Chern Forms on Mapping Spaces CHERN FORMS ON MAPPING SPACES

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CHERN FORMS ON MAPPING SPACES

JEAN-PIERRE MAGNOT

Laboratoire de Mathematiques Appliquees

Universite Blaise Pascal (Clermont II)

Complexe Universitaire des Cezeaux

63177 Aubiere Cedex, France.

[email protected]

New address:

Institut fur Angewandte Mathemetik

Abt. fur Wahrsheinlichkeitstheorie und Mathematische Statistik

Wegelerstr. 6

D-53155 Bonn

[email protected]

Abstract. We state a Chern-Weil type theorem which is a generalization of a

Chern-Weil type theorem for Fredholm structures stated by Freed in [4]. Usingthis result, we investigate Chern forms on based manifold of maps Mapb(M,N)following two approaches, the first one using the Wodzicki residue, and the sec-ond one using renormalized traces of pseudo-differential operators along the

lines of [1], [19], [20], [14]. We specialize to the case N = G to study currentgroups. Finally, we apply these results to a class of holomorphic connections

on the loop group H1/2b (S1, G). In this last example, we precise Freed’s con-

struction [5] on the loop group: the cohomology of the first Chern form of anyholomorphic connection in the class considered is given by the Kahler form.

key words : Construction of Chern-Weil forms; manifolds of maps; loop groups;renormalized traces; pseudo-differential operators.

MSC 2000: 58D15

Contents

1. Introduction 22. Infinite dimensional frame bundles and adjoint bundles 43. Chern forms in infinite dimensions 64. The geometric framework on based groups and manifolds of maps 84.1. Definitions and basic properties of manifolds of maps 84.2. Weights and Hs metrics 104.3. The case of a parallelizable target manifold 124.4. On current groups 125. Fields of functionals induced by Wodzicki residue and renormalized

traces 135.1. Wodzicki residue and renormalized traces 13

1

2 JEAN-PIERRE MAGNOT

5.2. Bracket property for trQ 145.3. The case of a trivial bundle 155.4. Fields of linear functionals 166. Chern-Weil type theorem on manifolds of maps 176.1. Chern forms defined with Wodzicki residue 176.2. Chern forms for weighted manifolds of maps 177. The loop group as a complex manifold 23Acknowledgements 25References 25

1. Introduction

The goal of this article is to propose a generalization of Chern-Weil forms onfinite dimensional manifolds to manifolds of maps and to the complexification ofthe loop group, for certain classes of connections. This extends some results of [14].

In finite dimensions, for k ∈ IN∗, the Chern-Weil forms of the type Chk(θ) =tr(Ωk), where Ω is the curvature of the connection θ, give rise to topological charac-teristic classes. Since the linear group GL( ICn) is not contractible, one can expectnon trivial topological classes in finite dimensions. Since our goal is to generalizethis construction to infinite dimensions, let us sketch the key points that makeChern forms topological invariants of the manifold.

After noticing that the Lie algebra gl( ICn) of GL( ICn) is itself an algebra, onecan see that the key features for the theory of Chern forms in finite dimensions arethe following:

1) Given a connection θ on a principal bundle P with curvature Ω, the formstr(Ωk) are closed and their cohomology class do not depend on the chosen connec-tion. In other words, the Chern maps

Chk : connections on P → 2k − forms on Mθ 7→ tr(Ωk)

induces a constant map from the set of connections on P to H2k(M). This value iscalled the k-th Chern class of the bundle P . A trivial bundle furnishes only Chernforms with vanishing cohomology classes. (in particular, a principal bundle withcontractible structure group is always trivial and thus has vanishing Chern classes)

2) The cohomology classes of the Chern forms 1(2π)k

tr(Ωk) are the pull-backs of

the cohomology classes of a space called classifying space via a map defined by theprincipal bundle P .

Let us now turn to infinite dimensions. The linear group GL(H) of an infinitedimensional separable Hilbert space H is contractible [13] so that every Hilbertvector bundle is trivial, and we expect characteristic classes to vanish if such objectscan be defined. So that, to expect non vanishing Chern classes, one needs first toconsider structure groups that are non contractible. The underlying idea of thisarticle is to replace GL(H) by a non contractible Lie group, in order to define Chernforms with non vanishing cohomology class by the approach 1). For this, we setG a (non contractible) infinite dimensional Lie group, with Lie algebra g, and letP be a principal bundle of basis M (infinite dimensional manifold) with structure

CHERN FORMS ON MAPPING SPACES 3

group G. We note AdP = P ×G g the adjoint bundle of P . We assume that g isembedded in some algebra a, and we replace the trace of matrices tr by some linearfunctionals

λ : a → IC.

We define this way a k-th Chern form (k ∈ IN∗):

Chλk : Connections on P → 2k − forms on M

θ 7→ Chλk(θ) = λ(Ωk),

where Ω is the curvature of θ. In order to follow the lines of 1), one needs to answerto the following questions:

(i) Does the map Chλk take values into closed forms ?

(ii) Is the cohomology class of Chλk(θ) independent on θ ?

If (i) is verified for any connection, and if (ii) is true, we can conclude as inthe finite dimensional case that 1) is fulfilled. Unfortunately, in the examples wedevelop in the present article, as well as in the example developed by D.Freed in[4] and [5],

- many Chern forms are exact, even if the structure group G is non contractible;however, there exists non vanishing Chern classes;

- there exists some examples of linear functional λ and of Lie groups G for whichthe k-th Chern form is a closed form only for some particular k;

- there exists some examples where we cannot show that the cohomology classof the k-th Chern form does not depend on the choice of the connection: we haveto restrict ourselves to a smaller class of connections to have this property.

Let us now describe with more details the contents of this article. We generalizeChern forms to an infinite dimensional setting in this article, namely when M is amanifold of maps and when G is a group of invertible pseudo-differential operatorsof order 0, by the approach 1), following [1], [19], [20].

We give a Chern-Weil type theorem in an abstract setting for some linear func-tional λ in section 2. This result is a generalization of a Chern-Weil type theoremfor Fredholm structures given by Freed in [4] for λ = tr. Then, we apply this re-sult to two types of linear functionals λ on pseudo-differential operators. We firsttake λ as the Wodzicki residue resW (which is the only trace on classical pseudo-differential operators up to a multiplicative constant). This leads to Chern formsresW (Ωk) with vanishing cohomology class for any integer k. Then, in section 6.2,following [1] and [19], we use renormalized traces trQ (which are extensions of theusual trace tr of trace-class operators to pseudo-differential operators) to defineweighted Chern forms for three bundles of frames and their connections, using thepreliminary results of section 5. These examples generalize Freed’s results in thecase of Chern forms of Levi-Civita connections on current groups [4], [5]: in thepresent context, more Chern forms are well-defined. We discuss their closeness, andinvestigate their dependence on the weight chosen. In two cases among the threeconsidered, the cohomology classes of weighted Chern-Weil forms vanish. In theother case, even though there are good reasons to expect them to give rise to nonvanishing cohomology classes, we actually do not know if all cohomology classes ofthese weighted Chern-Weil forms vanish. Finally, at the end of section 6.2.2 and insection 7, we give two examples where we cannot prove that the cohomology class ofthe first Chern form is independent on the choice of the connection. In Theorem 6


of section 6.2.2, we show that the first Chern form of the H1-Levi-Civita connectionof the current group of the torus is vanishing, even if we are not able to prove thatall first Chern forms are closed. In section 7, we discuss the loop group viewed as acomplex manifold, extending a result of [1] and defining a class of connections thatgenerates a non vanishing first Chern class. This last example shows the necessityof Theorem 1, since Freed’s Chern-Weil type theorem was not adapted to conclude.

We preferred in this article to restrict the class of connections (and the class ofweights), working in the spirit of Freed’s work. This approach, that led us to anon vanishing first Chern class, is then fully justified. Moreover, this provides anexample which seems to be useful for further developments of this work, as a wayto point out where the problems are.

The approach 2), which is to define the classifying space of the group G consid-ered, led S. Paycha and S. Rosenberg [20] to produce some results concerning theclassifying space of principal bundles with structure group the group of classicalpseudo-differential operators of order 0, and to consider only linear functional thatyield traces on the algebra of classical pseudo-differential operators.

All these possible generalizations of the finite dimensional Chern-Weil theory tothe infinite dimensional setting show that the situation is still unclear in the generalcase and requires further investigations.

2. Infinite dimensional frame bundles and adjoint bundles

Let us first recall the definition of a vector bundle in the sense of Milnor [18]:

Definition 1. Let E and M be manifolds modelled on locally convex topologicalvector spaces F0 and F1. Let π : E → M be a smooth function. Let F be locallyconvex vector space. Then (E, π,M) is a vector bundle with fiber F and of basisM if and only if:(i) there in an open covering Uii∈ IN of M and a family of smooth diffeomorphisms(called local trivializations of E) Φi : π

−1(Ui) → Ui × Fi∈ IN.(ii) ∀i ∈ IN, π Φi = π|Ui

.

(iii) Given i, j ∈ IN, the transition maps Φj Φ−1i : Ui ∩ Uj × F are linear

maps in the second variable (in particular, for any x ∈ Ui ∩ Uj, the linear map

Φj Φ−1i (x, .) belong to GL(F ), the group of the units of the algebra L(F ) ).

A family of local trivializations indexed on an open covering of the basis M iscalled a vector bundle atlas.

Let us now give the definition of vector bundle with structure group G.

Definition 2. Let π : E → M be a vector bundle in the sense of Milnor [18]with typical fiber F , and let G be a Lie group acting freely and smoothly on F .Let Φii∈I be a vector bundle atlas. If, for any i, j ∈ IN, the transition mapτi,j = Φj Φ−1

i defines a smooth map

Ui ∩ Uj → G

x 7→ Φj Φ−1i (x, .),

then π : E → M is called a vector bundle with structure group G.

Remark : If G acts smoothly on F , its Lie algebra g also acts smoothly on F bytaking the derivative of the action in the G-variable.


Proposition 1. Let Φii∈ IN be a vector bundle atlas of the bundle π : E → Mwith structure group G as in Definition 2. Then, we define

FrG(E) =⨿x∈M

ϕx : F → Ex|∀i such that x ∈ Ui, ∃gi ∈ G; Φ−1i (x, .) = ϕx(gi.(.)).

F rG(E) is called bundle of G-frames of E, or bundle of frames when no con-fusion is possible. This is a principal bundle based on M , with structure groupG.

Notice that the definition of FrG(E) carries no ambiguity because the transitionmaps τi,j are G-valued.

Proof : One gets a local trivialization of FrG(E) with the collection Φi : π−1(Ui) →

EM ×Gi∈I by:

Φi(ϕx) = (x,Φi(x, ϕx(.))). ⊔⊓Classical examples of this construction are the bundles of frames GL(E), O(E)

and U(E) when E is a vector bundle with typical fiber an Hilbert space H. Withthis setting, there is a one to one correspondence between the connections on thebundle ofG-frames and the covariant derivatives on E that read locally as∇ = d+θion each local trivialization Φi, where each θi is a local g-valued 1-form. See e.g.[11], section 37.27.

We can also define the bundle of g-homomorphisms (resp. G-homomorphisms)of E by :

Homg(E) = Φi(x, .) u Φi(x, .)−1, such that u ∈ g

HomG(E) = Φi(x, .) u Φi(x, .)−1, such that u ∈ G

Let us now recall some properties of the bundle Ad(P ). We define the adjointbundle AdP as the quotient (P ×g)/G = P ×G g, where G acts by coadjoint actionon g. Ad(P ) is a vector bundle with basis M and with typical fiber g. Then, aG-invariant n-form on TP that vanishes on the vertical vectors reads as an elementof Ωn(M,Ad(P )). This is the case for the curvature of a connection which is a2-form with values in Ad(P ).

Moreover, if we consider the last bundle of frames, a section of Homg(E) fur-nishes a section of Ad(P ), i.e. a G- invariant map FrG(E) → g, the followingway:

Lemma 1. There is a one to one correspondence between sections of Homg(E)and G-invariant functions on FrG(E) with values in g ( i.e. G-invariant 0-formson FrG(E) with values in g ).

Proof : Let u be a section of Homg(E). Let Φ ∈ FrG(E), and let x = π(Φ). Wedefine

vΦ = Φ−1 ux Φ.The map v : Φ 7→ vΦ is smooth. Let g ∈ G, we have

vΦ.g = g−1 Φ−1 ux Φ g = Adg−1(vΦ).

Hence, v is G-invariant, and hence induces a section of Ad(P ). The correspondenceu 7→ v is obviously one to one. ⊔⊓


3. Chern forms in infinite dimensions

Let P be an (infinite dimensional) principal bundle, with basis M and withstructure group G. All the objects considered are smooth objects. As we mentionedin the introduction, the first technical tool used to define Chern forms in finitedimension is that gl( ICn) is an algebra. This condition is not always fulfilled in aninfinite dimensional abstract setting. So that, we work with an additional algebraa with the following properties:

- g ⊂ a as a Lie algebra;- The coadjoint action Ad : g×G → g extends to a smooth action Ad : a×G → a

(we note invariantly Ad the action on g or on a since it carries no ambiguity) thatis compatible with the structure of algebra of a.

Under these assumptions, we can define:

Ad(P, a) = P ×G a = (P × a)/G.

We also need the following objects: let i ⊂ a be a two sided ideal of a which isstable under the action of G on a. In sake of coherence of notations, we set i0 = a

Then, we can define the bundles Ad(P, i) and Ad(P, il) the same way as thebundle Ad(P, a). Notice also that a connection on P induces a covariant derivationon Ad(P ), Ad(P, a), Ad(P, i) and Ad(P, il).

Definition 3. λ : Ad(P, a) → IC is a smooth field of linear functionals if andonly if

(i) λ : Ad(P, a) → IC is a smooth map(ii) ∀x ∈ M , λx : Ad(P, a)x → IC is a linear functional.

Notice that we have not assumed the functionals λx, for x ∈ M , to be tracials,because the examples that we shall study have the following weaker property.

Let us fix i an ideal of a as before.

Definition 4. Let us fix k ∈ IN∗. A field of linear functionals

λ : Ad(P, a) → IC

has the k-filtered trace property if and only if, for any x ∈ M , for any integersl, l′ ∈ IN such that l + l′ ≥ k, for any u ∈ Ad(P, il) and for any v ∈ Ad(P, il

′),

λx([u, v]) = 0.

Definition 5. Let k be an integer, let λ be a smooth field of linear functionals withthe k-filtered trace property. We note by C(λ, i) an affine space of connections ofP with underlying vector space Ω1(M,Ad(P, i)) and with i- valued curvature.

Lemma 2. Let k be an integer, i an ideal of a, and let λ be a smooth field of linearfunctionals with the k-filtered trace property. Let α ∈ Ω∗(M,Ad(P, ik)), then, forany connections θ and θ′ on P with covariant derivatives ∇ and ∇′,

λ(∇′α)) = λ(∇α).

Proof. η = θ′−θ, since it vanishes on vertical vectors, induces a 1-form on M withvalues in Ad(P ). Hence, λ(∇′α))−λ(∇α) = λ([η, α]) = 0 since λ has the k-filteredtrace property. ⊔⊓

Let us now give the Chern-Weil type theorem announced:


Theorem 1. Let us fix i an ideal of a and let λ be a smooth field of linear functionalson Ad(P, a) that has the filtered trace property for a fixed k ∈ IN∗. Let C(λ, i) bea class of connections as in Definition 5. We also assume that, for any differentialform α ∈ Ω∗(M,Ad(P, ik)), and for any connection θ ∈ C(λ, i) with covariantderivative ∇,

d (λ(α)) = λ (∇α) .

Then,• Let θ a connection in C(λ, i), and let Ω be the curvature of ∇. Then, for l ≥ k,the 2l-form λ(Ωl) is well defined and closed.• Let θ0 and θ1 be two connections in C(λ, i) with curvature Ω0 and Ω1, then,

for l ≥ k, λ(Ωl1)− λ(Ωl

0) is exact.

Proof.• Since θ ∈ C(λ, i), we find dλ(Ωl) = λ(∇Ωl).By the Bianchi identity , ∇Ω = 0, hence dλ(Ωl) = 0.• We can follow the proof of the finite dimensional case [12]:Let η = θ1 − θ0 ∈ Ω1(M,AdP ), let θt = θ0 + tη, and let Ωt be its curvature. We

notice that η is i-valued.Then dΩt

dt = ∇tη ∈ Ω∗(M,Ad(P, i)). Using the k-filtered trace property of λ, wehave

d

dtλ(Ωl

t) =l−1∑i=0

λ(Ωit

dΩt

dtΩl−i−1

t ) = lλ(Ωl−1t

dΩt

dt)

= lλ(Ωl−1t ∇tη) = lλ(∇t(Ω

l−1t η)) (using Bianchi identity)

= d(lλ(Ωl−1t η))

Hence, the form λ(Ωl1)− λ(Ωl

0) =

∫ 1

0

d

dtλ(Ωl

t)dt = d

∫ 1

0

lλ(Ωl−1t η)dt is exact. ⊔⊓

Let us now state, as a corollary, the case where λ is a field of traces on a, i.e.setting i = a.

Corollary 1. Let λ : Ad(P, a) → IC be a field of traces, and assume that, for anyconnection on P with covariant derivative ∇, and for any α ∈ Ω∗(M,Ad(P, u)),

d (λ(α)) = λ(∇α).

Then,(i) ∀l ∈ IN∗, λ(Ωl) is closed(ii) The cohomology class of λ(Ωl) does not depend on the choice of the connectionon P .

This corollary is a straightforward consequence of Theorem 1. One can see thatit is analogous to the finite dimensional case. We can also state the followingproposition, which is a simple application of Theorem 1

Proposition 2. Let P be a principal bundle that has a flat connection. Then, forany field of functionals λ with the k-filtered trace property and any class C(i) inwhich there is a flat connection, all the Chern forms defined by Theorem 1 have avanishing cohomology class.

In particular, this is the case when P = M × G and when Corollary 1 can beapplied.


4. The geometric framework on based groups and manifolds of maps

We now want to apply these results to manifolds of maps. Let M be a connectedRiemannian compact manifold of dimension m without boundary , x0 a fixed pointin M , N a compact connected Riemannian manifold of dimension n without bound-ary, y0 a fixed point in N . We embed N in IR2n+1. Let G be a compact Lie groupwith Lie algebra g embedded in U(n) for some n large enough.

We do not consider here the framework of [4] which is related to the theory ofgroups of operators on Hilbert spaces, but we the one given in [19] and [1], where thestructure groups under consideration are some groups of classical pseudo-differentialoperators, that enables us to choose many different fields of linear functionals.

An exposition of basic facts on pseudo-differential operators can be found in [6].Let E be a smooth vector bundle over a compact manifold without boundary M. Wedenote by PDO(M,E) ( resp. PDOk(M,E), resp. Cl(M,E), resp. Clk(M,E),resp. Clodd(M,E), resp. Clkodd(M,E), resp. Ell(M,E), resp. Ellk(M,E)) thespace of pseudo-differential operators (resp. pseudo-differential operators of orderk, resp. classical pseudo-differential operators, resp. classical pseudo-differentialoperators of order k, resp. odd class classical pseudo-differential operators, resp.odd class classical pseudo-differential operators of order k, resp. elliptic classi-cal pseudo-differential operators, resp. elliptic classical pseudo-differential oper-ators of order k) acting on smooth sections of E. We denote by Cl∗(M, IKn),

Cl∗odd(M, IKn), Cl0,∗(M, IKn), Cl0,∗odd(M, IKn), Cl0,∗p−odd(M, IKn) the groups of the

units of the algebras Cl(M, IKn), Clodd(M, IKn), Cl0(M, IKn), Cl0odd(M, IKn),

Cl0p−odd(M, IKn) = Cl0odd(M, IKn) + Cl−(p)(M, IKn). Notice that these groups ofthe units are CBH Lie groups, and belong to a wider class of such groups that isstudied in [7].

We also denote by Cl−(p),∗(M, IKn) the group of invertible pseudo-differentialoperators of the type Id + A, where A ∈ Cl−p(M, IKn). Notice that, here, thenotation Cl−(p),∗ could appear misleading. This is why we feel the need to precisethat this is not the groups of the units of a unital algebra: this is only a regularLie group, with Lie algebra Cl−p(M, IKn).

Odd class classical operators were introduced by M.Kontsevich and S.Vishik in[9], [10]. One can also find their properties in [2]. We also recall what is an oddclass pseudo-differential operator in section 5.1.

On Cl(M,E) we use the topology described in [9], which is also carefully de-scribed in [2]. This topology ensures the convergence of partial symbols at anyorder, and also the convergence of smooth kernels of smoothing operators. It in-duces a Frechet structure on Clk(M,E), but not on Cl(M,E) =

∪k∈ZZ Clk(M,E).

We also consider families of pseudo-differential operators that are non classical, butthese families of non classical pseudo-differential operators are always derived froma family of classical pseudo-differential operators. For these families of non classicalpseudo-differential operators, we assume the convergence of the families of classicalpseudo-differential operators they are derived.

4.1. Definitions and basic properties of manifolds of maps. The main refer-ence used here for the notion of manifolds of maps is [3], see also [11], [5]. Let E bea smooth vector bundle over a compact connected manifold without boundary M ofdimension m. Let C∞

0 (M,E) the set of smooth sections σ of E such that σ(x0) = 0.


When E is trivial i.e. E ≃ M ×V , we use C∞0 (M,V ) instead of C∞

0 (M,E). LetK = IR or IC, and let N be a smooth IK Riemannian manifold without boundary.Let C∞

b (M,N) (”b” for based) be the set of smooth maps f : M → N such thatf(x0) = y0 and C∞

b (M,G) the set of smooth maps f : M → G such that f(x0) = ewhere e is the neutral element of G. Then, C∞

b (M,N) is a smooth Frechet manifold,and C∞

b (M,G) is a smooth Frechet Lie group.Let us now describe these manifolds. The connected components of C∞

b (M,N)are given by the based homotopy classes of based maps f : M → N . Note thattwo connected components of C∞

b (M,N) need not be modeled on the same vectorspace. One example is provided by the based loop space C∞

b (S1, N), when N is theMobius band. That is the reason why we shall restrict ourselves to the study of theconnected component of the constant loop in C∞

b (M,N) when necessary.However, if f and g are in the same arcwise connected component of C∞

b (M,N),the vector bundles f∗TN and g∗TN are isomorphic.

- If N is parallelizable, choosing a parallelization of N (i.e. a global section of thebundle of frames) amounts to choosing a canonical map If to identify M× IKn withf∗TN , and hence to identify smooth sections of M × IKn with smooth sections off∗TN .

- If N is not parallelizable, we fix fiberwise the identification If between f∗TNand an extra vector bundle E. In all this article, we do not require further assump-tions on If in that case.

Let us now define local charts on C∞b (M,TN). Let f ∈ C∞

b (M,N). We definethe map Expf : C∞

0 (M,f∗TN) → C∞b (M,N) defined by Expf (v) = expf(.)v(.)

where exp is the exponential map on N. Then Expf is a smooth local diffeomor-

phism. Restricting Expf to a C∞ - neighborhood Uf of the 0-section of f∗TN , wedefine a diffeomorphism, setting

(Expf )|Uf: Uf → Vf = Expf (Uf ) ⊂ C∞

b (M,N).

Then, setting Uf = I−1f Uf , we can define a chart Ξf on Vf by:

Ξf (g) = (I−1f (Expf )

−1

|Uf)(g) ∈ Uf ⊂ C∞

b (M,E).

Given f, g in C∞b (M,N) such that Vf,g = Vf ∩ Vg = 0, we compute the changes

of charts Ξf,g from Uff,g = ΞfVf,g to Ug

f,g = ΞgVf,g. Let u ∈ Uff,g, v = (Ξf )−1(u) ∈

Vf,g.

Ξf,g(u) = Ξg (Ξf )−1(u) = (I−1g (Expg)

−1 Expf If )(u).Since, ∀x ∈ M , the transition maps

Ξf,g(u)(x) = (I−1g (expg(x))−1 expf(x) If )(u(x))

are smooth, (Vf ,Ξf , Uf )f∈C∞

b (M,N) is a smooth atlas on C∞b (M,N). Moreover, let

w ∈ C∞0 (M,E), setting v = (Ξf )−1(u), the evaluation of the differential at x ∈ M

reads :

DuΞf,g(w)(x) = (I−1

g Dv(x)(expg(x))−1 Du(x)(expf(x) If ))(w(x)).

Hence, for u ∈ C∞, DuΞf,g is a multiplication operator acting on smooth sections

of E for any isomorphism If and Ig we can choose. Since If and Ig are fixed, the


family u 7→ DuΞf,g is a smooth family of 0- order differential operators. As a

consequence we have the following

Proposition 3. TC∞b (M,N) is a vector bundle with structure group the group of

invertible differential operators of order 0. As a consequence, we can define thefollowing bundles of frames and other related bundles (we skip in the notations theterm C∞

b (M,N) that should appear everywhere for homogeneity of notations withthe last section, since it carries here no ambiguity):

Structure groups Cl∗(M, IKn) Cl∗odd(M, IKn) Cl0,∗(M, IKn)Frame bundles FrCl FrCl∗odd

FrCl0,∗

Adjoint bundles AdCl AdClodd AdCl0

Bundles of g- HomCl HomClodd HomCl0

homomorphisms

and also

Structure groups Cl0,∗odd(M, IKn) Cl0,∗p−odd(M, IKn) DO0,∗(M, IKn)

Frame bundles FrCl0,∗oddFrCl0,∗p−odd

FrDO0,∗

Adjoint bundles AdCl0oddAdCl0p−odd

AdDO0

Bundles of g- HomCl0oddHomCl0p−odd

HomDO0

homomorphisms

Note that the Lie algebras of the Lie groups considered here are also associativealgebras. On each of these bundles there exists a connection, since the followingconnections are connections on any of these bundles.

Connections on TC∞b (M,N) induced by connections on N : Let f ∈ TC∞

b (M,N),

X ∈ TfC∞b (M,N), c ∈ C∞(]− 1; 1[;C∞

b (M,N)) such that c(0) = f , ( ddtc(t))|t=0 =

X, and Y a smooth section of TC∞b (M,N). Given ∇N a covariant derivative on N,

we can define a covariant derivative ∇ev∗N on TC∞b (M,N) by the following local

formula: in the chart Ξf on TC∞b (M,N) , given the exponential chart on N in

f(x) ∈ N and ∇N = d + θN the decomposition of ∇N on the exponential charton N based of f(x) , (∇ev∗N

X Y )(x) = ( ddtYc(t))|t=0 + θNX(x)Y (x). By this formula,

one can see that the connection 1-form on TC∞b (M,N) is given pointwise by the

connection 1-form on N. Moreover, the curvature Ωev∗N of ∇ev∗N is given pointwiseby the curvature ΩN of ∇N . Hence, we have the following :

Proposition 4. ∀f ∈ C∞b (M,N), ∀X,Y ∈ C∞

0 (M,f∗TN), Ωev∗N (X,Y ) is adifferential operator of order 0.

4.2. Weights and Hs metrics. Let π : E → M a Riemannian (resp. Hermitian)vector bundle over M . We call a weight an elliptic injective pseudo-differentialoperator of positive order on the set of smooth sections C∞(M,E), such that thereis an angle of vertex 0 which contains the spectrum of the principal symbol of A.This is a sufficient condition for the existence of the operators Az

θ and logθA [24].Note that, if Q is a weight and C ∈ Cl∗(M,E), C−1QC is also a weight, since thespectrum is unchanged. Thus we can define fields of weights on TC∞

b (M,N), alsocalled a weight on TC∞

b (M,N) for short, as a smooth section (Qf )f∈C∞b (M,N) of

HomCl such that, ∀f ∈ C∞b (M,N), Qf is a weight on C∞

b (M,f∗TN). Let (., .)Nbe the metric on a Riemannian manifold N. The (weak) L2 metric on TC∞

b (M,N)is given fiberwise by the (weak) L2 product on C∞

0 ((M,f∗TN)


(X,Y )L2,f =

∫M

(X(x), Y (x))Ndx,

where dx is the Riemannian volume form on M . Let us define fields of positiveoperators with respect to this (weak) metric. Given (Qf )f∈C∞

b (M,N) a self-adjoint

positive weight on TC∞b (M,N) of order 2s, the family

(X,Y )Hs,f = (QfX,Y )L2,f

is anHs metric on TC∞b (M,N). Defining a positive weight of order 2s on TC∞

b (M,N)boils down to defining a smooth section of Ell2s(C∞

b (M,N)), which is also a fieldof L2 self-adjoint, positive, injective (unbounded) operators on TC∞

b (M,N).

We claim that a positive weight can be always defined on TC∞b (M,N). On loop

spaces C∞b (S1, N), we can define a smooth field of injective elliptic differential

operators, induced by a connection∇ onN. [27]. Let ξ be the unit vector field on S1,we define the elliptic injective 1-order differential operator acting on C∞

0 (S1, f∗TN)defined on local coordinates on S1 by

∇f

dt=

d

dt+ θξ

where θ is the pull-back of the connection 1-form of ∇ by f . The family ∇f

dt

∗ ∇f

dtis a weight of order 2 on any TC∞

b (M,N). Generalizing this construction, we canbuild a smooth field of differential operators of order 2 on TC∞

b (M,N):

Proposition 5. Given a smooth vector field X on f∗TN , we define tr∇f ∗∇f (X)as the only vector field on f∗TN such that

∀Y ∈ C∞0 (M,f∗TN),

∫M

(tr∇f ∗∇f (X)(x), Y (x))Ndx =

∫M

(∇fX(x),∇fY (x))N,Mdx

where (., .)N,M is the metric induced by T ∗M ⊗ f∗TN by the metrics on M and

N . tr∇f ∗∇f is a smooth field of elliptic differential operators of order 2.

Proof. We use the same method as before to give an expression of ∇fX. LettingSuppX be in the domain U of a local trivialization of f∗TN , let (ξ1, ..., ξm) be asmooth section of the bundle of orthonormal frames O(U), we decompose

∇fX =m∑i=1

ξ∗i (∇fξiX).

Recall that, given a ⊗ A, b ⊗ B ∈ C∞(T ∗M ⊗ f∗TN), ((a ⊗ A, b ⊗ B))N,M =(a, b)T∗M (A,B)N .

So that, (∇fX(x),∇fY (x))N,M =∑m

i=1(∇fξiX)(x), (∇f

ξiY )(x))N .

Hence,

∫U

(∇fX(x),∇fY (x))N,Mdx =

m∑i=1

∫U

(∇fξi

∗∇f

ξiX)(x), Y (x))Ndx.

Then yields the result, since differential operators are local (see [6]). ⊔⊓

As a consequence of Lemma 1, a field of weights defines also a smooth section ofAdCl that has fiberwise the same properties.


4.3. The case of a parallelizable target manifold. If N is parallelizable, thereexists a smooth section of the bundle of frames GL(N), i.e. a smooth map ΦN :N → GL(N) that such that πN ΦN = IdN , where πN : GL(N) → N is thebundle projection. Then, for each f ∈ C∞

b (M,N), we define Φ(f) = ΦN f . Themap f 7→ Φ(f) is well defined for any f , and since it is defined point wise, Φ is asmooth global section of the bundle of frames FrDO0,∗ , and hence Φ is also a globalsection of the bundles FrCl, FrCl∗odd

, FrCl0,∗ , FrCl0,∗oddand FrCl0,∗p−odd

. Let us note

Φ = (e1, ..., en) in the notes, or Φ = (ei)i∈ INn for short. Using these identifications,a pseudo-differential operator A acting on C∞

0 (M,f∗TN) = C∞0 (M, IK)⊗ IKn can

be decomposed as a matrix A = (Aij)i,j∈Nn of pseudo-differential operators acting

on C∞0 (M, IK), namely, if X(x) =

∑i∈ INn

Xfi (x)ei(f(x)),

(AX)(x) =∑

i∈ INn

(∑

j∈ INn

(AijXj)(x))ei(f(x)).

We now define scalar operators.

Definition 6. We call an operator (resp. a field of operators) scalar if and only

if it can be written in the form A = A ⊗ Id IKn , where A is an operator (resp.a smooth family of operators) on C∞

0 (M, IK). We call a scalar weight a weight

Q = Q⊗ Id IKn , where Q is a weight on C∞0 (M, IK).

Remark : ([1], from an original idea of [5]) Let f ∈ C∞b (M,N). Under a fixed

trivialization of N , a pseudo-differential operator A acting on TC∞b (M,N)f =

C∞b (M, IKn) = C∞

b (M, IK) ⊗ IKn can be decomposed as a matrix (Aij)i,j∈Nn ofpseudo-differential operators. The contraction operator

tr IKn : A ∈ Cl(M, IKn) 7→n∑

i=1

Aii ∈ Cl(M, IK).

defines a smooth vector bundle morphism, from Cl(C∞b (M,N)) (resp. Cla(C∞

b (M,N)),resp. Clodd(C

∞b (M,N))) to the trivial bundle of basis C∞

b (M,N) and with typicalfiber Cl(M, IK) (resp. Cla(M, IK), resp. Clodd(M, IK)). Notice also that, if A isscalar, then

A =1

ntr IKn(A), hence A =

1

ntr IKn(A)⊗ Id IKn .

In this context, we can also define another principal bundle of frames.

Definition 7. Let p ∈ IN∗. Assume that N is parallelizable and Φ is as above.Then, we define

FrΦCl−(p),∗ = Φ(f).g; f ∈ C∞b (M,N); g ∈ Cl−(p),∗.

F rΦCl−(p),∗ is obviously a trivial principal bundle with structure group Cl−(p),∗.

4.4. On current groups. Let us now consider C∞b (M,G), where G is a compact

Lie group. Let us take ΦG as the trivialisation induced by the left multiplication ofG. Then, we get an easy computation of the Levi-Civita covariant derivatives forHs-metrics [5] : Let Q a left-invariant field of weights on TC∞

b (M,G), which weidentify with the operator on C∞

b (M, g). For the left-invariant Hs metric definedby Q

(X,Y )Hs,f = (QfX,Y )L2,f ,


there exists a Levi-Civita covariant derivative ∇, which reads ∇ = d+ θ on a fixedleft-invariant orthonormal frame as

(1) θXY =1

2adXY +Q−1adXQY −Q−1adQXY .

Note that, if Q = ∆s (or more generally if Q is scalar),

θX = adX +AX .

where AX is a 1-form that takes values in classical pseudo-differential operators oforder -1. Hence, the connection that corresponds to the covariant derivative ∇ is aconnection on FrΦCl−1,∗ .

5. Fields of functionals induced by Wodzicki residue andrenormalized traces

5.1. Wodzicki residue and renormalized traces. Recall that, throughout allthe article, the topology of pseudo-differential operators acting on smooth sectionsof a vector bundle over M is the topology used in

[9],[10], [2], which is a Frechet topology when considering pseudo-differential op-erators of limited order Clk(M,E). Let Q be a weight of order q and A be a classicalpseudo-differential operator on a finite rank vector bundle over the manifold M ofdimension m, and with total symbol σ(A) (defined locally). Then, the functiontr (AQ−s) , defined for s large, extends to the whole plane to a meromorphic func-tion with a simple pole at 0, with residue 1

q resWA, where resW is the Wodzicki

residue ([26], see also [8]). Let us recall some properties of the Wodzicki residue:

Proposition 6.(i) The Wodzicki residue resW is a trace on the algebra of classical pseudo-differentialoperators Cl(M,E), i.e. ∀A,B ∈ Cl(M,E), resW [A,B] = 0.(ii) (local formula for the Wodzicki residue) Moreover, if m = dimM and A ∈Cl(M,E),

resWA =1

(2π)n

∫M

∫|ξ|=1

trσ−m(x, ξ)dξdx

where σ−m is the (-m) positively homogenous part of the symbol of A. In particular,resW does not depend on the choice of Q.(iii) Let λ : Cl(M,E) → IC be trace. If dimM ≥ 2, then ∃k ∈ IC, λ = kresW .

Following the notations of [1], we define the renormalized trace.

Definition 8. trQA = lims→0(tr(AQ−s)− 1qs resWA).

If A is trace class, trQ(A) = tr(A). The functional trQ is of course not a trace onCl(M,E). Notice also that, if A and Q are pseudo-differential operators acting onsections on a real vector bundle E, they also act on E ⊗ IC. The Wodzicki residueres and the renormalized traces trQ have to be understood as functional defined onpseudo-differential operators acting on E ⊗ IC. In order to compute trQ[A,B] andto differentiate trQA, in the topology of classical pseudo-differential operators, weneed the following ([1], see also [16] for the first point):

Proposition 7. (i) Given two (classical) pseudo-differential operators A and B,given a weight Q,

(2) trQ[A,B] = −1

qresW (A[B, logQ]).


(ii) Given a differentiable family At of pseudo-differential operators, given a differ-entiable family Qt of weights of constant order q,

(3)d

dt

(trQtAt

)= trQt

(d

dtAt

)− 1

qresW

(At(

d

dtlogQt)

).

The following ”covariance” property of trQ ([1], [19]) will be useful to definerenormalized traces on bundles of operators,

Proposition 8. Under the previous notations, if C is a classical elliptic injective

operator of order 0, trC−1QC

(C−1AC

)is well-defined and equals trQA.

Wemoreover have specific properties for weighted traces of a more restricted classof pseudo-differential operators (see [9],[10],[1]), called odd class pseudo-differentialoperators following [9],[10] :

Definition 9. A classical pseudo-differential operator A is called odd class if andonly if

∀n ∈ZZ, ∀(x, ξ) ∈ T ∗M,σn(A)(x,−ξ) = (−1)nσn(A)(x, ξ).

Such a definition is consistent for pseudo-differential operators on smooth sec-tions of vector bundles, and applying the local formula for Wodzicki residue, onecan prove [1]:

Proposition 9. If M is an odd dimensional manifold, A and Q lie in the odd class,then f(s) = tr(AQ−s) has no pole at s = 0. Moreover, if A and B are odd classpseudo-differential operators, trQ ([A,B]) = 0 and trQA does not depend on Q.

5.2. Bracket property for trQ.

Definition 10. Let E be a vector bundle over M, Q a weight and a ∈ZZ. We define:

AQa = B ∈ Cl(M,E); [B, logQ] ∈ Cla(M,E).

Theorem 2.(i) AQ

a ∩ Cl0(M,E) is an subalgebra of Cl(M,E) with unit.

(ii) Let B ∈ Ell∗(M,E), B−1AQa B = AB−1QB

a .

(iii) Let A ∈ Clb(M,E), and B ∈ AQ−dimM−b−1, then trQ[A,B] = 0.

(iv) For a < −dimM2 , AQ

a ∩Cl−dimM

2 (M,E) is an algebra on which the renormalizedtrace is a trace (i.e. vanishes on the brackets).

Proof.(i): Clearly, AQ

a is a closed vector subspace of Cl(M,E). In order to check thestability under multiplication, one computes, for A,B ∈ AQ

a ∩ Cl0(M,E),

[AB, logQ] = ABlogQ− (logQ)AB =

(ABlogQ−A(logQ)B) + (A(logQ)B − (logQ)AB) = A[B, logQ] + [A, logQ]B.

Since A and B are of order 0, [A, logQ] and [B, logQ] are of order 0. Since A andB ∈ AQ

a , [A, logQ] and [B, logQ] are of order a. This shows that [AB, logQ] is oforder min(a, 0). On the other hand, since [Id, logQ] = 0, so that AQ

a ∩ Cl0(M,E)is an unitary algebra.


(ii): Let A ∈ AQa ,

[B−1AB, log(B−1QB)] = [B−1AB,B−1(logQ)B] = B−1[A, logQ]B.

Since [A, logQ] is of order a, [B−1AB, log(B−1QB)] is of order a, and henceB−1AB ∈AB−1QB

a .

(iii): If A ∈ Clb(M,E), and B ∈ AQ−dimM−b−1, A[B, logQ] is of order −dimM −

1, and hence trace class. Since the Wodzicki residue vanishes on trace class opera-tors, it follows that

trQ[A,B] = −1

qresW (A[B, logQ]) = 0.

(iv) is a simple consequence of (i) and (iii). ⊔⊓

5.3. The case of a trivial bundle. Let us now give a formula for trQ(A) whenQ is scalar and E = M × V . Recall that Q is scalar if and only if it can be writtenas Q = Q⊗ IdV where Q is a weight on C∞

0 (M, IC).

Proposition 10. [1] trQ(A) = trQ(tr IKn(A)).

We now produce non trivial examples of operators that are in AQa when Q is

scalar, and secondly we give a formula for some non vanishing renormalized tracesof a bracket.

Lemma 3. Let Q be a weight on C∞0 (M,V ) and let B be a classical pseudo-

differential operator of order b. If B or Q is scalar, then [B, logQ] is a classicalpseudo-differential operator of order b− 1.

Proof. It is a classical result that [B, logQ] is a classical pseudo-differential operatorof order b. In order

to show that it is of order b− 1, we show that σb([B, logQ]) = 0.If Q is a classical weight of order q, logQ is a logarithmic ( hence a non classi-

cal) pseudo-differential operator, and its total symbol locally reads σlogQ(x, ξ) =qlog|ξ| + σC(x, ξ) ( σC is a classical symbol with non positive order). Since B isclassical, σ ([B, logQ]) is equivalent, up to smoothing symbols to∑

α∈ INm

Dαξ σ(B)Dα

xσ(logQ)−Dαξ σ(logQ)Dα

xσ(B)

= q[σ(B), log|ξ|] +∑

α∈ INm;|α|>0

Dαξ σ(B)Dα

xσC −Dαξ (qlog|ξ|+ σC)D

αxσ(B)

=∑

α∈ INm;|α|>0

Dαξ σ(B)Dα

xσC − (qDαξ log|ξ|+Dα

ξ σC)Dαxσ(B).

Let us compute the homogeneous symbol of order b of [B, logQ]

σb[B, logQ](x, ξ) = [σbB(x, ξ), σ0logQ(x, ξ)]

where σ0logQ denotes the homogenous part of order 0 of σC , hence• ifQ and thus logQ is scalar, i.e. logQ = (logQ)⊗IdV , we have σb[B, logQ](x, ξ) =

σ0logQ(x, ξ)[σbB(x, ξ), IdV ] = 0.

• if B is scalar,σb[B, logQ](x, ξ) = σbB(x, ξ)[IdV , σ0logQ(x, ξ)] = 0.In the both cases we find that the (classical) symbol of [B, logQ] is of order b−1.

⊔⊓Let us now give a closed two-sided ideal of Cl0(M,V ) contained in AQ

a .


Proposition 11. Let Q be a scalar weight on C∞0 (M,V ). Then

Cla+1(M,V ) ⊂ AQa .

Consequently,(i) if ord(A) + ord(B) = −dimM, trQ[A,B] = 0.(ii) when M = S1, if A and B are classical pseudo-differential operators, if A iscompact and B is of order 0, trQ[A,B] = 0.

Proof. If A ∈ Cla+1(M,V ), by lemma 3, [A, logQ] is of order (a + 1) − 1 = a.Hence, A ∈ AQ

a .Let us now prove the two consequences:(i): If ord(A) + ord(B) = −dimM, ord(A) = −ord(B) − dimM, hence A ∈

AQ−ord(B)−dimM−1. Hence, we apply theorem 2, (iii)

(ii): If A is a compact classical pseudo-differential operator, A ∈ Cl−1. We apply(i) to S1 (that is dimS1 = 1). ⊔⊓

One could wonder whether an inclusion Clb(M,V ) ⊂ AQa holds for b > a + 1.

We know that AQa is unitary, and hence contains pseudo-differential operators of

order 0, but easy calculations, based on techniques of calculus on formal symbols,show that b = a + 1 is the greatest order b such that Clb(M,V ) ⊂ AQ

a . Namely,σb−1([B, logQ]) does not vanish in general. Let us now give a formula for trQ[A,B]when ord(A)+ord(B) = −dimM+1 which shows that trQ[A,B] does not generallyvanish in this case.

Lemma 4. Let Q be a scalar weight on C∞0 (M,V ), and A, B two pseudo-differential

operators of orders a and b on C∞0 (M,V ), such that a+ b = −m+1 (m = dim M).

Then

trQ[A,B] = −1

qresW (A[B, logQ]) = − 1

q(2π)n

∫M

∫|ξ|=1

tr(σa(A)σb−1([B, logQ])).

Proof : [B, logQ) is of order b − 1, so A[B, logQ] is of order a+b-1. If a + b =−m + 1, A[B, logQ] is of order -m, so that its partial symbol of order -m is itsprincipal symbol. Then, using the local formula for res (Proposition 6),

resW (A[B, logQ]) = − 1

(2π)n

∫M

∫|ξ|=1

tr(σ−m(A[B, logQ]))

= − 1

(2π)n

∫M

∫|ξ|=1

tr (σa(A)σb−1([B, logQ])) , this yields the result. ⊔⊓

5.4. Fields of linear functionals. Let us first remark that λ = resW , whereresW is Wodzicki residue, is a trace on Cl(M, IKn), and hence, since for any C ∈Cl∗(M, IKn) and A ∈ Cl(M, IKn), resW (C−1AC) = resW (A), we can pull-back theWodzicki residue to the bundle AdCl, and hence to all the adjoint bundles we haveconstructed before. Since the renormalized traces trQ have not the same propertiesas resW , we cannot define a priori a field of renormalised traces on AdCl in such astraightforward way. But we have shown that there exists a field of weights as asmooth section of HomCl. Hence, by Lemma 1, a field of weights is also a smoothsection of AdCl. Hence, by Proposition 8, it is possible to define a smooth field ofrenormalized traces on the vector bundle AdCl defined by the weight chosen.


6. Chern-Weil type theorem on manifolds of maps

6.1. Chern forms defined with Wodzicki residue. We get the following result,trying to apply Corollary 1 of the Chern-Weil type theorem.

Theorem 3. Let θ be a connection on covariant derivative on TC∞b (M,N), of

curvature Ω. The 2k-forms resW (Ωk) are exact for k ∈ IN∗.

Proof.(Ωev∗N

)ktakes values into differential operators of order 0. Hence, using

the local formula for the Wodzicki residue, resW

((Ωev∗N

)k)= 0. We get the

result, applying corollary 1 to λ = resW . ⊔⊓

6.2. Chern forms for weighted manifolds of maps. Since all the forms resW (Ωk)are exact on manifolds of maps, following [19] and [1], we now use instead the linearfunctionals trQ, where Q is a weight on TC∞

b (M,N). We need to decide on whichbundles of frames we can define some classes of connections C(λ, i) and some fieldsof weights Q for which we can apply Theorem 1, with λ = trQ and i a suitableideal. We summarize the main results of this section in the tables below. We buildthree classes of connections that correspond to three following contexts to whichwe can apply theorem 1 with λ = trQ. Recall that m is the dimension of the sourcemanifold M and n is the dimension of the target manifold N . We decide in thissection to work in the following three contexts, and we get the following results,where k is the constant of definition of the k-filtered property and of theorem 1:

case A case B case CManifold M odd dimensional − odd dimensionalManifold N − N = G, Lie group parallelizable

k 1 −[−m

l

]1

P = FrG FrCl0,∗oddFrΦCl−k,∗ FrΦ

Cl0,∗m,odd

Ad(P ) AdCl0oddAdΦCl−k AdΦ

Cl0m,odd

Ad(P, a) idem AdΦCl0 idemAd(P, i) idem AdΦCl−l idem

odd class, odd class,Weight Q left-invariant, d logQ

of order − 1,scalar andQ scalar

C(trQ, i) Codd C−l Codd−m

Vanishing of When the corres- every Chern class every Chern classChern classes -ponding Chern classes vanishes vanishes

of N vanish

First note that [19], for any weight Q on TC∞b (M,N), for any differential form

α ∈ Ω∗(C∞b (M,N), Cl0), we can define the differential form trQα using the ”covari-

ance property” of trQ (see Proposition 8). Then, let us recall the general formulafor d(trQα):


Proposition 12. [1] Given a pseudo-differential n-form α on TC∞b (M,N), a

pseudo-differential connection ∇, ∇ = d+ θ locally , and for any weight Q,

(4) d(trQ(α)) = trQ(∇α) +(−1)n+1

qresW (α(∇logQ)).

Moreover, 1q resW (α(∇logQ)) reads in local coordinates as

(5)1

qresW (α(∇logQ)) = trQ([α, θ]) +

1

qresW (α(d logQ))

Let us first describe the three classes of connections announced and show thatthey have the desired properties in order to apply Theorem 1.

A : The class of connections Codd.We set P = FrCl0,∗odd

and Codd is the space of connections on P .

Proposition 13. If θ ∈ Codd, Q is a weight of odd class, and M is odd-dimensional,if α ∈ Ω∗(C∞

b (M,N), AdCl0odd), we have d(trQα) = trQ (∇α).

Proof. Since α, Q and θ are odd class in local coordinates, (dQ)Q−1 and [α, θ] areodd class in local coordinates, hence, since M is odd dimensional,

resW (αdlogQ) = 0

by the local formula of the Wodzicki residue and trQ[α, θ] = 0 by Proposition 9.Then yields the result, using 6.2. ⊔⊓

As an example of connections lying in such classes, we can give the connectionsobtained by pull back of a connection on N , and also the Levi-Civita connectionson the groups C∞

b (M,G) given by the formula 1, when Q is odd class (for example,when Q = ∆s, s ∈ IN∗).

B : The class C−l on C∞b (M,G) for l > 0.

We set P = FrΦCl−l,∗ and C−l the set of connections on P . In this case, we neednot to assume anything on the dimension of M . We fix m = dimM and

k = −[−m

l] =

m/l if m/l ∈ IN∗,the smaller integer bigger that m/l if m/l ∈ IN∗.

.

We have Ad(P, ik) ⊆ AdCl−m . We write ∇ = d + θ on the trivialization Φ definedbefore by the left action of the group. We also fix a left-invariant scalar weight Q.

Proposition 14. Let α ∈ Ω∗(C∞b (M,N), Ad(P )) and let θ be a connection on

FrCl0,∗ . Then

dtrQα = trQ (∇α) .

Note that we get a stronger result than the one needed in order to apply Theorem1 .

Proof. We show that resW (α(∇logQ)) = 0.• d logQ = 0 since Q is left-invariant.• Applying Proposition 11 with a = −1, [θ, logQ] is a 1-form with values in Cl−1.Hence, α(∇logQ) = α(dlogQ) + α[θ, logQ] ∈ Ω∗(C∞

b (M,N), Cl−m−1(M, g)).Since the Wodzicki residue vanishes on classical pseudo-differential operators thatare trace class, resW (α(∇logQ)) = 0. ⊔⊓


C : The class Codd−m. We now define a class of connections that combines Codd

and C−m, and show that there exists weights for which it has the desired properties,when M is odd dimensional and N is parallelizable. We fix k = 1. Given a fixedtrivialization of N, we fix Q as an odd class scalar weight on TC∞

b (M,N) such thatdlogQ is of order -1 in this trivialization. Recall that we set P = FrCl0,∗m,odd

. Notice

that, by Propositions 9 and 11, trQ is a field of traces on AdCl0m−oddWe define Codd

−m

as the connections on P .

Proposition 15. Let ∇ ∈ Codd−m and α ∈ Ω∗(C∞

b (M,N), AdCl0m,odd), dtrQα =

trQ(∇α).

Proof. Let α ∈ Ω2(C∞b (M,N), Ad(P ). Then α decomposes as α = β + γ, with

β ∈ Ω∗(C∞b (M,N), Cl−m), and γ ∈ Ω∗(C∞

b (M,N), Cl0odd).

• Let us show that resW (α (dlogQ)) = 0.d logQ ∈ Cl−1

odd hence αd logQ = βd logQ+γd logQ, with βd logQ ∈ Ω∗(C∞b (M,N), Cl−m−1),

and γd logQ ∈ Ω∗(C∞b (M,N), Cl−1

odd). The Wodzicki residue vanishes on Cl−m and

Cl−1odd, thus

resW (α (d logQ)) = 0.

• Let us show that trQ[α, θ] = 0.We write, locally, θ = η + λ, where η ∈ Ω1(Uf , Cl−m), and λ ∈ Ω1(Uf , Cl0odd).

Then[α, θ] = [β, η] + [β, λ] + [γ, η] + [γ, λ] .

- Since Q is odd class, trQ [γ, λ] = 0.- Since Q is scalar, trQ [β, η] = 0, trQ [β, λ] = 0 and trQ [γ, η] = 0. Hence,

trQ[α, θ] = 0. Then yields the result, using . ⊔⊓

We can now state the following Chern-Weil type theorem :

Theorem 4. In the contexts A, B and C, if Ω is the curvature of a connectionin C(trQ, i), the Chern forms trQ(Ωl) for l ≤ k are closed and their cohomologyclasses do not depend on the chosen connection.

The cohomology classes of these Chern forms apparently depend on the weightQ. Let us now deal with the dependence on the weight .

6.2.1. Dependence on the weight. We investigate only the dependence of the weightfor the case A, since the results of the next sections will deal with the other cases,When investigating the dependence on the weight, we assume that M is odd dimen-sional. We show that the characteristic classes defined in the case A of Theorem 4do not depend on the choice of the weight Q.

Proposition 16. Let ∇ be a connection in Codd, with curvature Ω. Then, giventwo odd class weights Q1 and Q2, ∀k ∈ IN∗, trQ1(Ωk) = trQ2(Ωk).

6.2.2. Vanishing of Chern forms. Let us first prove that the vanishing of a Chernclass of N ensures the exactness of the corresponding Chern) form on TC∞

b (M,N)in the case A.

Theorem 5. Assume that M is odd dimensional. Let j be an integer such that thek-th Chern class of N vanishes. Then for any θ in Codd with curvature Ω, if Q isan odd class weight on TC∞

b (M,N), then trQ(Ωj) is exact.


Proof. It is sufficient to show this for a chosen connection ∇ ∈ Codd and a chosenweight on M × ICn since the renormalized trace does not depend on the choice ofthe odd class weight. The idea is to pull back Chern forms of N on C∞

b (M,N).

• Let us first consider Chern forms on N . Let θN be a connection on N. Wehave tr((ΩN )j) is exact. Hence, there exists a (2k-1) form α on N such thatdα = tr((ΩN )j). We can write α as

α = αtr(1

nId ICn).

Hence, setting

∀x ∈ N, ∀X1, ..., X2k−1 ∈ TxN, β(X1, ..., X2k−1) =1

nα(X1, ..., X2k−1).IdTxN

we have α = tr(β). So that tr(ΩN )k = dα = d(tr(β)) = tr(∇Nβ), where ∇N is thecovariant derivative on TN defined by θN .

• From the form β on N we now build its pull-back βev∗N : ∀x ∈ M,∀X1, ..., X2k−1 ∈ TfC

∞b (M,N),

βev∗N (X1, ..., X2k−1)f (x) = β(X1(x), ..., X2k−1(x)).

• Let us now give the computation of trQ((Ωev∗N )k)f in terms of βev∗N . Recallthat (section 2), when N is not parallelizable, we restrict ourselves to the study ofthe connected component that contains the constant based loop. Let us fix, first,f in this connected component. f∗TN ∼ M × ICn. Hence, there is an identificationbetween TC∞

b (M,N)f and C∞b (M, IC)⊗ ICn which we now fix as long as we stay

on the fiber over f . Since (Ωev∗N ) takes values in odd class pseudo-differentialoperators, trQ((Ωev∗N )k)f does not depend on the choice of the weight Q amongodd class weights. Since TC∞

b (M,N)f = C∞b (M, IC) ⊗ ICn, we choose an odd

class weight which is also scalar : Q = ∆⊗ Id ICn . Then, applying Proposition 10,

trQ(Ωev∗N )kf = tr∆(tr ICn(Ωev∗N )kf ).

Then, since tr ICn((Ωev∗N )k) is given point wise by tr((OmegaN )k) = tr(∇Nβ),

tr ICn(∇ev∗Nβev∗N ) = (tr ICn(Ωev∗N )kf .

Hence, trQ(Ωev∗N )kf = trQ(∇ev∗Nβev∗N ).

This equality depends neither on the choice of the odd class weight, nor on the choiceof the trivialization of TC∞

b (M,N)f . So, we can state it, not only fiberwise, butalso with respect to any field of odd class weight Q on TC∞

b (M,N).Applying Proposition 13, we get :

trQ(Ωev∗N )kf = trQ(∇ev∗Nβev∗N ) = d(trQ(βev∗N )). ⊔⊓We have also the following, which is a simple consequence of Proposition 2:

Proposition 17. The Chern forms defined in the cases B and C are exact.

Hence, in the contexts A, B, C, one can expect to find non vanishing Chernclasses only when the target manifold N has non vanishing Chern classes. In-tuitively, the non vanishing Chern classes should appear in examples where thebundle of multiplication operators DO0(C∞

b (M,N)) is not trivial. However, evenif we expect that the set up A can give rise to non vanishing characteristic classes,we actually do not know how to prove it.


Moreover, even if we define more Chern classes than in [4] and [5] on currentgroups C∞

b (M,G), we cannot avoid to notice that our approach still deals with thecohomology classes of only some Chern forms on current groups C∞

b (M,G). In par-ticular, the first Chern forms on the bundle FrCl0 are not completely understood,even if we can give for example the following result:

Theorem 6. Let M = T 2 be the 2-dimensional torus, and let ∆ = −∑2

i=1d2

dx2i.

Let Ω be the curvature of the H1-Levi-Civita connection (1) with Q = ∆. Then,tr∆(Ω) = 0.

Proof. Let θ be the H1-Levi-Civita connection. We have, for X ∈ C∞0 (T 2, g),

θX = 12

adX +∆−1adX∆−∆−1ad∆X

. Let X,Y ∈ C∞

b (M, g).

trδ(Ω)(X,Y ) = tr∆([θX , θY ]− θ[X,Y ]

).

Using Proposition 8, we have

tr∆(θ[X,Y ]) = tr∆(tr ICn(ad[X,Y ])−

1

2∆−1tr ICn(ad∆([X,Y ]))

)= 0.

Thus, tr∆(Ω)(X,Y ) = tr∆ ([θX , θY ]) decomposes into the following sum:

tr∆(1

4([adX , adY ] + [∆−1adX∆,∆−1adX∆])

)+tr∆

(1

4([adX ,∆−1adY ∆] + [∆−1adX∆, adY ])

)+tr∆

(−1

4([adX +∆−1adX∆,∆−1ad∆Y ] + [∆−1ad∆X , adY +∆−1adY ∆])

)+tr∆

(1

4[∆−1ad∆X ,∆−1ad∆Y ]

).

Let us study line by line this expression.• We calculate first line:

[∆−1adX∆,∆−1adX∆] = ∆−1[adX , adY ]∆.

By Proposition 8, we have

tr∆(1

4([adX , adY ] + [∆−1adX∆,∆−1adX∆])

)=

1

2tr∆

(ad[X,Y ]

)= 0.

• Let us now turn to the second line. We have

tr∆(1

4([adX ,∆−1adY ∆] + [∆−1adX∆, adY ])

)= tr∆

(1

4([adX∆−1, adY ∆] + [adX∆, adY ∆

−1]

)= tr∆

(1

4([adX∆−1, adY ∆]− [adY ∆

−1, adX∆])

).

Let us compute the partial symbols of the operators inside the brackets. We noticethat adX has only a non vanishing partial symbol of order 0, σ0(adX) = adX , thatdoes not depend on the ξ variable, ∆ has only a non vanishing partial symbol oforder 2, σ2(∆) = ||ξ||2, which does not depend on the x-variable, ∆−1 has only anon vanishing partial symbol of order -2, an easy calculation on formal symbols showthat σ−2(∆

−1) = ||ξ||−2, which does not depend on the x-variable, and we have


also σ(log∆) = log(||ξ||2). Then, adX∆−1 and adY ∆−1 (resp. adX∆ and adY ∆)

have only a partial symbol of order −2 (resp. of order 2). We can now calculate,setting ξ = (ξ1, ξ2) and x = (x1, x2) and applying equation 2 of Proposition 7:

tr∆([adX∆−1, adY ∆]

)= − 1

8π2

∫T 2

∫||ξ||=1

trσ−2(adX∆−1[adY ∆, log∆])dξdx

= − 1

8π2

∫T 2

∫||ξ||=1

tr

∑i+j−|α|=−2

(−i)|α|

|α|!Dα

ξ σi(adX∆−1)Dαxσj([adY ∆, log∆])

dξdx

= − 1

8π2

∫T 2

∫||ξ||=1

tr(σ−2(adX∆−1)σ0([adY ∆, log∆])

)dξdx

+i

8π2

∫T 2

∫||ξ||=1

tr

2∑j=1

∂ξjσ−2(adX∆−1)∂xjσ1([adY ∆, log∆])

dξdx.

We calculate separately the two lines.∫T 2

∫||ξ||=1


)dξdx

=

∫T 2

∫||ξ||=1

tr(adX2(ξ22 − ξ21)ad∂2

x1Y − 4ξ1ξ2ad∂2

x1,x2Y + 2(ξ21 − ξ22)ad∂2

x2Y

)dξdx.

Since ∫||ξ||=1

(4ξ1ξ2) =

∫||ξ||=1

((ξ2 + ξ1)2 − (ξ2 − ξ1)

2)

is proportional to ∫||ξ||=1

(ξ21 − ξ22) = 0,

we get ∫T 2

∫||ξ||=1


)dξdx = 0.

By easy calculations, one has σ1([adY ∆, log∆]) =∑2

k=1 −2iξkad∂xkY , this gives

2∑j=1

∂ξjσ−2(adX∆−1)∂xjσ1([adY ∆, log∆]) =4i

||ξ||42∑

j,k=1

ξjξkadXad∂xj∂xk

Y .

We already know that∫||ξ||=1

ξ1ξ2dξ = 0, which leads to the following:

tr∆([adX∆−1, adY ∆]

)=

−i

2π2

∫T 2

∫||ξ||=1

2∑j=1

ξ2j tr(adXad∂2

xjY

)dξdx

=

(i

2π2

∫||ξ||=1

ξ21

)(∫T 2

tr(adXad∂2

xjY

)dx

)= C(X,Y )H1

0.

Then, one gets the result:


tr∆(1

4([adX∆−1, adY ∆]− [adY ∆

−1, adX∆])

)=

C

4((X,Y )H1

0− (Y,X)H1

0) = 0.

• Let us now calculate the third line:

[adX +∆−1adX∆,∆−1ad∆Y ]

and

[∆−1ad∆X , adY +∆−1adY ∆])

are brackets of a pseudo-differential operator of order 0 with a pseudo-differentialoperator of order −2. Since ∆ is a scalar operator, by Proposition 11, the renor-malized traces of the two brackets vanish. Hence, the third line vanishes.

• The fourth line is the renormalized trace of the bracket of two Hilbert-Schmidt operators. Since the renormalized trace equals to the usual trace on traceclass operators, this line vanishes.

Gathering the four lines, we obtain tr∆(Ω)(X,Y ) = 0. ⊔⊓

In the next section, we deal with an example for which it is known that the firstChern form does not vanish.

7. The loop group as a complex manifold

Let us now give another application to the complexification of the loop group.In the following, we denote G IC the complex Lie group with Lie algebra g IC = g⊗ IC.An interesting metric is the H1/2 metric for the weight |D| = | d

dθ | (D is the Dirac

operator ddθ , which is invertible on based loops). It is homogeneous Kahler [21]. Its

symplectic form is left invariant, and is given on the Lie algebra by

ω(X,Y ) = (DX,Y )L2 =

∫S1

(DX(x), Y (x))dx.

The associated almost complex structure J is derived from the scalar pseudo-differential operator ϵ(D) = −iD

|D| acting on C∞0 (S1; g IC). The complex manifold

of C∞b (S1, G) will be noted X IC. Its complex tangent space is given by V1, the

eigenspace corresponding to the eigenvalue 1 of ϵ(D). Let V−1 be the eigenspacecorresponding to the eigenvalue -1 of ϵ(D). An operator A on C∞

b (S1, g IC) can bewritten as a matrix operator on V1 ⊕ V−1 of the form(

A++ A−+

A+− A−−

)whereA++ (resp. A+−, resp. A−+, resp. A−−) are the obvious restrictions of the

operator A: A++ : V1 → V1, A+− : V1 → V−1, A−+ : V−1 → V1, A−− : V−1 → V−1.A++ is the holomorphic part of the operator A. Noting p+ = 1

2 (Id + −iD|D| )

the orthogonal projection on V1 and p− = 12 (Id − −iD

|D| ) the orthogonal projection

on V−1, we identify A++ with the operator p+Ap+ =

(A++ 00 0

). Since p+ is

a scalar pseudo-differential operator, we say that an operator B on V1 is pseudo-

differential if and only if

(B 00 0

)is a pseudo-differential operator. Alternatively,

we realize X IC as a quotient of a complex group [5], and hence give its complex


structure. Let P the subgroup of C∞b (S1, G IC) which extends to holomorphic maps

from |z| ≥ 1 to G IC.Then

X IC = C∞b (S1, G IC)/P.

This defines pseudo-differential operators TC∞b (S1, G) IC as pseudo-differential op-

erators on the complex bundle TC∞b (S1, G IC). In [1], adapting a result of [5] in

terms of renormalized traces, given ∇ the H12 Levi-Civita covariant derivative on

the loop group with curvature Ω, we have shown that

tr|D|(Ω1,0) = −iω.

In other words, the loop group is Kahler-Einstein. In [5], Freed shows the nonvanishing of the De Rham cohomology of ω. We define here a class of connectionsthat contains the H

12 Levi-Civita connection, and we show that we can apply the

Chern-Weil type theorem for this class of connections. Then, we show that this nonvanishing first Chern class expresses the non triviality of the holonomy bundle of aconnection. For this, let us say briefly the way of defining these holonomy bundles.The group of the units of the algebra of classical pseudo-differential operators oforder 0 is a group of exponential type [7], or in other terms of type I in the terminol-ogy of Robart [23]. Hence, following the results of [15], we can build the holonomybundle Hθ of any connection θ. Moreover, the Lie algebra of the structure groupof the holonomy bundle is spanned by the curvature elements. We apply theseconstructions to the following connections: since we are not working on a Lie groupbut on a flag manifold, we need to take under consideration connections that arenot induced by left invariant connections on C∞

b (S1, G IC), but some of those thatare compatible with the almost complex structure and the action of the group P .

Definition 11. Let θ be a connection on FrCl0,∗ with covariant derivative ∇.θ ∈ CJ

−1 if and only if(i) (θX)++ − (adX)++ is of order -1.(ii) ∇J = 0(iii) θ is an invariant connection under the action of P

These conditions ensures that the connections in CJ−1 reduce to connections on

X IC. Notice that CJ−1 is obviously an affine space of connections.

Proposition 18. If the connection θ is in CJ−1, then the associated curvature on

the holomorphic bundle Ω1,0 is of order -1.

Proof. Let ∇ = d + θ be a connection in the class CJ−1. p+ is a classical pseudo-

differential operator of order 0, hence Ω1,0, the curvature of ∇++ = d + θ++, is2-form with values into classical pseudo-differential operators of order 0. In orderto show that it is Cl−1-valued, let us compute its symbol of order 0. We haveσ0(p+) =

12 (1 +

ξ|ξ| )Idg.

Hence, σ0((θX)++) =1

2(1 +

ξ

|ξ|)Idg ICadX

1

2(1 +

ξ

|ξ|)Idg IC =

1

2(1 +

ξ

|ξ|)adX .

Remarking that σ0((θX)++) is left-invariant,

σ0Ω1,0(X,Y ) = σ0((θX)++)σ0((θY )++)− σ0((θY )++σ0((θX)++)− σ0((θ[X,Y ])++)

=1

2(1 +

ξ

|ξ|)(adXadY − adY adX − ad[X,Y ]) = 0.


Thus, Ω1,0(X,Y ), which is a classical pseudo-differential operator of non-positiveorder, is of order -1. ⊔⊓

As a consequence, given a connection θ ∈ CJ−1, the Lie algebra of the structure

group of the holonomy bundle Hθ++ is a Lie subalgebra of Cl−1. Hence, we cangive the following:

Theorem 7. Let Q be a scalar weight on H1/2b (S1, g IC). Let θ ∈ CJ

−1, and let Hθ++

be its Holonomy bundle over X IC.(i) ∀θ ∈ CJ

−1, dtrQ((Ω1,0)n

)= 0∀n

(ii) Given two any connections θ0 and θ1 ∈ CJ−1, the differential forms trQ

((Ω1,0

1 ))−

trQ((Ω1,0

0 ))are exact.

(iii) for any connection θ ∈ CJ−1, H

θ++ is non trivial.

Proof.(i) and (ii): We claim that theorem 1 can be applied here, with C(λ, i) = CJ

−1,

and with Q scalar weight. For this, given θ ∈ CJ−1,

• its holomorphic curvature is Cl−1-valued.• If θ0 and θ1 are two connections in CJ

−1,

θ0 − θ1++ = θ0 − ad− θ1 + ad++ ∈ Cl−1

Hence, the affine space of connections CJ−1 has an underlying vector space that is

contained in the vector space of Cl−1-valued forms.• Moreover, using Proposition 14, we have for any connection θ in CJ

−1 with hor-

izontal derivation ∇ and for any Cl−1-valued form α, dtrQ(α++) = trQ(∇++α++).Hence, since trQ has the filtered trace property for i = Cl−1, a = Cl0 and k = 1(by Proposition 11), we can apply Theorem 1.

(iii): Let θ ∈ CJ−1, with holomorphic curvature Ω1,0. The first Chern form

tr∆(Ω1,0) has the same cohomology class as the Kahler form, since the Kalherconnection is in CJ

−1. Hence, tr∆(Ω1,0) has non trivial cohomology class.

The Holonomy algebra of θ++ is a Lie subalgebra of Cl−1, and tr∆ is a trace inCl−1. We can apply Corollary 1 to Hθ++ and also Proposition 2, to conclude. ⊔⊓

Acknowledgements

I would like to thank Sylvie Paycha, who gave me her interest on this subject,for the constant help provided, and also Claude Roger, Steve Rosenberg, TilmannWurzbacher for useful comments. This paper has been completed during a post-doctoral fellowship at the University of Bonn (Germany), supported by a BourseLavoisier, and I would also like to thank warmly Sergio Albeverio.

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