Institute for Mathematical Physics on the Analyticity of the Bivariant Jlo Cocycle

. The goal of this note is to outline a proof that, for any ℓ ≥ 0, the JLO bivariant cocycle associated with a family of Dirac type operators along a smooth ﬁbration 𝑀 → 𝐵 over the pair of algebras ( 𝐶 ∞ ( 𝑀 ) ,𝐶 ∞ ( 𝐵 )), is entire when we endow 𝐶 ∞ ( 𝑀 ) with the 𝐶 ℓ +1 topology and 𝐶 ∞ ( 𝐵 ) with the 𝐶 ℓ topology. As a corollary, we deduce that this cocycle is analytic when we consider the Fr´echet smooth topologies on 𝐶 ∞ ( 𝑀 ) and 𝐶 ∞ ( 𝐵 ).


Preliminaries and notations
It has been generally thought that the Atiyah-Singer families index theorem should be expressible in terms of bivariant cyclic theory. From the authors' point of view this question arose from a discussion at Oberwolfach in 2004 (we thank Masoud Khalkhali and Alain Connes for comments). A number of partial results in this direction have been obtained ( [11]). Finding the right bivariant framework for this problem and for the related question of the bivariant version of the Connes-Moscovici residue cocycle has presented an obstacle. In [10] we have found an appropriate formalism. In this note we define a bivariant JLO cocycle in terms of which we can reformulate the local families index theorem [6,5]. We prove that our bivariant cocycle is analytic in the sense of the formalism introduced by Meyer [10], when one considers the ℓ topology on the algbera of functions on the base, and the ℓ+1 topology on the algebra of functions on the total space of the fibration. We thank Ralf Meyer for his helpful comments on a previous version of this work.
Thus we shall consider a fibration → → of closed manifolds endowed with smooth metrics. As can be seen from the proofs of the present paper, the main result remains true in larger categories like manifolds with corners [9] or Heisenberg manifolds [12], but we will not give the details here. The dimension of the fibers is denoted by and the dimension of the base is ′ . We assume for simplicity that the fibers of our fibration are odd dimensional, so is odd. We also fix a hermitian vector bundle → whose fibres are modules over the Clifford algebra of the fiberwise tangent bundle = Ker( * ). There is a Clifford homomorphism of algebra bundles

MOULAY-TAHAR BENAMEUR AND ALAN L. CAREY
We endow with a Clifford connection ∇ and consider the fiberwise Dirac operator associated with this connection. can be regarded as a family of elliptic operators along the fibers parametrized by the elements of the base manifold ; i.e., = ( ) ∈ where : ∞ ( , | ) → ∞ ( , | ) is an essentially self adjoint operator [5]. We choose the horizontal distribution so that = ⊕ and for any ∈ , * , : The dual vector bundle * * can be identified with a subbundle of * and allows us to define the restriction projection : * → * * , which extends to the exterior powers and yields The space ∞ ( , ⊗ Λ * * ) is clearly a module over the algebra Ω * of differential forms on the base manifold . For any integer ℎ ∈ ℤ, we denote by Ψ ℎ ( | , ) the space of 1-step polyhomogeneous classical pseudodifferential operators of order ℎ, acting along the fibres of : → , see [14,1]. The local coefficients of such operators are thus smooth in the base variables and the space Ψ ℎ ( | ; ) is a module over the algebra ∞ ( ) of smooth functions on . We also set ℎ ( | , ; Λ * ) = Ψ ℎ ( | , )⊗Ω * , for the space of order ℎ classical fiberwise pseudodifferential operators with coefficients in differential forms on the base . See, for instance, [3]. In the sequel and as usual Moreover, the curvature operator ∇ 2 of ∇ is a fiberwise first order differential operator with coefficients in This is a classical lemma that we proved in [2]. We quote another well-known property of the quasi-connection ∇ for later use: for any ∈ ℎ ( | , ; Λ ) the commutator ∂( ) := [∇, ] is well-defined and belongs to ℎ ( | , ; Λ +1 ). In particular, the commutator operator ∂ preserves −∞ ( | , ; Λ ).
Integration over the fibers with respect to the fiberwise volume form, composed with the pointwise trace, yields : −∞ −→ Ω * ( ). Following [13] we introduce an extra Clifford variable having degree 1 and central in the graded sense (i.e. graded commuting with all the operators). Hence we replace the algebra −∞ by its extension −∞ [ ], for instance. Recall that the fibers of our fibration are odd dimensional. We extend and the commutator The proof uses that is a graded trace and that ∘ ∂ = ∘ [3]. We now consider the superconnection defined by = ∇ + , and Definition 1.4. Following [5] we will use the notation − 2 to denote the semigroup, that is, the solution to the heat equation associated with 2 , given by the following finite perturbative sum It is well known that − 2 belongs to −∞ ( | , ), therefore, − 2 belongs to −∞ ( | , ; Λ * ).

The main theorem
Let ∈ ∞ ( | , ; Λ * )[ ] and with Δ( ) being, as before, the -simplex we set [8,15] It is proved in [2] that this formula defines a multilinear functional when the sum of the pseudodifferential orders of the 's is ≤ .
Definition 2.1. The bivariant JLO cochain is defined by the sequence ( ) given for ( 0 , ⋅ ⋅ ⋅ , ) ∈ ∞ ( ) +1 by the formula For any ℓ ≥ 0 the semi-norms define the ℓ topology on ∞ ( ). We denote by Σ ℓ the bornology which is given by the bounded sets for these semi-norms. We also denote by Σ ℓ the corresponding bornology on ∞ ( ). We introduce, to denote the operator ∘ and the bornology Σ Ω ℓ on the algebra Ω * ( ) given on Ω ( ) by the bounded sets for the semi-norms where 0 ≤ ≤ ℓ and , vector fields on .
We shall use the formalism of analytic cyclic homology for bornological algebras due to R. Meyer. Recall from [10] that the universal differential graded algebra Ω ∞ ( ) is endowed with the analytic bornology Σ ,ℓ generated by the sets < > ( ) ∞ where describes the bounded subsets of ∞ ( ) for the bornology Σ ℓ recalled above. We are now in a position to state the main result. We restrict to the odd case for simplicity.
We similarly denote by Σ ∞ the bornology associated with the Fréchet topology on ∞ ( ) or on ∞ ( ).

Main steps of the proof
The proof is lengthy and so we only explain here the main steps, referring to [2] for the details. Recall that we restrict to the odd case. We first prove the easy part of the theorem, namely the algebraic cyclic cocycle condition, see also [15,11].
This lemma is a consequence of a bivariant generalization of Quillen's cochain formalism [13] but we gave another direct proof in [2]. Now, we concentrate on the heart of the theorem, that is the analyticity of the JLO cocycle. The proof is first reduced to the proof of Proposition 3.2 below. Let us introduce some notation. We set = {(1, 0, 0); (0, 1, 0); (0, 0, 1)}. For ∈ we denote by ( ) the -th component of , = 1, 2, 3. We shall write ( ) in a given expression to mean that we take into account only when ( ) = 1. For instance For any ≥ 0, = ( 0 , ⋅ ⋅ ⋅ ) ∈ ℕ +1 and = ( 1 , ⋅ ⋅ ⋅ , ) ∈ , we define an ∑ , We first show that the JLO cocycle can be expanded as a finite sum of cochains of the form , . Then we need to prove the following Proposition 3.2. The bihomogeneous family ( , ) , , is a bounded morphism from (Ω ∞ ( ), Σ ,ℓ+1 ) to (Ω * ℓ ( ), Σ Ω ℓ ). The proof of this proposition is tedious and is carried out for = 0 and = 0 first, then for = 0 and general , and eventually for the general case. For a vector field on , we denote by˜ the horizontal (i.e. valued) vector field on satisfying * ˜ = . The relation ˜ ∘ + ∘ ∂ = 0 is used to reduce to simpler estimates. In order to carry out the proof we prove and use Lemmas 3.3, 3.4, 3.5 and Proposition 3.6 below.

Lemma 3.3. [2]
For any ≥ 0, we have that are both finite, where the 's are vector fields on .
Proof. The operator [∂ 1 ⋅ ⋅ ⋅ ∂ ]( 2 ) is a second order vertical differential operator, with smooth coefficients. Therefore, the norms of all of the operators involved are well-defined. The method of proof of this lemma is to use compactness to reduce to estimates in terms of local coordinates where the proof may be seen to be straightforward. □ The following two lemmas are proved by similar methods in [2]. ∥( where 1 , 2 , 1 , ⋅ ⋅ ⋅ , are vector fields on that we view through their horizontal lifts. We denote for ≥ 0 by ∥ ∥ the norm sup 0≤ ≤ ( ).
Lemma 3.5. For any ≥ 0, there exists a constant ≥ 0 such that for any ∈ ∞ ( ) and any vector fields of norm ≤ 1.
Theorem 2.2 implies that if is continuous with respect to the ℓ topology, then the sequence is an analytic cyclic cocycle over the algebra ∞ ( ) when endowed with the bornology Σ ℓ+1 , and hence over its Banach completion ℓ+1 ( ). Using the comparison of the analytic theory with Connes' entire theory as carried out in [10], we deduce that also defines an entire cyclic cocycle over the Banach algebra ℓ+1 ( ). Now, that any can be represented by such is classical.
The other main consequence of Theorem 2.2 is of course that the Chern character of the Toeplitz index can be expressed as a cohomology class on of a convergent series in a ℓ topology of Ω * ( ) [2].