#### 6.1. Chern Forms in Infinite Dimensional Setting

Let P be a principal bundle, of basis M and with structure group G. Let $\mathfrak{g}$ be the Lie algebra of G. Recall that G acts on P, and also on $P\times \mathfrak{g}$ by the action $((p,v),g)\in (P\times \mathfrak{g})\times G\mapsto (p.g,A{d}_{{g}^{-1}}\left(v\right))\in (P\times \mathfrak{g})$. Let $AdP=P{\times}_{Adg}=(P\times \mathfrak{g})/G$ be the adjoint bundle of P, of basis M and of typical fiber $\mathfrak{g}$, and let $A{d}^{k}P={\left(AdP\right)}^{\times k}$ be the product bundle, of basis M and of typical fiber ${\mathfrak{g}}^{\times k}$.

**Definition 32.** Let k in ${\mathbb{N}}^{*}$. We define ${\mathfrak{Pol}}^{k}\left(P\right)$, the set of smooth maps $A{d}^{k}P\to \mathbb{C}$ that are k-linear and symmetric on each fiber, equivalently as the set of smooth maps $P\times {\mathfrak{g}}^{k}\to \mathbb{C}$ that are k-linear symmetric in the second variable and G-invariants with respect to the natural coadjoint action of G on ${\mathfrak{g}}^{k}.$

Let $\mathfrak{Pol}\left(P\right)={\u2a01}_{k\in \mathbb{N}*}\mathfrak{Pol}\left(P\right)$.

Let

$\mathcal{C}\left(P\right)$ be the set of connections on

P. For any

$\theta \in \mathcal{C}\left(P\right)$, we denote by

$F\left(\theta \right)$ its curvature and

${\nabla}^{\theta}$ (or ∇ when it carries no ambiguity) its covariant derivation. Given an algebra

A, In this section, we study the maps, for

$k\in {\mathbb{N}}^{*}$,

where

$Alt$ denotes the skew-symmetric part of the form. Notice that, in the case of the finite dimensional matrix groups

$G{l}_{n}$ with Lie algebra

${\mathfrak{gl}}_{n}$, the set

$\mathfrak{Pol}\left(P\right)$ is generated by the polynomials

$A\in {\mathfrak{gl}}_{n}\mapsto \text{tr}\left({A}^{k}\right),$ for

$k\in 0,...,n$. This leads to classical definition of Chern forms. However, in the case of infinite dimensional structure groups, most situations are still unknown and we do not know how to define a set of generators for

$\mathfrak{Pol}\left(P\right).$**Lemma 33.** Let $f\in {\mathfrak{Pol}}^{k}\left(P\right).$ Then**Proof.** Let us notice first that

f is symmetric. Let

$v\in \mathfrak{g},$ and

${c}_{t}$ a path in

G such that

${\left\{\frac{d}{dt}{c}_{t}\right\}}_{t=0}=v.$ Let

${a}_{1},...,{a}_{k}\in {\mathfrak{g}}^{k}$.

Since

f in

G-invariant, we get:

**Lemma 34.** Let $f\in {\mathfrak{Pol}}^{k}\left(P\right)$ such that f, as a smooth map $P\times {\mathfrak{g}}^{k}\to \mathbb{C},$ satifies ${d}^{M}f=0$ on a system of local trivializations of $P.$ Then, the maptakes values into closed forms on P. Moreover,- (i)
it is vanishing on vertical vectors and defines a closed form on M.

- (ii)
the cohomology class of this form does not depend on the choice of the chosen connexion θ on P.

**Proof.** The proof runs as in the finite dimensional case, see e.g., [

30]. First, it is vanishing on vertical vectors and

G-invariant because the curvature of a connexion vanishes on vertical forms and is

G-covariant for the coadjoint action. Let us now fix

$f\in {\mathfrak{Pol}}^{k}\left(P\right)$. We compute

$df\left(F\right(\theta ),...,F(\theta \left)\right).$ We notice first that it vanishes on vertical vectors trivially. Let us fix

${Y}_{1}^{h},...,{Y}_{2k}^{h},{X}^{h}$ $2k+1$ horizontal vectors on

P at

$p\in P$. On a local trivialization of

P around

p, these vectors read as:

where

$\tilde{\theta}$ stands here for the expression of

θ in the local trivilization, and

${Y}_{1},...,{Y}_{2k},X$ $2k+1$ tangent vectors on

M at

$\pi \left(p\right)\in M.$ We extend these vector fields on a neighborhood of

p- -
by the action of G in the vertical directions

- -
setting the vectors fields constant on $U\times p$, where U is a local chart on M around $\pi \left(p\right)$.

Then, we have:

since

$F\left(\theta \right)$ is vanishing on vertical vectors.

Then, on a local trivialization with the notations defined before (the sign

$Alt$ is omitted for easier reading), and writing

${d}^{M}$ for the differential of forms on any open subset of

M,

and then, using Lemma 33,

Then, by Bianchi identity, we get that:

This proves (i) Then, following e.g., [

30], if

θ and

${\theta}^{\prime}$ are connections, fix

$\mu ={\theta}^{\prime}-\theta $ and

${\theta}_{t}=\theta +t\nu $ for

$t\in [0;1].$ We have:

Moreover,

μ is

G-invariant and vanishes on vertical vectors. Thus,

Integrating in the

t-variable, we get:

Even if these computations are local, the two sides are global objects and do not depend on the chosen trivialization, which ends the proof. ☐

**Important Remark.** The condition

${d}^{M}f=0$ is a

**local** condition, checked in an (adequate) system of trivializations of the principal bundle, because it has to be checked on the vector bundle

$Ad{\left(P\right)}^{\times k}.$ This is in particular the case when we can find a 0-curvature connection

θ on

P such that:

In that case, since the structure group

G is regular, we can find a system of local trivializations of

P defined by

θ and such that, on any local trivialization,

${\nabla}^{\theta}={d}^{M}$ (see e.g., [

23,

40] for the technical tools that are necessary for this).

This technical remark can appear rather unsatisfactory first because it restricts the ability of application of the previous lemma, secondly because we need have a local (and rather unelegant) condition. This is why we give the following theorem, from Lemma 34.

**Theorem 35.** Let $f\in \mathfrak{Pol}\left(P\right)$ for which there exists $\theta \in \mathcal{C}\left(P\right)$ such that $[{\nabla}^{\theta},f]=0.$ We shall note this set of polynomials by ${\mathfrak{Pol}}_{reg}\left(P\right).$ Then, the map:takes values into closed forms on P. Moreover,- (i)
it is vanishing on vertical vectors and defines a closed form on M.

- (ii)
The cohomology class of this form does not depend on the choice of the chosen connexion θ on P.

Moreover, $\forall (\theta ,f)\in \mathcal{C}\left(P\right)\times {\mathfrak{Pol}}_{reg}\left(P\right),[{\nabla}^{\theta},f]=0.$

**Proof.** Let

$f\in {\mathfrak{Pol}}_{reg}\left(P\right)$ and let

$\theta \in \mathcal{C}\left(P\right)$ such that

$[{\nabla}^{\theta},f]=0.$ Let

${\theta}^{\prime}\in \mathcal{c}\left(P\right)$ and let

$\nu ={\theta}^{\prime}-\theta \in {\Omega}^{1}(M,\mathfrak{g}).$ Let

$({\alpha}_{1},...,{\alpha}_{k})\in {\left({\Omega}^{2}(M,\mathfrak{g})\right)}^{k}.$Then,

$\forall (\theta ,f)\in \mathcal{C}\left(P\right)\times {\mathfrak{Pol}}_{reg}\left(P\right),[{\nabla}^{\theta},f]=0.$ By the way,

$\forall {\theta}^{\prime}\in \mathcal{C}\left(P\right),$Applying this to

${\alpha}_{1}=...={\alpha}_{k}=F\left({\theta}^{\prime}\right),$ we get:

by Bianchi identity. Thus

$Ch(f,{\theta}^{\prime})$ is closed. Then, mimicking the end of the proof of Lemma 34, we get that the difference

$Ch(f,\theta )-Ch(f,{\theta}^{\prime})$ is an exact form, which ends the proof.

**Proposition 36.** Let $\varphi :{\mathfrak{g}}^{k}\to \mathbb{C}$ be a $k-$linear, symmetric, $Ad-$invariant form. Let $f:P\times {\mathfrak{g}}^{k}\to \mathbb{C}$ be the map induced by $\varphi $ by the formula: $f(x,g)=\varphi \left(g\right).$ Then $f\in {\mathfrak{Pol}}_{reg}.$

**Proof.** Obsiously, $f\in \mathfrak{Pol}.$ Let $\phi :U\times G\to P$ and ${\phi}^{\prime}:U\times G\to P$ be a local trivialisations of P, where U is an open subset of $M.$ Then there exists a smooth map $g:U\to G$ such that ${\phi}^{\prime}(x,{e}_{G})=\phi (x,{e}_{G}).g\left(x\right).$ Then we remark that ${\phi}^{*}f={\phi}^{\prime *}f$ is a constant map on horizontal slices since ϕ is Ad-invariant. Moreover, since ${\phi}^{*}f$ in a constant (polynomial-valued) map on $\phi (x,{e}_{G})$ we get that $[{\nabla}^{\theta},f]=0$ for the (flat) connection θ such that $T\phi (x,{e}_{G})$ spans the horizontal bundle over U. ☐

#### 6.2. Application to $Emb(M,N)$

Mimicking the approach of [

6], the cohomology classes of Chern-Weil forms should give rise to homotopy invariants. Applying Theorem 35, we get:

**Theorem 37.** The Chern-Weil forms $C{h}^{f}$ is a ${H}^{*}\left(B(M,N)\right)-$valued invariant of the homotopy class of an embedding, $\forall k\in {\mathbb{N}}^{*}.$

When

$M={S}^{1}$,

$Emb({S}^{1},N)$ is the space of (parametrized) smooth knots on

N, and

$B({S}^{1},N)$ is the space of non parametrized knots. Its connected components are the homotopy classes of the knots, through classical results of differential topology, see e.g., [

41]. We now apply the material of the previous section to manifolds of embeddings. For this, we can define invariant polynomials of the type of those obtained in [

6] (for mapping spaces) by a field of linear functionnal

λ with “good properties” that ensures that:

This approach is a straightforward generalization of the description of Chern-Weil forms on finite dimensional principal bundles where polynomials are generated by functionnals of the type $A\mapsto \text{tr}\left({A}^{k}\right)$ (tr is the classical trace) but as we guess that we can consider other classes of polynomials for spaces of embeddings. In this paper, let us describe how to replace the classical trace of matrices tr by a renormalized trace ${\text{tr}}^{Q}.$ In the most general case, it is not so easy to define a family of weights $f\in Emb(M,N)\mapsto {Q}_{f}$ which satisfy the good properties. Indeed, we have two examples of constructions which match the necessary assumptions for ${\mathfrak{Pol}}_{reg}$ when $M={S}^{1}$, and the first one is derived from the following example:

#### Knot Invariant Through Kontsevich and Vishik Trace

The Kontsevich and Vishik trace is a renormalized trace for which

$t{r}^{Q}\left([A,B]\right)=0$ for each differential operator

$A,B$ and does not depend on the weight chosen in the odd class. For example, one can choose

$Q=Id+{\nabla}^{*}\nabla $, where ∇ is a connection induced on

${\mathcal{N}}_{f}$ by the Riemannian metric, as described in [

6]. It is an order 2 injective elliptic differential operator (in the odd class), and the coadjoint action of

$Aut\left({\mathcal{N}}_{f}\right)$ will give rise to another order 2 injective elliptic differential operator [

7]. When

$Q=Id+{\nabla}^{*}\nabla ,$ this only changes ∇ into another connection on

$E.$ Thus, setting:

we have:

Let us now consider a connected component of

$B(M,N),$ i.e., a homotopy class of an embedding among the space of embeddings. We apply now the construction to

$M={S}^{1}.$ The polynomial:

is

$Diff\left({S}^{1}\right)-$invariant, and gives rise to an invariant of non oriented knots,

i.e., a Chern form on the base manifold:

by theorem 37. This approach can be extended to invariant of embeddings, replacing

${S}^{1}$ by another odd-dimensional manifold.