On a class of closed cocycles for algebras of non-formal, possibly unbounded, pseudodifferential operators

In this article, we consider algebras $\mathcal{A}$ of non-formal pseudodifferential operators over $S^1$ which contain $C^\infty(S^1),$ understood as multiplication operators. We apply a construction of Chern-Weil type forms in order to get $2k-$closed cocycles. For $k=1,$ we obtain a cocycle on the algebra of (maybe non classical) pseudodifferential operators with the same cohomology class as the Schwinger cocycle on the algebra of Classical pseudodifferential operators, previously extended and studied by the author on algebras of the same type.


Introduction
We present here a construction of a family of cocycles on the Lie algebra of maybe unbounded, maybe non classical, non formal pseudodifferential opeartors P DO(S 1 , V ).Cocycles on algebras of pseudodifferential operators have been studied from the viewpoint of algebras of formal symbols, see e.g.[11,14] for cocycles related to our study, or algebras of non-formal but classical pseudo-differential operators, see e.g.[31].In the last study, the notion of renormalized trace plays an important role, as well as in e.g. the works [4,5,26] where these cocycles are shown to be linked with anomalies in physics via differential geometric considerations.
In our works [17,19,20], we made more precise the link between various aspects: • the Kravchenko-Khesin cocycle [14] on formal pseudodifferential operators over S 1 • the index cocycle on the restricted linear group GL res defined in [30] • the approach by Radul [31] • the Schwinger cocycle [33], see e.g.[4,26], and we showed [20] that the Schwinger cocycle and the index cocycle could be extended to the algebra P DO(S 1 , V ) of maybe unbounded, maybe on classical, non formal pseudodifferential operators over S 1 .
We come back to this program in the present work by adding a new idea to our investigations: integrate the classical formulas for Chern-Weil forms trΩ k in order to define (closed) 2k−cocycles on P DO(S 1 , V ).For this task, the classical Bianchi identity is obtained algebraically on the Lie algebra P DO(S 1 , V ) and the connection that we consider is with values in smoothing operators.This enables us to consider tr as the classical trace of trace class operators, even if technical steps of our investigations require zeta-renormalized traces along the lines of [29,34].In this framework, classical computations of Chern-Weil forms apply and we show that trΩ k is a 2k−cocycle on P DO(S 1 , V ) for k ∈ N * .
For k = 1, even if our connection gives a slightly different 2−cocycle trΩ from the one studied in [20], we show that this 2−cocycle related to the cohomology class of the Schwinger cocycle.We have to remark that we are here able to prove that these two cocycles have the same cohomology class (up to a factor) on the algebra of classical pseudodifferential operators, but we have no information of this kind for P DO(S 1 , V ) (except that trΩ is not a coboundary), except that it is not a coboundary.

Preliminaries
There are multiple frameworks that call, under the name pseudodifferential operator, various objects which may not be (true) operators acting on sections of a finite dimensional complex vector bundle.These last ones are called formal, while the others are called non formal.Moreover, there exists restricted classes of pseudodifferential operators: classical, log-polyhomogeneous among others.For the sake of clarity and for a comprehensive exposition, we start by a non-technical presentation of the operators that we consider, and a more rigorous description will follow.

1.1.
A panoramic overview on pseudodifferential operators.We specialize below to the trivial complex vector bundle S 1 × V in which V is a d−dimensional complex vector space.The following definition appears in [3, Section 2.1].
Definition 1.1.The graded algebra of differential operators acting on the space of smooth sections C ∞ (S 1 , V ) is the algebra DO(S 1 , V ) generated by: where ∇ is a smooth connection on E and X is a smooth vector field on S 1 .
We assign the order 0 to smooth function multiplication operators.The derivation operators have the order 1.We denote by DO k (S 1 , V ),k ≥ 0, the differential operators of order less or equal than k.The algebra DO(S 1 , V ) is filtered by the order.It is a subalgebra of the algebra of classical pseudo-differential operators Cl(S 1 , V ) that we describe shortly hereafter, focusing on its necessary aspects.This is an algebra that contains, for example, the square root of the Laplacian where ∆ = − d 2 dx 2 is the positive Laplacian and Γ is a contour around the spectrum of the Laplacian, see e.g.[32,29] for an exposition on contour integrals of pseudodifferential operators.Cl(S 1 , V ) contains also the inverse of Id + ∆, and all smoothing operators on L 2 (S 1 , V ).
Wider classes of pseudo-differential operators can be also considered.An example of frequent use remains on the real powers of the Laplacian ∆ α , where α ∈ R, defined through contour integrals like ∆ 1/2 already mentioned.One can define the same way log ∆.Pseudodifferential operators (maybe non-classical) are linear operators acting on C ∞ (S 1 , V ) which reads locally as where σ ∈ C ∞ (T * S 1 , M n (C)) satisfying additional estimates on its partial derivatives which will be given in a detailed form next section and f means the Fourier transform of f .
Another special class of pseudo-differential operators is also of great interest.This is the set of smoothnig pseudo-differential operators.They are equivalently: • classical pseudo-differential operators with order −∞, that is, they are in the set of classical pseudo-differential operators k for any k ∈ Z.
which explains the terminology.
These operators from an ideal in any algebra of pseudo-differential operators.Quotienting by this ideal, we obtain algebras of formal pseudo-differential operators.
1.2.Rigorous approach to pseudodifferential operators.Basic facts on pseudodifferential operators defined on a vector bundle E → S 1 can be found e.g. in [8].
Let us now recall them in a rigorous exposition, as brief as possible, concentrating our exposition on pseudo-differential operators over S 1 which is equipped with an atlas in which changes of coordinates are affine maps.This is made possible through the structure of abelian group of S 1 , where the atlas considered is the atlas obtained through the exponential map.We set S 1 = {z ∈ C | |z| = 1}.We shall use for convenience the smooth atlas A of S 1 defined as follows: Associated to this atlas, we fix a smooth partition of the unit {s 0 ; s 1 }.We identify each of these functions with its associated multiplication operator when necessary.An operator A : C ∞ (S 1 , C) → C ∞ (S 1 , C) can be described in terms of 4 operators A scalar pseudo-differential operator of order o is an operator We note by P DO(S 1 , V ) the space of maybe non classical, maybe unbounded, pseudodifferential operators acting on C ∞ (S 1 , V ).A pseudo-differential operator of order o is called classical if and only if its symbols σ have an asymptotic expansion σ(x, ξ) ∼ |ξ|→+∞ o j=−∞ σ j (x, ξ), where the maps σ j : S 1 × R * → C, called partial symbols, are j-positively homogeneous, i.e.
The class of log-polyhomogeneous pseudo-differential operators can be also described this way.These are operators in Cl(S 1 , V )[log(∆)], which inherits also a second degree, from the evaluation of Cl(S 1 , V )−polynomials at log(∆).
Pseudodifferential operators can be also described by their kernel which is off-diagonal smooth.Pseudodifferential operators with infinitely smooth kernel (or "smoothing" operators).
The quotient F Cl(S 1 , V ) = Cl(S 1 , V )/P DO −∞ (S 1 , V ) of the algebra of pseudo-differential operators by P DO −∞ (S 1 , V ) forms the algebra of formal classical pseudo-differential operators.
Remark 1.2.Through identification of F Cl(S 1 , V ) with the corresponding space of formal symbols, the space F Cl(S 1 , V ) is equipped with the natural locally convex topology inherited from the space of formal symbols.A formal symbol σ k is a smooth function in C ∞ (T * S 1 \ S 1 , M n (C)) which is k−homogeneous (for k > 0)), and hence with an element of C ∞ (S 1 , M n (C)) 2 evaluating σ k at ξ = 1 and ξ = −1, see [19,25].
the vector space F Cl d (S 1 , V ) is a Fréchet space, and hence is a locally convex topological algebra.
1.3.The splitting with induced by the connected components of T * S 1 \ S 1 .. In this section, we define two ideals of the algebra F Cl(S 1 , V ), that we call This decomposition is explicit in [10, section 4.4., p. 216], and we give an explicit description here following [17,19].Definition 1.3.Let σ be a partial symbol of order o on E.Then, we define, for (x, ξ) are clearly smooth algebra morphisms (yet non-unital morphisms) that leave the order invariant and are also projections (since multiplication on formal symbols is expressed in terms of pointwise multiplication of tensors).
Since p + is a projection, we have the splitting Let us give another characterization of p + and p − .The operator D = −i d dx splits C ∞ (S 1 , C n ) into three spaces : -its kernel E 0 , built of constant maps -E + , the vector space spanned by eigenvectors related to positive eigenvalues -E − , the vector space spanned by eigenvectors related to negative eigenvalues.The L 2 −orthogonal projection on E 0 is a smoothing operator, which has null formal symbol.By the way, concentrating our attention on thr formal symbol of operators, we can ignore this projection and hence we work on E + ⊕ E − .When dealing with non-formal operators, we shall set p + = p E+ .The following elementary result will be useful for the sequel.Lemma 1.5.[17,19] Let p E+ (resp.p E− ) be the projection on From this, we have the following result.
Proposition 1.6.[17,19] Notation.For shorter notations, we note by A ± = p ± (A) the formal operators defined from another viewpoint by We now turn to maybe non classical pseudo-differentail operators, along the lines of [20]: 1.4.Renormalized traces of classical pseudodifferential operator.S 1 × V is equiped this an Hermitian products < ., .>, which induces the following L 2 -inner product on C ∞ (S 1 , V ) : where dx is the Riemannian volume.
Definition 1.8.Q is a weight of order s > 0 on E if and only if Q is a classical, elliptic, self-adjoint, positive pseudo-differential operator acting on smooth sections of E.
Recall that, under these assumptions, the weight Q has a real discrete spectrum, and that all its eigenspaces are finite dimensional.For such a weight Q of order q, one can define the complex powers of Q [32], see e.g.[4] for a fast overview of technicalities.The powers Q −s of the weight Q are defined for Re(s) > 0 using with a contour integral, where Γ is a contour around the real positive axis.Let A be a log-polyhomogeneous pseudo-differential operator.The map ζ(A, Q, s) = s ∈ C → tr (AQ −s ) ∈ C , defined for Re(s) large, extends on C to a meromorphic function [16].When A is classical, ζ(A, Q, .)has a simple pole at 0 with residue 1 q res W A, where res W is the Wodzicki residue ( [35], see also [10]).Notice that the Wodzicki residue extends the Adler trace [2] on formal symbols.Following textbooks [29,34] for the renormalized trace of classical operators, we define Definition 1.9.Let A be a log-polyhomogeneous pseudo-differential operator.The finite part of ζ(A, Q, s) at s = 0 is called the renormalized trace tr If A is trace class acting on L 2 (S 1 , C k ), tr Q (A) = tr(A).The functional tr Q is of course not a trace.In this formula, it appears that the Wodzicki residue res W (A).
Proposition 1.10.(i) The Wodzicki residue res W is a trace on the algebra of classical pseudo-differential operators Cl(S 1 , E), i.e. ∀A, In particular, res W does not depend on the choice of Q.
Since tr Q is a linear extension of the classical trace tr of trace-class operators acting on L 2 (S , V ), it has weaker properties.Let us summarize some of them which are of interest for our work following first [4], completed by [22] for the third point.
• Given two (classical) pseudo-differential operators A and B, given a weight Q, • Under the previous notations, if C is a classical elliptic injective operator or a diffeomorphism, tr C −1 QC C −1 AC is well-defined and equals tr Q A.
Since tr Q is not tracial, let us give one more property on the renormalized trace of the bracket, from e.g.[24].Proposition 1.12.
Moreover, we can push further the property on P DO(S 1 , V ) : Since A is smoothing, [A, B] is also smoothing and hence trace class.Therefore, the following expression makes sense and we can compute (since we are commuting operators of order -2 at most, which are trace class anyway): = tr(BA).

A family of cocycles
We consider the Lie algebra cohomology of P DO(S 1 , V ) and of its Lie subalgebras.
Definition 2.1.We define on P DO(S 1 , V ) In order to understand better our construction, we have to precise that the reader has to understand θ as a 1-form with values in linear maps on P DO(S 1 , V ), in order to understand better the link of this construction with the theory of connection 1-forms.This linear map is only a composition operator, in the spirit of the connection 1-forms described in [24].
Therefore, we consider the curvature Ω of the connection θ, that is, which is, like θ a , is an operator of coomposition on the left, i.We now consider Ω k (a 1 , ..., a 2k ) = Ω ∧ ... ∧ Ω(a 1 , ..., a 2k ) = s k (a 1 , ..., a 2k ) • (.). with Since s is with values in smoothing operators, so is the 2k−form s k .By slight abuse of notations, we define the 2k−form with values in C : (2.1) where tr is the trace of trace-class operators on L 2 (S 1 , V ).Since Ω(a, b) = s(a, b)•(.) is the composition on the left by a smoothing operator, we will now understand tr(Ω k ), and the similar expressions, along the lines of (2.1).This is the main property to get the following theorem Proof.We compute directly the coboundary and reduce the computation to Therefore, This result remains valid for any Lie subalgebra A ⊂ P DO(S 1 , V ).Let us now give a key elementary lemma about non-exactness, already applied in [20]: Lemma 2.5.Let A be a Lie subalgebra of P DO(S 1 , V ).Let c be a cocycle on A. Let B be a commutative Lie subalgebra of A, i.e. [B, B] = {0}.If c is non vanishing on B, then c is not exact.
3. trΩ is cohomologous to the Schwinger cocycle on Cl(S 1 , V ).Theorem 3.1.On Cl(S 1 , V ), trΩ has the same cohomology class as 1  2 c s , where c s is the Schwinger cocycle.By the way it has non-trivial Lie algebra cohomology class on P DO(S 1 , V ).The last term thus can be identified with the pull-back of the Kravchenko-Khesin-Radul cocycle on Cl(S 1 , V ) so that it has the same cohomology class as the Schwinger cocycle following [19].
If trΩ was a coboundary on P DO(S 1 , V ), it would be also a coboundary on Cl(S 1 , V ).So that, trΩ has non-trivial Hochschild cohomology class on P DO(S 1 , V ).
We have here to remark that the full comparison of trΩ with 1  2 c s remains an open question, because the correspondence is only established on Cl(S 1 , V ).

Conclusion
The family of cocycles on P DO(S 1 , V ) that we produced show that there can exist some 2k−cocycles on P DO(S 1 , V ).Beside the 2−cocycle trΩ which is cohomologous to the well-known Schwinger cocycle on Cl(S 1 , V ) we get the following open question: Prove that there exists other non-trivial cocycles in our family.
More generally and algebraically, the full description of the Hochschild cohomoloy of various Lie subalgebras of P DO(S 1 , V ) (especially those with some unbounded operators) needs to be investigated.From a geometric viewpoint, the meaning of the higher Chern-Weil forms that we describe here, intrinsically liked with the sign of the Dirac operator, carry interpretations that can be only heuristic since the classical differential geometry (with atlases) fail to apply, but the seem intrinsically linked with the integrable almost complex structure described in [25] in the context of formal classical pseudo-differential operators.