symbol equal to σ ( D ) − 1 . Then a direct computation sho ws that D − 1 is an inv erse of D mo dulo o perato rs of negativ e order. Remark 1.1. The set of a ll sym b ols is an algebra with resp ect t o the pro duct (1.3). This alg ebra is actually the algebra of matrices, whose entries are elemen ts of the crossed pro duct (e.g., see [18]) o f the a lgebra C ∞ ( S ∗ M ) of smo oth functions o n S ∗ M a nd the group Z . The la tter algebra is denoted by C ∞ ( S ∗ M ) ⋊ Z . In the presen t pap er, w e only consider algebraic crossed pro ducts, whose elemen ts ha v e at most a finite nu m b er o f nonzero comp onen ts. 4 2 Chern characters for crosse d pro ducts 2.1 Chern c haracter Let g : X → X b e a diffeomorphism of a smo oth closed manifold X and E ∈ V ect ( X ) b e a v ector bundle . W e recall some f a cts o f the theory of noncomm utativ e differen tial forms (e.g., see [1, 2, 19]). Noncomm utativ e differenatial forms. Let Λ( X ) b e the algebra o f differen tial fo rms on X with smo oth co efficien ts. F ollowing [2], w e define the space Λ( X , End E ) Z of non- c ommutative forms on X . This space consists of finite sequences a = { a ( k ) } , a ( k ) ∈ Λ( X ) ⊗ Hom( g k ∗ E , E ) , deg a = max k deg a ( k ) , whic h w e repres en t as op erators a = X k a ( k ) T k : Λ( X , E ) → Λ( X , E ) , where for ω ∈ Λ ( X, E ) we set T ω := g ∗ ω ∈ Λ( X , g ∗ E ). This op erator interpretation endo w es the space Λ( X , End E ) Z with the algebra structure . Namely , the pro duct of tw o forms a = P k a ( k ) T k , b = P l b ( l ) T l is the form ab = X k ,l a ( k ) g k ∗ ( b ( l )) T k + l . The subalgebra of forms of zero de gree is denoted by C ∞ ( X , End E ) Z . Remark 2.1. F or a trivial bundle 1 n of rank n , w e ha v e Λ( X , End 1 n ) Z ≃ Λ( X , Mat n ( C )) ⋊ Z . More generally , supp ose w e are giv en a g - bundle E . This means that the mapping g : X → X is extende d to a fib erwise-linear ma pping e g = α g ∗ : E → E , where α : g ∗ E → E is an isomorphism of v ector bundles. In this case we ha v e an isomorphism of algebras Λ( X , End E ) Z ≃ Λ( X , End E ) ⋊ Z , P k a ( k ) T k 7− → P k  a ( k ) T k ( αT ) − k  e T k . Here e T = αT : Λ( X , E ) → Λ( X , E ) (2.1) is the action of the shift op erator on the sections of E , while Λ( X , End E ) ⋊ Z is the crossed pro duct for the s hift opera t o r (2.1). 5 Graded t r ace. W e define a graded trace o n noncomm utativ e forms ranging in Haefliger forms on the manifold. T o this end , we first recall necessary facts ab out Haefliger forms and cohomo lo gy (see [12]). In the de Rham complex (Λ( X ) , d ), consider the sub complex ((1 − g ∗ )Λ( X ) , d ). Definition 2.1. The sp a c e of Haefliger forms o n X is the quotient space Λ( X ) / (1 − g ∗ )Λ( X ) and is denoted by Λ ( X/ Z ). The cohomology of the quotien t complex (Λ( X ) / (1 − g ∗ )Λ( X ) , d ) is Haeflige r c o h o molo gy of X with resp ect to the diffeomorphism g and is denoted b y H ( X/ Z ). Example 2.1. It is clear fr o m the definition, that Haefliger forms are automatically g -inv ariant. Moreo v er, if g N = I d , then the sp ectral decomp osition with resp ect to g sho ws that the Haefliger complex is isomorphic to the comple x (Λ( X ) g , d ) of g - in v ar ian t forms. Therefore, Haefliger cohomology in this case is isomorphic to the cohomology of the quotien t X/ Z N of X by the action of the group generated by the diffeomorphis m g . This g ives an explanation for o ur notation for Haefliger cohomo lo gy . Consider the mapping τ E : Λ( X , End E ) Z − → Λ( X/ Z ) , P k ω ( k ) T k 7− → tr E ( ω (0)) , (2.2) where Λ( X/ Z ) is t he space of Haefliger f o rms on X , while tr E : Λ( X , End E ) → Λ ( X ) is the tra ce of an endomorphism of E . The mapping (2.2) for the tr ivial bundle E = X × C n will b e denoted simply by τ . Lemma 2.1. The mappi n g (2.2) is a gr ade d tr ac e on the algebr a Λ( X , End E ) Z , i.e., τ E  [ ω 1 , ω 2 ]  = 0 , whenever ω 1 , ω 2 ∈ Λ( X , End E ) Z wher e [ , ] stands for the sup er c omm utator [ ω 1 , ω 2 ] = ω 1 ω 2 − ( − 1) deg ω 1 deg ω 2 ω 2 ω 1 . Pr o of. The pro of is straightforw a rd: τ E ( ω 1 T k ω 2 T − k ) = tr E ( ω 1 g k ∗ ( ω 2 )) = tr g − k ∗ E  g − k ∗ ( ω 1 g k ∗ ( ω 2 ))  = tr g − k ∗ E (( g − k ∗ ω 1 ) ω 2 ) = ( − 1) deg ω 1 deg ω 2 τ E ( ω 2 T − k ω 1 T k ) . (2.3) Here ω 1 ∈ Λ( X, Hom( g k ∗ E , E )) and ω 2 ∈ Λ( X, Hom( g − k ∗ E , E )) . The first and the last equalities fo llow from the definition of the pro duct of noncomm utative forms; the second equalit y follow s from the prop erties of Haefliger forms; t he third equalit y follows from the prop erties o f the induced mapping. 6 Noncomm utativ e connection and curv ature form. W e choose a connection in E ∇ E : Λ( X , E ) − → Λ( X , E ) . Giv en a pro jection p ∈ C ∞ ( X , End E ) Z , we define a differen tial op erator of order one ∇ := p ∇ E p : Λ( X , E ) − → Λ( X , E ) . (2.4) A no n c ommutative c onne ction for a pro j ection p is the sum p ( ∇ E + ω ) p of op erator (2.4) and an op erator of m ultiplicatio n b y pω p , where ω ∈ Λ 1 ( X , End E ) Z . Lemma 2.2. F or a nonc ommutative c onne ction ∇ is one has an e quality dτ E ( A ) = τ E ([ ∇ , A ]) , whenever A ∈ Λ( X , End E ) Z is such that pA = A = Ap . Pr o of. 1 . Since τ E is a graded trace, w e see that the righ t-hand side of t he equalit y do es not dep end on the choice of ∇ . Therefore, b elow w e a ssume that ∇ is defined as in (2.4) . 2. F or a trivial bundle with the trivial connection ∇ = p · d · p w e obtain τ ([ p · d · p, A ]) = τ ( pdpA + pdA − d pA ) = τ ( pdA ) = τ ( dA ) = dτ ( A ) . (Here w e use the identities ( d A ) p + A ( dp ) = dA and ( dp ) A + p dA = dA , whic h are obta ined b y differentiation o f Ap = A = pA .) 3. Let us realize a non trivial ve ctor bundle E as a subbundle in the trivial bundle X × C n . Then the left-hand side of the desired equalit y is equal to dτ E ( A ) = dτ ( A ) , whereas the righ t-hand side is equal to τ ([ p ∇ E p, A ]) = τ ([ p ∇ C n p, A ]) = τ ([ p · d · p, A ]) = dτ ( A ) . (In the trivial bundle X × C n w e c ho ose a conne ction ∇ C n equal to the direct sum of the connection in E and some connection in its orthogona l complemen t.) Prop osition 2.1. F or any nonc ommutative c onne ction ∇ the op er ator ∇ 2 : Λ( X , E ) − → Λ( X , E ) is an op er ator of multiplic ation by a 2 -form. This form is denote d by Ω ∈ Λ 2 ( X , End E ) Z . (2.5) and is c al le d the curvatur e form of the nonc ommutative c onne ction ∇ . 7 Pr o of. Let us em b ed E as a subbundle of the trivial bundle X × C n . Then the direct sum of the noncomm utativ e connec tion for p , and some noncomm utative connection for 1 − p is a noncomm utativ e connection of t he form ∇ C n = d + ω , where ω is a matrix-v alued noncomm utativ e 1- form on X . Giv en a section u = pu , a direct computation enables us to compute the curv ature form ( p ∇ p ) 2 u = ( p ∇ C n p ) 2 u = ( p ( d + ω ) p ) 2 u = ( p ( d + ω )) 2 u = = ( pdpdp − ω dp + ω pω + dpω + pdω ) u. Chern c haracter. Definition 2.2. The Chern c har acter form of a pro jection p ∈ C ∞ ( X , End E ) Z is the Haefliger form c h p := τ E  p exp  − Ω 2 π i  ∈ Λ ev ( X/ Z ) , where Ω is the curv ature form (2.5). Prop osition 2.2. The form ch p i s close d and its Haefliger c ohomolo gy cl a ss d o es not dep end on the choi c e o f a n onc ommutative c onne ction and is determine d by the c lass of pr oje ction p in the gr oup K 0 ( C ∞ ( X , End E ) Z ) . Remark 2.2. Here the K 0 -group of C ∞ ( X , End E ) Z is b y definition the Gr o thendiec k group of homotop y classes of matrix pro jections with en tries in this alg ebra. Remark 2.3. Strictly sp eaking, t o define the Chern character on the K -group, we need to consider arbitrary matr ix pro jections ov er C ∞ ( X , End E ) Z , while w e considered only scalar pro jections. How ev er, matrix pro jections can b e considered as elemen ts o f the algebra C ∞ ( X , End( E ⊗ C k )) Z . Therefore, w e do not consider the matrix case to av oid excess iv ely complicated notation. Pr o of. 1 . By Lemma 2.2, the form c h p is closed. Indeed, we ha v e dτ E (Ω k ) = τ E  [ ∇ , Ω k ]  = 0 , since [ ∇ , ∇ 2 k ] = 0 . 2. Let us sho w that t he cohomology class of c h p do es not dep end on the choice of the noncomm utativ e connection ∇ . Let ∇ 0 , ∇ 1 b e t wo noncomm utativ e connections for p . Then their difference is an op erator of m ultiplication by a noncomm utativ e 1- f orm: α := ∇ 1 − ∇ 0 . Consider the homotop y o f noncomm utativ e connections ∇ t = (1 − t ) ∇ 0 + t ∇ 1 . Then we hav e d dt ∇ t = p ( ∇ 1 − ∇ 0 ) p = pαp. Henc e d dt τ E ( ∇ 2 k t ) = k τ E  d dt ∇ 2 t  ∇ 2 k − 2 t  = k τ E  ∇ t , d dt ∇ t  ∇ 2 k − 2 t  = = k τ E  ∇ t ,  d dt ∇ t  ∇ 2 k − 2 t  = d τ E  k  d dt ∇ t  ∇ 2 k − 2 t  . 8 In tegrating t his expression o v er t ∈ [0 , 1], w e obtain: τ E ( ∇ 2 k 1 ) − τ E ( ∇ 2 k 0 ) = dω ′ , (2.6) where ω ′ is some differen tial fo rm. Equalit y (2.6) means that the forms ch p defined in terms of t w o connections ∇ 1 and ∇ 0 , are cohomologous. 3. Let p t , t ∈ [0 , 1] b e a smo oth homotopy of pro jections connecting p 0 to p 1 . W e wan t to show tha t the difference of the corresp onding Chern forms is an exact form. By item 2 of the presen t pro of, it suffices to consider the case, whe re p a cts in a tr ivial bundle with the tr ivial connection ∇ E = d . In this case, the f unctiona l τ E is a differen tial graded trace. Hence, homotop y in v ariance of the Chern form in Haefliger cohomology follo ws from the standard computatio ns (e.g., see [2]). By this prop osition, w e obta in a w ell-defined mapping (Chern c ha r a cter) K 0 ( C ∞ ( X , End E ) Z ) c h − → H ev ( X/ Z ) ⊗ C . Example 2.2. Let E = X × C N b e the trivial bundle and ∇ E = d . Then the no ncommu- tativ e connection is equal to ∇ p = pdp , it s curv ature form is equal to Ω p = ( ∇ p ) 2 = pd pdp . Hence, the Chern ch aracter form is giv en b y the standard form ula c h p = tr [ p exp( − dpdp/ 2 π i )] 0 , (2.7) where ω 0 denotes the co efficien t at T 0 = 1 and tr is the matrix trace. 2.2 Twisted Chern c haracter (Chern c haracter with co efficien ts in a v ector b u ndle) Giv en a diffeomorphism g : X → X as a bov e, we defined Chern c haracter on the K -g r oup K 0 ( C ∞ ( X ) ⋊ Z ). Supp ose now , we are also giv en a g -bundle E ∈ V ect( X ) (i.e., there is an exten tion o f the diffeomorphism g : X → X to a fib erwise isomorphism E → E ). Then on the same K -group we can define Chern ch aracter twisted b y E . T o define this t wisted Che rn c haracter, consider t he algebra homomorphism β : C ∞ ( X ) ⋊ Z − → C ∞ ( X , End E ) ⋊ Z , P k a ( k ) T k 7− → P k ( a ( k ) ⊗ 1 E ) e T k , where e T : C ∞ ( X , E ) → C ∞ ( X , E ) is the shift operato r of E (see (2.1)). Definition 2.3. The twiste d Chern cha r acter is the comp o sition of mappings: c h E : K 0 ( C ∞ ( X ) ⋊ Z ) β ∗ − → K 0 ( C ∞ ( X , End E ) ⋊ Z ) c h − → H ev ( X/ Z ) . The following prop osition is ob vious. Prop osition 2.3. The c omp osition of mappings K 0 ( C ∞ ( X ) ) → K 0 ( C ∞ ( X ) ⋊ Z ) c h E → H ev ( X/ Z ) is e qual to [ p ] 7→ (c h Im p )(ch E ) , wher e Im p ∈ V ect ( X ) is the ve ctor bund le define d by pr oje ction p , while c h is the Chern cha r acter of a v e ctor bund le. 9 2.3 Chern c haracter of an elliptic sym b ol Class of sym b ol in K -theory . Let D b e an elliptic op erator of the form (1.1). T o its sym b ol σ ( D ) w e no w assign an elemen t in K -theory . T o this end, w e extend the diffeomorphism ∂ g : S ∗ M → S ∗ M to a diffeomorphism S ∗ M × S 1 → S ∗ M × S 1 that acts as iden tit y along S 1 . This action defines a crossed pro duct denoted b y C ∞ ( S ∗ M × S 1 ) ⋊ Z . Consider the pro jection P = { P ( ϕ ) } P ( ϕ ) =         I N cos 2 ϕ σ ( D ) sin ϕ cos ϕ σ ( D ) − 1 sin ϕ cos ϕ I N sin 2 ϕ  , f or ϕ ∈ [0 , π / 2] ,  I N cos 2 ϕ I N sin ϕ cos ϕ I N sin ϕ cos ϕ I N sin 2 ϕ  , for ϕ ∈ [ π / 2 , 2 π ] , where ϕ is the co ordinate on the circle. Note that the co efficien ts of P are piecewise smo oth functions of ϕ . Let us define the elemen t [ σ ( D )] = [ P ] ∈ K 0 ( C ∞ ( S ∗ M × S 1 ) ⋊ Z ) , (2.8) where [ P ] is the equiv alence class of the smo othed family of pro jections {P ( ϕ ) } in a neigh b orho o d of submanifolds ϕ = 0 a nd ϕ = π / 2. Chern character. Giv en an elliptic sym b ol and a g -bundle E ∈ V ect( X ), it follows fr o m the constructions of t he previous subsection that w e hav e the twis ted Chern ch aracter c h E [ σ ( D )] ∈ H ev ( S ∗ M / Z ) (2.9) in Haefliger cohomology . In Sec. 3, we define one sp ecial t wisting bundle useful for the index form ula. Let us now obtain one prop erty of t he Chern character ( 2.9) tha t simplifies its computation. Let t = ϕ/ 2 π b e the coordina t e a long on the torus S ∗ M × S 1 . Lemma 2.3. One has c h E [ σ ( D )] = tr E exp  − ∇ 2 tor 2 π i  ∈ H ∗ (( S ∗ M × S 1 ) / Z ) , (2.10) wher e the op er a tor ∇ tor = dt ∂ ∂ t + t ∇ + (1 − t ) σ − 1 ∇ σ , σ = β ( σ ( D )) , (2.11) is define d b y an arbitr ary nonc om mutative c onne ction ∇ in E , and ∇ 2 tor is the curvatur e form. Pr o of. W e ha v e [ σ ( D )] = [ P ] (see (2.8)). Consider the isomorphism U ϕ : Im P (0 ) − → Im P ( ϕ ) , ϕ ∈ [0 , 2 π ] . 10 U ϕ =         I N cos ϕ σ ( − sin ϕ ) σ − 1 sin ϕ I N cos ϕ  , if ϕ ∈ [0 , π / 2] ,  cos( ϕ − π / 2) − sin( ϕ − π / 2) sin( ϕ − π / 2) cos( ϕ − π / 2)   0 − σ σ − 1 0  , if ϕ ∈ [ π / 2 , 2 π ] . W e use this isomorphism and op erator (2.11) to define the noncommutativ e connection ∇ ′ = ( P ( ϕ ) U ϕ ) ∇ tor ( U − 1 ϕ P ( ϕ )) , (2.12) for the pr o jection P = {P ( ϕ ) } on the cylinder S ∗ M × [0 , 2 π ]. W e claim that this expression defines a connection on the to r us S ∗ M × S 1 . T o pro ve this, w e need to c hec k that the co efficien ts of the connection a t ϕ = 0 and ϕ = 2 π are compatible. W e hav e at ϕ = 0 ∇ ′ | ϕ =0 =  1 0 0 0  σ − 1 ∇ σ  1 0 0 0  =  σ − 1 ∇ σ 0 0 0  , while a t ϕ = 2 π w e obtain ∇ ′ | ϕ =2 π =  1 0 0 0   σ − 1 0 0 σ  ∇  σ 0 0 σ − 1   1 0 0 0  =  σ − 1 ∇ σ 0 0 0  . Therefore, we obta in the equality ∇ ′ | ϕ =0 = ∇ ′ | ϕ =2 π , i.e., the co efficien ts of the connection are compatible and, therefore, ∇ ′ is a w ell-defined connection on the t o rus S ∗ M × S 1 . The p o w ers o f the curv ature form of this connection are equal to ( ∇ ′ ) 2 N = ( P ( ϕ ) U ϕ ) ∇ 2 N tor ( U − 1 ϕ P ( ϕ )) , N ≥ 1 . Hence, we ha v e τ E ( ∇ ′ 2 N ) = τ E ( ∇ 2 N tor ) , i.e., w e obtain the desired equalit y (2.10). 3 The e qu ation c h y = Td x Let x b e a complex v ector bundle o ver some space Z . Consider the equation c h y = Td x, (3.1) where Td x is the T o dd clas s of x . Since the Chern ch aracter defines a rational isomor- phism K 0 ( Z ) ⊗ Q ≃ H ev ( Z ) ⊗ Q , Eq. (3.1) has a unique solution y ∈ K 0 ( Z ) ⊗ Q , whic h w e denote for brevit y b y ψ ( x ). Moreo v er, t he mapping x 7→ ψ ( x ) defines an op eration ψ : K 0 ( Z ) ⊗ Q − → K 0 ( Z ) ⊗ Q in K -theory with rational co efficien ts. This op eration is multiplicativ e: ψ ( a + b ) = ψ ( a ) ψ ( b ) (this follo ws from the m ultiplicative prop erty of the T o dd class) and stable: ψ ( a + 1 ) = ψ ( a ) . 11 According to a theorem of A t iyah [20] an y stable op eration in K -theory is a formal p o w er series in G rothendiec k op erations γ j , j = 1 , 2 , ... with rational co efficien ts. In addition, an y m ultiplicativ e op eration is determined by a fo rmal p o w er series f ( x ) = 1 + X k ≥ 1 a k x k as fo llows: 1) the infinite pro duct f ( x 1 ) f ( x 2 ) ... is represen ted as a symmetric f ormal p ow er series in v ariables x 1 , x 2 , ... . He nce, this pro duct is expressed as a formal p ow er series in terms of elemen tary symmetric functions σ 1 ( x 1 , x 2 , ... ) , σ 2 ( x 1 , x 2 , ... ) , ... Y j f ( x j ) = P ( σ 1 , σ 2 , ... ); 2) no w a m ultiplicative op eration for f is obtained if w e replace the elemen t a ry sym- metric functions b y G rothendiec k op erations P ( γ 1 , γ 2 , ... ) . F or the op eration ψ , the corresp onding f o rmal p o wer series is computed in the follo wing prop osition. Prop osition 3.1. The multiplic ative op e r ation ψ is define d by the series ψ ( x ) = ln(1 + x ) x (1 + x ) = 1 + ∞ X n =1 ( − 1) n +1 n ( n + 1) x n . Pr o of. T o compute the co efficien ts of the desired series f , let us take Z = CP N as N → ∞ . W e hav e K ( CP N ) = Z [ x ] / { x N +1 = 0 } , x = [ ε ] − [ 1 ], where ε is the tautological line bundle o v er the pro jectiv e space, and 1 is the trivial line bundle. W e ha v e c h x = e u − 1 , where u = [ CP 1 ] ⊂ H 2 ( CP N ) = Z is the g enerator. Let us use the metho d of undetermined coefficien ts. Let ψ ( x ) = X k ≥ 0 c k x k , c 0 = 1 . Then the equation c h ψ ( x ) = Td x is written as X k c k ( e u − 1) k = u 1 − e − u . Changing the v ariable by the rule e u − 1 = t , this giv es the desired function ψ ( t ) = (1 + t ) ln(1 + t ) t . 12 Note that Grothendiec k op erations can b e expres sed in terms of o peratio ns of direct sum, tensor pro duct and ex terior p o w ers. Therefore, if E is a g -bundle, then ψ ( E ) (as a virtual bundle with rational co efficien ts) can also be considered as a g -bundle. A direct computation give s the follow ing explicit expressions for the op eration ψ on spaces Z of small dimens ion. Prop osition 3.2. The op er ation ψ is e qual to (her e n = rk E ) dim Z ≤ 3 : ψ ( E ) = 1 + E − n 2 ; dim Z ≤ 5 : ψ ( E ) = 3 n 2 − 19 n + 24 24 + ( − 3 n + 13 ) 12 E − 1 6 E ⊗ E + 7 12 Λ 2 E . Pr o of. The series ψ ( x ) define s the symmetric formal p ow er series Y j (1 + x j ) Y j ln(1 + x j ) x j ≡ AB , A = 1 + σ 1 + σ 2 + . . . . Let us expres s the term B in terms o f elemen tary symmetric functions. W e hav e B = Y  1 − x j 2 + x 2 j 3 − x 3 j 4 + . . .  = = 1 − 1 2 X x j + 1 3 X x 2 j + 1 4 X i dim Z . This implies that the terms denoted by do t s in Eq. (3.3) are actually equal to zero pro vided tha t dim Z ≤ 7. Using this remark, w e obtain the fo llo wing expressions for the op eratio n ψ : dim Z ≤ 3: ψ ( E ) = 1 + ( E − n ) / 2. dim Z ≤ 5: ψ ( E ) = 1 + E − n 2 + − 2( E 2 − 2 nE + n 2 ) + 7( E + Λ 2 E − nE + n ( n − 1) / 2 ) 12 = = 3 n 2 − 19 n + 24 24 + ( − 3 n + 13 ) 12 E − 1 6 E ⊗ E + 7 12 Λ 2 E (3.4) and so on. 4 Index theorem The complexification of the cotangen t bundle w ill b e denoted by T ∗ C M = T ∗ M ⊗ C . 14 Theorem 4.1. L et D b e an el liptic op er ator. Then its index is e qual to ind D = Z S ∗ M × S 1 c h ψ ( T ∗ C M ) [ σ ( D )] , (4.1) wher e the op er ation ψ : K 0 ( M ) → K 0 ( M ) ⊗ Q was define d in Se c. 3, and c h stands for the twiste d Chern char acter de fi ne d in Se c. 2. The righ t -hand sid e of Eq. (4.1) will be referred to as the top o lo gic al i n dex of D and denoted b y ind top D . Prop osition 4.1. F or an el lip tic ψ DO D , the top olo gic al inde x is e qual to the top olo gic al index of A tiyah and Singer (se e [22]). Pr o of. In our situation w e ha v e equalities c h ψ ( T ∗ C M ) [ σ ( D )] = c h[ σ ( D )] c h ψ ( T ∗ C M ) = ch[ σ ( D )] Td( T ∗ C M ) . Here the second equalit y follo ws f r o m Prop osition 2.3 and the fact that D is a ψ D O. The second equalit y follows fro m t he definition of ψ . These equalities show that the top ological index in this case is equal t o ind top D = Z S ∗ M × S 1 c h[ σ ( D )] Td( T ∗ C M ) . The last expression is actually the A tiy ah–Singer index fo rm ula for the index of a pseu- do differen tia l op erator D . Theorem 4.1 will b e pro v ed in subsequen t sections. Here w e giv e the sc heme of the pro of. 1. First in Sec. 5 w e giv e a r eduction of the op erator (1.1) to some sp ecial t wo-term op erator, whose index is equal to the index of a certain elliptic ψ DO on a special smo oth closed manifold: the mapping torus of the diffeomorphism g : M → M . 2. Then w e pro v e in Sec. 6 that the index of this ψ DO on the to r us (computed using the A tiy ah–Singer index form ula) is equal to the top ological index of the op erator (1.1) on M . 5 Reduction of a noncomm utativ e op erato r to a ψ DO on a closed manifold 5.1 Reduction to a sp ecial t w o-term op erators In this subsection we obtain a reduction (stable homotop y) of the op erator (1.1) to an op erator of the same t yp e, but of a simpler form. Giv en matrix pro jections p, q ∈ C ∞ ( S ∗ M , Mat N ( C )) , p 2 = p, q 2 = q , o v er S ∗ M , w e c ho ose some ψ DOs with sym b ols equal to p, q and denote them b y P , Q . 15 Definition 5.1. A sp e cial two-term op er ator is an op erator of the form D = QD 0 T P + ( 1 − Q ) D 1 (1 − P ) : H s ( M , C N ) − → H s ( M , C N ) , (5.1) where H s is a Sobolev space, while D 0 : H s ( M , C N ) − → H s ( M , C N ) , D 1 : H s ( M , C N ) − → H s ( M , C N ) , are ψ DOs of or der zero suc h that their symbols define v ector bundle isomorphisms σ ( D 0 ) : ( ∂ g ) ∗ Im p − → Im q , σ ( D 1 ) : Im(1 − p ) − → Im(1 − q ) , (5.2) o v er S ∗ M . These v ector bundles are defined as the ranges of pro j ections p , q , 1 − p, 1 − q . A sp ecial t w o-term op erator is elliptic in the sense o f definition 1.2. Moreo v er, an almost-in v erse operato r can b e defined b y the form ula D − 1 = P T − 1 D − 1 0 Q + (1 − P ) D − 1 1 (1 − Q ) . A homotopy of elliptic op erator s is a family { D t } , t ∈ [0 , 1] of elliptic op erators suc h that the families of their co efficien ts are piecewise smo oth functions of the parameter t and the num b er of nonzero comp onen ts of the family and its almost inv erse family are uniformly b ounded. Tw o elliptic op erators are stably ho m otopic if there exists a homotopy b et wee n their dir ect sums with iden tity op erators acting in se ctions of some bundles . Prop osition 5.1. (cf. [23, 24]) The fol lowing statements hold: 1. An arbitr ary el li p tic op er ator is stably homotopic to some sp e cial two-term op er ator. 2. An arbitr ary sp e ci a l two-term op er ator c an b e r e duc e d by a stable ho motopy an d dir e ct sum with op er ator T ⊕ T ⊕ ... ⊕ T to a dir e ct sum of an el liptic ψ DO a n d a sp e cia l two-term op er a tor of the form D = P D 0 T P + ( 1 − P ) : H s ( M , C N ) − → H s ( M , C N ) , (5.3) i.e., in (5.1) one c an supp ose that Q = P and D 1 = 1 . Pr o of. 1 . Indeed, a direct computation sho ws t hat the homotopy defined in t he pap er [4] giv es the desired result, i.e., the ho mo t op y preserv es ellipticit y and w e obtain a sp ecial t w o -term o perato r at the end of the homotop y . 2. By (5.2), w e hav e the ve ctor bundle isomorphism Im(1 − p ) ≃ Im(1 − q ) . This implies that [Im p ] = [Im q ] ∈ K ( S ∗ M ). If the ranks of the pro jections are large enough (this can be ac hiev ed b y a direct sum of the special t w o -term op erator and some op erator of the f orm T ⊕ T ⊕ ... ⊕ T ), then there exists a v ector bundle isomorphism a : Im q → Im p . Consider an elliptic ψ DO D ′ with the sym b ol σ ( D ′ ) = a ⊕ ( σ ( D 1 )) − 1 : Im q ⊕ Im(1 − q ) − → Im p ⊕ Im (1 − p ) . Then we obtain the factorization D = D 0 T P + D 1 (1 − P ) = ( D ′ ) − 1 ( D ′ D 0 T P + ( 1 − P )) mo dulo compact op erators. This prov es t he prop osition, since a comp osition o f op erator s is stably homotopic to their direct sum. 16 5.2 Reduction to a b oundary v alue problem Let us consider the elliptic sp ecial t wo-term op erator D = D 0 T P + ( 1 N − P ) : C ∞ ( M , C N ) − → C ∞ ( M , C N ) . (5.4) Recall that the ellipticit y conditio n in this case means that the sym b ol of D 0 defines an isomorphism σ ( D 0 ) : ( ∂ g ) ∗ Im σ ( P ) − → Im σ ( P ) (5.5) of v ector bundles ov er S ∗ M . Here the v ector bundles are defined b y the sym b ol of P . On the cylinder M × [0 , 1] with co ordina t es x and t consider the b oundary v alue problem (see [13, 14])       ∂ ∂ t + (2 P − 1 N ) p ∆ M  u = f 1 , u ∈ H s ( M × [0 , 1] , C N ) D 0 T P u | t =0 − u | t =1 = f 2 , f 1 ∈ H s − 1 ( M × [0 , 1] , C N ) , f 2 ∈ H s − 1 / 2 ( M , C N ) , (5.6) where ∆ M is the Laplace op erator defined b y a metric on M . This bo undary v alue problem, denoted for brevit y b y ( D , B ), is elliptic and one has (in op.cit. ) ind D = ind( D , B ) . (5.7) 5.3 Homotop y of the b oundary condition Metho ds o f the theory of b oundary v alue problems (e.g., see [25, 26]) enable one to simplify the b oundary op erator in Eq. (5.6) using homotopies of elliptic b oundary v alue problems. Namely , we start with the r otation homotopy P ( ϕ ) =  (cos 2 ϕ ) P (cos ϕ sin ϕ ) g − 1 ∗ ( D − 1 0 P ) (cos ϕ sin ϕ ) g − 1 ∗ ( P D 0 ) (sin 2 ϕ ) g − 1 ∗ ( P )  , ϕ ∈ [0 , π / 2] , connecting t he almost pro jections P ⊕ 0 and 0 ⊕ g − 1 ∗ ( P ). F or all ϕ ∈ [0 , π / 2] the op erator P ( ϕ ) is an almost-pro jection, i.e., its sym b ol is a pro jection. T o c hec k this prop ert y , it is useful t o represen t this homotop y in the form P ( ϕ ) = U ϕ  P 0 0 0  U − 1 ϕ (5.8) in terms of the family of almost-in ve rtible op erators U ϕ =  (cos ϕ ) P ( − sin ϕ ) g − 1 ∗ ( D − 1 0 P ) (sin ϕ ) g − 1 ∗ ( P D 0 ) (cos ϕ ) g − 1 ∗ ( P )  +  1 N − P 0 0 1 N − g − 1 ∗ ( P )  . Here w e hav e an equality U − 1 ϕ = U − ϕ mo dulo compact op erators. Then w e define the homotop y of op erators D ( ϕ ) =  (cos ϕ ) D 0 g ∗ ( P ) (sin ϕ ) P 0 0  =  D 0 g ∗ ( P ) 0 0 0  g ∗ ( U − 1 ϕ ) . 17 Finally , w e define the homotop y of b oundary v alue problems on the cy linder       ∂ ∂ t + [2 P ( ϕχ ( t )) − 1 2 N ] p ∆ M  U = F 1 , D ( ϕ ) T U | t =0 − U | t =1 = F 2 , (5.9) and denote this homotop y b y ( D ϕ , B ϕ ). Here the unkno wn function and the right-hand sides b elong to the spaces U ∈ H s ( M × [0 , 1] , C 2 N ) , F 1 ∈ H s − 1 ( M × [0 , 1] , C 2 N ) , F 2 ∈ H s − 1 / 2 ( M , C 2 N ), and χ ( t ) is a smo oth nonincreasing function eq ual to 1 if t ≤ 1 / 3 a nd 0 if t ≥ 2 / 3. Lemma 5.1. The homotopy (5.9) c onsists of el liptic b oundary value pr oblem s. Pr o of. (cf. [13 ]). The b oundary conditio n in (5.9) relates the v alues of U at t = 0 and t = 1, i.e., it is a nonlo cal condition. Nonlo cal b oundary v alue problems of this t yp e were considered in [2 7 ]. Let us sho w that this problem is elliptic in the sense o f the cited pap er. Indeed, let us reduce the nonlo cal problem to a lo cal problem in a neighborho o d of the b oundary of the cylinder. T o this end w e in tro duce the follo wing unknow n functions V and W : V ( t ) = T U ( t ) , W ( t ) = U (1 − t ) . (5.10) In a neigh b orho o d of the boundary the system (5 .9) is w ritten in t he following equiv- alen t form              T  ∂ ∂ t + [2 P ( ϕχ ( t )) − 1 2 N ] p ∆ M  T − 1 V ( t ) = T F 1 ( t ) ,  − ∂ ∂ t + [2 P (0) − 1 2 N ] p ∆ M  W ( t ) = F 1 (1 − t ) , D ( ϕ ) V | t =0 − W | t =0 = F 2 . 0 ≤ t < 1 / 2 (5.11) Note that the system (5.11) is already lo cal. The first tw o equations of the system are elliptic. Let us sho w that the b oundary v a lue problem (5.11) is elliptic, i.e., it satisfies the Shapiro–L o patinskii condition (e.g., see [25 ]). T o prov e this, we consider the Calderon bundle [25] (see also [27]) L + ⊂ S ∗ M × C 4 N of the main op erator in (5.11). A direct computation sho ws that this bundle is equal to L + = Im  ( ∂ g ) ∗ σ ( P ( ϕ ))  ⊕ Im  1 2 N − σ ( P (0))  = Im  ( ∂ g ) ∗ σ ( U ϕ P (0) )  ⊕ Im  1 2 N − σ ( P (0))  . (5.12) Here the second equality follows f rom (5.8). The Shapiro–Lopat inskii condition for the problem (5.11) is equiv alen t to the require- men t that the sym b ol of the b oundary op erator, i.e., the v ector bundle homomorphism L + − → S ∗ M × C 2 N , ( V , W ) 7− → σ ( D ( ϕ )) V − W (5.13) 18 is an isomorphism. This requ iremen t is satisfied in our case, since (5.1 2) implies that W ∈ Im [1 2 N − σ ( P (0))] , and the mapping σ ( D ( ϕ )) = σ  D 0 g ∗ ( P ) 0 0 0  g ∗ ( U − 1 ϕ )  : Im  ( ∂ g ) ∗ σ ( U ϕ P (0) )  − → Im  σ ( P (0 ))  is an isomorphism of v ector bundles by (5.8) and ( 5 .5). So, the Shapiro– Lopatinskii condition is satisfied and the problem (5.1 1) is elliptic. Hence, (5.9 ) defines a F redholm op erato r . It follows from Lemma 5.1 and Eq. (5.7) that ind D = ind( D 0 , B 0 ) = ind( D π / 2 , B π / 2 ) . Let us no w consider the b o undary v alue problem           ∂ ∂ t + [2 P ( π 2 χ ( t )) − 1 2 N ] p ∆ M ,h ( t )  U = F 1 ,  0 1 1 0  T U | t =0 − U | t =1 = F 2 , (5.14) where ∆ M ,h ( t ) is the Laplace o p erator on M for a family of metrics h ( t ) smo othly dep end- ing o n t . Lemma 5.2. The pr obl e m (5.1 4) is el liptic and its index is e qual to the inde x of the pr oblem ( D π / 2 , B π / 2 ) . Pr o of. It suffic es to sho w that the linear homotopy connecting thes e tw o boundary v alue problems pr eserv es ellipticit y . First, the linear homotopy b et wee n the main op erators of (5.1 4) and (5.9) consists of elliptic op era t o rs. This follo ws from the fa ct that the op erators differ only b y metrics defining the Laplace o p erator s. Second, for the linear ho mo t o p y b etw een the pro blems (5.14 ) and (5.9 ) , the Calderon bundle L + is constant. Moreo v er, for t he b oundary v alue problems in this homotopy the correspo nding fa mily of v ector bundle homomorphisms (5.13) also do es not change. Hence, the linear homotopy consists of elliptic problems . 5.4 Reduction to a ψ DO on the torus Let M × g S 1 b e the torus of the diffe omorphism g . Recal that the torus of a diffeomorphism is a closed smo oth manifold obtained from the cylinder M × [0 , 1] b y iden tifying its bases with a “t wist” defined b y the diffeomorphism g : M × g S 1 = M × [0 , 1] .  ( x, 0 ) ∼ ( g − 1 ( x ) , 1 )  . (5.15) 19 Consider the family of metrics h ( t ) in (5.14) suc h that one has h ( t ) =  h, if t < 1 / 3 , g ∗ h, if t > 2 / 3 , where h is some fixed metric. This family is a smoot h family of metrics in the fib ers of the bundle M × g S 1 . The problem (5.1 4) defined b y this family of metrics is denoted b y ( D ′ , B ′ ). The op erator D ′ defines an elliptic ψ DO on the torus M × g S 1 : D ′ 0 = ∂ ∂ t + [2 P ( π 2 χ ( t )) − 1 2 N ] p ∆ M ,h ( t ) : C ∞ ( M × g S 1 , E ) − → C ∞ ( M × g S 1 , E ) , (5.1 6) where E ∈ V ect( M × g S 1 ) stands for the v ector bundle with the total space E = ( M × [0 , 1] × C 2 N ) / { ( x, 0 , v 1 , v 2 ) ∼ ( g − 1 ( x ) , 1 , v 2 , v 1 ) } . (5.17) Prop osition 5.2. One has ind( D ′ , B ′ ) = ind D ′ 0 . Pr o of. 1 . Since the op erator of b oundary condition in ( D ′ , B ′ ) is surjectiv e, w e see that the index o f the b oudnary v alue pro blem is equal to the index of the same b oundary v alue problem but with ho mo g eneous b oundary condition. This condition has the form  0 1 1 0  T U | t =0 − U | t =1 = 0 , i.e., it coincides with the condition o f contin uity of the function U ( t ), considered as a section o f the bundle E o v er the torus M × g S 1 . 2. Since the b oundary condition is actually the con tin uity condition, the remaining part of the pro of is standard and we omit it (e.g., see [13]). 6 Comparison of top o logical indices 6.1 Computation of the index of ψ DO on the torus using A tiy ah– Singer form ula The o perato r D ′ 0 is an operato r of the form D ′ 0 = ∂ ∂ t + A ( t ) , i.e., it is defined b y a family A = { A ( t ) } , t ∈ [0 , 1] of op erators on the sections M × { t } of the torus. Moreov er, the fa mily consists of elliptic op erators and the corresp onding family of sym b ols σ ( A ( t ))( x, ξ ) = [2 σ ( P ( t ))( x, ξ ) − 1] | ξ | t (here | ξ | t is the nor m of a co v ector with respect to a family of metrics h ( t )) has a real sp ectrum a t each p oin t ( x, ξ ). It follo ws that the p ositiv e sp ectral subspace of the sym b ol σ ( A ( t ))( x, ξ ) (by definition this subspace is generated b y the eigen ve ctors with p ositiv e 20 eigen v alues) is just the space Im σ ( P ( t ))( x, ξ ). Hence, the f amily of p ositiv e sp ectral subspaces defines a smooth v ector bundle o v er the torus S ∗ M × g S 1 . Denote this v ector bundle b y σ + ( A ) ∈ V ect( S ∗ M × g S 1 ) . (6.1) The follo wing lemma (cf. Theorem 7 .4 in [15]) expresses the index of o p erator D ′ 0 in terms of the bundle (6.1). Lemma 6.1. One has ind D ′ 0 = Z S ∗ M × g S 1 c h[ σ + ( A )] Td( T ∗ C M × g S 1 ) . (6.2) Pr o of. T o mak e the pap er self-con tained, w e give the pr o o f of this fact. 1. The A tiy ah–Singer index formula for D ′ 0 has the form ind D ′ 0 = Z T ( M × g S 1 ) c h [ σ ( D ′ 0 )] Td( T C ( M × g S 1 )) . (6.3) Here and below in the pro of w e iden tify the tangen t a nd con tagen t bundles using some metric. 2. W e ha v e the decomp osition T ( M × g S 1 ) = ( T M × g S 1 ) ⊕ 1 into directions p erp en- dicular and parallel to the generator of the torus. Thus , w e get Td( T C ( M × g S 1 )) = Td( T C M × g S 1 ) . 3. D enote the comp osition of embeddings S M × g S 1 ⊂ T M × g S 1 ⊂ T ( M × g S 1 ) b y i . The normal bundle of this em b edding is, ob viously , a direct sum of t wo one-dimensional trivial bundles. Moreo v er, o ne has [ σ ( D ′ 0 )] = i ! [ σ + ( A )] ∈ K 0 ( T ( M × g S 1 )) , (6.4) where i ! : K 0 ( S M × g S 1 ) → K 0 ( T ( M × g S 1 )) is the direct image mapping corresponding to the em b edding i (see [20]). Applying the Riemann–Ro c h–A tiy ah– Hirzebruch form ula [28] to (6.4), we obtain c h[ σ ( D ′ 0 )] = c h i ! [ σ + ( A )] = i ∗ c h[ σ + ( A )] (the normal bundle of the em b edding i is trivial, hence the T o dd class is equal to one). 4. Substituting the fo rm ulas obtained in items 2 and 3 of the pro of in Eq. ( 6 .3), we obtain ind D ′ 0 = p ∗  i ∗ (c h σ + ( A ) Td( T C M × g S 1 )  = p 0 ∗  c h σ + ( A )) Td( T C M × g S 1 )  , where p ∗ and p 0 ∗ stand fo r G ysin maps in cohomology (in tegration o v er the fundamen tal cycle), induce d b y the pro jections p : T ( M × g S 1 ) → pt and p 0 : S M × g S 1 → pt . The pro of of the lemma is complete. 21 On the cylinder consider the v ector bundle Im σ ( P ) ∈ V ect( S ∗ M × [0 , 1]) and iden tify the fibers of this bundle ov er the compo nents o f the b oundary using the mapping σ ( D 0 ) (see (5 .4)) as f o llo ws V =  ( x, ξ , t, v ) v ∈ Im σ ( P )( x, ξ )  /  ( x, ξ , 0 , v ) ∼ (( ∂ g ) − 1 ( x, ξ ) , 1 , σ ( D 0 )(( ∂ g ) − 1 ( x, ξ )) v )  . (6.5) This space is a v ector bundle V ∈ V ect( S ∗ M × g S 1 ) ov er the torus S ∗ M × g S 1 . Lemma 6.2. One has an isomorphism of ve ctor bund les over S ∗ M × g S 1 : σ + ( A ) ≃ V . (6.6) Pr o of. The pull- ba c ks o f the bundles σ + ( A ) and V to the cy linder S ∗ M × [0 , 1] are equal to Im p  π 2 χ ( t )  and Im p (0) , where p ( ϕ ) = σ ( P ( ϕ )) . The f o rm ula u − 1 π 2 χ ( t ) : Im p  π 2 χ ( t )  − → Im p (0) , wh ere u ϕ = σ ( U ϕ ) (6.7) defines a v ector bundle isomorphism on the cylindeer. This mapping is we ll defined b y Eq. ( 5.8). Let us v erify that the isomorphism (6.7) of v ector bundles ov er the cylinder extends b y con tin uit y t o an isomorphism of bundles on the torus. T o prov e this, it suffices to sho w that the diagram Im p ( π / 2) u − 1 π/ 2 / /   Im p (0 )   Im p (0 ) u − 1 0 / / Im p (0 ) (6.8) is commutativ e. Here the horizon tal mappings are just the restrictions of the isomorphism u − 1 π 2 χ ( t ) at t = 0 (upp er row) and at t = 1 (low er row), while the vertical mappings are just iden tifications of v ector bundles on the b oundary of the cylinde. Recall that these iden t ification mappings are defined in (5.17) and (6.5) and giv e bundles on the torus. Let us pro v e that (6.8) is a commutativ e diagram. W e ha ve u 0 = 1 2 N , u − 1 π / 2 = σ  1 N − P g − 1 ∗ ( D − 1 0 P ) − g − 1 ( P D 0 ) 1 N − g − 1 ∗ ( P )  , p (0) = σ  P 0 0 0  , p ( π / 2) = σ  0 0 0 g − 1 ∗ ( P )  . Let ( x, ξ , 0 , v ) ∈ Im p ( π / 2). This means that ( x, ξ ) ∈ S ∗ M and v ∈ Im p (( ∂ g ) − 1 ( x, ξ )). Then, passing the diag ram (6.8) thro ug h the low er left corner, w e obtain the eleme n ts ( x, ξ , 0 , v )   (( ∂ g ) − 1 ( x, ξ ) , v , 0) / / ((( ∂ g ) − 1 ( x, ξ ) , v ) . (6.9) 22 If we now pass the diagram through the righ t upp er corner, w e obtain the elemen ts ( x, ξ , 0 , v ) / / ( x, ξ , ∗ σ ( D 0 ) − 1 (( ∂ g ) − 1 ( x, ξ )) v )   (( ∂ g ) − 1 ( x, ξ ) , v ) . (6.10) Since the elemen ts obt a ined in the righ t lo w er corner in (6.9) and (6.10) ar e equal, the diagram (6.8) is comm utativ e. Hence, (6.7) defines a n isomorphism on the cylinder, and this isomorphism defines t he desired isomorphism (6.6) on the torus. 6.2 Comparison of the top ological indices of the ψ DO on the torus and of the original op erator Consider the equalities ind D = ind D ′ 0 = Z S ∗ M × g S 1 c h[ σ + ( A )] Td( T ∗ C M × g S 1 ) = Z S ∗ M × g S 1 c h V Td( T ∗ C M × g S 1 ) = Z S ∗ M × g S 1 c h V c h ψ ( T ∗ C M × g S 1 ) = Z S ∗ M × g S 1 c h( V ⊗ ψ ( T ∗ C M × g S 1 )) . (6.11) Here the first equalit y follo ws from the res ults of Sec. 5. The second follo ws from Lemma 6 .1 and t hir d fo llows from Lemma 6 .2. The fourt h equalit y is just the defini- tion of op eration ψ . T o complete the pro of of index theorem 4.1, it suffices to show that the top ological index ind top D of the op erator (5.4) is equal to the right-hand side in (6.11). Note that, generally sp eaking, the elemen t ψ ( T ∗ C M × g S 1 ) ∈ K ( M × g S 1 ) ⊗ Q is a virtual bundle, i.e., a linear com bination of v ector bundles with rational co efficien ts (see Sec. 3). T o simplify the notation, w e shall assume that ψ ( T ∗ C M × g S 1 ) is a v ector bundle, thus omitting the corresp onding sum and co efficien t s. Let F = V ⊗ ψ ( T ∗ C M × g S 1 ) for brevit y . The sections of F are just sections F = { F ( x, ξ , t ) } o f the bundle Im p ⊗ ψ ( T ∗ C M ) ( here p = σ ( P )) on the cylinder S ∗ M × [0 , 1] suc h that σ ( D 0 )( ∂ g ) ∗ F | t =0 = F | t =1 . (6.12) This statement is v erified b y a direc t computation using Eq. (6.5). Let us compute the Chern c haracter of F using formalism of connec tions. Let ∇ p = p ∇ ψ p b e a connection in the bundle (Im p ) ⊗ ψ . Here ψ stands for the bundle ψ ( T ∗ C M ) for brevit y , while ∇ ψ is a connec tion in this bundle. Then the formula ∇ ′ tor = d t ∂ ∂ t + t ∇ p + (1 − t )[ σ ( D 0 )( ∂ g ) ∗ ] − 1 ∇ p [ σ ( D 0 )( ∂ g ) ∗ ] (6.13) 23 defines the connection in F : ∇ ′ tor : Λ( S ∗ M × g S 1 , F ) − → Λ( S ∗ M × g S 1 , F ) . A direct computat io n using (6.12) shows that this connection is well defined. The Chern c haracter form of F is defined b y the class ical form ula c h F = tr  p exp  − ∇ ′ tor 2 2 π i  . (6.14) On the other hand, consider the original op erator (5.4). By Lemma 2.3 we ha v e for this operato r (and E = ψ ( T ∗ C M )) c h ψ ( T ∗ C M ) [ σ ( D )] = tr  exp  − ∇ 2 tor 2 π i  , (6.15) where the noncomm utat iv e connection is equal to ∇ tor = d t ∂ ∂ t + t ∇ + (1 − t )( σ ( D ) − 1 ) ∇ ( σ ( D )) (6.16) and is ex pressed in terms of some connection ∇ in t he bundle C N ⊗ ψ o v er S ∗ M . Let us no w defin e ∇ as ∇ = p ∇ ψ p + (1 − p ) ∇ ψ (1 − p ) (6.17) and recall that σ ( D ) = σ ( D 0 ) T p + (1 − p ) , σ ( D ) − 1 = ( σ ( D 0 ) T ) − 1 p + (1 − p ) (6.18) (see (5 .4)). Substituting the ex pressions (6.17) and (6.18) in Eq. (6.16), we obtain ∇ tor = dt ∂ ∂ t + tp ∇ ψ p + (1 − p ) ∇ ψ (1 − p ) + (1 − t )( σ ( D 0 ) T ) − 1 p ∇ ψ p ( σ ( D 0 ) T ) = = p ∇ ′ tor p + (1 − p )( dt ∂ ∂ t + ∇ ψ )(1 − p ) . This implies that t he curv ature f orm is equal to ( ∇ tor ) 2 = p ∇ ′ tor 2 p + [(1 − p ) ∇ ψ (1 − p )] 2 . Hence the Chern c haracter forms for the connections ∇ ′ tor and ∇ tor differ b y a form that do es not con tain dt . Hence, the in tegra ls of these Chern character forms are equal: Z S ∗ M × [0 , 1] tr  p exp  − ( ∇ tor ) 2 2 π i  = Z S ∗ M × [0 , 1] tr  p exp  − ( ∇ ′ tor ) 2 2 π i  , i.e., w e obtain the des ired equalit y Z S ∗ M × g S 1 c h( V ⊗ ψ ( T ∗ C M × g S 1 )) = Z S ∗ M × [0 , 1] c h ψ ( T ∗ C M ) [ σ ( D )] = ind top D . The pro of of the index theorem 4.1 is no w complete. 24 7 Index form ula in cyclic cohomology In this section w e giv e an inte rpretation of the index for mula (4.1) in terms of cyclic cohomology (se e [1, 29] and for cyclic cohomology of crossed pro ducts [30 – 32]). 7.1 Equiv arian t Ch ern c haracter 1. Chern ch aracter in cyclic c ohomology . Let E ∈ V ect( X ) b e a v ector bundle o v er a smo oth closed orien ted manifo ld X , dim X = n . W e fix a connection ∇ E in E and define fo llo wing [17] the m ultilinear functionals Char k ( E , ∇ E ; a 0 , a 1 , ..., a k ) = = ( − 1) ( n − k ) / 2 (( n + k ) / 2)! X i 0 + i 1 + ... + i k =( n − k ) / 2 Z X tr E  a 0 θ i 0 ∇ ( a 1 ) θ i 1 ∇ ( a 2 ) . . . ∇ ( a k ) θ i k  0  (7.1) k = n, n − 2 , n − 4 , . . . (cf. Jaffe–Lesnievski–Osterw alder formula [33]). Here fo r a non- comm utativ e form ω by ω 0 w e denote the co efficien t a t T 0 = 1, θ = ∇ 2 E is the cu rv ature of the connection, while the op erator ∇ : Λ( X , End E ) Z → Λ( X , End E ) Z is defined as ∇ ( ω ) = ∇ ω − ( − 1) deg ω ω ∇ or more explicitly ∇ ( X k ω k T k ) = X k  ∇ E ω k − ( − 1) deg ω k ω k g k ∗ ( ∇ E )  T k , where the expres sion ∇ E ω k − ( − 1) deg ω k ω k g k ∗ ( ∇ E ) is an op erator of m ultiplication b y a 1-form. It follows from [17] that the collection of functionals { Char k ( E , ∇ E ) } defines a cyclic co cycle o v er the algebra C ∞ ( X , End E ) Z , a nd the class of this co cycle in p eriodic cyclic cohomology Char( E ) = [ { Char k ( E , ∇ E ) } ] ∈ H P ∗ ( C ∞ ( X , End E ) Z ) do es not depend on the c hoice of connection ∇ E . Example 7.1. Let E b e a flat bundle, i.e., θ = 0. Then the Chern c haracter ( 7 .1) has only one nonzero comp onen t Char n ( E ) that is equal to Char n ( E , ∇ E ; a 0 , ..., a n ) = 1 n ! Z X tr E ( a 0 ∇ ( a 1 ) ∇ ( a 2 ) . . . ∇ ( a n )) 0 , dim X = n. (7.2) 2. Relation to the Chern character in Haefliger cohomology . Prop osition 7.1. One has a c ommutative diagr am K 0 ( C ∞ ( X , End E ) Z ) c h u u k k k k k k k k k k k k k k C n h· , Char( E ) i ' ' P P P P P P P P P P P P P P H ev ( X/ Z ) R X / / C , (7.3) 25 wher e C n = (2 π i ) − n/ 2 , c h is the Chern char acter fr om Subse c. 2.1 , R X stands for the inte gr al, and h· , ·i is the p airing of the K 0 -gr oup with cyclic c oh omolo gies. This p airing is define d by the formula h [ p ] , [ ϕ ] i = X k ( − 1) k (2 k )! k ! ϕ 2 k ( p − 1 / 2 , p, . . . , p ) , (7.4) wher e [ p ] ∈ K 0 ( C ∞ ( X , End E ) Z ) , [ ϕ ] ∈ H P ev ( C ∞ ( X , End E ) Z ) , and the cyclic c o c ycle ϕ is extende d to matrix elemen ts in the usual way ϕ l ( m 0 ⊗ a 0 , m 1 ⊗ a 1 , . . . , m l ⊗ a l ) = tr( m 0 m 1 . . . m l ) ϕ l ( a 0 , a 1 , . . . , a l ) . Pr o of. 1 . Let us mak e an additional construction. Namely , w e em b ed the triangle (7.3) in the diagram K 0 ( C ∞ ( X , Mat N ( C )) Z ) c h x x q q q q q q q q q q q q q q q q q q q q q q q q q q q C n h· , Char( X × C N ) i $ $ I I I I I I I I I I I I I I I I I I I I I I I I I I K 0 ( C ∞ ( X , End E ) Z ) O O c h t t h h h h h h h h h h h h h h h h h h C n h· , Char( E ) i * * T T T T T T T T T T T T T T T T T T T T H ev ( X/ Z ) R X / / C (7.5) Here the v ertical mapping K 0 ( C ∞ ( X , End E ) Z ) → K 0 ( C ∞ ( X , Mat N ( C )) Z ) is induce d by the embedding E ⊂ X × C N in the trivial bundle. 2. W e claim that the left and the right triangles of the diag r am (7.5) are comm utativ e. Indeed, let us prov e the commutativit y of the left tria ng le (the commutativit y of the rig h t triangle is o btained similarly). Supp ose that E = Im q ⊂ X × C N , where q is a pro j ection in the trivial bundle. Then we ha v e an isomorphism C ∞ ( X , End E ) Z = q  C ∞ ( X , Mat N ( C )) Z  q . In particular, to a pr o jection p o v er C ∞ ( X , End E ) Z w e assign a pro jection p ′ o v er the algebra C ∞ ( X , Mat N ( C )) Z . Th us, w e hav e c h[ p ] = [c h( p, ∇ E )] = [c h( p ′ , q ∇ E q + (1 − q ) d ( 1 − q ))] = ch[ p ′ ] . Here the first and last equalities follow from t he definition of the Chern character, and the equalit y in the middle follo ws from the equalit y of the corresp onding differen tia l forms. 3. The p erimeter of the diagra m (7.5 ) is also a comm utativ e triangle. Indeed, in the trivial bundle, let us choose the flat connection defined b y the exterior differen tia l d . Then b y Example 2.2 for a pro j ection p o v er C ∞ ( X , Mat N ( C )) Z w e obtain Z X c h[ p ] = 1 ( n/ 2)!  − 1 2 π i  n/ 2 Z X tr  p ( dpdp ) n/ 2  0 . On the other hand, it follows f rom the form ula obtained in the Example 7 .1 that h [ p ] , Char( X × C N ) i = ( − 1) n/ 2 ( n/ 2)! Z X tr  p ( dpdp ) n/ 2  0 . 26 W e see that the la st tw o express ions differ only by the factor C n = (2 π i ) − n/ 2 . This prov es that the p erimeter of the diagram (7.5) is commu tativ e. 4. In the dia gram (7.5), w e pro v ed the comm utativit y of all the t r iangles, except for the low er triangle. Hence, the lo w er triangle is comm utativ e. The pro of of the prop osition is complete. 3. Equiv arian t Chern c haracter [17]. The e quivari a nt Chern char acter of a g - bundle E on X Ch( E ) ∈ H P ∗ ( C ∞ ( X ) ⋊ Z ) is defined as Ch( E ) := β ∗ Char( E ) , where β ∗ : H P ∗ ( C ∞ ( X , End E ) ⋊ Z ) − → H P ∗ ( C ∞ ( X ) ⋊ Z ) is the mapping induced b y the homomorphism of alg ebras β : C ∞ ( X ) ⋊ Z − → C ∞ ( X , End E ) ⋊ Z ; X k ω k T k 7→ X k ( ω k ⊗ 1 E ) e T k . There is an analo gue of the comm utativ e diagram (7.3 ) fo r the equiv ariant Chern c harac- ter. Namely , one has K 0 ( C ∞ ( X ) ⋊ Z ) c h E v v m m m m m m m m m m m m m C n h· , Ch( E ) i & & N N N N N N N N N N N N H ev ( X/ Z ) R X / / C . (7.6) 7.2 Index form ula in cyclic cohomology Giv en a g -bundle E ∈ V ect( X ) o v er a smo oth closed orien ted manifold X , w e define the equiv arian t T odd class T o dd( E ) ∈ H P ∗ ( C ∞ ( X ) ⋊ Z ) as T o dd( E ) := Ch( ψ ( E )) , where ψ is the op eration in ratio nal K -theory defined in Sec. 3. 1. Index form ula. Theorem 7.1. F or an el liptic op er a tor D one has an index formula ind D = (2 π i ) − n h [ σ ( D )] , T o dd( π ∗ T ∗ C M ) i , dim M = n, (7.7) wher e π : S ∗ M × S 1 − → M is the pr oje ction and the br ackets h , i stand for the p airin g of K -the ory with c yclic c ohomolo gy (se e (7.4) ). Pr o of. The index formula (4.1) g iv es us ind D = Z S ∗ M × S 1 c h ψ ( π ∗ T ∗ C M ) [ σ ( D )] . (7.8) 27 Using the comm utativ e diagram (7.6) and the definition of the equiv ariant T o dd class, w e can rewrite the right-hand side in (7.8) in the desired form Z S ∗ M × S 1 c h ψ ( π ∗ T ∗ C M ) [ σ ( D )] = (2 π i ) − n h [ σ ( D )] , Ch( ψ ( π ∗ T ∗ C M )) i = = (2 π i ) − n h [ σ ( D )] , T o dd( π ∗ T ∗ C M ) i . 2. A sp ecial case. Supp ose that the T o dd class Td( T ∗ C ( M × g S 1 )) of the complexifi- cation o f the cotangen t bundle of the twis ted torus is equal to one. Then it turns out that in this case one can replace the class T o dd( π ∗ T ∗ C M ) in (7.7) simply by the transv erse fundamen tal class of t he manifold S ∗ M × S 1 in the sense of [1 9]. This enables one to write the index form ula in the form ind D = ( n − 1)! (2 π i ) n (2 n − 1)! Z S ∗ M tr( σ − 1 dσ ) 2 n − 1 0 , σ = σ ( D ) . (7.9 ) The index form ula (7.9) is a corollary of the follo wing mor e g eneral statemen t. Prop osition 7.2. Supp ose that in the ho m olo gy class Poinc ar´ e dual to the T o d d class Td( T ∗ C M × g S 1 ) ther e exists a r epr esentative 1 of the form ω 7− → z   Z S 1 ω   , ω ∈ Λ( S ∗ M × g S 1 ) , wher e z is a close d g -invarian t curr e nt on S ∗ M . Then the e quivariant T o dd class in the index formula (7.7) c a n b e r epl a c e d by the c ol le ction of cyclic c o cycles with the c o m p onents ( a 0 , ..., a 2 k ) 7− → (2 π i ) n − k (2 k )! z ( a 0 da 1 da 2 ...da 2 k ) , k = 0 , 1 , ..., n. Pr o of. The pro of of this prop osition is similar t o the pro of o f index form ula (7.7). The main difference is tha t instead o f equalit y (6.11) one use s equalities of t he form ind D = ind D ′ 0 = Z S ∗ M × g S 1 (c h V ) Td ( T ∗ C M × g S 1 ) = z   Z S 1 c h V   . (7.10) Example 7.2. Supp ose that Td( T ∗ C M × g S 1 ) = 1. Then w e can take the curren t z of degree 2 n − 1 defined by integration o v er S ∗ M . Then Prop osition 7.2 giv es the index form ula (7.9) (aft er a standard in tegration o v er S 1 ). This remark applies, fo r instance, in the case of elliptic o perato rs for a diffeomorphism of the sphere in the connected comp onen t of the iden t it y (see [1 0, 11]). 1 Here the homolog y gr oup is trea ted in terms of closed de Rham currents (see [34]). 28 Prop osition 7.2 can b e applied if g is an isometry . In this case we define the curren t z b y the formula z ( ω ) = Z S ∗ M ω ∧ Td( T ∗ C M ) , ω ∈ Λ( S ∗ M ) , where Td( T ∗ C M ) is the differen tial form represen ting the T o dd class using a g -inv ariant metric. In this case, w e obtain the inde x f o rm ula first prov ed in [2]. 8 Examples. Remarks 8.1 Example. Op erators on the torus T 3 The index form ula (7.7), despite its compact and elegan t form, often leads to serious com- putational difficulties, w hen one really needs to compute the index of a sp ecific op erator. T o solve this problem, it is sometimes useful to simplify the form ula so that the simplified form ula could really b e use d to compute the desired n um b er. In this subsection, we exibit a pro cedure of this form for a relativ ely simple op erator related to the D irac op erator. The answ er w e obtain is quite suitable to obtain explicit n umerical expression for the index of the problem. 1. Consider the to rus T 3 = R 3 / 2 π Z 3 with co ordinates x = ( x 1 , x 2 , x 3 ) and t he diffeo- morphism 2 g : T 3 − → T 3 , g   x 1 x 2 x 3   =   2 1 0 1 1 0 0 0 1     x 1 x 2 x 3   . Consider the Dirac o perato r on T 3 3 X j =1 c j  − i ∂ ∂ x j  : C ∞ ( T 3 , C 2 ) − → C ∞ ( T 3 , C 2 ) , (8.1) where c 1 =  0 1 1 0  , c 2 =  0 − i i 0  , c 3 =  1 0 0 − 1  stand for Pauli matrices . The Dirac op erator is elliptic and self-a dj o in t in the space L 2 . Therefore, it has a discrete real sp ectrum, while the eigen v alues hav e finite m ultiplicities. Consider the p ositiv e sp ectral pro jection for the D irac op erator, i.e., the orthogonal pro- jection on the subspace gene rated by eigenfunctions of the Dirac op erator s with p ositiv e eigen v alues. This pro jection is denoted b y P and is a ψ DO (se e [36]) of o rder zero. Theorem 8.1. L et f ∈ C ∞ ( T 3 , Mat N ( C )) b e a function r anging in invertible matric es. Then the op er ator D = ( f ⊗ 1)(1 ⊗ P ) T (1 ⊗ P ) + 1 ⊗ (1 − P ) : H s ( T 3 , C N ⊗ C 2 ) − → H s ( T 3 , C N ⊗ C 2 ) , (8.2) 2 This diffeomo rphism is the “Ar no ld’s ca t map” [35] ac ting along x 1 , x 2 and the identit y map along x 3 . 29 wher e T = g ∗ is the shift op er ator for g , is F r e dholm fo r al l s and its index is e qual to ind D = 1 (2 π i ) 2 3! Z T 3 tr( f − 1 d f ) 3 . (8.3) Let us write the operator (8.2) simply as D = f P T P + 1 − P omitting the tensor pro ducts. 2. Let us prov e that D is elliptic. T o this end, w e first compute the sym b ol of P . The sym b ol of the Dirac op erator (8.1) is eq ual to c ( ξ ) = c 1 ξ 1 + c 2 ξ 2 + c 3 ξ 3 ∈ Mat 2 ( C ) , where ξ = ( ξ 1 , ξ 2 , ξ 3 ) stand f or v ariables dual to x . In what follows , it is useful to write the following Clifford iden tity (e.g., see [37]): c ( ξ ) c ( ξ ′ ) v + c ( ξ ′ ) c ( ξ ) v = 2( ξ , ξ ′ ) v , ξ , ξ ′ ∈ R 3 , v ∈ C 2 , (8.4) where in the righ t-hand side of (8.4) w e ha v e inner pro duct of v ectors. In particular, Eq. ( 8.4) implies t hat the matrix p ( ξ ) = 1 + c ( ξ ) 2 , | ξ | = 1 is a rank one pro jection. Moreo v er, this pro jection is just the p ositiv e sp ectral pro jection of the sym b ol c ( ξ ) of the Dirac o perato r. W e extend this function to a degree zero homogeneous function in ξ . Then t he results of the pap er [36] giv e the equalit y σ ( P ) = p. W e are no w ready to prov e tha t D is elliptic. Consider the mapping u ( ξ ) = p ( ξ ) : Im( ∂ g ) ∗ p ( ξ ) − → Im p ( ξ ) . (8.5) W e claim that the mapping ( 8 .5) is in v ertible (cf. [13]). Indeed, let us consider the con v erse, i.e., suppose that for some ξ w e ha v e a nonzero v ector v ∈ Im( ∂ g ) ∗ p ( ξ ) = Im p ( g − 1 ξ ) such that p ( ξ ) v = 0 . (8.6) In terms of Clifford m ultiplication, condition (8.6) is written as: c ( ξ ) v = − v , c  g − 1 ξ | g − 1 ξ |  v = v . Substituting these t w o fo r m ulas in (8.4), w e get − 2 v = 2  ξ , g − 1 ξ | g − 1 ξ |  v or cos( ξ , g − 1 ξ ) = − 1 , (8.7) 30 i.e., the v ectors ξ and g − 1 ξ form an angle equal to π . But this can not b e true, sin ce the matrix g − 1 has no negativ e eigen v alues. This contradiction sho ws that the mapping (8.5) is an isomorphism. Denote b y U − 1 a ψ DO on T 3 suc h that the restriction of its sym b ol to the subspace Im p ( ξ ) coinc ides with u ( ξ ) − 1 . W e claim that the op erator B = T − 1 f − 1 U − 1 P + 1 − P is an almost inv erse of D (i.e., in v erse up to op erators o f negat ive order). Indeed, for example, let us compute the comp osition o f sym b ols: σ ( D ) σ ( B ) = ( f pT p + 1 − p )( T − 1 u − 1 pf − 1 + 1 − p ) = = f pT pT − 1 u − 1 pf − 1 + (1 − p ) + (1 − p ) T − 1 u − 1 pf − 1 = = f p [( ∂ g ) ∗ p ] u − 1 pf − 1 + (1 − p ) + (1 − p ) T − 1 T pT − 1 u − 1 pf − 1 = = f pf − 1 + (1 − p ) + (1 − p ) pT − 1 u − 1 pf − 1 = p + 1 − p = 1 . (8.8 ) The equality σ ( B ) σ ( D ) = 1 is obtained similarly . Th us, D is elliptic and F redholm b y Theorem 1.1. 3. By the index theorem 4.1 the a na lytic index of D is equal to the top ological index of its sym b ol. Let us compute the top ological index of σ ( D ). The sym b ol σ ( D ) has the factorization σ ( D ) = σ 0 σ 1 , σ 0 = pf p + 1 − p, σ 1 = pT p + ( 1 − p ) , (8.9 ) in to tw o elliptic sym b o ls, where σ 0 do es not con tain shift T . Let us compute the top ologi- cal indices of these sym b ols. The index of σ 0 coincides with the Atiy ah–Singer to polo gical index a nd is equal to (see [3 8 , 39]) ind top σ 0 = Z S ∗ T 3 c h[ f ] c h(Im p ) Td( T ∗ C T 3 ) , where [ f ] ∈ K 1 ( T 3 ) is the class o f f in the o dd K -gr oup. F urt her, w e get Z S ∗ T 3 c h[ f ] c h (Im p ) Td( T ∗ C T 3 ) = Z S ∗ T 3 c h[ f ] c h(Im p ) = Z T 3 c h[ f ] Z S 2 c h(Im p ) = C Z T 3 tr( f − 1 d f ) 3 , where C = ((2 π i ) 2 3!) − 1 . Here w e first noted that the ta ngen t bundle of the torus is trivial and replaced the T o dd class b y one. Then, w e used the decomp osition S ∗ T 3 = T 3 × S 2 with t he co ordinates x, ξ on the factors. Moreov er, since f depends only on x , and p dep ends only on ξ , the inte gral o v er T 3 × S 2 is just the pro duct of an in tegral ov er T 3 and an inte gral o v er S 2 . In the next to the last equalit y , the Chern c haracter in the first factor is represen ted by a differen tial form and w e noted t hat Im p is the Bott bundle on S 2 (see [20] and [40]) a nd one has Z S 2 c h Im p = 1 . 31 So, w e obtain ind top σ 0 = 1 (2 π i ) 2 3! Z T 3 tr( f − 1 d f ) 3 . (8.10) Prop osition 8.1. One has ind top σ 1 = 0 . Pr o of. By Eq. (7.9), the top ological index of σ 1 is equal to ind top σ 1 = 2! (2 π i ) 3 5! Z S ∗ M tr( σ − 1 1 dσ 1 ) 5 0 . (8.11) Let us compute this in tegral. The sym b ol σ 1 ∈ C ∞ ( S ∗ T 3 , Mat 2 ( C )) ⋊ Z is constan t in x . Th us, its differen tial dσ 1 ∈ Λ 1 ( S ∗ T 3 , Mat 2 ( C )) ⋊ Z do es not con tain differen tials dx j . The pro duct σ − 1 1 dσ 1 also ha s no differen tials dx j . This uses the fact that g is a linear diffeomorphism. The same reasoning sh o ws that the form ( σ − 1 1 dσ 1 ) 5 (see (8 .11)) also do es not con tain differen tials dx j . On the other hand, the degree of this form is equal t o fiv e. Therefore, this form is identically zero. Th us, ( 8 .11) implies that the top olog ical index of σ 1 is zero. The form ula (8.3) no w follows fr om Eq. ( 8 .10) and Prop osition 8.1. This completes the pro of of Theorem 8.1. 4. Let us giv e a direct pro of of Theorem 8.1 . Namely , let us first prov e that D is elliptic. One has a n equality (cf. (8 .9)) mo dulo compact operat ors D = D 0 D 1 , D 0 = f P + (1 − P ) , D 1 = P T P + (1 − P ) . Here D 0 is an elliptic ψ D O and its index (computed b y the A tiy ah–Singer f o rm ula) is equal to the righ t-hand side in (8.10) (see the ab o v e computation). Thus , to prov e Theorem 8.1, it suffices to show that D 1 is a F redholm op erator of index zero. Prop osition 8.2. The op er ator D 1 = P T P + (1 − P ) is invertible. Pr o of. Let us represen t functions on the torus as F ourier series P k a k e i ( k, x ) , where k = ( k 1 , k 2 , k 3 ) ∈ Z 3 , and ( k , x ) = k 1 x 1 + k 2 x 2 + k 3 x 3 . In this notation, we ha v e P X k a k e i ( k, x ) ! = X k 6 =0 a k p ( k ) e i ( k, x ) , ( 8 .12) where p ( k ) is the v a lue of the function p ( ξ ) at ξ = k . In this represen tation, the shift op erator is equal to T X k a k e i ( k, x ) ! = X k a k e i ( gk ,x ) , 32 since g has a symmetric matrix. L et U − 1 b e the op erator defined b y a form ula of the for m (8.12) using the sym b ol u − 1 ( ξ ). In this case t he mappings P : Im T P T − 1 → Im P and U − 1 : Im P → Im T P T − 1 are in v erse to eac h other. It follow s that the op erator B = T − 1 U − 1 P + 1 − P is the inv erse of D 1 . The pro of s of these statemen ts are similar to the computation (8.8). 8.2 Remark. Sp ecial t w o-term op erators as op erators in sub- spaces Let us giv e here a metho d of computing the index of sp ecial tw o-term op erators using elliptic theory in subspaces defined by pseudodiff erential pro jections ( see [41, 42]). W e write a sp ecial t w o-term o perato r (5.1) as D = QD 0 T P + ( 1 − Q ) D 1 (1 − P ) : C ∞ ( M , C N ) − → C ∞ ( M , C N ) . (8.13) Without loss of generalit y , we can assume that P and Q are pro jections P 2 = P , Q 2 = Q . In this case, the opera t or D is a direct sum D = QD 0 T P ⊕ (1 − Q ) D 1 (1 − P ) of op erators acting in subspaces define d b y the pro jections P, Q, 1 − P , 1 − Q . Using this decomp osition, we can compute the index of D . Indeed, we get ind D = ind[ QD 0 T P : Im P → Im Q ] + ind[(1 − Q ) D 1 (1 − P ) : Im(1 − P ) → Im (1 − Q )] = = ind[ D 0 : Im g ∗ P → Im Q ] + ind[ D 1 : Im(1 − P ) → Im (1 − Q )] . (8.14) Here in the last eq ualit y we used the fact that T defines an isomorphism of the ra nges o f pro j ections P and g ∗ P = T P T − 1 . An application of the index form ulas obtained in the pap ers [41, 42] to the op erators in (8 .14) giv es an index form ula for D . T o form ulate the result, consider the in volution α : T ∗ M → T ∗ M , α ( x, ξ ) = ( x, − ξ ) and for a ψ DO A let α ∗ A denote an y ψ D O with the sym b ol α ∗ σ ( A ). Prop osition 8.3. L et a sp e cial two-term op er a tor ( 8 .13) b e el liptic, the manifold M b e o dd-dimen sional, an d the pr o j e ctions P , Q b e even, i.e., they satisfy the c ondition α ∗ σ ( P ) = σ ( P ) , α ∗ σ ( Q ) = σ ( Q ) . Then one has an e quality ind D = 1 2 ind  D 0 ( α ∗ ( D 0 ) − 1 ) Q + D 1 ( α ∗ ( D 1 ) − 1 )(1 − Q )  , (8.15) wher e the op er a tor in the squar e br ackets is an el lip tic ψ DO on M . 33 Pr o of. 1 . Application of the index form ula from the pap er [41] to the op erators D 0 : Im g ∗ P → Im Q and D 1 : Im(1 − P ) → Im (1 − Q ) gives us ind( D 0 : Im g ∗ P → Im Q ) = 1 2 ind  D 0 ( α ∗ ( D 0 ) − 1 ) Q + (1 − Q )  + d ( g ∗ P ) − d ( Q ) , (8.16) ind( D 1 : Im(1 − P ) → Im(1 − Q )) = = 1 2 ind  Q + D 1 ( α ∗ ( D 1 ) − 1 )(1 − Q )  + d (1 − P ) − d (1 − Q ) , (8.17) where d is the homotopy inv a r ia n t of ev en pseudo differen t ia l pro jections constructed in [41]. Adding the last tw o expressions (8.16) and (8.17), we obtain the fo llowing expres- sion for the index of D : ind D = 1 2 ind  D 0 ( α ∗ ( D 0 ) − 1 ) Q + D 1 ( α ∗ ( D 1 )) − 1 )(1 − Q )  + + ( d ( g ∗ P ) + d ( 1 − P ) ) − ( d ( Q ) + d (1 − Q )) . (8.18) The cited pap er con tains the followin g properties of the functional d : d ( Q ) + d (1 − Q ) = 0 and d ( g ∗ P ) = d ( P ) . Hence, the last t w o terms in Eq. (8.18) are equal to zero and w e obtain the desired index form ula (8.15). There is a n analog of this prop osition for so-called o dd pro jections on ev en-dimensional manifolds (see [42]). 8.3 Remark. A generalization of the notion of ellipticit y In [7, 24], a different condition of ellipticit y o f op erators (1.1) is used. This condition do es not require that the n um b er of nonzero comp onen ts of the in v erse sym b ol is finite. In this situation, the sym b o l is naturally an elemen t of the C ∗ -crossed pro duct C ( S ∗ M ) ⋊ Z (see [18]) of t he a lg ebra of contin uous sym b ols o n S ∗ M b y the action of the diffeomorphism g , and the ellipticit y is just the in vertibilit y in this C ∗ -crossed pro duct. On the other hand, it was sho wn in the pap ers [4, 1 3] that an elliptic op erator in this sense is stably homotopic t o an op erator elliptic in the sense o f Definition 1.2. This implies that to obtain an index formula for this class of op erators, it suffices to extend the cyclic co cycle T o dd ∈ H P ∗ ( C ∞ ( S ∗ M × S 1 ) ⋊ Z ), see (7.7) to some lo cal a lg ebra A such that C ∞ ( S ∗ M ) ⋊ Z ⊂ A ⊂ C ( S ∗ M ) ⋊ Z or, in more in v arian t form, to define a class T o dd ∈ H P ∗ ( A ) that is the pull-bac k of the class T o dd ∈ H P ∗ ( C ∞ ( S ∗ M × S 1 ) ⋊ Z ) under the em b edding C ∞ ( S ∗ M × S 1 ) ⋊ Z ⊂ A . Suc h exten tions are kno wn fo r man y interes ting classes of diffeomorphisms, for example see [4, 19, 4 3 , 44]. Therefore, w e obtain an index form ula of the ty p e (7.7) for op erators elliptic in the sense of [24] for these clas ses o f diffeomorphisms. 34 References [1] A. Conne s. Nonc omm utative ge ometry . Academic Press Inc., San Diego, CA, 1994. [2] V. E. Nazaikinskii, A. Y u. Savin, and B. Y u. Sternin. El liptic the ory a nd nonc om- mutative ge ometry . Birkh¨ aus er V erlag, Basel, 2008. [3] A. Y u. Sa vin and B.Y u. Sternin. 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