Zeta Functions, Excision in Cyclic Cohomology and Index Problems
aa r X i v : . [ m a t h . K T ] S e p ZETA FUNCTIONS, EXCISION IN CYCLIC COHOMOLOGY AND INDEXPROBLEMS
RUDY RODSPHON
Abstract.
The aim of this paper is to show how zeta functions and excision in cyclic coho-mology may be combined to obtain index theorems. In the first part, we obtain a local indexformula for "abstract elliptic pseudodifferential operators" associated to spectral triples. Thisformula is notably well adapted when the zeta function has multiple poles. The second partis devoted to give a concrete realization of this formula by deriving an index theorem on thesimple, but significant example of Heisenberg elliptic operators on a trivial foliation, which arein general non-elliptic but hypoelliptic. The last part contains a discussion on manifolds withconic singularity, more precisely about the regularity of spectral triples in this context.
Keywords.
Cyclic cohomology, K-theory, Index theory, Pseudodifferential operators
MSC.
Introduction
Several years ago, Connes and Moscovici obtained in [4] a general index formula given in termsof residues of zeta functions, working with the so-called spectral triples . A major advance wasmade since this formalism enlarges index theory to the more general context of the transversegeometry of foliations, where the interesting pseudodifferential operators are hypoelliptic withoutnecessary being elliptic, while remaining Fredholm. Let us be a little more precise on this generalformula. Connes and Moscovici constructed a Residue Cocycle on the algebra of the spectral triple,cohomologous to the Chern-Connes character in the ( B , b ) -complex of Connes. This cocycle hasthe feature of being "local", contrary to the representative of the Chern-Connes character obtainedby changing the "Dirac operator" D to the pseudodifferential operator F = D | D | − , which involvesthe operator trace, see [2] or [3]. Here, "local" means that the cohomology class of the ResidueCocycle remains unchanged if the "Dirac operator" is perturbed by a smoothing operator. Theinteresting fact is that this happens because the Residue Cocycle is given by residues of zetafunctions. Local index formulas are then deduced from a pairing between this cocycle and theK-theory of the algebra.In the spirit of the techniques developed by Connes and Moscovici, we give an abstract localindex formula of a different flavour, which turns out to be useful to calculate the index of abstractelliptic pseudodifferential operators, in a sense to be defined. The formula is also given by a residueof a zeta function, but there is one important difference in that the cyclic cocycles concerned aredefined not only on an "algebra of smooth functions", as in the Connes-Moscovici formula, butdirectly on the algebra of formal symbols of the pseudodifferential operators considered. We thenillustrate on a simple but interesting example how such a formula may amount to topologicalindex formulas, and in the end, discuss on the case of manifolds with conic singularity. Let usgive an overview of the paper.Section 1 serves to recall some material about Higson’s formalism developed in [7], concerningalgebras of abstract differential operators and their relation with spectral triples, in particularregular ones. Following [14], this allows to develop an abstract pseudodifferential calculus and anotion of ellipticity which covers many interesting examples. We shall focus on the example ofConnes and Moscovici on foliations, involving the Heisenberg pseudodifferential calculus. he aim of Section 2 is to study the index theory in this context. More precisely, we constructa cyclic 1-cocycle on algebras of abstract pseudodifferential operators which generalizes the Radulcocycle defined for any closed manifold M , introduced by Radul in [13]. The two importantingredients to construct this cocycle are, on the one hand, that the zeta function of a (classical)pseudodifferential operator on M has a meromorphic extension to the complex plane, whose set ofpoles is at most simple and discrete. This allows the use of the Wodzicki-Guillemin residue. On theother hand, one uses the pseudodifferential extension and excision in periodic cyclic cohomologyto push the trace on regularizing operators on M , viewed as a cyclic 0-cocycle, to a cyclic 1-cocycleon the algebra of formal symbols on M . The remarkable fact on using the Wodzicki-Guilleminresidue is that it handles all the analytic issues, which will allow us to adopt an algebraic viewpointin most of the paper. Excision in periodic cyclic cohomology then gives a local index formula forelliptic pseudodifferential operators, by compatibility with excision in K-theory.This construction is then extended to the abstract setting recalled in Section 1, and we obtaina cyclic 1-cocycle which generalizes the Radul cocycle in contexts where the zeta function exhibitsmultiple poles. Theorem . Let Ψ ( ∆ ) be an algebra of abstract pseudodifferential operators on a Hilbertspace H , and consider the pseudodifferential extension → Ψ − ∞ ( ∆ ) → Ψ ( ∆ ) → S = Ψ/Ψ − ∞ → Suppose that the pole at zero of the zeta function is of order p > . Then, the cyclic 1-cocycle ∂ [ Tr ] ∈ HP ( S ) , where Tr denotes the operator trace on H , is represented by the followingfunctional, that we also call the Radul cocycle : c ( a , a ) = Z − a δ ( a ) − ! Z − a δ ( a ) + . . . + (− ) p − p ! p Z − a δ p ( a ) where δ ( a ) = [ log ∆ , a ] and δ k ( a ) = δ k − ( δ ( a )) is defined by induction. The r denotes the"order of ∆ " The R − k are "higher Wodzicki-Guillemin residues" defined in Proposition 1.10.In Section 3, we show on an example how the results of the previous section may lead toindex theorems, in the spirit of the Atiyah-Singer theorem. The example we work on is that ofa trivial foliation R p × R q , dealing with the Heisenberg pseudodifferential calculus. Even if thisexample is simple, it is also relevant for three reasons : Firstly, it allows to deal with hypoelliptic(non-elliptic) operators. Secondly, one can see how this leads to a purely algebraic approach ofindex theory ; analytic details are handled by the Wodzicki residue trace. Thirdly, the philosophyof the construction given is useful to understand how to adapt the techniques developed in [11]to treat for example the general case of foliations on closed manifolds (whose leaves are notnecessarily compact). One interesting perspective is to obtain an index formula in the contextof the transverse geometry of foliations, leading to a different approach as those of Connes andMoscovici in [5].When dealing with the Radul cocycle, the main obstacle is that the formulas arising are,except in low dimensions, rather complicated. It is not obvious at all to obtain directly an indexformula which depends only on the principal symbol. To cope with this difficulty, the general ideais to construct ( B , b ) -cocycles of higher degree which are cohomologous to the Radul cocycle inthe ( B , b ) -bicomplex. These ( B , b ) -cocycles are shown to be more easily computable in the highestdegree, for a reason that will be understood later. We give two ways of constructing these cocycles.In the first construction, we introduce homogeneous ( B , b ) -cocycles on regularizing operators, inmany points similar to the cyclic cocycles associated to Fredholm modules given by Connes.The game still consists in pushing them to (inhomogeneous) ( B , b ) -cocycles on the algebra of eisenberg (formal) symbols, using a zeta function regularisation of the trace and excision. Thesecond construction involves Quillen’s cochain theory in [12]. The interest of using this formalismstands in the way we obtain the desired cocycles. Indeed, we do not have to go through thealgebra of regularizing operators, so this method is completely algebraic.The context is a trivial foliation R p × R q of R n . Let S ( R n ) be the associated algebra ofHeisenberg formal symbols of order , and denote by σ : S ( R n ) → C ∞ ( S ∗ H R n ) the principal symbol map. Here, S ∗ H R n denotes the "Heisenberg cosphere bundle", which is definedin Section 1.6. Then, the main result of the section can be stated as follows : Theorem . The Radul cocycle is ( B , b ) -cohomologous to the homogeneous ( B , b ) -cocycle on S ( R n ) defined by ψ − ( a , . . . , a − ) = (− ) n ( i ) n Z S ∗ H R n σ ( a ) dσ ( a ) . . . dσ ( a − ) As an immediate corollary, we obtain the following index theorem.
Theorem . Let P ∈ M N ( Ψ ( R n )) a Heisenberg elliptic pseudodifferential operator offormal symbol u ∈ GL N ( S ( R n )) , and [ u ] ∈ K ( S ( R n )) its (odd) K-theory class. Then, wehave a formula for the Fredholm index of P : Ind ( P ) = Tr ( Ind [ u ]) = − ( n − )!( i ) n ( − )! Z S ∗ H R n tr ( σ ( u ) − dσ ( u )( dσ ( u ) − dσ ( u )) n − )) Section 4 is a discussion on manifolds with conic singularity, and spectral triples associated. Inthis direction, note the work of Lescure in [8], where spectral triples associated to conic manifoldsare constructed. This construction has the notable feature that the zeta function associated hasdouble order poles. The algebra considered in the spectral triple is the algebra of smooth functionsvanishing to infinite order in a neighbourhood of the conic point, with a unit adjoined. Thus, manyinformations are lost in the differential calculus, e.g the abstract algebra of differential operatorsassociated to the spectral triple cannot contain all the conic differential operators. Therefore, itis natural to ask if one can refine the choice of the algebra. Actually, we shall see that obtaininga regular spectral triple on such spaces inevitably leads us, in a certain manner, to erase thesingularity. However, looking at this example gives a good picture of what happens when theregularity of the spectral triple is lost. The abstract Radul cocycle of Theorem 0.1, and thus theindex formulas are no more local, because the terms killed by the residue in presence of regularitycannot be neglected in that case. We refer the reader to the concerned section for the differentdefinitions and notations.
Theorem . Let M be a conic manifold, i.e a manifold with boundary endowed witha conic metric, and let r be a boundary defining function. Let ∆ be the "conic laplacian" ofExample 4.11. Then, the Radul cocycle associated to the pseudodifferential extension → r ∞ Ψ − ∞ b ( M ) → r − Z Ψ Z b ( M ) → r − Z Ψ Z b ( M ) /r ∞ Ψ − ∞ b ( M ) → is given by the following non local formula : c ( a , a ) = ( Tr ∂ , σ + Tr σ )( a [ log ∆ , a ]) − Tr ∂ , σ ( a [ log ∆ , [ log ∆ , a ]])++ Tr ∂ a X k = a ( k ) ∆ − k ! + i Tr (cid:18) Z λ − z a ( λ − ∆ ) − a ( N + ) ( λ − ∆ ) − N − (cid:19) dλ or a , a ∈ Ψ Z b ( M ) /r ∞ Ψ − ∞ b ( M ) This approach yields another point of view on the eta invariant, the notable fact is that it issuitable also for pseudodifferential operators, and not only for Dirac operators. It might be aninteresting problem to compare the formulas obtained with the usual eta invariant.
Acknowledgements.
The author wishes to warmly thank Denis Perrot for sharing his in-sights, for his advices and constant support. He also thanks Thierry Fack for interesting dis-cussions, and relevant remarks which helped to improve preliminary versions of this paper. Theauthor is also grateful to Mathias Pétréolle for sharing some technical tips.1.
Abstract Differential Operators and Traces
In this part, we recall the Abstract Differential Operators formalism developed by Higson in [7] tosimplify the proof of the Connes-Moscovici local index formula [4]. This is actually another wayof defining regular spectral triples. For details, the reader may refer to [7] or [14].1.1.
Abstract Differential Operators.
Let H be a (complex) Hilbert space and ∆ a unbounded,positive and self-adjoint operator acting on it. To simplify matters, we suppose that ∆ has acompact resolvent.We denote by H ∞ the intersection of all these Sobolev spaces. H ∞ = ∞ \ k = dom ( ∆ k ) Definition . An algebra D ( ∆ ) of abstract differential operators associated to ∆ is analgebra of operators on H ∞ fulfilling the following conditions(i) The algebra D ( ∆ ) is filtered, D ( ∆ ) = ∞ [ q = D q ( ∆ ) that is D p ( ∆ ) · D q ( ∆ ) ⊂ D p + q ( ∆ ) . We shall say that an element X ∈ D q ( ∆ ) is an abstractdifferential operator of order at most q . The term differential order will be often used for theorder of such operators.(ii) There is a r > 0 ("the order of ∆ ") such that for every X ∈ D q ( ∆ ) , [ ∆ , X ] ∈ D r + q − ( ∆ ) .To state the last point, we define, for s ∈ R , the s - Sobolev space H s as the subspace dom ( ∆ s/r ) of H , which is a Hilbert space when endowed with the norm k v k s = ( k v k + k ∆ s/r v k ) (iii) Elliptic estimate. If X ∈ D q ( ∆ ) , then, there is a constant ε > 0 such that k v k q + k v k > ε k Xv k , ∀ v ∈ H ∞ Having Gärding’s inequality in mind, the elliptic estimate exactly says that ∆ should be thoughtas an "abstract elliptic operator" of order 1. It also says that any differential operator X of order q can be extended to a bounded operator form H s + q to H s . This last property will be useful todefine pseudodifferential calculus in this setting.One example to keep in mind is the case in which ∆ is a Laplace type operator on a closedRiemannian manifold M . Here, r = and D ( ∆ ) is simply the algebra of differential operators,the H s are the usual Sobolev space and we have an elliptic estimate. In fact, the definition above s an abstraction of this example, but it can be adapted to many more situations, for instance thecase of foliations, on which we shall focus more in detail.1.2. Correspondence with spectral triple.
Let ( A , H , D ) a spectral triple (cf. [4] or [7]). Onemay construct a algebra of abstract differential operators D = D ( A , D ) inductively as follows : D = algebra generated by A and [ D , A ] D = [ ∆ , D ] + D [ ∆ , D ] ... D k = k − X j = D j · D k − j + [ ∆ , D k − ] + D [ ∆ , D k − ] Let δ be the unbounded derivation ad | D | = [ | D | , . ] on B ( H ) . The spectral triple is ( A , H , D ) issaid regular if A , [ D , A ] are included in T ∞ n = dom δ n . The following theorem of Higson makesthe bridge between algebras of abstract differential operators and spectral triples. Theorem . (Higson, [7]). Suppose that A maps H ∞ into itself. Then, the spectraltriple ( A , H , D ) is regular if and only if the elliptic estimate of Definition 1.1 holds. Regularity in spectral triples may be viewed an assumption allowing to control some asymptoticexpansions of "pseudodifferential operators", as we shall see in the next paragraph from theperspective of the elliptic estimate.1.3.
Zeta Functions.
Let D ( ∆ ) be an algebra of abstract differential operators. For z ∈ C , onedefines the complex powers ∆ − z of ∆ using functional calculus : ∆ − z = Z λ − z ( λ − ∆ ) − dλ where the contour of integration is a vertical line pointing downwards separating and the (dis-crete) spectrum of ∆ . This converges in the operator norm when Re ( z ) > 0 , and using thesemi-group property, all the complex powers can be defined after multiplying by ∆ k , for k ∈ N large enough. Moreover, since ∆ has compact resolvent, the complex powers of ∆ are well definedoperators on H ∞ .We will suppose that there exists a d > such that for every X ∈ D q ( ∆ ) , the operator X∆ − z extends to a trace-class operator on H for z on the half-plane Re ( z ) > q + dr . The zeta function of X is ζ X ( z ) = Tr ( X∆ − z/r ) The smallest d verifying the above property is called the analytic dimension of D ( ∆ ) . In thiscase, the zeta function is holomorphic on the half-plane Re ( z ) > q + d . We shall say that D ( ∆ ) hasthe analytic continuation property if for every X ∈ D ( ∆ ) , the associated zeta function extendsto a meromorphic function of the whole complex plane.There properties are set for all the section, unless if it is explicitly mentioned.These notions come from properties of the zeta function on a closed Riemannian manifold M :it is well-known that the algebra of differential operators on M has analytic dimension dim M and the analytic continuation property. Its extension to a meromorphic function has at mostsimple poles at the integers smaller that dim M . In the case where M is foliated, the dimensionof the leaves appears in the analytic dimension when working in the suitable context. Hence, thezeta function provide informations not only on the topology of M , but also on its the geometricstructure, illustrating the relevance of this abstraction. .4. Abstract Pseudodifferential Operators.
Let D ( ∆ ) an algebra of abstract differentialoperators of analytic dimension d . To define the notion of pseudodifferential operators, we need amore general notion of order, not necessary integral, which covers the one induced by the filtrationof D ( ∆ ) . Definition . An operator T : H ∞ → H ∞ is said to have pseudodifferential order m ∈ R if for every s > , it extends to a bounded operator from H m + s to H s . In addition, we requirethat operators of analytic order stricly less than − d are trace-class operators.That this notion of order covers the differential order is due to the elliptic estimate, as alreadyremarked in Section 1.1. The space of such operators, denoted Op ( ∆ ) , forms a R -filtered algebra.There is also a notion of regularizing operators which are, as expected, the elements of the (two-sided) ideal of operators of all order. Remark . Higson uses in [7] the term "analytic order", but as the examples we deal within the paper are about pseudodifferential operators, we prefer the term pseudodifferential order.
Example . For every λ ∈ C not contained in the spectrum of ∆ , the resolvent ( λ − ∆ ) − has analytic order r . Moreover, by spectral theory, its norm as an operator between Sobolev spacesis a O ( | λ | − ) .The following notion is due to Uuye, cf. [14]. We just added an assumption on the zeta functionwhich is necessary for what we do. Definition . An algebra of abstract pseudodifferential operators is a R -filtered subalgebra Ψ ( ∆ ) of Op ( ∆ ) , also denoted Ψ when the context is clear, satisfying ∆ z/r Ψ m ⊂ Ψ Re ( z )+ m , Ψ m ∆ z/r ⊂ Ψ Re ( z )+ m and which commutes, up to operators of lower order, with the complex powers of ∆ , that is ,for all m ∈ R , z ∈ C [ ∆ z/r , Ψ m ] ⊂ Ψ Re ( z )+ m − Moreover, we suppose that for every P ∈ Ψ m ( ∆ ) , the zeta function ζ P ( z ) = Tr ( P∆ − z/r ) is holomorphic on the half-plane Re ( z ) > m + d , and extends to a meromorphic function of thewhole complex plane. We shall denote by Ψ − ∞ = \ m ∈ R Ψ m Of course, this is true for the algebra of (classical) pseudodifferential operators on a closed mani-fold. We shall recall later what happens in the example of Heisenberg pseudodifferential calculuson a foliation, as described by Connes and Moscovici in [4].We end this part with a notion of asymptotic expansion for abstract pseudodifferential operators.This can be seen as "convergence under the residue".
Definition . Let T and T α ( α in a set A ) be operators on Ψ . We shall write T ∼ X α ∈ A T α f there exists c > 0 and a finite subset F ⊂ A such that for all finite set F ′ ⊂ A containing F , themap z → Tr ( T − X α ∈ F ′ T α ) ∆ z/r ! is holomorphic in a half-plane Re ( z ) > − c (which contains z = ). Example . Suppose that that for every
M > 0 , there exists a finite subset F ⊂ A suchthat T − X α ∈ F T α ∈ Ψ − M Then, T ∼ P α ∈ A T α In this context, asymptotic means that when taking values under the residue, such infinite sums,which have no reason to converge in the operator norm, are in fact finite sums. Thus, this will allowus to disregard analytic subtleties and to consider these sums only as formal expansions withoutwondering if they converge or not. In other words, this notion allows to adopt a completelyalgebraic viewpoint. To this effect, the following lemma is crucial.
Lemma . (Connes-Moscovici’s trick, [4, 7]) Let Q ∈ Ψ ( ∆ ) be an abstract pseudodifferen-tial operator. Then, for any z ∈ C , we have (1.1) [ ∆ − z , Q ] ∼ X k > (cid:18) − zk (cid:19) Q ( k ) ∆ − z − k where we denote Q ( k ) = ad ( ∆ ) k ( Q ) , ad ( ∆ ) = [ ∆ , . ] . Two important facts.
Firstly, remark that the pseudodifferential order of terms in thesum are decreasing to − ∞ , so that the difference between [ ∆ − z , Q ] and the sum becomes moreand more regularizing as the number of terms grows.Secondly, and more importantly, this is essentially the elliptic estimate, or the regularity of thespectral triple, which implies this property. Then, if the sum in the lemma above is not asymptoticin the sense defined, the elliptic estimate cannot hold. In terms of spectral triple, this means it isnot regular. Proof.
For z ∈ C of positive real part large enough, one proves, using Cauchy formulas andreasoning by induction, that the following identity holds (cf [7], Lemma 4.20) :(1.2) ∆ − z Q − Q∆ − z = N X k = (cid:18) − zk (cid:19) Q ( k ) ∆ − z − k + Z λ − z ( λ − ∆ ) − Q ( N + ) ( λ − ∆ ) − N − dλ By the elliptic estimate, the integral term in the right hand-side has pseudodifferential orderord Q + ( N + ) r − N − − ( N + ) r = ord ( Q ) − r − N − , which can therefore be made as smallas we want by taking N large. This proves the lemma in the case where Re ( z ) > 0 . The generalcase follows from the analytic continuation property. (cid:3) Higher traces on the algebra of abstract pseudodifferential operators.
We give inthis paragraph a simple generalization of the Wodzicki residue trace, when the zeta function ofthe algebra D ( ∆ ) has poles of arbitrary order. Actually, this was already noticed by Connes andMoscovici (see [4]). roposition . Let Ψ ( ∆ ) an algebra of abstract pseudodifferential operators, followingthe context of the previous paragraphs. Suppose that the associated zeta function has a poleof order p > in . Then, the functional p Z − P = Res z = z p − Tr ( P∆ − z/r ) defines a trace on Ψ ( ∆ ) . Proof.
Let P , Q ∈ Ψ ( ∆ ) . Then, for Re ( z ) ≫ , we can use the trace property on commutatorsto write :Tr ([ P , Q ] ∆ − z/r ) = Tr ( P ( Q − ∆ − z/r Q∆ z/r ) ∆ − z/r ) Hence, using the analytic continuation property, we have p Z −[ P , Q ] = Res z = z p − Tr ( P ( Q − ∆ − z/r Q∆ z/r ) ∆ − z/r ) By Lemma 1.9, ∆ − z/r Q − Q∆ − z/r ∼ X k > (cid:18) − z/rk (cid:19) Q ( k ) ∆ − k · ∆ − z/r so that, p Z −[ P , Q ] = Res z = X k > z p − Tr (cid:18)(cid:18) − z/rk (cid:19) Q ( k ) ∆ − k · ∆ − z/r (cid:19) The sum is finite : Indeed, the order of Q ( k ) ∆ − k is ord ( Q ) − k , so the terms in the sum abovebecome holomorphic at z = when k is large enough, and vanish when taking values under theresidue. Finally, the finite sum remaining vanishes since the zeta function has at most a pole oforder p at . (cid:3) If k < p , then R − k is no more a trace in general, but one has an explicit relation expressingthe commutators, cf. [4].1.6. The example of Connes and Moscovici.
Heisenberg pseudodifferential calculus on foliations.
Let M be a foliated manifold ofdimension n , and let F be the integrable sub-bundle of the tangent bundle T M of M which definesthe foliation. We denote the dimension of the leaves by p , and by q = n − p their codimension.For the moment, we work in distinguished local charts. Let ( x , . . . , x n ) a distinguished localcoordinate system of M , i.e, the vector fields ∂∂x , . . . , ∂∂x p (locally) span F , so that ∂∂x p + , . . . , ∂∂x n are transverse to the leaves of the foliation. Connes and Moscovici constructed in [4] an algebraof generalized differential operators using Heisenberg calculus, whose main idea is that : • The vector fields ∂∂x , . . . , ∂∂x p are of order 1. • The vector fields ∂∂x p + , . . . , ∂∂x n are of order 2.The Heisenberg pseudodifferential calculus consists in defining a class of smooth symbols σ ( x , ξ ) on R nx × R nξ which takes this notion of order into account. To this end, they set | ξ | ′ = ( ξ + . . . + ξ + ξ + + . . . + ξ ) h α i = α + . . . + α p + p + + . . . n for every ξ ∈ R n , α ∈ N n . efinition . A smooth function σ ( x , ξ ) ∈ C ∞ ( R nx × R nξ ) is a Heisenberg symbol of order m ∈ R if σ is x -compactly supported, and if for every multi-index α , β , one has the followingestimate | ∂ βx ∂ αξ σ ( x , ξ ) | ( + | ξ | ′ ) m − h α i To such a symbol σ of order m , one associates its left-quantization, which is the following operator P : C ∞ ( R n ) → C ∞ ( R n ) , Pf ( x ) = ( ) n Z R n e i x · ξ σ ( x , ξ ) ˆ f ( ξ ) dξ We shall say that P is a Heisenberg pseudodifferential operator of order m , and denote the classof such operators by Ψ mH ( R n ) . The Heisenberg regularizing operators, whose class is denoted by Ψ − ∞ ( R n ) , are those of arbitrary order, namely Ψ − ∞ ( R n ) = \ m ∈ R Ψ mH ( R n ) The reason why there is no H -subscript is that the Heisenberg regularizing operators are exactlythe regularizing operators of the usual pseudodifferential calculus, i.e the operators with smoothSchwartz kernel.Actually we shall restrict to the smaller class of classical Heisenberg pseudodifferential opera-tors . For this, we first define the Heisenberg dilations λ · ( ξ , . . . , ξ p , ξ p + , . . . , ξ n ) = ( λξ , . . . , λξ p , λ ξ p + , . . . , λ ξ n ) for any non-zero λ ∈ R and non-zero ξ ∈ R n .Then, a Heisenberg pseudodifferential operator P ∈ Ψ mH ( R n ) of order m is said classical if itssymbol σ has an asymptotic expansion(1.3) σ ( x , ξ ) ∼ X j > σ m − j ( x , ξ ) where σ m − j ( x , ξ ) ∈ S m − jH ( R n ) are Heisenberg homogeneous , that is, for any non zero λ ∈ R , σ m − j ( x , λ · ξ ) = λ m − j σ m − j ( x , ξ ) The ∼ means that for every M > 0 , there exists an integer N such that σ − P Nj = σ m − j ∈ S − MH ( R n ) .To avoid an overweight of notations, we shall keep the notation Ψ H to refer to classical elements.Another important point is the behaviour of symbols towards composition of classical pseudo-differential operators. Of course, if P , Q ∈ Ψ H ( R n ) are Heisenberg pseudodifferential operatorsof symbols σ P and σ Q , PQ is also a Heisenberg pseudodifferential operator of order at mostord ( P ) + ord ( Q ) , and its the symbol σ PQ is given by the following asymptotic expansion calledthe star-product of symbols, given by the formula(1.4) σ PQ ( x , ξ ) = σ P ⋆ σ Q ( x , ξ ) ∼ X | α | > (− i ) | α | α ! ∂ αξ σ P ( x , ξ ) ∂ αx σ Q ( x , ξ ) Note that the order of each symbol in the sum is decreasing while | α | is increasing.We define the algebra of Heisenberg formal classical symbols S H ( R n ) as the quotient S H ( R n ) = Ψ H ( R n ) /Ψ − ∞ ( R n ) Its elements are formal sums given in (1.3), and the product is the star product (1.4). Note thatthe ∼ can be replaced by equalities when working at a formal level.We now deal with ellipticity in this context. A Heisenberg pseudodifferential operator is said Heisenberg elliptic if it is invertible in the unitalization S H ( R n ) + of S H ( R n ) . One can showthat this is actually equivalent to say that its Heisenberg principal symbol , e.g the symbol of igher degree in the expansion (1.3) is invertible on R nx × R nξ r { } . An adaptation of argumentsfrom classical elliptic regularity shows that the elliptic estimate holds in this case. A remarkablespecificity of these operators is that they are hypoelliptic, but not elliptic in general. Nevertheless,they remain Fredholm operators between Sobolev spaces relative to this context. The interestedreader should consult [1] for details. Example . The following operator, also called sub-elliptic sub-laplacian, ∆ H = ∂ + . . . + ∂ p + ∂ p + + . . . + ∂ n has Heisenberg principal symbol σ ( x , ξ ) = | ξ | ′ , and is therefore Heisenberg elliptic. However, itsusual principal symbol, as an ordinary differential operator, is ( x , ξ ) → P pi = ξ , so ∆ H is clearlynot elliptic.Finally, Heisenberg pseudodifferential operators behaves well towards distinguished charts change.Therefore, Heisenberg pseudodifferential calculus can be defined globally on foliations by using apartition of unity. Then, for a foliated manifold M , we denote by Ψ mH ( M ) the algebra of Heisenbergpseudodifferential operators on M .It is not very difficult to verify the required assumptions of Definition 1.6. However, what concernsthe zeta function is not obvious.1.6.2. Residue Trace on Foliations.
We now recall these results, proved by Connes and Moscoviciin [4].
Theorem . (Connes - Moscovici, [4]) Let M be a foliated manifold of dimension n , p be the dimensions of the leaves, and P ∈ Ψ m ( M ) be a Heisenberg pseudodifferential operatorof order m ∈ R . Let ∆ the sub-elliptic sub-laplacian defined in Example 1.12, that we extendglobally to M by using a partition of unity. Then, the zeta function ζ P ( z ) = Tr ( P∆ − z/4 ) is holomorphic on the half-plane Re ( z ) > m + p + , and extends to a meromorphic functionof the whole complex plane, with at most simple poles in the set { m + p + , m + p + − , . . . } Remark . The analytic dimension of the algebra of Heisenberg differential operators isthen p + . The p is the dimension of the leaves, the " " is the degree of the vector fields transverseto them.The meromorphic extension of the zeta function given by this theorem allows the construction ofa Wodzicki-Guillemin trace on S H ( M ) = Ψ H ( M ) /Ψ − ∞ ( M ) . Theorem . (Connes - Moscovici, [4]) The Wodzicki residue functional Z − : S H ( M ) − → C , P → Res z = Tr ( P∆ − z/4 ) is a trace. It is the unique trace on S H ( M ) , up to a multiplicative constant. Moreover, for P ∈ Ψ H ( M ) , we have the following formula, only depending on the symbol σ of P . (1.5) Z − P = ( ) n Z S ∗ H M ι L (cid:18) σ −( p + ) ( x , ξ ) ω n n ! (cid:19) Here, S ∗ H M is the Heisenberg cosphere bundle, that is, the sub-bundle S ∗ H M = { ( x , ξ ) ∈ T ∗ M ; | ξ | ′ = } is the generator of the Heisenberg dilations, ι stands for the interior product and ω denotes thestandard symplectic form on T ∗ M . Remark . All these results still holds for Heisenberg pseudodifferential operators actingon sections of a vector bundle E over M : In this case, the symbol σ −( p + ) ( x , ξ ) above is at eachpoint ( x , ξ ) an endomorphism acting on the fibre E x , and (1.5) becomes : Z − P = ( ) n Z S ∗ H M ι L (cid:18) tr ( σ −( p + ) ( x , ξ )) ω n n ! (cid:19) where tr denotes the trace of endomorphisms.2. The Radul cocycle for abstract pseudodifferential operators
Abstract index theorems.
We begin with another abstract setting. Let A be an associativealgebra over C , possibly without unit, and I an ideal in A . The extension → I → A → A/I → gives rise to the following excision diagram, relating algebraic K-theory and periodic cyclic ho-mology(2.1) K alg ( A/I ) Ind / / ch (cid:15) (cid:15) K alg ( I ) ch (cid:15) (cid:15) HP ( A/I ) ∂ / / HP ( I ) The vertical arrows are respectively the odd and even Chern character.We still denote ∂ : HP ( I ) → HP ( A/I ) the excision map in cohomology. As mentioned in [10],for [ τ ] ∈ HP ( I ) , [ u ] ∈ K ( A/I ) , one has the equality :(2.2) h [ τ ] , ch Ind [ u ] i = h ∂ [ τ ] , ch [ u ] i One should have in mind that the left hand-side is an "analytic index", and think about the righthand-side as a "topological index".The construction of a boundary map ∂ in cohomology associated to an extension is standard.If [ τ ] ∈ HP ( I ) is given by a hypertrace τ : I → C , i.e a linear map satisfying the condition τ ([ A , I ]) = , then let us recall how to compute ∂ [ τ ] ∈ HP ( A/I ) . To begin, choose a lift e τ : A → C of τ , such that e τ is linear (in general, this is not a trace), and a linear section σ : A/I → A suchthat σ ( ) = , after adjoining a unit where we have to. Then, ∂ [ τ ] is represented by the followingcyclic cocycle : c ( a , a ) = b e τ ( σ ( a ) , σ ( a )) = e τ ([ σ ( a ) , σ ( a )]) where b is the Hochschild coboundary recalled in Section 3.1.2.2. The generalized Radul cocycle.
We can finally come to the main theorem of this sec-tion. Let D ( ∆ ) be an algebra of abstract differential operators and Ψ be an algebra of abstractpseudodifferential operators. We consider the extension → Ψ − ∞ → Ψ → S → where S is the quotient Ψ/Ψ − ∞ . The operator trace on Ψ − ∞ is well defined, and Tr ([ Ψ − ∞ , Ψ ]) = . heorem . Suppose that the pole in zero of the zeta function is of order p > . Then,the cyclic 1-cocycle ∂ [ Tr ] ∈ HP ( S ) is represented by the following functional : c ( a , a ) = Z − a δ ( a ) − ! Z − a δ ( a ) + . . . + (− ) p − p ! p Z − a δ p ( a ) where δ ( a ) = [ log ∆ , a ] and δ k ( a ) = δ k − ( δ ( a )) is defined by induction. We shall call thiscocycle as the generalized Radul cocycle . Here, the commutator [ log ∆ , a ] is defined as the non-convergent asymptotic expansion(2.3) [ log ∆ , a ] ∼ X k > (− ) k k a ( k ) ∆ − k where a ( k ) has the same meaning as in Lemma 1.9. This expansion arises by first using functionalcalculus :log ∆ = i Z log λ ( λ − ∆ ) − dλ and then, reproducing the same calculations made in the proof of Lemma 1.9 to obtain the formula(cf. [7] for details). In particular, note that log ∆ = log ∆ .Another equivalent expansion possible, that we will also use, is the following(2.4) [ log ∆ , a ] ∼ X k > (− ) k k a [ k ] ∆ − k/r where a [ ] = [ ∆ , a ] , and a [ k + ] = [ ∆ , a [ k ] ] . Before giving the proof of the result, let us givea heuristic explanation of how to get this formula. We first lift the trace on Ψ − ∞ to a linear map e τ on Ψ using a zeta function regularization by "Partie Finie" : e τ ( P ) = Pf z = Tr ( P∆ − z/r ) for any P ∈ Ψ . The "Partie Finie" Pf is defined as the constant term in the Laurent expansion ofa meromorphic function. Let Q ∈ Ψ be another pseudodifferential operator. Then, we havePf z = Tr ([ P , Q ] ∆ − z/r ) = Res z = Tr (cid:18) P · Q − ∆ − z/r Q∆ z/r z ∆ − z/r (cid:19) by reasoning first for z ∈ C of sufficiently large real part to use the trace property, and thenapplying the analytic continuation property. Then, informally we can think of the complexpowers of ∆ as ∆ z/r = e log ∆ · z/r = + zr log ∆ + . . . + ! (cid:16) zr (cid:17) p ( log ∆ ) p + O ( z p + ) which after some calculations, gives the expansion ( Q − ∆ − z/r Q∆ z/r ) ∆ − z/r = zδ ( Q ) − z ( Q ) + . . . + (− ) p − z p p ! δ p ( Q ) + O ( z p + ) Proof.
Let P , Q ∈ Ψ be two abstract pseudodifferential operators. The beginning of theproof is the same as the heuristic argument given above, so we start from the equalityPf z = Tr ([ P , Q ] ∆ − z/r ) = Res z = Tr (cid:18) P · Q − ∆ − z/r Q∆ z/r z ∆ − z/r (cid:19) = Res z = Tr P · X k > (cid:18) − z/rk (cid:19) Q ( k ) ∆ − k · ∆ − z/r The second equality comes from Lemma 1.9. hen, let X be an indeterminate. As power series over the complex numbers with indeterminate X , we remark that for any z ∈ C , one has X k > (cid:18) − z/rk (cid:19) X k = (( + X ) − z/r − ) On the other hand, we have, for q ∈ N ,ad ( log ∆ ) q ( Q ) = q [ log ∆ , [ ..., [ log ∆ , Q ]]] ∼ q X k > q X k + ... + k q = k (− ) k k . . . k q Q ( k ) ∆ − k Using once more the indeterminate X , one has X k > q X k + ... + k q = k (− ) k k . . . k q X k = X l > (− ) l X l l = log ( + X ) q thus obtaining X q > (− ) q − q ! z q − r q log ( + X ) q = (( + X ) − z/r − ) This proves that the coefficients of Q ( k ) ∆ − k in the sums X k > (cid:18) − zk (cid:19) Q ( k ) ∆ − k , X q > (− ) q − q ! z q − r q X k > q X k + ... + k q = k (− ) k k . . . k q Q ( k ) ∆ − k are the same, hence the result follows. (cid:3) Applying the pairing (2.2), we have a local index theorem.
Example . Let M be a closed foliated manifold with integrable sub-bundle F ∈ T M , ∆ the sub-elliptic sub-laplacian of Example 1.12 sand take Ψ ( ∆ ) = Ψ H ( M ) the algebra of (classical)Heisenberg pseudodifferential operators on M , Ψ − ∞ ( ∆ ) = Ψ − ∞ ( M ) the ideal of regularizing op-erators. The quotient Ψ/Ψ − ∞ is the algebra S H ( M ) of full classical Heisenberg symbols. A traceon Ψ − ∞ ( M ) is given by(2.5) τ ( K ) = Tr ( K ) = Z M k ( x , x ) d vol ( x ) where k is the Schwartz kernel of K . Then, using the residue defined in Theorem 1.15 and applyingTheorem 2.1, ∂ [ τ ] is represented by the following cyclic 1-cocycle on S H ( M ) :(2.6) c ( a , a ) = Z − a [ log | ξ | ′ , a ] With a slight abuse of notation, we denoted by log | ξ | ′ the symbol of ∆ . We emphasize thatthe product of symbols is the star-product defined in (1.4), but we omit the notation ⋆ .Remark that log | ξ | ′ is a log-polyhomogeneous (Heisenberg) symbol and is not classical. But using(2.4), it is clear that its commutator with any element of S H ( M ) is. Note also that the cocycle isdefined on the symbols rather that on the operators, but this does not matter since the residuekills the smoothing contributions. In particular, only a finite number of terms of the star-productare involved. This is exactly what we meant when we said that the Wodzicki residue handlesanalytic issues in the introduction.This cocycle was first introduced by Radul in [13] in the context of closed manifold, withoutconsidering foliations, as a 2-cocycle over the Lie algebra of formal symbols on the manifold. The adul cocycle also may appear from a Partie Finie regularization of the zeta function, so we keepthe same name for the cocycle 2.6 obtained in this more general setting.From this cocycle, we then get an index formula for Heisenberg elliptic pseudodifferential opera-tors. Indeed, if P is such an operator of formal symbol u ∈ S H ( M ) , and Q a parametrix of P inthe Heisenberg calculus, of formal symbol u ′ ∈ S H ( M ) , then, the Fredholm index of P is given byInd ( P ) = c ( u , u ′ ) As we can see, the Radul cocycle is given by a Wodzicki residue, and is hence local. However, itseems to be an unattainable task to get an index formula in terms of the principal symbol onlysince by (1.5), we have to find the symbol of order −( p + ) of u [ log | ξ | ′ , u ′ ] . At first sight, manyterms of the formal expansions of u and u ′ , as well as many of their higher derivatives, seem tobe involved. We shall see in next section a way to overcome this difficulty.3. A computation of the Radul cocycle
At first sight, the latter index formula obtained is local in the sense that it is given as a residueformula, a little in the spirit of that of Connes and Moscovici. However, as already noted inExample 2.2, the formula obtained is rather involved.This section is devoted to show how one may recover an interesting index formula from the Radulcocycle, working on the simplest foliation possible. For all this section, even if it is not explicitlymentioned, we consider R n as a trivial foliation R p × R q , where p < n and q = n − p , andconsider the associated classical Heisenberg pseudodifferential operators Ψ ( R n ) of order 0.Our goal is to show that the Radul cocycle (2.6) on S ( R n ) is cohomologous in HP ( S H ( R n )) to simple inhomogeneous ( B , b ) -cocycles of higher degree, making the computation of the indexproblem easier. We shall always use coordinates adapted to the foliation R p × R q .We shall give two ways of constructing these cocycles. Before beginning these constructions, webriefly recall how to define the ( B , b ) -bicomplex.3.1. The ( B , b ) -bicomplex. Let A be an associative algebra over C . For k > , denote by CC k ( A ) the space of ( k + ) -linear forms on the unitalization A + of A such that φ ( a , . . . , a k ) = when a i = for some i > . Then, define the differentials B : CC k + ( A ) → CC k ( A ) , b : CC k ( A ) → CC k + ( A ) by the formulas Bφ ( a , . . . , a k ) = k X i = (− ) ik φ ( , a i , . . . , a k , a , . . . , a i − ) bφ ( a , . . . , a k + ) = k X i = (− ) i φ ( a , . . . , a i − , a i a i + , a i + , . . . , a k + )+ (− ) k + φ ( a k + a , . . . , a k ) hat is, B = b = . Moreover, B and b anticommute, which allows to define the ( B , b ) -bicomplex... ... .... . . B / / CC ( A ) B / / b O O CC ( A ) B / / b O O CC ( A ) b O O . . . B / / CC ( A ) B / / b O O CC ( A ) b O O . . . B / / CC ( A ) b O O Then, the periodic cyclic cohomology HP • ( A ) is the cohomology of the total complex. Moreprecisely, it is the cohomology of the 2-periodic complex. . . B + b / / CC even ( A ) B + b / / CC odd ( A ) B + b / / CC even ( A ) B + b / / . . .where CC even ( A ) = CC ( A ) ⊕ CC ( A ) ⊕ . . .CC odd ( A ) = CC ( A ) ⊕ CC ( A ) ⊕ . . .Hence, there are only an even and an odd periodic cyclic cohomology groups, respectively denotedHP ( A ) and HP ( A ) . Remark . Sometimes, authors consider the total differential B − b instead of B + b .3.2. General context.
Recall from Section 1.5 that the residue trace of a Heisenberg pseudodif-ferential operator P ∈ Ψ H ( R n ) of symbol σ is given by(3.1) Z − P = ( ) n Z S ∗ H R n ι L (cid:18) σ −( p + ) ( x , ξ ) ω n n ! (cid:19) where σ −( p + ) is the Heisenberg homogeneous term of order −( p + ) in the asymptotic ex-pansion of σ , ω = P i dx i dξ i is the standard symplectic form on T ∗ R n = R nx × R nξ , and L is thegenerator of the Heisenberg dilations, given by the formula L = p X i = ξ i ∂ ξ i + n X i = p + ξ i ∂ ξ i Note that in this example, the sub-elliptic sub-laplacian has not a compact resolvent since wework on R n . However, the results in Section 1.6.2 on the Wodzicki residue still holds because weconsider pseudodifferential operators which have compact support.We first extend the trace on Ψ − ∞ ( R n ) given in (2.5) to a graded trace on the graded algebra Ψ − ∞ ( R n ) ⊗ Λ • T ∗ R n , using a Berezin integral :Tr ( K ⊗ α ) = α [ ] Tr ( K ) where K ∈ Ψ − ∞ ( R n ) , and α [ ] is the coefficient of the form dx . . . dx n dξ . . . dξ n in α (thewedges are dropped to simplify notations). Here, we emphasize once more that T ∗ R n is seen asthe vector space R nx × R nξ . Therefore Λ • T ∗ R n stands for the exterior algebra of the vector space T ∗ R n = R nx × R nξ , and not for the vector bundle of exterior powers of the cotangent bundle, asusual. oreover, the Wodzicki residue trace on Ψ H ( R n ) is given by a zeta function regularisation of thistrace. Therefore, the latter procedure also extends the Wodzicki residue trace to a graded trace onthe graded algebra Ψ H ( R n ) ⊗ Λ • T ∗ R n . The latter descends to a graded trace on S H ( R n ) ⊗ Λ • T ∗ R n .The composition law of pseudodifferential operators, or the star-product of symbols for the latter,are extended to these algebras just by imposing that they commute to elements of the exterioralgebra.Remark also that the following commutation relations hold [ x i , ξ j ] = i δ i , j , [ x i , x j ] = [ ξ i , ξ j ] = where we denote i = √ − . In short, ad ( x i ) and ad ( ξ i ) are respectively the differentiation ofsymbols with respect to the variables ξ i and x i .Finally, let F be the multiplier on S H ( R n ) ⊗ Λ • T ∗ R n defined by F = X i ( x i dξ i + ξ i dx i ) As the two following lemmas might indicate, this operator will play a role rather similar tooperators usually denoted by F when dealing with finitely summable Fredholm modules. Thedifference is that this F here is not the main object of study, and acts more as an intermediatetowards the main result. Lemma . F is equal to i ω , where ω is the standard symplectic form on T ∗ R n . Inparticular, F commutes to every element in S H ( R n ) ⊗ Λ • T ∗ R n . Lemma . For every symbol a ∈ S H ( R n ) , one has [ F , a ] = i da = i X i (cid:18) ∂a∂x i dx i + ∂a∂ξ i dξ i (cid:19) The proof of both lemmas follows from a simple computation, just using the commutation relationsmentioned above. Another important property of the multiplier F , easy to verify, is the following Lemma . For every a ∈ S H ( R n ) ⊗ Λ • T ∗ R n , we have Z −[ F , a ] = Construction by excision.
The previous lemma shows that it may be relevant to con-sider the following cyclic cocycles on Ψ − ∞ ( R n ) , inspired of Connes’ cyclic cocycles associated toFredholm modules (see [2] or [3]).(3.2) φ ( a , ..., a ) = k ! i k ( )! Tr (cid:18) a [ F , a ] . . . [ F , a ] ⊗ ω n − k n ! (cid:19) for k n . Therefore, we obtain the following result, very similar to that of Connes. Proposition . The periodic cyclic cohomology classes of the cyclic cocycles φ areindependant of k . Proof.
Set(3.3) γ + ( a , . . . , a + ) = ( k + )! i k + ( + )! Tr (cid:18) a F [ F , a ] . . . [ F , a + ] ⊗ ω n − k n ! (cid:19) It is then a straightforward calculation to verify that ( B + b ) γ + = φ − φ + , which showsthe result. (cid:3) t this stage, we are not very far from being done. To obtain the desired cyclic cocycles on thealgebra S R n ⊗ Λ • T ∗ R n from those previously constructed, it suffices to push the latter usingexcision in periodic cyclic cohomology. Indeed, as we have the pseudodifferential extension → Ψ − ∞ ( R n ) → Ψ ( R n ) → S R n → we look at the image of the ( B , b ) -cocycles φ under the boundary map ∂ : HP ( Ψ − ∞ ( R n )) − → HP ( S R n ) Thanks to this, the cocycles (3.2) involving the operator trace, which are highly non local, will beavoided and transferred to cocycles involving the Wodzicki residue.To compute the image of the the cocycles (3.2) under the excision map ∂ , we lift the cocycles φ on Ψ − ∞ ( R n ) to cyclic cochains e φ ∈ CC • ( Ψ ( R n )) using a zeta function regularization, e φ ( a , ..., a )= k ! i k ( )! + X i = Pf z = Tr (cid:18) a [ F , a ] . . . [ F , a i ] ∆ − z/4 [ F , a i + ] . . . [ F , a ] ⊗ ω n − k n ! (cid:19) For k = , we already know that ∂ [ φ ] is represented by the Radul cocycle c ( a , a ) = Z − a δa where δa = [ log | ξ | ′ , a ] .Now, let k ∈ N . Then, the usual construction of the boundary map in cohomology associated toan extension gives that ∂ [ φ ] is represented by the inhomogeneous ( B , b ) -cocycle ( B + b ) e φ = ψ − + φ + ∈ CC − ( Ψ ( R n )) ⊕ CC + ( Ψ ( R n )) where ψ − = B e φ and φ + = b e φ are given by(3.4) ψ − ( a , . . . , a − )= k ! i k ( )! − X i = (− ) i + Z − (cid:18) a [ F , a ] . . . [ F , a i ] δF [ F , a i + ] . . . [ F , a − ] ⊗ ω n − k n ! (cid:19) (3.5) φ + ( a , . . . , a + )= k ! i k ( + )! + X i = (− ) i − Z − (cid:18) a [ F , a ] . . . [ F , a i − ] δa i [ F , a i + ] . . . [ F , a + ] ⊗ ω n − k n ! (cid:19) where we define ψ − as zero. φ is precisely the Radul cocycle. For the clarity of the exposition,the calculations will be detailed later in Appendix A. Then, we have : Theorem . The Radul cocycle c is cohomologous in the ( B , b ) -complex, to the ( B , b ) -cocycles ( ψ − , φ + ) , for all k n . Indeed, usual properties of boundary maps in cohomology automatically ensures this result. Asa matter of fact, one can be more precise and give explicitly the transgression cochains allowingto pass from one cocycle to another. For this, we lift the transgression cochain γ given in (3.3) to he ( B , b ) -cochain e γ ∈ CC • ( Ψ H ( R n )) , using the same trick as before : e γ + = ( k + )! i k + ( + )! + (cid:20) Pf z = Tr (cid:18) a ∆ − z/4 F [ F , a ] . . . [ F , a + ] ⊗ ω n − k − n ! (cid:19) + + X i = Pf z = Tr ( a F [ F , a ] . . . [ F , a i ] ∆ − z/4 [ F , a i + ] . . . [ F , a + ] ⊗ ω n − k − n ! ! and the term i = of the sum means Pf z = Tr ( a F∆ − z [ F , a ] . . . , [ F , a + ] ⊗ ω n − k − n ! ) in the righthand-side. Proposition . The inhomogeneous ( B , b ) -cochains e φ − e φ + − ( B + b ) e γ + = γ − γ ′ + ∈ CC ( Ψ ( R n )) ⊕ CC + ( Ψ ( R n )) for k n , viewed as cochains on S H ( R n ) , are transgression cochains between ( ψ − , φ + ) and ( ψ + , φ + ) , that is, ( ψ − + φ + ) − ( ψ + + φ + ) = ( B + b )( γ − γ ′ + ) Moreover, one has (3.6) γ ( a , . . . , a )= k ! i k + ( + )! X i = (− ) i Z − (cid:18) a F [ F , a ] . . . [ F , a i ] δF [ F , a i + ] . . . [ F , a + ] ⊗ ω n − k − n ! (cid:19) (3.7) γ ′ ( a , . . . , a ) = Z − (cid:18) a δa [ F , a ] . . . [ F , a ] ⊗ ω n − k n ! (cid:19) + k ! i k ( + )! X i = (− ) i − Z − (cid:18) a F [ F , a ] . . . [ F , a i − ] δa i [ F , a i + ] . . . [ F , a ] ⊗ ω n − k n ! (cid:19) That e φ − e φ + − ( B + b ) e γ + gives a transgression cochain comes once again from the con-struction of a boundary map in cohomology associated to a short exact sequence. Once more, thecalculations leading to these formulas are given in Appendix A.3.4. Construction with Quillen’s Algebra Cochains.
The interest about Quillen’s theory ofcochains here is that the ( B , b ) -cocycles we want to get are obtained purely algebraically, sincewe do not need to pass first through ( B , b ) -cocycles on the algebra of regularizing operators. Forthe convenience of the reader, we briefly recall this formalism, and let him report to the originalpaper [12] or the Appendix B for more details.3.4.1. Preliminaries.
Let A an associative algebra over C with unit. The bar construction B of A is the differential graded coalgebra B = L n > B n , with B n = A ⊗ n for n > with coproduct ∆ : B → B ⊗ B∆ ( a , . . . , a n ) = n X i = ( a , . . . , a i ) ⊗ ( a i + , . . . , a n ) The counit map η is the projection onto A ⊗ = C , and the differential is b ′ : b ′ ( a , . . . , a n + ) = n X i = (− ) i − ( a , . . . , a i a i + , . . . , a n + ) which is defined as the zero map on B and B . These operations confer a structure of differentialgraded coalgebra to B . bar cochain of degree n on A is a n -linear map over A with values in an algebra L . Thesecochains form a complex denoted Hom ( B , L ) , whose differential is given by δ bar f = (− ) n + fb ′ for f ∈ Hom n ( B , L ) . Moreover, one has a product on Hom ( B , L ) : If f and g are respectivelycochains of degrees p and q , it is given by fg ( a , . . . , a p + q ) = (− ) pq f ( a , . . . , a p ) g ( a p + , . . . , a p + q ) Therefore, Hom ( B , L ) has a structure of differential graded algebra.We next define Ω B and Ω B , ♮ to be the following bicomodules over B : Ω B = B ⊗ A ⊗ B , Ω B , ♮ = A ⊗ B Here, the ♮ in exponent means that Ω B , ♮ is the cocommutator subspace of Ω B . Thanks to this,one can show that the differential δ bar induced on Ω B , ♮ is in fact the Hochschild boundary, anddeduce that the complex ( Hom ( Ω B , ♮ , C ) , b ) is isomorphic to the Hochschild complex ( CC • ( A ) , b ) of A , with degrees shifted by one.We recall Quillen’s terminology. Let L be a differential graded algebra. Elements of Hom ( Ω B , L ) will be called Ω -cochains , and those in Hom ( Ω B , ♮ , L ) as Hochschild cochains . Recall also thatthe bar cochains are the elements of Hom ( B , L ) . Important fact.
A cochain f of this kind has three degrees : a A -degree as a multilinearmap over A , a L degree and a total degree f , which is sum. This is the one which will be considered.The map ♮ : Ω B , ♮ → Ω B , defined by the formula ♮ ( a ⊗ ( a , . . . , a n )) = n X i = (− ) i ( n − ) ( a i + , . . . , a n ) ⊗ a ⊗ ( a , . . . , a i ) induces a map from Hochschild cochains to bar cochains. If we have a (graded) trace τ : L − → C ,we then obtain a morphism of complexes τ ♮ : Hom ( Ω B , L ) − → Hom ( Ω B , ♮ , C ) f → τ ♮ ( f ) = τf ♮ Return to the initial problem.
We can now return to our context. Let A be the algebra S ( R n ) of Heisenberg formal symbols on R n = R p × R q , and B the bar construction of A . Also,let L be the graded algebra S ( R n ) ⊗ Λ • T ∗ R n . The product on these algebras is the star-productof symbols, twisted with the product on the exterior algebra. The injection ρ : A − → L is a homomorphism of algebras. As a consequence, ρ should be viewed as a 1-cochain of "curvature"zero, e.g δ bar ρ + ρ = . We introduce a formal parameter ε of odd degree such that ε = , andshall actually work in the extended algebraHom ( B , L )[ ε ] = Hom ( B , L ) + ε Hom ( B , L ) The role of that ε is to kill the powers of log | ξ | ′ which are not classical symbols, and to keep onlyits commutator with other symbols.Now, denote ∇ = F + ε log | ξ | ′ , and ∇ = F + ε [ log | ξ | ′ , F ] the square of ∇ , and introduce the"connection" ∇ + δ bar + ρ . The fact that this operator does not belong to the algebra above isnot a problem, since we shall only have interest in its "curvature", which is well defined, K = ∇ + [ ∇ , ρ ] = F + ε [ log | ξ | ′ , F ] + [ F + ε log | ξ | ′ , ρ ] nd its action on Hom ( B , L )[ ε ] with commutators. Here, we emphasize that the commutatorsinvolved are in fact graded commutators . Let τ be the graded trace on Hom ( B , L )[ ε ] ⊗ Λ • T ∗ R n given by τ ( x + εy ) = Z − y It turns out that the cocycles (3.4) and (3.5) constructed using excision in the previous sectionare obtained by considering the even cochain θ = τ ( ∂ρ · e K ) ∈ Hom ( Ω B , ♮ , C ) where ∂f · g is defined, for f , g ∈ Hom ( Ω B , L ) of respective degrees and n − , by the followingformula : ( ∂f · g ) ♮ ( a ⊗ ( a , . . . , a n )) = (− ) | g | f ( a ) g ( a , . . . , a n ) The calculation of θ becomes easier if one remarks that e K = e F · e [ F , ρ ]+ ε [ log | ξ | ′ , F + ρ ] as F = i ω is central in L . Then, this easily provides that θ = P k ( θ ′ + θ ′′ ), where(3.8) θ ′ = i n − k + ( − )! − X i = Z − (cid:18) ∂ρ · [ F , ρ ] i − δρ [ F , ρ ] − − i ⊗ ω n − k + ( n − k + )! (cid:19) (3.9) θ ′′ = i n − k ( )! − X i = Z − (cid:18) ∂ρ · [ F , ρ ] i δF [ F , ρ ] − − i ⊗ ω n − k ( n − k )! (cid:19) Evaluating on elements of A , this gives :(3.10) θ ′ ( a , . . . , a − )= i n − k + ( − )! − X i = (− ) i Z − (cid:18) a [ F , a ] . . . [ F , a i − ] δa i [ F , a i + ] . . . [ F , a − ] ⊗ ω n − k + ( n − k + )! (cid:19) (3.11) θ ′′ ( a , . . . , a − )= i n − k ( )! − X i = (− ) i + Z − (cid:18) a [ F , a ] . . . [ F , a i ] δF [ F , a i + ] . . . [ F , a − ] ⊗ ω n − k ( n − k )! (cid:19) The signs above not appearing in the cochains (3.8) and (3.9) occur since the a i , δρ and δF areodd.As announced earlier, we observe that θ ′ and θ ′′ are up to a certain constant term the cochains φ − and ψ − obtained in (3.4) and (3.5). The difference in signs is due to Quillen’s formalism,which considers the total differential B − b , see Remark B.4. Unfortunately, each component of θ = θ ′ + θ ′′ of θ is not a ( B , b ) -cocycle, but taking the entire cochain θ into account, this is.To prove this, it only suffices to check that all the things we defined have the good algebraicproperties to fit into Quillen’ proof. This is the content of the following lemma, which is actuallya "Bianchi identity" with respect to the "connection" ∇ + δ bar + ρ . Lemma . (Bianchi identity.) We have ( δ bar + ad ρ + ad ∇ ) K = ( δ bar + ad ρ + ad ∇ ) e K = ,where ad denotes the (graded) adjoint action. emark . The thing which guarantees this identity is that [ ∇ , ∇ ] = . Then, the proof isthe same as that given in the paper of Quillen, [12], Section 7. Thanks to this lemma, we directlyknow that ( B − b ) θ = , by adapting the arguments of [12], Sections 7 and 8. For the convenienceof the reader, we recalled these arguments in Appendix B. This result can be refined, and we getthe same results as those obtained using excision. Theorem . The inhomogeneous Hochschild cochains θ ′′ − θ ′ + ∈ Hom ( Ω B , ♮ , C ) ⊕ Hom + ( Ω B , ♮ , C ) for k n , define a ( B , b ) -cocycle. Proof.
Introduce a parameter t ∈ R , and consider the following family of curvatures ( K t ) : K t = ∇ , t + [ tF + ε log | ξ | ′ , ρ ] where ∇ , t = F + ε [ log | ξ | ′ , tF ] . Because the identity [ ∇ , ∇ , t ] still holds, we have a Bianchiidentity ( δ bar + ad ρ + ad ∇ ) K t = Thus, the Hochschild cochain θ t = τ ♮ ( ∂ρ · e K t ) ∈ Hom ( Ω B , ♮ , C )[ t ] satisfies the relation ( B − b ) θ t = for every t ∈ R , where we denote by R [ t ] the polynomials withcoefficients in an algebra R . Therefore, this relation also holds for every k , for the coefficient of t k . This coefficient is the cochain θ ′′ + θ ′ + , thus, θ ′′ − θ ′ + defines a ( B , b ) -cocycle. (cid:3) Denote by Ω = [ F , ρ ] + ε [ log | ξ | ′ , ρ + F ] . The cochains which cobounds these cocycles (up to modifyeach of them by a constant term depending on their degrees) may be obtained rather easily byusing suitable linear combinations of pairs of bar cochains ( µ , µ + ) , where µ is given by : µ k = τ ∂ρ · e F k ! k X i = Ω i FΩ k − i ! Doing this gives transgression formulas in the spirit of those obtained in Proposition 3.7.3.5.
Index theorem.
Now we know that the Radul cocycle on S R n c ( a , a ) = Z − a δa with δa = [ log | ξ | ′ , a ] , is cohomologous to the inhomogeneous ( B , b ) -cocycle ψ − + φ + ∈ CC − ( S ( R n )) ⊕ CC + ( S ( R n )) recalling that, ψ − ( a , . . . , a − ) = i n ( )! − X i = (− ) i + Z − a [ F , a ] . . . [ F , a i ] δF [ F , a i + ] . . . [ F , a − ] φ + ( a , . . . , a + ) = i n ( + )! + X i = (− ) i − Z − a [ F , a ] . . . [ F , a i − ] δa i [ F , a i + ] . . . [ F , a + ] t suffices to compute ψ − + φ + to obtain an index theorem. To begin, we first notice thatby Lemma 3.3, we may rewrite the cocycles above as(3.12) ψ − ( a , . . . , a − ) = i − i n ( )! − X i = (− ) i + Z − a da . . . da i δFda i + . . . da − (3.13) φ + ( a , . . . , a + ) = i − i n ( + )! + X i = (− ) i − Z − a da . . . da i − δa i da i + . . . da + The construction of the Wodzicki residue to Λ • T ∗ R n -valued symbols in the Paragraph 3.2 imposesthat the R − selects only the coefficient associated to the volume form dx . . . dx n dξ . . . dξ n . In(3.13), this coefficient must be a sum of terms of the form ∂b ∂x . . . ∂b n ∂x n ∂b + ∂ξ . . . ∂b ∂ξ n for someHeisenberg symbols b , . . . , b n of order 0 . Such terms have Heisenberg pseudodifferential order −( p + ) .However, in (3.13), there is in each sum an additional factor of the form δa i , which is a symbolof degree − . Hence, the symbols appearing in the formula are at most of Heisenberg order −( p + + ) , and vanishes because of (3.1).The formula for the cocycle (3.12) also reduces to a more simple one, but which is in generalnon-zero. A simple computation gives that δF = i p X i = ξ dξ i | ξ | ′ + n X i = p + ξ i dξ i | ξ | ′ Then, we proceed as we did to obtain the formula (3.13). The coefficient on dx . . . dx n dξ . . . dξ n of the symbols in (3.12) must be of the form(i) ∂b ∂x . . . ∂b n ∂x n ∂b + ∂ξ . . . ξ | ξ | ′ . . . ∂b ∂ξ n if i p ,(ii) ∂b ∂x . . . ∂b n ∂x n ∂b + ∂ξ . . . ξ i | ξ | ′ . . . ∂b ∂ξ n if p + i n where in each point, the term depending on | ξ | ′ replaces the term ∂b + i ∂ξ i . In all case, these termsare of order −( p + ) . Thus, if we denote the Heisenberg principal symbol by σ : S ( R n ) → C ∞ ( S ∗ H R n ) the symbol of order −( p + ) of a da . . . da i δFda i + . . . da − is σ ( a ) dσ ( a ) . . . dσ ( a i ) δFdσ ( a i + ) . . . dσ ( a − ) = (− ) i δFσ ( a ) dσ ( a ) . . . dσ ( a − ) We emphasize that the latter product is no more the star-product but the usual product offunctions.The vector field L = P pj = ξ j ∂ ξ j + P nj = p + ξ j ∂ ξ j on T ∗ R n is the generator of the Heisenbergdilations. This implies that ι L dσ ( a i ) = dσ ( a i ) · L = since the a i are symbols of order . Using(3.1), and observing that ι L δF = i , we obtain ψ − ( a , . . . , a − ) = (− ) n ( i ) n ( − )! Z S ∗ H R n σ ( a ) dσ ( a ) . . . dσ ( a − ) So, we have proved the following theorem
Theorem . The Radul cocycle is ( B , b ) -cohomologous to the homogeneous ( B , b ) -cocycle on S H ( R n ) defined by ψ − ( a , . . . , a − ) = ( i ) n ( − )! Z S ∗ H R n σ ( a ) dσ ( a ) . . . dσ ( a − ) rom this theorem and the pairing (2.2), given for any [ φ ] ∈ HP ( S H ( R n ) and u ∈ K ( S H ( R n ) bythe formula h [ φ ] , u i = X k > (− ) k k !( φ + ⊗ tr )( u , u − , . . . , u , u − ) one has the following index theorem for Heisenberg elliptic pseudodifferential operators of order ,which only depends on the principal symbol. Here, working in the framework of cyclic cohomologyis convenient because we can directly pass from scalar symbols to matrices thanks to Moritaequivalence. Theorem . Let P ∈ M N ( Ψ ( R n )) a Heisenberg elliptic pseudodifferential operator ofsymbol u ∈ GL N ( S ( R n )) , and [ u ] ∈ K ( S ( R n )) its (odd) K-theory class. Then, we have aformula for the Fredholm index of P : Ind ( P ) = Tr ( Ind [ u ]) = − ( n − )!( i ) n ( − )! Z S ∗ H R n tr ( σ ( u ) − dσ ( u )( dσ ( u ) − dσ ( u )) n − )) Discussion on manifolds with conical singularities
Studying index theory on manifolds with singularities is actually one of the motivations for study-ing a residue index formula adapted to cases where the zeta function exhibits multiple poles. It isindeed known for many years that zeta functions of differential operators on conic manifolds havedouble poles, see for example the paper of Lescure [8]. In the pseudodifferential case, even triplepoles may occur, see [6].We shall first recall briefly what we need from the theory of conic manifolds, e.g pseudodifferentialcalculus, residues and results on the associated zeta function. This review part essentially followsthe presentation of [6].4.1.
Generalities on b-calculus and cone pseudodifferential operators.
In our context,manifolds with conical singularities are just manifolds with boundary with an additional structuregiven by a suitable algebra of differential operators.More precisely, let M be a compact manifold with (connected) boundary, and r : M → R + be aboundary defining function, e.g a smooth function vanishing on ∂M and such that its differential isnon zero on every point of ∂M . We work in a collar neighbourhood [ , ) r × ∂M x of the boundary,the subscripts are the notations for local coordinates. Definition . A Fuchs type differential operator P of order m is a differential operatoron M which can be written in the form P ( r , x ) = r − m X j + | α | m a j , α ( r , x )( r∂ r ) j ∂ αx in the collar [ , ) r × ∂M x . The space of such operators will be denoted r − m Diff mb ( M ) .Diff mb ( M ) are the b -differential operators of Melrose’s calculus for manifolds with boundary. Wenow recall the associated small b -pseudodifferential calculus Ψ b ( M ) .Let M be the b -stretched product of M , e.g the manifold with corners whose local charts aregiven by the usual charts on M r ∂M , and parametrized by polar coordinates over ∂M in M .More precisely, writing M × M near r = r ′ = as M ≃ [ , ] r × [ , ] r ′ × ∂M this means that we parametrize the part [ , ] r × [ , ] r ′ in polar coordinates r = ρ cos θ , r ′ = ρ sin θ or ρ ∈ R + , θ ∈ [ , π/2 ] . The right and left boundary faces are respectively the points where θ = and θ = π/2 .Let ∆ b the b -diagonal of M , that is, the lift of the diagonal in M . Note that ∆ b is in factdiffeomorphic to M , so that any local chart on ∆ b can be considered as a local chart on M . Definition . The algebra of b -pseudodifferential operators of order m , denoted Ψ mb ( M ) ,consists of operators P : C ∞ ( M ) → C ∞ ( M ) having a Schwartz kernel K P such that(i) Away from ∆ b , K P is a smooth kernel, vanishing to infinite order on the right and leftboundary faces.(ii) On any local chart of M intersecting ∆ b of the form U r , x × R n such that ∆ b ≃ U × { } ,and where U is a local chart in the collar neighbourhood [ , ) r × ∂M x of ∂M , we have K P ( r , x , r ′ , x ′ ) = ( ) n Z e i ( log ( r/r ′ ) · τ + x · ξ ) a ( r , x , τ , ξ ) dτ dξ where a ( y , ν ) , with y = ( r , x ) and ν = ( τ , ξ ) , is a classical pseudodifferential symbol oforder m , plus the condition that a is smooth in the neighbourhood of r = .Remark that log ( r/r ′ ) should be singular at r = r ′ = if we would have considered kernels definedon M . Introducing the b -stretch product M has the effect of blowing-up this singularity.The algebra of conic pseudodifferential operators is then the algebra r − Z Ψ Z b ( M ) . The opposedsigns in the filtrations are only to emphasize that r ∞ Ψ − ∞ b ( M ) is the associated ideal of regularizingoperators.To such an operator A = r − p P ∈ r − p Ψ mb , we define on the chart U the local density ω ( P )( r , x ) = (cid:18) Z | ν | = p − n ( r , x , τ , ξ ) ι L dτdξ (cid:19) · drr dx where ν = ( τ , ξ ) and L is the generator of the dilations.It turns out (but this is not obvious) that this a priori local quantity does not depend on thechoice of coordinates on M , and hence, define a globally defined density ω ( P ) , smooth on M , thatwe call the Wodzicki residue density . Unfortunately, the integral on M of this density does notconverge in general, as the boundary introduces a term in in the density. However, we canregularize this integral, thanks to the following lemma. Here, Ω b denote the bundle of b -densitieson M , that is, the trivial line bundle with local basis on the form ( dr/r ) dx . The following lemmafrom Gil and Loya is proved in [6]. Lemma . Let r − p u ∈ C ∞ ( M , Ω b ) , and p ∈ R . Then, the function z ∈ C → Z M r z u is holomorphic on the half plane Re z > p , and extends to a meromorphic function with onlysimple poles at z = p , p − , . . . . If p ∈ N , Its residue at z = is given by (4.1) Res z = Z M r z u ( r , x ) drr dx = ! Z ∂M ∂ pr ( r p u ( r , x )) r = dx Applying this regularization to the Wodzicki residue density is useful to many "residues traces"that we immediately study. races on conic pseudodifferential operators. We first begin by defining different algebrasof pseudodifferential operators, introduced by Melrose and Nistor in [9]. The main algebra thatwe shall consider is A = r − Z Ψ Z ( M ) = [ p ∈ Z [ m ∈ Z r − p Ψ m ( M ) which clearly contains the algebra of Fuchs type operators. The ideal of regularizing operators is I = r ∞ Ψ − ∞ ( M ) = [ p ∈ Z [ m ∈ Z r − p Ψ m ( M ) and this explains why we note the two filtrations by opposite signs in A . Consider the followingquotients I σ = r ∞ Ψ Z ( M ) /I , I ∂ = r Z Ψ − ∞ ( M ) /I Here, I σ should be thought as an extension of the algebra of pseudodifferential operators in theinterior of M , whereas I ∂ are regularizing operators up to the boundary. We finally define A ∂ = A /I σ , A σ = A/I ∂ , A ∂ , σ = A/ ( I ∂ + I σ ) Definition . Let P ∈ r − p Ψ m ( M ) be a conic pseudodifferential operator, with p , m ∈ Z .According to Lemma 4.3, define the functionals Tr ∂ , σ , Tr σ to beTr ∂ , σ ( P ) = Res z = Z M r z ω ( P )( r , x ) drr dx = ! Z ∂M ∂ pr ( r p ω ( P )( r , x )) r = dx (4.2) Tr σ ( P ) = Pf z = Z M r z ω ( P ) drr dx (4.3)where Pf denotes the constant term in the Laurent expansion of a meromorphic function. Remark . Using Lemma 4.3, one can show that Tr ∂ , σ ( P ) does not depend on the choiceof the boundary defining function r . This is not the case for Tr σ ( P ) , but its dependence on r canbe explicitly determined, cf. [6].The "Partie Finie" regularization of a trace does not give in general a trace, and this is indeedthe same for the functional Tr σ ( P ) acting on these algebras, the obstruction to that is preciselythe presence of the boundary. However, by definition, Tr σ ( P ) clearly defines an extension ofthe Wodzicki residue for pseudodifferential operators, one can expect that it is a trace on I σ = r ∞ Ψ Z ( M ) /I . Theorem . (Melrose - Nistor, [6, 9]) Tr σ is, up to a multiplicative constant, the uniquetrace on the algebra I σ By Lemma 4.3 and the definition above, the defect of Tr σ to be a trace is precisely measured byTr ∂ , σ ( P ) , which can therefore be viewed as a restriction of the Wodzicki residue to the boundary ∂M . Then, the following proposition seems natural. Theorem . (Melrose - Nistor, [6, 9]) Tr ∂ , σ is, up to a multiplicative constant, the uniquetrace on the algebras A ∂ , A σ and A ∂ , σ These two traces may be seen as "local" terms, since they only depend on the symbol of thepseudodifferential operator considered. The first can be seen as a trace on interior of M , thesecond is related to the boundary ∂M . There is one last trace to introduce, less easy to deal withbecause this one is not local. ix a holomorphic family Q ( z ) ∈ r αz Ψ βzb ( M ) , with α , β ∈ R , such that Q is the identity at z = .Take P ∈ r − p Ψ mb , with p , m ∈ Z and let ( PQ ( z )) ∆ be the restriction to the diagonal ∆ of M ofthe Schwartz kernel of PQ ( z ) . Melrose and Nistor noticed in [9] that ( PQ ( z )) ∆ is meromorphic in C , with values in r αz − p C ∞ ( M ) with possible simple poles in the set (cid:12) − n − mβ , − n − m + , . . . (cid:13) Definition . Let P ∈ r − p Ψ mb be a conic pseudodifferential operator. Then, we defineTr ∂ ( P ) = ! Z ∂M ∂ pr ( r p Pf z = ( PQ ( z )) ∆ ) r = dx If p is not an integer, then, Tr ∂ ( P ) is defined to be . Remark . Tr ∂ ( P ) depend on the choice of the operator Q , but the dependence can beexplicitly determined, see [9].There is an interpretation of Tr ∂ analogous to those of Tr ∂ , σ : If the order of P is less than thedimension of M , then Tr ∂ ( P ) is a kind of L of P restricted to the boundary. This is precisely thecontent of the following result. Theorem . (Melrose - Nistor, [6, 9]) Tr ∂ ( P ) is, up to a multiplicative constant, theunique trace on the algebra I ∂ = r Z Ψ − ∞ ( M ) /I Heat kernel expansion and zeta function.
Now, let ∆ ∈ r − Diff ( M ) be fully elliptic , or parameter elliptic with respect to a parameter α . We refer to [6] for the definition, what weneed to know is just that this condition ensures the existence of the heat kernel e − t∆ of A , andthat operators of the type P∆ − z , with P ∈ r − p Ψ mb , are of trace-class on r α − m L ( M ) for z in thehalf-plane Re z > max { m + n2 , p2 } , n = dim M . Example . As usual, we work in a collar neighbourhood of M . Then, the operator(4.4) ∆ = (cid:18) ( r∂ r ) − ∆ ∂M + ( n − ) + a (cid:19) where a > 1 , is and α = , is an example of such an operator. See [6] for more details.Then, the traces introduced in the previous paragraph gives the coefficients of the expansion ofTr ( Pe − t∆ ) . Theorem . (Gil - Loya, [6]) Under the conditions above, we have Tr ( Pe − t∆ ) ∼ t → X k > a k t ( k − p ) /2 + ( b k + β k log t ) t k + ( c k + γ k log t + δ k ( log t ) ) t ( k − m − n ) /2 where β k = C k ( Tr σ + Tr ∂ )( P∆ k ) γ k = C ′ K Tr ∂ , σ ( P∆ k − m − n ) δ k = C ′′ k Tr ∂ , σ ( P∆ k − m − n ) C k , C ′ K , C ′′ k are explicit (but not of interest for us). n particular, the coefficient of log t is − Tr σ ( P ) − Tr ∂ ( P ) − Tr ∂ , σ ( P ) and the coefficient of ( log t ) is − Tr ∂ , σ ( P ) Using a Mellin transform, we can writeTr ( P∆ − z/2 = ( z/2 ) Z ∞ t z − Tr ( Pe − t∆ ) dt and knowing, that z → R ∞ t z − Tr ( Pe − t∆ ) dt is entire, the asymptotic expansion of the previousproposition gives the following corollary on the zeta function. Corollary . The zeta function z → Tr ( P∆ − z/2 ) is holomorphic in the half-plane Re z > max { m + n , p } , and extends to a meromorphic function with at most triple poles,whose set is discrete. At z = , there are simple and double poles only, which are respectivelygiven by the terms of log t and ( log t ) in the heat kernel expansion of Tr ( Pe − t∆ ) . Spectral triple and regularity.
In this paragraph, we want to investigate if Fuchs typeoperators on conic manifolds can define an abstract algebra of differential operators, so that thelocal index formula we gave in the first section applies.We start with a conic manifold. Let M be a manifold with connected boundary, with boundarydefining function r , endowed with the algebra of Fuchs type differential operators. The points (i),(ii), (iii) of Definition 1.1 are verified, if for example we take for ∆ the fully-elliptic operator oforder 2 given in Example 4.4, and require that the order is given by the differential order. Moregenerally, working locally in a collar neighbourhood [ , ) r × ∂M x of the boundary ∂M , elementarycalculations shows that(4.5) [ r p Diff mb ( M ) , r p ′ Diff m ′ b ( M )] ⊂ r p + p ′ Diff m + m ′ − ( M ) and as we shall see, the fact that the order in r does not decrease is the problem.Let us denote by r p C ∞ ( ∂M ) (find a better notation ...) the subalgebra of C ∞ ( M ) of functions f which have an asymptotic expansion f ( r , x ) ∼ r p f p ( x ) + r p + f p + ( x ) + . . .in a neighbourhood of r = . Here, the ∼ means that the rest of such an expansion is of the form r N f N ( r , x ) , with f N bounded in the collar [ , ) × ∂M . The case p = actually corresponds tothe smooth functions on the collar.For the algebra of the spectral triple, it seems a good choice to look for a candidate among theseclasses of functions. But doing so, the formula of Lemma 1.9 is no more asymptotic in the senseof Definition 1.7. Indeed, if b ( r , x ) = r p for p ∈ N , the observation (4.5) shows that the terms b ( k ) are in r p − Diff kb ( M ) , but by the properties of the zeta function given in the Corollary 4.13,the function z → Tr ( b ( k ) ∆ − k − z ) is holomorphic for Re ( z )+ k > max (cid:10) n + k2 , − p2 (cid:11) , which is equivalent to Re ( z ) > max (cid:8) n − k2 , − p2 (cid:9) .Hence, if p > , the function above is in general not holomorphic at when N goes to infinity. Inother terms, the spectral triple we may construct will be not regular, and local index formulas ofConnes-Moscovici, or those given at the beginning cannot be applied directly. As we have seen,the main problem is due to the fact that there are two notions of order : The differential order, hich is local, and "the order in r ", which is not, and comes form the presence of the boundary ∂M .However, we may recover some interesting informations on M from the zeta function. Note forinstance that the higher residue R − defined in Proposition 1.10 gives the trace Tr ∂ , σ . R − is,modulo some constant terms, the sum of the three functionals Tr ∂ , σ , Tr σ , Tr ∂ , which illustratesthat it is no more a trace on the algebra of conic pseudodifferential operators. The next paragraphis a discussion on index theory.4.3. A local index formula.
The formula of Theorem 2.1 cannot be applied directly since we arenot in the context of regular spectral triples. However, there are always some relevant informationsto get on index theory.Let M be a manifold with boundary, seen as a conic manifold, and consider the extension → r ∞ Ψ − ∞ b ( M ) → r − Z Ψ Z b ( M ) → r − Z Ψ Z b ( M ) /r ∞ Ψ − ∞ b ( M ) → Here, by an elliptic pseudodifferential operator P ∈ r − Z Ψ Z b ( M ) , we shall mean that P is invertiblein the quotient A = r − Z Ψ Z b ( M ) /r ∞ Ψ − ∞ b ( M ) . Being fully elliptic is an extra condition on theindicial or normal operator, which guarantees that P is Fredholm between suitable spaces. Weshall not enter into these details : What we want to investigate is just the pairing given in theparagraph (2.2). In particular, if P is fully elliptic, then the pairing really calculates a Fredholmindex.Now, let P , Q ∈ r − Z Ψ Z b ( M ) . We can still follow the "Partie Finie" argument given in the proof ofTheorem 2.1, so that we still have the Radul cocycle c ( P , Q ) = Pf z = Tr ([ P , Q ] ∆ − z ) Res z = Tr (cid:18) P · (cid:18) Q − ∆ − z Q∆ − z z (cid:19) ∆ − z (cid:19) As we already said, the Connes-Moscovici’s formula in Lemma 1.9 is no more asymptotic, butfrom an algebraic viewpoint, the (1.2) still holds. So, for any integer N , which will be thoughtlarge enough, we have Q − ∆ − z Q∆ − z = N X k = Q ( k ) ∆ − k + i Z λ − z ( λ − ∆ ) − Q ( N + ) ( λ − ∆ ) − N − dλ We now take advantage of the fact that the traces Tr σ and Tr ∂ , σ vanishes when the differentialorder of the operators is less that the dimension of M . We then have the following result. Theorem . Let M be a conic manifold, i.e a manifold with boundary endowed witha conic metric, and let r be a boundary defining function. Let ∆ be the "conic laplacian" ofExample 4.11. Then, the Radul cocycle associated to the pseudodifferential extension → r ∞ Ψ − ∞ b ( M ) → r − Z Ψ Z b ( M ) → r − Z Ψ Z b ( M ) /r ∞ Ψ − ∞ b ( M ) → is given by the following non local formula : c ( a , a ) = ( Tr ∂ , σ + Tr σ )( a [ log ∆ , a ]) − Tr ∂ , σ ( a [ log ∆ , [ log ∆ , a ]])++ Tr ∂ a X k = a ( k ) ∆ − k ! + i Tr (cid:18) Z λ − z a ( λ − ∆ ) − a ( N + ) ( λ − ∆ ) − N − (cid:19) dλ for a , a ∈ Ψ Z b ( M ) /r ∞ Ψ − ∞ b ( M ) n the right hand-side, the first line consists in local terms only depending on the symbol of P ,the second line gives the non local contributions.If P ∈ r − Z Ψ Z b ( M ) is an elliptic operator, so that P defines an element in the odd K-theory group K alg ( A ) , and Q an inverse of P modulo A , we then obtain a formula for the index of P . The secondline of the formula above should be a part of the eta invariant (when it is defined). A perspectivemay be to investigate how to compare these different elements in order to get another definitionof the eta invariant, suitable not only for Dirac operators but also for general pseudodifferentialoperators. Appendix A. Computations of Section 3.1
We give here the details of the different computations allowing to derive the different formulas ofSection 3.A.1.
Cocycles formulas.
Recall that e φ ( a , . . . , a )= k ! i k ( )! + X i = Pf z = Tr (cid:18) a [ F , a ] . . . [ F , a i ] ∆ − z/4 [ F , a i + ] . . . [ F , a ] ⊗ ω n − k n ! (cid:19) Formula (3.4).
We compute ψ − = B e φ B e φ ( a , ..., a − )= k ! i k ( )! + X i = Pf z = Tr h(cid:16) [ F , a ] . . . [ F , a i ] ∆ − z/4 [ F , a i + ] . . . [ F , a − ]− [ F , a − ][ F , a ] . . . [ F , a i − ] ∆ − z/4 [ F , a i ] . . . [ F , a − ] + . . . +(− ) − [ F , a ] . . . [ F , a i + ] ∆ − z/4 [ F , a i + ] . . . [ F , a − ][ F , a ] (cid:17) ⊗ ω n − k n ! (cid:21) Then, by the graded trace property, one can remark that all the terms of the sum P = . . . aresimilar, so, this sum equals ( + ) times the term i = . B e φ ( a , ..., a − )= k ! i k ( )! Pf z = Tr h(cid:16) [ F , a ] . . . [ F , a − ] ∆ − z/4 − [ F , a − ][ F , a ] . . . [ F , a − ] ∆ − z/4 + . . . + (− ) − [ F , a ] . . . [ F , a − ][ F , a ] ∆ − z/4 (cid:17) ⊗ ω n − k n ! (cid:21) = k ! i k ( )! − X i = Pf z = Tr (cid:18) [ F , a ] . . . [ F , a i ] ∆ − z/4 [ F , a i + ] . . . [ F , a − ] ⊗ ω n − k n ! (cid:19) here we used the graded trace property in the second equality. Then, writing [ F , a ] = Fa − a F ,using the fact that F anticommutes with the [ F , a i ] and the graded trace property again, we obtain B e φ ( a , ..., a − )= k ! i k ( )! − X i = Pf z = Tr (cid:16) a [ F , a ] . . . [ F , a i ]((− ) − i ∆ − z/4 F − (− ) i F∆ − z/4 )[ F , a i + ] . . . [ F , a − ] ⊗ ω n − k n ! (cid:19) = k ! i k ( )! − X i = (− ) i + Res z = Tr (cid:18) a [ F , a ] . . . [ F , a i ] [ F , ∆ − z/4 ] z [ F , a i + ] . . . [ F , a − ] ⊗ ω n − k n ! (cid:19) From Theorem 2.1, or, to be more precise, the part of the proof allowing to pass from the PartieFinie to the residue, we finally obtain B e φ ( a , ..., a − )= k ! i k ( )! − X i = (− ) i + Z − (cid:18) a [ F , a ] . . . [ F , a i ] δF [ F , a i + ] . . . [ F , a − ] ⊗ ω n − k n ! (cid:19) = ψ − ( a , . . . , a − ) (cid:3) Formula (3.5).
We now compute φ + = b e φ . As [ F , . ] is an derivation on S H ( R n ) , thefollowing equality may be observed easily b e φ ( a , ..., a + ) = k ! i k ( + )! X i = (− ) i Pf z = Tr (cid:16) a [ F , a ] . . . [ F , a i ][ a i + , ∆ − z/4 ][ F , a i + ] . . . [ F , a + ]) ⊗ ω n − k n ! (cid:19) Again, from the proof of Theorem 2.1, we finally have b e φ ( a , ..., a + )= k ! i k ( + )! + X i = (− ) i − Z − (cid:18) a [ F , a ] . . . [ F , a i − ] δa i [ F , a i + ] . . . [ F , a + ] ⊗ ω n − k n ! (cid:19) = φ + ( a , ..., a + ) (cid:3) A.2.
Transgression formulas.
We now give the details of the computations needed to obtainthe formulas of Proposition 3.7. Recall that e γ + ( a , . . . , a + )= ( k + )! i k + ( + )! + (cid:20) Pf z = Tr (cid:18) a ∆ − z/4 F [ F , a ] . . . [ F , a + ] ⊗ ω n − k − n ! (cid:19) + + X i = Pf z = Tr (cid:18) a F [ F , a ] . . . [ F , a i ] ∆ − z [ F , a i + ] . . . [ F , a + ] ⊗ ω n − k − n ! (cid:19) where the term i = of the sum means Pf z = Tr (cid:16) a F∆ − z [ F , a ] . . . , [ F , a + ] ⊗ ω n − k − n ! (cid:17) . ormula (3.6). We compute B e γ + ( a , . . . , a ) . By the graded trace property, applyingthe operator B to each term of e γ + yields the same contribution. As there are ( + ) terms,we have B e γ + ( a , . . . , a ) = ( k + )! i k + ( + )! Pf z = Tr (cid:18) F [ F , a ] . . . [ F , a ]+ F [ F , a ][ F , a ] . . . [ F , a − ] + . . . + F [ F , a ] . . . F [ F , a ][ F , a ]) ∆ − z/4 ⊗ ω n − k − n ! (cid:19) Writing ( k + )!( + )! =
12 k !( + )! , knowing that F anticommutes to the [ F , a i ] and that F = i ω iscentral, developing F [ F , a ] and finally using the graded trace property, we obtain B e γ + ( a , . . . , a )= k ! i k + ( + )! · X i = Pf z = (cid:18) ( a F − Fa F )[ F , a ] . . . ∆ − z/4 . . . [ F , a ]) ⊗ ω n − k − n ! (cid:19) Once again using that F = i ω , we can write e φ ( a , . . . , a )= k ! i k + ( + )! X i = Pf z = Tr (cid:18) a F [ F , a ] . . . [ F , a i ] ∆ − z/4 [ F , a i + ] . . . [ F , a ] ⊗ ω n − k − n ! (cid:19) hence, ( e φ − B e γ + )( a , . . . , a )= k ! i k + ( + )! · X i = Pf z = (cid:18) ( a F + Fa F )[ F , a ] . . . ∆ − z/4 . . . [ F , a ] ⊗ ω n − k − n ! (cid:19) = k ! i k + ( + )! · X i = Pf z = a F [ F , a ] . . . ((− ) i F∆ − z/4 − (− ) − i ∆ − z/4 F ) . . . [ F , a ] ⊗ ω n − k − n ! (cid:19) Finally, we obtain ( e φ − B e γ + )( a , . . . , a )= k ! i k + ( + )! X i = (− ) i Z − (cid:18) a F [ F , a ] . . . δF . . . [ F , a ] ⊗ ω n − k − n ! (cid:19) = γ ( a , . . . , a ) (cid:3) Formula (3.7).
We now calculate b e γ + . Writing a F = −[ F , a ] + Fa and using thederivation property of [ F , . ] , b e γ + ( a , . . . , a + )= − e φ + ( a , . . . , a + )+ ( k + )! i k + ( + )! (cid:20) Pf z = (cid:18) a [ a , ∆ − z/4 ][ F , a ] . . . [ F , a + ] ⊗ ω n − k − n ! (cid:19) + + X i = (− ) i Pf z = (cid:18) a F [ F , a ] . . . [ a i + , ∆ − z/4 ][ F , a ] . . . [ F , a + ] ⊗ ω n − k − n ! (cid:19) inally, ( e φ + + b e γ + )( a , . . . , a + )= ( k + )! i k + ( + )! (cid:20) Z − (cid:18) a δa [ F , a ] . . . [ F , a + ] ⊗ ω n − k − n ! (cid:19) + + X i = (− ) i − Z − (cid:18) a F [ F , a ] . . . δa i . . . [ F , a + ] ⊗ ω n − k − n ! (cid:19) = γ + ( a , . . . , a + ) (cid:3) Appendix B. Complements on Section 3.2
For the convenience of the reader, we recall here Quillen’s picture of ( B , b ) -cocycles and how it isused to obtain Theorem 3.10 from the Bianchi identity of Lemma B.5.B.1. More on Quillen’s formalism.
Let A be an associative algebra over C , and B be the barconstruction of A . Recall that Ω B and Ω B , ♮ are the following bicomodules over B : Ω B = B ⊗ A ⊗ B , Ω B , ♮ = A ⊗ B Theorem
B.1 . One has a complex of period . . . ∂ / / B β / / Ω B , ♮ ∂ / / B β / / . . . with ∂ = ∂ ♮ : Ω B , ♮ → B , where ♮ : Ω B , ♮ → Ω B , ∂ : Ω B → B , β : B → Ω B , ♮ are defined by thefollowing formulas : ♮ ( a ⊗ ( a , . . . , a n )) = n X i = (− ) i ( n − ) ( a i + , . . . , a n ) ⊗ a ⊗ ( a , . . . , a i ) ∂ ( a , . . . , a p − ) ⊗ a p ⊗ ( a p + , . . . , a n ) = ( a , . . . , a n ) ∂ ( a ⊗ ( a , . . . , a n )) = n X i = (− ) i ( n − ) ( a i + , . . . , a n , a , a , . . . , a i ) β ( a , . . . , a n ) = (− ) n − a n ⊗ ( a , . . . , a n − ) − a ⊗ ( a , . . . , a n ) As Quillen shows in [12], it turns out that the 2-periodic complex constructed above is exactly theLoday-Quillen cyclic bicomplex with degrees shifted by one, and is therefore equivalent to Connes ( B , b ) -bicomplex. The shift of the degrees makes that elements of the algebra A become odd inthe bar construction, while they are even in the cyclic bicomplex.Now, let L be a differential graded algebra. The maps ∂ and β of the periodic complex inducesmaps from bar cochains to Hochschild cochains (with values in L ) and conversely by pull-back.The following formula is a key step. Lemma
B.2 . Let f , g ∈ Hom ( B , L ) be bar cochains. Then, we have β ( τ ♮ ( ∂f · g )) = − τ ([ f , g ]) We carry a purely computational proof, because of the way we introduced Quillen’s formalism. Amore elegant and conceptual proof is given in Quillen’s article [12], paragraph 5.2. The proof ofthis lemma is based on the following formula,(B.1) ( ∂f · g ) ♮ ( a ⊗ ( a , . . . , a n )) = X n − p
Let f and g be bar cochains of respective degrees p and n − p . By definition, β ( τ ♮ ( ∂f · g )) = τ ( ∂f · g ) ♮ β , and using (B.1), so, β ( τ ♮ ( ∂f · g ))( a , . . . , a n )= τ ( ∂f · g ) ♮ (((− ) n − a n ⊗ ( a , . . . , a n − ) − a ⊗ ( a , . . . , a n ))= τ X n − p
B.3 . Let θ ∈ Hom ( Ω B , ♮ , C ) be a Hochschild cochain, and η ∈ Hom ( B , C ) be thebar cochain defined by η k ( a , . . . , a k ) = θ ( , a , . . . , a k ) Suppose that for each k , we have δ bar η k = (− ) k βθ k + , δ bar θ k + = (− ) k ∂η k + and that in addition, θ n + ( a , a , . . . , a n ) = if a i = , for i > .Then, for all k , Bθ k + = bθ k − . Remark
B.4 . This means that if we redefine signs correctly in θ , we obtain a ( B , b ) -cocyclein our sign conventions. .2. Complements on Remark 3.9.
We give here the details of Quillen’s arguments. The onlything we have done towards the original paper [12] is to mix the arguments of Sections 7 and 8.
Lemma
B.5 . (Bianchi identity.) We have ( δ bar + ad ρ + ad ∇ ) K = ( δ bar + ad ρ + ad ∇ ) e K = ,where ad denotes the (graded) adjoint action. Proof.
Let D be the derivation δ bar + ad ρ + ad ∇ . It suffices to check that D ( K ) = , theother equality will follow in virtue of the differentiation formula D ( e K ) = Z e ( − s ) K D ( K ) e sK ds We first remark that [ ∇ , ∇ ] = , using that ε commutes (in the graded sense) with elements ofHom ( B , L ) and that ε = . Furthermore δ bar ∇ = since δ bar vanishes on 0-cochains. Therefore, D ( K ) = ( δ bar + ad ρ + ad ∇ )( ∇ + [ ∇ , ρ ])= δ bar [ ∇ , ρ ] + [ ρ , [ ∇ , ρ ]] + [ ρ , ∇ ] + [ ∇ , [ ∇ , ρ ]]= [ ∇ , ρ ] + ρ [ ∇ , ρ ] − [ ∇ , ρ ] ρ + [ ρ , ∇ ] + [ ∇ , ρ ]= The result is proved. (cid:3)
According to Theorem B.3, let us define the bar cochain η ∈ Hom ( B , C ) : η − ( a , . . . , a − ) = θ ( , a , . . . , a + ) Also remark that η = τ ( e K ) . Proposition
B.6 . The bar and Hochschild cochains η and θ satisfies the relations δ bar η = ± βθ , δ bar θ = ± ∂η The ± means that the sign is positive in the even case and negative in the odd case. Proof.
For the first formula of the proposition, we have δ bar η = δ bar ( τ ( e K )) = τ ( δ bar e K ) = τ ( δ bar e K + [ ∇ , e K ]) = − τ ([ ρ , e K ]) = ± β ( τ ♮ ( ∂ρ · e K )) The second equality uses the trace property of τ , the third is the Bianchi identity of the lemmaabove, and the last one is Lemma B.2.For the second formula, first recall that δ bar ρ + ρ = . Then, one has : δ bar ( τ ♮ ( ∂ρ · e K )) = τ ♮ ( ∂ (− ρ ) e K − ∂ρ · δ bar e K ) = τ ♮ ([ ρ , ∂ρ · e K ]) = τ ♮ (( ρ · ∂ρ + ∂ρ · ρ ) e K − ∂ρ · [ ρ , e K ]) = τ ♮ ([ ∇ , ∂ρ · e K ]) = τ ♮ ( ∂ [ ∇ , ρ ] e K − ∂ρ · [ ∇ , e K ]) Adding these three equations, using Bianchi identity and δ bar ρ + ρ = yields δ bar ( τ ♮ ( ∂ρ · e K )) = τ ♮ ( ∂ [ ∇ , ρ ] e K ) = τ ♮ ( ∂K · e K ) The last equality follows from the definition of K . Moreover, ∂ ( e K ) = τ ♮ ( ∂e K ) = Z τ ♮ ( e ( − t ) K · ∂K · e tK ) dt = τ ♮ ( ∂K · e K ) where last equality stands because of the trace property. This concludes the proof. (cid:3) Hence, Theorem B.3 shows that θ gives rise to a ( B , b ) -cocycle (up to changing signs). The samearguments may be used to complete the proof of Theorem 3.10. eferences [1] R. Beals and P.C. Greiner. Calculus on Heisenberg manifolds , volume 119 of
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Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43, Bd du 11 novembre1918, 69622 Villeurbanne Cedex, France
E-mail address : [email protected]@math.univ-lyon1.fr