On elliptic differential operators with shifts: II. The cohomological index formula

On Elliptic Differen tial Op erators with Shifts I I. The Cohomologi ca l Index F orm ula V. E. Nazaik inskii, A. Y u. Sa vin, and B. Y u. Ster n i n In tro duction This pap er is a con tin uation o f [1], where w e ha v e studied a general class of (pseudo)differential op erators with nonlo cal co efficien ts, referred to as op era- tors w ith shifts, and obtained a lo cal index formula (i.e., a fo rm ula expressing the index as t he in tegral of a differen tial form explicitly determined by the principal sym bo l of t he op erator ) for matrix elliptic op era t o rs of this kind. In the presen t pap er w e finish the business b y establishing a cohomo lo gical in- dex form ula of Atiy ah–Singer t ype for elliptic differen tial op erators with shifts acting b etw een section space s of arbitrary ve ctor bundles. The k ey step is the construction o f closed g raded traces o n certain differen tia l algebras o v er the sym b ol alg ebra for this class of op erators. W e do not f o rmally assume the reader to b e fa miliar with [1] as far as definitions a r e concerned but freely use the results obtained there. W e also do not repro duce the discussion o f general mot iv ations for this researc h, whic h, as well as the bibliog r a ph y , can b e found in [1]. Ac kno wledgemen ts. The researc h w a s supp orted in part b y RFBR gran ts nos. 05-01-00982 and 06- 01-00098 and DF G grant 436 RU S 1 1 3/849/0-1 r “ K - theory a nd noncommutativ e g eometry of stratified manifolds.” The authors thank Professor Schrohe and Leibniz Univ ersit¨ at Hanno ver for kind hospitality . 1 Elliptic op erators with s hifts 1.1 Pseudo differen tial op erators with shifts The group Γ . Let M b e a compact orien ted Riemannian manifold with- out b oundary , and let Γ b e a coun table dense subgroup o f a L ie group Γ of orien tation-preserving isometries of M . The natura l a ction o f Γ on functions 1 on M will b e denoted b y T , so that [ T g u ]( x ) = u ( g − 1 ( x )) , x ∈ M . W e assume tha t Γ satisfies the fo llo wing tw o conditions: 1. ( Po lynomial gr owth .) The group Γ is finitely generated, and the n um b er of distinct elemen ts o f Γ represe ntable b y w ords of length ≤ k in some finite system of generators grows at most p olynomially in k . In what follows , w e fix some system of g enerato r s and denote by | g | the minim um length of w o rds represen ting g ∈ Γ. 2. ( D iophantine pr op erty .) Let fix( g ) b e the set of fixed p oin ts of g ∈ Γ. The estimate dist( g ( x ) , x ) ≥ C | g | − N dist( x, fix( g )) holds for some N , C > 0 and for all x ∈ M and g ∈ Γ. Here dist( x, fix( g )) is the Riemannian distance b etw een x a nd the set fix( g ), a nd b y con- v ention we set dist( x, fix( g )) = 1 if fix ( g ) is empt y . Matrix op erators. Matrix pseudodifferen tial op erator s with shifts, ı.e., ΨDO with shifts acting on v ector functions on M , can b e describ ed as follo ws. (F or more detail, see [1], where also further bibliogr a phical references can b e found.) A matrix ΨDO of order m with shifts has the form D = X g ∈ Γ T g D g , (1) where D g is a classical ΨDO of order m on M and the opera t o rs D g rapidly deca y as | g | → ∞ in the natural F r´ eche t top ology on the set of m th-order ΨDO. Op erators on sections of v ector bundles. Pseudodifferential op erators with shifts acting on sections of vec tor bundles are an easy generalization of matrix op erators. T o define them, one should lo calize in to neigh b orho o ds where the bundles are trivial. The only differen ce with the case of pseudo differ- en tia l op era t o rs without shifts is that our op erators are no longer lo cal, so w e cannot lo calize into a neigh b orho o d of the diagona l; hence t w o neighborho o ds, instead of one, in the subseq uen t argumen t. Let E and F b e finite-dimensional complex vector bundles on M . A linear op erator D : C ∞ ( M , E ) − → C ∞ ( M , F ) (2) is called an m th-order ΨDO with shifts if for any trivializations of E and F ov er some neigh b orho o ds U E , U F ⊂ M , respective ly , and any functions 2 ϕ ∈ C ∞ 0 ( U E ) and ψ ∈ C ∞ 0 ( U F ) the op erator ψ D ϕ is a n m th-order matrix ΨDO with shifts of the form (1 ) . W e p oint out that no action of Γ on the bundles E and F is needed in this definition. The linear space of m th-order pseudo differen tia l op erators (2) with shifts will be denoted b y Ψ m ( E , F ) Γ . If E , F and H are three vec tor bundles on M , then the mu ltiplication o f op erators induces a w ell-defined bilinear mapping Ψ m ( E , F ) Γ × Ψ m ′ ( F , H ) Γ − → Ψ m + m ′ ( E , H ) Γ . Just as for matrix op erators, one readily prov es that an m th-order ΨD O with shifts is a con tin uous o perato r of order m in the Sob olev spaces of sections of E and F . 1.2 Sym b ol, ellipticit y , and F redholm prop ert y Sym b ol: the matrix case. First, let us recall what happ ens in case the bundles E a nd F are trivial. F or the n × n ′ matrix op erator (1), the sym b ol is defined b y the form ula σ ( D ) = X g ∈ Γ T ∂ g σ ( D g ) : L 2 ( S ∗ M , C n ) − → L 2 ( S ∗ M , C n ′ ) , (3) where t he co differen tial ∂ g : S ∗ M → S ∗ M is the map induced b y g (it acts as g along the base and as ( ( dg ) ∗ ) − 1 in the fib ers of S ∗ M ). Sym b ol: the general case. If the o perato r (2) is a usual pseudo differen tial op erator, then its sym b ol is a bundle homomorphism π ∗ E → π ∗ F , where π : S ∗ M → M is t he natural pro jection. F or pseudo differen tial op erators with shifts, whic h a re highly nonlo cal, this is no longer the case, a nd their sym b ols are defined as homo mo r phisms of section spaces of the bundles π ∗ E and π ∗ F rather than of the bundles themselv es. Definition 1. The symb ol of the op erator (2) is the op erator σ ( D ) : L 2 ( S ∗ M , π ∗ E ) − → L 2 ( S ∗ M , π ∗ F ) (4) suc h that for an y trivializations of E and F ov er some neighborho o ds U E , U F ⊂ M , resp ectiv ely , and an y functions ϕ ∈ C ∞ 0 ( U E ) a nd ψ ∈ C ∞ 0 ( U F ) the operato r ψ σ ( D ) ϕ is the sym b ol o f the op erator ψ D ϕ . 3 One can readily v erify that the sym b ol of a ΨDO with shifts is w ell defined. The space of sym b ols of ΨDO with shifts a cting b et w een section spaces of v ector bundles E and F will b e denoted by C ∞ ( S ∗ M , Hom( E , F )) Γ . F or E = F , w e use the no t ation C ∞ ( S ∗ M , End( E )) Γ , and for scalar sy m b ols write C ∞ ( S ∗ M ) Γ , just as in the first part of the pap er. A generalization of the argumen t giv en there shows that C ∞ ( S ∗ M , End( E )) Γ is a lo cal subalgebra of the C ∗ -algebra B L 2 ( S ∗ M , E ). Hence if a sym b ol σ ∈ C ∞ ( S ∗ M , Hom( E , F )) Γ is inv ertible (as an op erator in L 2 ), then one necessarily has σ − 1 ∈ C ∞ ( S ∗ M , Hom( F , E )) Γ . Definition 2. An op erator D ∈ Ψ m ( E , F ) Γ is said to b e el liptic if its sym b ol σ ( D ) is in v ertible. As usual, o ne has the finiteness theorem. Theorem 3 (the finiteness theorem) . An op er ator D ∈ Ψ m ( E , F ) Γ is F r e d- holm if and only if its symb ol is invertible. 2 The ind e x t h eorem In this sec tion we obtain a cohomolog ical index formula for elliptic op erators D ∈ Ψ m ( E , F ) Γ . First, w e shall in tro duce the elemen ts that o ccur in this form ula. 2.1 Some ob jects asso ciated with the group Γ W e represen t the group Γ as the disjoin t union Γ = G g 0 h g 0 i of conjugacy classes and arbitrar ily fix an elemen t, g 0 , in each conjugacy class h g 0 i . In what follows, the sym b ol g 0 is inv a riably used to denote t his fixed represen tative . By C g 0 w e denote the cen tralizer of g 0 in Γ: C g 0 = { h ∈ Γ : hg 0 h − 1 = g 0 } . This is a closed Lie subgroup of Γ. F or eac h g ∈ h g 0 i , consider the set Γ g 0 ,g of elemen ts h ∈ Γ conjugating g 0 with g , that is, satisfying hg 0 h − 1 = g . 4 Clearly , Γ g 0 ,g is a left coset of C g 0 in Γ and, as suc h, has a w ell- define d nor- malized Haar measure dh induced b y that o n C g 0 . If the g r o up Γ acts on a compact manif o ld X , then by X g w e denote the set of fixed p oin ts of an elemen t g ∈ Γ. This is a C ∞ submanifold of X consisting of finitely many comp onen ts (p ossibly of v ario us dimensions). 2.2 The T o dd class The T odd class Td( T M ⊗ C ; Γ) of the complexified ta ng en t bundle of M with resp ect t o the action of Γ is an elemen t of the g roup Q g 0 H ev ( M g 0 , C ) . (The pro duct is ta k en ov er represen tativ es of all conjugacy classes in Γ.) The g 0 th comp onen t o f the T o dd class is defined by the formula Td( T M ⊗ C ; Γ)( g 0 ) = Td( T ∗ M g 0 ⊗ C ) c h λ − 1 ( N M g 0 ⊗ C )( g 0 ) ∈ H ev ( M g 0 , C ) (5) (This form w as apparen tly first in tro duced b y Atiy ah and Singer in [2]; fol- lo wing Baum a nd Connes [3], we refer to it a s the “T o dd class.”) Let us mak e some explanations concerning this formula. The nume rator is the usual T o dd class of the complexified tangent bundle of M g 0 . Next, λ − 1 ( N M g 0 ) is the (virtual) v ector bundle λ − 1 ( N M g 0 ) = Λ eve n ( N M g 0 ) − Λ odd ( N M g 0 ) comp osed of the exterior p o w ers of N M g 0 , and ch λ − 1 ( N M g 0 ⊗ C )( g 0 ) is the Chern c haracter of the bundle λ − 1 ( N M g 0 ) ⊗ C lo calized at the elemen t g 0 . Recall that it is defined as follo ws. Since the ma pping g 0 preserv es the metric, it follows tha t the restriction of the differen tial dg 0 to the normal bundle N M g 0 is a w ell-defined automorphism of this bundle. Let Ω b e the curv ature form of some dg 0 -in v ariant connection on λ − 1 ( N M g 0 ) (e.g., of the connection induced b y the restriction of the Riemannian connection on T M to N M g 0 ). The lo calized Chern c har a cter c h λ − 1 ( N M g 0 ⊗ C )( g 0 ) ∈ H ev ( M g 0 , C ) is defined as the cohomology class of the form c h λ − 1 ( N M g 0 ⊗ C )( g 0 ) = tr  dg ∗ 0 exp  − 1 2 π i Ω  . (Here tr stands for the tr a ce in the fib ers of a vec tor bundle.) 2.3 The Chern c haracter of the sym b ol Let t he group Γ a ct on a compact manifold X . 5 Differen t ial forms and graded traces ov er the algebra C ∞ ( X ) Γ . Let E ∈ V ect( X ) b e a v ector bundle. By Λ ∗ ( X , End E ) Γ ⊂ B L 2 ( X , Λ ∗ ( X ) ⊗ E ) w e denote the subalgebra o f elemen ts A of the form A = X g ∈ Γ ω g , where the ω g ha v e the following prop ert y: for an y t w o functions ψ and ϕ with supp orts in neigh b orho o ds where E is trivialized, one has ω g = T g a g , where T g ω := ( g ∗ ) − 1 ω and a g are some differen tia l forms on X rapidly deca ying in t he C ∞ F r´ ec het top ology a s | g | → ∞ . W e define a mapping τ : Λ ∗ ( X , End E ) Γ − → M g 0 Λ ∗ ( X g 0 ) (6) (the sum is tak en o v er represen tativ es of a ll conjugacy classe s in Γ) by setting τ  X g ∈ Γ ω g , g 0  = X g ∈h g 0 i Z Γ g 0 ,g h ∗ ( ω g   X g ) dh. This is w ell define d. Indeed, g | X g = id, and so the op erator ω g can b e restricte d to X g , the restriction b eing an End E -v alued differen tial form on X g . The t r a ce tr in the last form ula is the fib erwise trace in End E . Lemma 4. The m apping τ is a gr ade d tr ac e on the a lgebr a Λ ∗ ( X , End E ) Γ in the sen se that τ  [ ω 1 , ω 2 ]  = 0 , for a l l ω 1 , ω 2 ∈ Λ ∗ ( X , End E ) Γ , wher e [ · , · ] is the sup e r c ommutator [ ω 1 , ω 2 ] = ω 1 ω 2 − ( − 1) deg ω 1 deg ω 2 ω 2 ω 1 . Chern c haracter of pro jections. No w w e shall define the Chern ch ar acte r c h : K 0 ( C ∞ ( X , End E ) Γ ) − → M g 0 H ev ( X g 0 , C ) (where K 0 ( A ) is the K -group of an op erator algebra A ). L et p b e a pr o jection o v er the algebra C ∞ ( X , End E ) Γ . ( T o make the subseque nt formulas shorter, 6 w e pretend that p is a pro jection in the algebra C ∞ ( X , End E ) Γ itself rather than in a matrix algebra o v er it.) W e tak e some connection ∇ E : Λ ∗ ( X , E ) − → Λ ∗ ( X , E ) in t he bundle E and define a first-order differen t ia l op erator with shifts, ∇ : Λ ∗ ( X , E ) − → Λ ∗ ( X , E ) , (7) b y the formula ∇ = p ∇ E p. (8) A stra ig h tforward computation sho ws that the following assertion is true. Lemma 5. T he op er ator Ω ≡ ∇ 2 : Λ ∗ ( X , E ) − → Λ ∗ ( X , E ) b elongs to Λ 2 ( X , End E ) Γ . The noncomm utative 2-form Ω is called the curvatur e fo rm corresponding to the pro jection p and the connection ∇ E . Definition 6. The Ch ern char acter of the cla ss [ p ] ∈ K 0 ( C ∞ ( X , End E ) Γ ) is the cohomolo gy class c h Γ [ p ] ∈ M g 0 H ev ( X g 0 , C ) of the differen tial fo rm c h Γ p := τ  e − Ω / 2 π i  ∈ M h g 0 i⊂ Γ Λ ev ( X g 0 ) . This is w ell defined. More precisely , the form ch Γ p is closed, and its cohomology class is indep enden t of the c hoice o f a connection in the bundle E and is uniquely determined b y the class of the pro jection p in the K -group K 0 ( C ∞ ( X , End E ) Γ ). The pro of is based on the identit y dτ ( A ) = τ ([ ∇ , A ]) , where A ∈ Λ ∗ ( X , End E ) Γ is an arbitr ary elemen t suc h that pA = A = Ap . 7 Chern cha racter of the sym b ol. Now let D : C ∞ ( M , E ) − → C ∞ ( M , F ) b e an elliptic op erator with shifts acting in sections of v ector bundles on M . T o define the Chern c hara cter of the sym b ol σ ( D ), w e introduce a pro jection and hence an elemen t in K -theory associated with the sym b ol. T o this end, w e mak e use of the bundle 2 B ∗ M = S ( T ∗ M ⊕ 1 ) of unit spheres in the vec tor bundle T ∗ M ⊕ 1 ov er M . Consider t he pro jection p o v er the algebra C (2 B ∗ M , End( E ⊕ F )) Γ defined b y the formula p ( ξ cos ψ , sin ψ ) = 1 2  (1 + sin ψ ) id E σ − 1 ( D )( ξ ) cos ψ σ ( D )( ξ ) cos ψ (1 − sin ψ ) id F  , (9) where ξ lies on the unit sphere in T ∗ M , so that ( ξ cos ψ , sin ψ ) just lies on the unit sphere in T ∗ M ⊕ 1 . R emark 7 . Note that in general the pro jection p is only con tin uous but not infinitely differen tiable at the p oin ts where cos ψ = 0. W e set [ σ ( D )] def = [ p ] ∈ K 0  C (2 B ∗ M , End( E ⊕ F )) Γ  . Note that K 0  C (2 B ∗ M , End( E ⊕ F )) Γ  = K 0  C ∞ (2 B ∗ M , End( E ⊕ F )) Γ  , since, as w as already men tioned ab o v e, C ∞ (2 B ∗ M , End( E ⊕ F )) Γ is a dense lo cal subalgebra of C (2 B ∗ M , End( E ⊕ F )) Γ . Hence w e obtain the cohomo lo gy class c h Γ [ σ ( D )] ∈ M g 0 H ev (2 B ∗ M g 0 , C ) , whic h will b e called the Chern char a cter of t he sym b ol σ ( D ). 2.4 Index theorem No w we are in p osition to state our main result. Theorem 8. L et D b e an el liptic o p er ator with shifts on the manifold M . Then the index of D is given by the formula ind D = h ch Γ [ σ ( D )] Td( T M ⊗ C ; Γ) , [2 B ∗ M ; Γ] i , (10) 8 wher e [2 B ∗ M ; Γ] = Y g 0 [2 B ∗ M g 0 ] ∈ Y g 0 H ev (2 B ∗ M g 0 ) is the fundamental class and angle br ackets denote the natur al p aring b etwe en c ohomolo g y and homolo gy. The pro of inv olv es extensiv e computat ions a nd g o es by reduction to the lo cal index form ula obta ined for elliptic op erators with shifts in the first part of this pap er. References [1] V. E. Nazaikinskii, A. Y u Sa vin, and B. Y u Sternin. On elliptic differen t ia l op erators with shifts, 2007. http://ar xiv.org/abs/0706.3511 . [2] M. F. A tiy ah and I. M. Singer. The index of elliptic op erators I II. A nn. Math. , 87 , 1968, 546–60 4. [3] Paul Ba um and Alain Connes. Chern character for discrete gr oups. In A fˆ ete of top olo gy , 1 9 88, pages 163–232. Academic Press, Bo ston, MA. 9