state, which explains more clearly the rule $a_{i,j}=a_{j,i}^{-1}.$ A pairwise
comparisons matrix is called consistent if and only if
$\forall(i,j,k),a_{i,j}a_{j,k}=a_{i,k},$ and one of the goals of applications
is more to minimize, in some sense, inconsistency, than to obtain strict
consistency. In order to evaluate inconsistency, one uses inconsistency
indicators, which are mostly Saaty’s [Saa1977] and Koczkodaj’s [Ko1993].

The link that we now describe in more details is new, after the preliminary
work [Y1999] that links gauge theories with foreign exchange markets.

###  5.1. A new discretization scheme for gauge theories based on [13]

The present approach of discretization is based on the following paradigm: the
convergence of the discretization for a fixed connection must be preserved, as
well as gauge covariance of the curvature in the discretization scheme.

Let $\pi:P\rightarrow M$ be a principal bundle of connected Riemannian base,
with structure group $G,$ equipped with a prescribed triangulation or
cubification $\tau.$ The canonical maps induced by $\pi$ on relevant objects
will be also noted by $\pi$ in the sequel when it carries no ambiguity. The
nodes of this triangulation or cubification are assumed indexed by
$\mathbb{N},$ noted by $(s_{n})_{\mathbb{N}}$ (the manifold $M$ can be non
compact). Recall that, for a fixed index $i_{1},...i_{n},$
$St(s_{i_{1}},...s_{i_{n}})$ is the domain described by the simplexes or the
cubes with nodes $s_{i}.$ We note by $\mathcal{C}$ the space of connections on
$P.$ Let $\theta\in\mathcal{C}.$ Fixing $s_{0}$ as a basepoint and
$p_{0}\in\pi^{-1}(s_{0}),$

  1. (1)

Let

$j=\min\left\\{i\in\mathbb{N}^{*}|s_{i}\in St(s_{0})\right\\}.$

We define $g_{0,j}=1$ and $p_{j}$ the endpoint of the horizontal path over
$[s_{0},s_{j}]$ with starting point $p_{0}.$ Let $I_{2}=\\{0;j\\}.$

  2. (2)

Assume that $I_{n}$ exists, and that,

     * •

$\forall i\in I_{n},$ we have constructed $p_{i}\in\pi^{-1}(s_{i})$ and

     * •

$\forall(i,j)\in I_{n}^{2},$ with $i<j$, $g_{i,j}$ is the holonomy of
$[s_{i},s_{j}]$, starting at $p_{i},$ i.e. $p_{j}.g_{i,j}$ is the endpoint of
the horizontal path over $[s_{i},s_{j}]$ with starting point $p_{i}.$

Let

$j=\min\left\\{i\in\mathbb{N}-I_{n}|s_{i}\in St(s_{k};k\in I_{n})\right\\}.$

We define

     * •

$k=\min\\{i\in I_{n}|s_{i}\in St(s_{j})\\}$ and let $p_{j}$ the endpoint of
the horizontal path over $[s_{i},s_{j}]$ starting at $p_{i}.$

     * •

for $i\in I_{n},$ $g_{i,j}$ is defined such as $p_{j}.g_{i,j}$ is the endpoint
of $[s_{i},s_{j}]$ starting at $p_{i}.$

     * •

$I_{n+1}=I_{n}\cup\\{j\\}.$

The discretization thus describes the holonomy of the connection along the
1-vertices. We have a first sequence $(p_{n})_{\mathbb{N}}$ which stands as a
slice of the pull-back $\left((s_{n})_{\mathbb{N}}\right)^{*}P$ and if
$\mathcal{K}_{1}$ is the 1-skeleton of $\tau,$ the family $(g_{i,j})_{i<j},$
expresses holonomy elements of the connection $\theta$ on the vertices of
$\mathcal{K}_{1}.$ The holonomy of a smooth path $\gamma,$ for a fixed
connection, can be approximated by the discretized holonomies computed along a
piecewise smooth path along the vertices of the triangulation, close enough to
$\gamma.$ In the sequel, we note by $|\sigma|$ the length (resp.
$n-$dimensinal the Haussdorf volume) of the 1-vertex (resp. the $n-$simplex or
$n-$cube) $\sigma.$ Comparing to Whitney’s discretization [Wh1957], this
scheme does not depend on any exterior trivialization of the principal bundle
$P.$ In the sequel, we work with triangulations but the same can be done for
cubifications. These discretizations are used to define the new discrete
analogs of connections and curvature. Let $(\tau_{n})$ be a sequence of
triangulations of $M$ such that $\tau_{n}\subset\tau_{n+1}$
(subtriangulations) and such that the length of 1-vertices converge uniformly
to $0.$ Then, for fixed $s\in\tau_{n}\subset...\subset\tau_{n+p}\subset...,$
we have the following

  * •

Let $X$ be a germ of a path $\gamma:t\rightarrow P$ on $P$ at the parameter
$t=0$ such that $\pi(\gamma)=\sigma_{n}$ is a 1-vertex of $\tau_{n}$ for $n$
large enough, and we assume with no loss of generality in the sequel that
$\gamma$ is parametrized by arc-length of $\tau_{n}.$ Then
$\pi(\gamma_{[0;|\sigma_{n+p}|]})=\sigma_{n+p},$ for $p\in\mathbb{N},$ and

$\theta(X)=\lim_{p\rightarrow+\infty}\frac{g(p)-I}{|\sigma_{p}|}$

where $g_{p}\in G$ is defined by
$\gamma(|\sigma_{n+p}|)=H_{\gamma(0)}\sigma_{n+p}(|\sigma_{n+p}|).g_{p}\in P.$

  * •

Let $X,Y$ be germs of paths $\gamma_{0},\gamma_{1}:t\rightarrow P$ on $P$ at
the parameter $t=0,$ with $\gamma(0)=\gamma^{\prime}(0),$ such that there
exists a 2-simplex $\sigma_{n}\in\tau_{n}$ for $n$ large enough, where
$\gamma$ and $\gamma^{\prime}$ project on 1-simplexes of $\partial\sigma_{n},$
and we also assume arc-length parametrisation as in the previous item. Then
$\partial\sigma_{n+p}$ is a piecewise smooth loop parametrized by arc-length,
staring from $\pi(\gamma(0))$ along $\pi(\gamma),$ and ending along
$\pi(\gamma^{\prime}).$ Then we have

$\Omega(X,Y)=C\lim_{p\rightarrow+\infty}\frac{Hol_{\gamma(0)}\partial\sigma_{n+p}(|\partial\sigma_{n+p}|)-I}{|\sigma_{p}|},$

where $C$ is a constant.

  * •

These two items correspond heuristically to a directional derivative of the
holonomy at the continuum limit.

Open problems: in the proposed approach, the “natural” continuum limit of the
quantized model does not involve cylindrical functions on a vector space of
connection forms equipped with a Lebesgue measure, but cylindrical functions
defined on products of unimodular groups aquipped with their Haar measure.
Even if there exists an obvious conceptual link between Haar measure and
Lebesgue measure, the two cylindrical approximations that are produced are not
a priori equivalent. We conclude this presentation conjecturing that the gauge
anomalies which can appear in classical discretized models may find an
expression in this measure defect.

###  5.2. Pairwise comparisons in decision making as a discrete Yang-Mills
theory, based on [9, 11, 14]

####  5.2.1. Changing the comparisons structure to arbitrary groups

Classicaly, the comparisons coefficients are $a_{i,j}$ are scaling
coefficients. This means that,if the PC matrix $A$ is consistent, given a
state $s_{k},$ we can recover all the other states $s_{j}$ by something
assimilated to scalar mutiplication:

$s_{j}=a_{j,k}s_{k}.$

In other words, even if the states $s_{j}$ are driven by more complex rules,
we reduce them to a “score” or an “evaluation” in $\mathbb{R}_{+}^{*}.$ The
states $s_{j}$ have to belong to a more complex state space $S,$ and in order
to have pairwise comparisons, a straightforard study shows that we define
matrices with coefficients in a group [14]. Let $I$ be a set of indexes and
let $(k,+,.,|.|)$be a field with absolute value and $V_{k}$ a normed
$k-$vector space.

######  Definition 5.1.

[11] Let $(G,.)$ be a group. A pairwise comparisons matrix is a matrix

$A=(a_{i,j})_{(i,j)\in I^{2}}$

such that

  1. (1)

$\forall(i,j)\in I^{2},a_{i,j}\in G.$

  2. (2)

$\forall(i,j)\in I^{2},\quad a_{j,i}=a_{i,j}^{-1}.$

  3. (3)

$a_{i,i}=1_{G}.$

We note by $PC_{I}(G)$ the set of pairwise comparisons matrices indexed by $I$
and with coefficients in $G.$ When $G$ is not abelian, there are two notions
of inconsistency covariantly or contravariantly consistent, which corresponds
merely to left or right actions. The two notions are dual [14]. Contravariant
consistency appears in the geometric realization of $PC_{I}(G)$ via the
holonomy of a connection on a simplex $\Delta:$

######  Proposition 5.2.

[11] If $G$ is exponential, the map

$\displaystyle\Omega^{1}(\Delta,\mathfrak{g})$ $\displaystyle\rightarrow$
$\displaystyle\\{\hbox{ PC matrices }\\}$ $\displaystyle\theta$
$\displaystyle\mapsto$ the holonomy matrix

is onto.

We note by $CPC_{I}(G)$ the set of consistent PC-matrices.

######  Definition 5.3.

A (non normalized, non covariant) inconsistency map is a map

$ii:PC_{I}(G)\rightarrow V_{k}$

such that $ii(A)=0$ if $A$ is consistent. Moreover, we say that $ii$ is
faithful if $ii(A)=0$ implies that $A$ is consistent.

In [9], we give arguments to consider only normalized inconsistency maps
called inconsistency indicators.

The main feature in applying this setting will be twofold,and these two points
are far to be systematically solved with the present work:

\- define a comparisons group $G$ for which we can get at least one comparison
coefficient $a_{i,j}$ between two states $s_{i}$ and $s_{j}$ (which means that
the $G-$action needs to be transitive),

\- evaluate (and compute!) inconsistency, if possible generalizing the
$\mathbb{R}_{+}^{*}$-setting, in a proper way to get safe decision making.
This second point is linked with multiscale analysis.

Examples from [14] highlight these features, such as when $G$ is a matrix
group $GL_{n}$ and when $G$ is an affine group. In this second example for
$G,$ we find applications to modelization of perspective and error in tunnel
building.

####  5.2.2. Algebraic properties, gauge group and generalization to graphs

The whole section is based on the work [14]

######  Proposition 5.4.

Any morphism of group $a:G\rightarrow G^{\prime}$ extends to a map
$\bar{a}:PC_{I}(G)\rightarrow PC_{I}(G^{\prime})$ by action on the
coefficients, and:

  * •

If $A\in PC_{I}(G)$ is consistent, then $\bar{a}(A)\in PC_{I}(G^{\prime})$is
consistent.

  * •

If $Ker(a)=\\{e_{G}\\},$ then $A\in PC_{I}(G)$ is consistent, if and only if
$\bar{a}(A)\in PC_{I}(G^{\prime})$

We call $G^{I}$ the gauge group of $G$. Then we get the following actions:

  * •

a left action $L:G^{I}\times PC_{I}(G)\rightarrow PC_{I}(G)$ defined, for
$(g_{i})_{I}\in G^{I}$ and $(a_{i,j})_{I^{2}}\in PC_{I}(G)$ by

$L_{(g_{i})_{I}}\left((a_{i,j})_{I^{2}}\right)=(b_{i,j})_{I^{2}}$

with

$b_{i,j}=\left\\{\begin{array}[]{ccl}1&\hbox{ if }&i=j\\\ g_{i}a_{i,j}&\hbox{
if }&i<j\\\ a_{i,j}g_{j}^{-1}&\hbox{ if }&i>j\\\ \end{array}\right.$

  * •

a right action $R:PC_{I}(G)\times G^{I}\rightarrow PC_{I}(G)$ defined, for
$(g_{i})_{I}\in G^{I}$ and $(a_{i,j})_{I^{2}}\in PC_{I}(G)$ by

$R_{(g_{i})_{I}}\left((a_{i,j})_{I^{2}}\right)=(b_{i,j})_{I^{2}}$

with

$b_{i,j}=\left\\{\begin{array}[]{ccl}1&\hbox{ if }&i=j\\\ a_{i,j}g_{j}&\hbox{
if }&i<j\\\ g_{i}^{-1}a_{i,j}&\hbox{ if }&i>j\\\ \end{array}\right.$

  * •

an adjoint action

$Ad_{(g_{i})_{I}}=L_{(g_{i})_{I}}\circ
R_{(g_{i})_{I}^{-1}}=R_{(g_{i})_{I}^{-1}}\circ L_{(g_{i})_{I}}$

  * •

a coadjoint action

$\left((a_{i,j})_{I^{2}},(g_{i})_{I}\right)\mapsto Ad_{(g_{i})_{I}^{-1}}.$

######  Theorem 5.5.

Consistent PC-matrices are the orbits of the PC-matrix $(1)_{I^{2}}$ with
respect to the adjoint action.

After remarking that $L$ and $R$ are effective actions, we have the following:

######  Theorem 5.6.

If $I=\mathbb{N}_{3},$ any orbit for the left action intersects $CPC_{3}(G).$
If $card(I)>3,$ there exists orbits for the left action which do not intersect
$CPC_{I}(G).$

Let us now turn to other properties inconsistency maps.

######  Definition 5.7.

Let $ii$ be an inconsistency map. It is called:

  * •

normalized if $\forall A\in PC_{I}(G),||ii(A)||\leq 1.$

  * •

Ad-invariant if $\forall A\in PC_{I}(G),\forall g\in
G^{I},ii\left(Ad_{g}(A)\right)=ii(A)$

  * •

norm invariant if $||ii(.)||$is Ad-invariant.

According to [KoSzy2015], we give now the following definition:

######  Definition 5.8.

An inconsistency indicator $ii$ on $PC_{I}(G)$ is a faithful, normalized
inconsistency map with values in $\mathbb{R}_{+}$ such that there exists an
inconsistency map $ii_{3}$ on $PC_{3}(G)$ that defines $ii$ by the following
formula

$ii(A)=\sup\left\\{ii_{3}(B)\ B\subset A;\,\,\,B\in PC_{3}(G)\right\\}.$

We remark here that since $ii$ is faithful, it is in particular (trivially)
$Ad-$invariant on $CPC_{I}(G),$ but we do not require it to be $Ad-$invariant.
Moreover, with such a definition, to show that $ii$ is $Ad-$invariant, it is
sufficient to show that $ii_{3}$ is $Ad-$invariant. However, we give the
example driven by Koczkodaj’s approach. This is already proved that $Kii_{3}$
generates an inconsistency indicator [KoSzy2015] and we complete this result
by the following property:

######  Proposition 5.9.

Let $n\geq 3.$ Koczkodaj’s inconsistency maps $Kii_{3}$ and $Kii_{n}$ generate
is $Ad-$invariant inconsistency maps on $PC_{n}(\mathbb{R}_{+}^{*}).$

####  5.2.3. Geometric aspects

We already mentionned the correspondence with holonomy. This correspondence is
exaclty the one described in section 5.1.

######  Theorem 5.10.

[14] If $G$ is a compact exponential finite dimensional Lie group, then for
each $(\tilde{g}_{i})_{i\in I}\in G^{I},$ there exists $g\in
C^{\infty}(\Delta,G)$ such that $g(s_{i})=g_{i}.$ Moreover, for any
contravariant PC matrix $A$ which is the holonomy matrix of a connection
$\theta,$ then $Ad_{(\tilde{g}_{i})_{i\in I}}(A)$ is the holonomy matrix of
the connection $g^{-1}dg+Ad_{g^{-1}}\theta.$

The gauge group also appears as a discretization of the classical gauge group
of a trivial principal bundle. We consider in this section a family of states
$(s_{i})_{I}$ such that any $s_{i}$ cannot be a priori compared directly with
any other $s_{j}.$ This leads us to consider a graph $\Gamma_{I}$ linking the
elements which can be compared. For example, in the previous sections,
$\Gamma_{I}$ was the $1-$skeleton of the simplex. For simplicity, we assume
that $\Gamma_{I}$ is a connected graph, and that at most one vertex connects
any two states $s_{i}$ and $s_{j}.$ We note this (oriented) vertex by
$<s_{i},s_{j}>,$ and the comparison coefficient by $a_{i,j}.$ By the way, we
get a pairwise comparisons matrix $A$ indexed by $I$ with “holes” (with
virtual $0-$coefficient) when a vertex does not exist, and for which
$a_{j,i}^{-1}=a_{i,j}.$

Let us summarize the main correspondences that we have highlighted:

discrete Yang-Mills formalism | Pairwise Comparisons (PC)
---|---
connection | PC matrix
flat connection | consistent PC matrix
curvature = loop holonomy | inconsistency
classical Yang-Mills functional | quadratic average of inconsistency
| on triads
“sup” Yang-Mills functional | Koczkodaj’s inconsistency indicator

##### Hierarchyless comparisons, “hearsay” evaluation and holonomy on a graph
[14]

In this model, the comparison between two states $s_{i}$ and $s_{j}$ can be
performed by any path between $s_{i}$ and $s_{j}$ of any length. This model
modelize the propagation of rumours, where validation of information is based
on hearsay results. With this approach, the capacity of propagation of an
evaluation is not controlled. We note by

$<s_{i_{1}},...,s_{i_{k}}>=<s_{i_{1}},s_{i_{2}}>\vee...\vee<s_{i_{k-1}},s_{i_{k}}>$

the composition of paths along vertices. By analogy with the holonomy of a
connection, we define:

######  Definition 5.11.

Let $s=s_{i}$ and $s^{\prime}=s_{j}$ be two states and let

$\mathcal{H}_{s,s^{\prime}}=\left\\{a_{i,i_{2}}...a_{i_{k-1},i}|<s,s_{i_{2}},...,s_{i_{k-1}},s^{\prime}>\hbox{
is a path from }s\hbox{ to }s^{\prime}\right\\}.$

We note by $\mathcal{H}_{s}$ the set $\mathcal{H}_{s,s}.$

By the way, we get the following properties, usual for classical holonomy and
with easy proof:

######  Proposition 5.12.

  1. (1)

Let $s$ be a state, then $\mathcal{H}_{s}$ is a subgroup of $G.$ We call it
holonomy group at $s.$

  2. (2)

Let $s$ and $s^{\prime}$ be two states. Then $\mathcal{H}_{s}$ and
$\mathcal{H}_{s^{\prime}}$ are conjugate subgroups of $G.$

  3. (3)

$\mathcal{H}_{s_{i},s_{j}}=a_{i,j}.\mathcal{H}_{s_{j}}=\mathcal{H}_{s_{i}}.a_{i,j}.$

######  Definition 5.13.

The PC matrix $A$ on the graph $\Gamma_{I}$ is consistent if and only if there
exists a state $s$ such that $\mathcal{H}_{s}=\\{1\\}.$

##### Ranking the trustworthiness of indirect comparisons

The main problem with hierarchiless comparisons of two states $s$ and
$s^{\prime}$ is that paths of any length give comparison coefficients which
cannot be distinguished. An indirect comparison, given by a path with 3
vertices, has the same status as a comparison involving a path with 100
vertices. This is why we need to introduce a grading on $H_{s,s^{\prime}}$
called order. This terminology will be justified by the propositions
thereafter.

######  Definition 5.14.

Let $s$ and $s^{\prime}$ be two states.

  * •

Let $\gamma$ be a path on $\Gamma_{I}$ from $s$ to $s^{\prime}$. The length of
$\gamma,$ noted by $l(\gamma),$ is the number of vertices of $\gamma,$ and by
$H(\gamma)$ its holonomy.

  * •

Let $h\in\mathcal{H}_{s,s^{\prime}}.$ The order of $h$ is defined as

$ord(h)=\min\left\\{l(\gamma)\left|H(\gamma)=h\right.\right\\}.$

As a trivial consequence of the triangular equality, and as a justification of
the terminology, we have:

######  Proposition 5.15.

Let $s,s^{\prime}$ a,d $s^{\prime\prime}$ three states. Let
$(h,h^{\prime})\in\mathcal{H}_{s,s^{\prime}}\times\mathcal{H}_{s^{\prime},s^{\prime\prime}}.$
Then

$ord(hh^{\prime})\leq ord(h)+ord(h^{\prime}).$

Left action, right action and adjoint action of $G^{I}$ extend straightway to
PC-matices on $\Gamma_{I}$ setting

$\forall g\in G,\quad g.0=0.g=0.$

Adapting the proof of Theorem 5.5 we get:

######  Theorem 5.16.

Let $A=(a_{i,j})_{(i,j)\in I^{2}}$ be a PC matrix on $\Gamma$ Then $A$ is
consistent if and only if there exists $(\lambda_{i})\in G^{I}$ such that

$a_{i,j}=\lambda^{-1}_{i}\lambda_{j}$

when $a_{i,j}\neq 0.$

##### Inconsistency maps ranked by trustworthiness [14]

Let $A$ be a PC matrix on $\Gamma_{I}.$ Inconsistency will be given here by
the holonomy of a loop. Let us recall that a trivial holonomy of a loop
$<s_{i_{1}},s_{i_{2}},...,s_{i_{k}},s_{i_{1}}>$ implies that

$a_{i_{1},i_{k}}\left(a_{i_{1},i_{2}}...a_{i_{k-1},i_{k}}\right)^{-1}=1.$

The principle of ranking inconsistency with loop lengthgives the following:

######  Definition 5.17.

Let $\mathcal{F}:G\rightarrow\mathbb{R}_{+}$ be a map such that $I(1)=0.$ Let
$s$ be a basepoint on $\Gamma_{I}.$ The ranked Koczkodaj’s inconsistency map
associated to $\mathcal{F}$ the map

$Kii_{\mathbb{N}}=\sum_{n\in\mathbb{N}}a_{n}X^{n}$

where

$a_{n}=\sup\left\\{\mathcal{F}\left(H(\gamma)\right)\left|\gamma\hbox{ is a
loop at }s\hbox{ and }\right.l(\gamma)=n\right\\}.$

One can easily see that $a_{n}$ generalize $Kii_{n},$ and $Kii_{\mathbb{N}}$
is a $\mathbb{R}[[X]]-$valued inconsistency map.

##### Holonomy versus distance

This section is based on [14]. In this section, $G=\mathbb{R}_{+}^{*}.$
Setting

$k_{i,j}=|log(a_{i,j})|$

we get another matrix, that we define as the distance matrix

$K=(k_{i,j})_{(i,j)\in I^{2}}.$

Notice that, if the coefficients of this matrix satisfy the triangle
inequality $\forall(i,j,l)\in I^{3},k_{i,l}\leq k_{i,j}+k_{j,l},$ we get a
curvature matrix for metric spaces [Gr1997]. Due to the absolute value, we
have the following:

######  Proposition 5.18.

Let $K$ be a non zero distance matrix on $\Delta_{n}.$ Let $N$ be the number
of non zero coefficients in $K.$ Then $N$ is even and there exist $2^{N/2}$
corresponding PC matrices.

Therefore, we have the following results:

######  Proposition 5.19.

Let $K$ be the distance matrix on $\Delta_{n}$ associated to a consistent PC
matrix $A,$ which is assumed to be non zero. Let $N^{\prime}$ be the number of
coefficients $k_{i,i+1}$ which are non zero. Then there exists
$2^{N^{\prime}}$ consistent PC matrices built with the coefficients
$k_{i,i+1},$ but only $2$ consistent ones, $A$ and its transposition.

Open problem: The basic evaluation of inconsistency in pairwise comparisons
[KoSza2010] seems to be less efficient than holonomy evaluation. A common
technical feature between the two approaches stems in potential analysis and
Lagrangian theory. Such an approach still has to be developped, in order for
example to include this part of decision theory as a part of quantum physics.

Back to inconsistency, the work led by physicists on discretized QFT may then
apply to information theory. But we highlight the following features:

  * •

What would be the meaning in information theory of the partition function? of
the continuum limit?

  * •

What other geometric quantities than holonomy could be adapted for information
theory?

  * •

Can there appear other ”good” inconsistency indicators from physics? Can
Feynman integration give an ”ideal” approach to many-criteria decision?

  * •

Can physics apply directly to image processing through inconsistency? to
faces/symbols recognition? to shape analysis?

##  Chapter 6 Pseudo-differential, Fourier integral operators with
applications to geometry

We now deal with non-formal pseudo-differential operators and Fourier integral
operators. These operators have also applications and we focuse on some
applications to the geometry of infinite dimensional manifolds.

###  6.1. On $\zeta-$renormalized traces

$E$ is equipped this an Hermitian products $<.,.>$, which induces the
following $L^{2}$-inner product on sections of $E$:

$\forall u,v\in
C^{\infty}(S^{1},E),\quad(u,v)_{L^{2}}=\int_{S^{1}}<u(x),v(x)>dx,$

where $dx$ is the Riemannian volume.

######  Definition 6.1.

[Pay2012, Sco2010] $Q$ is a weight of order $s>0$ on $E$ if and only if $Q$ is
a classical, elliptic, admissible pseudo-differential operator acting on
smooth sections of $E$, with an admissible spectrum.

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 $Q$ [See1967],
see e.g. [21, Le1998, Asa2000, Pay2001, Sco2010]. The powers $Q^{-s}$ of the
weight $Q$ are defined for $Re(s)>0$ using with a contour integral,

$Q^{-s}=\int_{\Gamma}\lambda^{s}(Q-\lambda Id)^{-1}d\lambda,$

where $\Gamma$ is an “angular” contour around the spectrum of $Q.$ Let $A$ be
a log-polyhomogeneous pseudo-differential operator. The map
$\zeta(A,Q,s)=s\in\mathbb{C}\mapsto\hbox{tr}\left(AQ^{-s}\right)\in\mathbb{C}$
, defined for $Re(s)$ large, extends on ${\mathbb{C}}$ to a meromorphic
function with a pole at $0$ ([Le1998]). When $A$ is classical, $\zeta(A,Q,.)$
has a simple pole at $0$ with residue ${1\over q}\hbox{\rm res}A$, where res
is the Wodzicki residue ([Wod1984], see also [Ka1989]). Notice that the
Wodzicki residue mimicks the Adler trace [Ad1979] on formal symbols.

######  Definition 6.2.

$tr^{Q}A=lim_{z\rightarrow 0}(\hbox{tr}(AQ^{-z})-{1\over qz}resA)$.

If $A$ is trace class, $\hbox{tr}^{Q}{(A)}=\hbox{tr}{(A)}$. The functional
$\hbox{tr}^{Q}$ is of course not a trace on $Cl(M,E)$. Notice also that, if
$A$ and $Q$ are pseudo-differential operators acting on sections on a real
vector bundle $E$, they also act on $E\otimes{\mathbb{C}}$. Before giving our
new developments, we need the following statements which are not so well
applied in various contexts

$\bullet$ Development presented in the PhD thesis and published in [21, 23],
and hence not presented for as new results for the habilitation:

The Wodzicki residue res and the renormalized traces $\hbox{tr}^{Q}$ have to
be understood as functional defined on pseudo-differential operators acting on
$E\otimes\mathcal{C}$. In order to compute $\hbox{\rm tr}^{Q}[A,B]$ and to
differentiate $\hbox{\rm tr}^{Q}A$, in the topology of classical pseudo-
differential operators, we need the following ([21] and references therein):

######  Proposition 6.3.

[21]

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

(6.1) $\hbox{\rm tr}^{Q}[A,B]=-{1\over q}\hbox{\rm res}(A[B,\log Q]).$

(ii) Given a differentiable family $A_{t}$ of pseudo-differential operators,
given a differentiable family $Q_{t}$ of weights of constant order q,

(6.2) ${d\over dt}\left(tr^{Q_{t}}A_{t}\right)=tr^{Q_{t}}\left({d\over
dt}A_{t}\right)-{1\over q}\hbox{\rm res}\left(A_{t}({d\over dt}\log
Q_{t})\right).$

The following ”covariance” property of $\hbox{tr}^{Q}$ ([21], [Pay2001]) will
be useful to define renormalized traces on bundles of operators,

######  Proposition 6.4.

Under the previous notations, if C is a classical elliptic injective operator
of order 0, $tr^{C^{-1}QC}\left(C^{-1}AC\right)$ is well-defined and equals
$\hbox{\rm tr}^{Q}A$.

We moreover have specific properties for weighted traces of a more restricted
class of pseudo-differential operators (see [KV1994],[KV1995],[21]), called
odd class pseudo-differential operators following [KV1994, KV1995] :

######  Definition 6.5.

A classical pseudo-differential operator $A$ is called odd class if and only
if

$\forall n\in\mathbb{Z},\forall(x,\xi)\in
T^{*}M,\sigma_{n}(A)(x,-\xi)=(-1)^{n}\sigma_{n}(A)(x,\xi).$

We note this class $Cl_{odd}.$

Such a definition is consistent for pseudo-differential operators on smooth
sections of vector bundles, and applying the local formula for Wodzicki
residue, one can prove:

######  Proposition 6.6.

[21]

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 class
pseudo-differential operators, $\hbox{\rm tr}^{Q}\left([A,B]\right)=0$ and
$\hbox{\rm tr}^{Q}A$ does not depend on $Q.$

This trace was first defined in the papers [KV1994] and [KV1995] by
Kontesevich and Vishik. We remark that it is in particular a trace on
$DO(M,E)$ when $M$ is odd-dimensional.

Let us now describe a class of operators which is, in some sense,
complementary to odd class:

######  Definition 6.7.

[23]

A classical pseudo-differential operator $A$ is called even class if and only
if

$\forall n\in\mathbb{Z},\forall(x,\xi)\in
T^{*}M,\sigma_{n}(A)(x,-\xi)=(-1)^{n+1}\sigma_{n}(A)(x,\xi).$

We note this class $Cl_{even}.$

We precised that $Cl_{even}$ and $Cl_{odd}$ are ”in some sense” complementary
because these are not supplementary vector spaces: $Cl_{odd}\cap
Cl_{even}=Cl^{-\infty}.$

######  Proposition 6.8.

[23]

$Cl_{even}\circ Cl_{odd}=Cl_{odd}\circ Cl_{even}=Cl_{even}$ and

$Cl_{even}\circ Cl_{even}=Cl_{odd}\circ Cl_{odd}=Cl_{odd}.$

Now, following [23], we explore properties of $\hbox{\rm tr}^{Q}$ on Lie
brackets.

######  Definition 6.9.

Let E be a vector bundle over M, Q a weight and $a\in\mathbb{Z}$. We define :

$\mathcal{A}^{Q}_{a}=\\{B\in Cl(M,E);[B,\log Q]\in Cl^{a}(M,E)\\}.$

######  Theorem 6.10.

[23]

(i) $\mathcal{A}^{Q}_{a}\cap Cl^{0}(M,E)$ is an subalgebra of $Cl(M,E)$ with
unit.

(ii) Let $B\in Ell^{*}(M,E)$, $B^{-1}\mathcal{A}^{Q}_{a}B=A^{B^{-1}QB}_{a}.$

(iii) Let $A\in Cl^{b}(M,E)$, and $B\in\mathcal{A}^{Q}_{-dimM-b-1}$, then
$\hbox{\rm tr}^{Q}[A,B]=0.$

(iv) For $a<-{dimM\over 2}$, $\mathcal{A}^{Q}_{a}\cap Cl^{-dimM\over 2}(M,E)$
is an algebra on which the renormalized trace is a trace (i.e. vanishes on the
brackets).

We now produce non trivial examples of operators that are in
$\mathcal{A}^{Q}_{a}$ when Q is scalar, and secondly we give a formula for
some non vanishing renormalized traces of a bracket.

######  Proposition 6.11.

[23]

Let Q be a scalar weight on $C^{\infty}_{0}(M,V)$. Then

$Cl^{a+1}(M,V)\subset\mathcal{A}^{Q}_{a}.$

Consequently,

  1. (1)

let B be a classical pseudo-differential operator of order $b$. Then $[B,\log
Q]$ is a classical pseudo-differential operator of order $b-1$.

  2. (2)

if $ord(A)+ord(B)=-dimM,$ $\hbox{\rm tr}^{Q}[A,B]=0$.

  3. (3)

when $M=S^{1}$, if A and B are classical pseudo-differential operators, if A
is compact and B is of order 0, $\hbox{\rm tr}^{Q}[A,B]=0$.

  4. (4)

Let Q be a scalar weight on $C^{\infty}_{0}(M,V)$, and A, B two pseudo-
differential operators of orders a and b on $C^{\infty}_{0}(M,V)$, such that
$a+b=-m+1$ (m = dim M). Then

$\hbox{\rm tr}^{Q}[A,B]=-{1\over q}\hbox{\rm res}\left(A[B,\log
Q]\right)=-{1\over
q(2\pi)^{n}}\int_{M}\int_{|\xi|=1}tr(\sigma_{a}(A)\sigma_{b-1}([B,\log Q])).$

Let $Q$ be a weight, the non-traciality of $\hbox{\rm tr}^{Q}$ defined a non-
vanishing cocycle (which is a coboundary in Hochschild cohomology) express as:

$(A,B)\in(Cl(M,E))^{2}\mapsto c^{Q}(A,B)=tr^{Q}[A,B],$

whiich is proportionnal to $res(A[B,logQ]).$ This cocycle is a generalization
to non formal pseudo-differential operators of the Kravchenko-Khesin-Radul
cocycle

$(a,b)\in(\mathcal{F}Cl(S^{1},\mathbb{C}))^{2}\mapsto
c_{KKR}(a,b)=\int_{S^{1}}\left(\sigma_{-1}\left(a[b,log\xi]\right)\right)_{\xi=1}dx.$

We notice that $c_{KKR}$ is not a coboundary [KK1991]. The two other cocycles
that I would like to mention are due to a splitting: we need for this a sign
operator $\epsilon,$ i.e. an operator such that $\epsilon^{2}=Id,$ see e.g.
[Mick1989]. The classical case is when $E$ is a Clifford bundle over $M$, and
$\epsilon$ is the sign of the Dirac operator [Mick1994], generalizing the
classical $M=S^{1}$ see e.g. [PS1988]. These works were motivated by [RSF1985,
SW1985, MR1988, Mick1988]. The splitting of $L^{2}(M,E)$ is given by the two
eigenvalues $-1$ and $1$ of $\epsilon$ (the kernel of the Dirac operator is
arbitrarily associated to the $-1$ or $1$ eigenspace). Let us note by $H_{-}$
and $H_{+}$, respectively, these two eigenspaces. The operators $A$ considered
split blockwise

$A=\left(\begin{array}[]{cc}A_{++}&A_{-+}\\\ A_{+-}&A_{--}\end{array}\right)$

under the decomposition $L^{2}(M,E)=H_{+}\oplus H_{-}=H.$ Defining the Banach
Lie group

$\displaystyle GL_{res}$ $\displaystyle=$ $\displaystyle\left\\{A\in
GL(H)|[\epsilon,A]\hbox{ is Hilbert-Schmidt}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{A\in GL(H)|A_{+-}\hbox{ and }A_{-+}\hbox{ are Hilbert-
Schmidt}\right\\},$

with Lie algebra

$\displaystyle L_{res}$ $\displaystyle=$ $\displaystyle\left\\{A\in
L(H)|[\epsilon,A]\hbox{ is Hilbert-Schmidt}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{A\in L(H)|A_{+-}\hbox{ and }A_{-+}\hbox{ are Hilbert-
Schmidt}\right\\}$

one can define [SW1985, PS1988, Mick1989], for $(A,B)\in(L_{res})^{2}$:

$\lambda(A,B)=\hbox{\rm tr}\left([A_{++},B_{++}]-[A,B]_{++}\right)$

and the Schwinger cocycle [Sch1959]:

$c_{S}(A,B)=\hbox{\rm tr}\left(\epsilon[\epsilon,A][\epsilon,B]\right).$

These two cocycles are proportional and non trivial, and they naturally
restrict to $L_{res}\cap Cl(M,E)$ [21].

####  6.1.1. A property, based on [6]

Let us now explore the action of $Diff(M)$ and of $Aut(E)$ on $tr^{Q}(A).$For
this, we get:

######  Lemma 6.12.

[6] Let $a\in\mathbb{Z}.$ Let $A\in Cl^{a}(M,E)$ and let $Q$ be a weight on
$E.$ Let $B$ be an operator on $C^{\infty}(M,E)$ such that

  1. (1)

$Ad_{B}(Cl^{a}(M,E))\subset Cl^{a}(M,E)$

  2. (2)

$Ad_{B}Q$ is a weight of the same order as $Q$

Then

  * •

$res(Ad_{B}A)=res(A)$

  * •

$tr^{Ad_{B}Q}(Ad_{B}A)=tr^{Q}(A).$

The properties 1,2 are true in particular for operators $B\in Aut(E).$

####  6.1.2. Some cocycles on $PDO(M,E)$, based on [2, 3]

If one wants to extend this construction of the index or of the Schwinger
cocycle to unbounded operators, one has to find a (non unitary) subalgebra
$\mathcal{I}$ in the algebra of operators $\mathcal{A}$ considered such that

\- $I^{2}$ is a set of trace-class operators

\- $[A,I]\subset I.$

Considering $\mathcal{A}=PDO(M,E),$ a natural choice for $\mathcal{I}$ is the
ideal $Cl^{-\infty}(M,E)$ [3]. But in that case, except when intrinsic
geometric structures enable to find a sign operator (e.g. from a Dirac
operator), it seems to be difficult to remain in an algebra of pseudo-
differential operators. More precisely:

######  Proposition 6.13.

[3] When $E=M\times\mathbb{C},$ If $\epsilon\in Cl^{0}(M,\mathbb{C}),$
$\epsilon=Id$ or $\epsilon=-Id$ up to a finite rank operator.

Moreover,

######  Proposition 6.14.

[3] The Schwinger cocycle $c_{S}$ on $\left(Cl(S^{1},\mathbb{C})^{\otimes
n}\right)_{res}$ related to the sign operator $\epsilon=(\epsilon(D))^{\otimes
n},$ where $\epsilon(D)$ is defined in the section $M=S^{1},$ is a non trivial
cocycle.

These results hae been obtained after those treating of the special case
$M=S^{1},$ principally given in [2] and completed in [3].

Open question: We notice that
$Cl((S^{1})^{n},\mathbb{C})\neq\left(Cl(S^{1},\mathbb{C})^{\otimes n}\right),$
and the question of an adequate choice of ”maximal” natural algebra of
operators for the Schwinger cocycle is still open when $M\neq S^{1}.$ The same
question arise in [BFR2018] where higher dimensional Kravchenko-Khesin
cocycles are investigated. A natural, but trivial, extension on
$Diff(M)-$pseudo-differential operators is given in [6], interesting results
being for $M=S^{1}.$

###  6.2. Specializing $M=S^{1}:$ a digression in algebraic structures, based
on [2, 6, 20]

The operator $D={-i}D_{x}$ splits $C^{\infty}(S^{1},\mathbb{C}^{k})$ into
three spaces :

\- its kernel $E_{0}$, made 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 following elementary result will be useful for the sequel, see [22] for
the proof, and e.g. [23, 3]:

######  Lemma 6.15.

(i) $\sigma(D)={\xi}$

(ii) $\sigma(|D|)={|\xi|}$ where
$|D|=\sigma\left(\int_{\Gamma}\lambda^{1/2}(\Delta-\lambda
Id)^{-1}d\lambda\right)$, with $\Delta=-D_{x}^{2}$.

(iii) $\sigma(D|D|^{-1})={\xi\over|\xi|}$, where $D|D|^{-1}=|D|^{-1}D$ is the
sign of D, since $|D|_{|E_{0}}=Id_{E_{0}}.$

(iv) Let $p_{E_{+}}$ (resp. $p_{E_{-}}$) be the projection on $E_{+}$ (resp.
$E_{-}$), then $\sigma(p_{E_{+}})={1\over 2}(Id+{\xi\over|\xi|})$ and
$\sigma(p_{E_{-}})={1\over 2}(Id-{\xi\over|\xi|})$.

Let us now define two ideals of the algebra $\mathcal{F}PDO$, that we call
$\mathcal{F}PDO_{+}$ and $\mathcal{F}PDO_{-}$, such that
$\mathcal{F}PDO=\mathcal{F}PDO_{+}\oplus\mathcal{F}PDO_{-}$. This
decomposition is implicit in [Ka1989], section 4.4., p. 216, for classical
pseudo-differential operators and we furnish the explicit description given in
[22], extended to the whole algebra of (maybe non formal, non classical)
pseudo-differential symbols here.

######  Definition 6.16.

Let $\sigma$ be a symbol (maybe non formal). Then, we define, for $\xi\in
T^{*}S^{1}-S^{1}$,

$\sigma_{+}(\xi)=\left\\{\begin{array}[]{ll}\sigma(\xi)&\hbox{ if $\xi>0$}\\\
0&\hbox{ if $\xi<0$}\\\ \end{array}\right.\hbox{ and
}\sigma_{-}(\xi)=\left\\{\begin{array}[]{ll}0&\hbox{ if $\xi>0$}\\\
\sigma(\xi)&\hbox{ if $\xi<0$}.\\\ \end{array}\right.$

At the level of formal symbols, we also define the projections:
$p_{+}(\sigma)=\sigma_{+}$ and $p_{-}(\sigma)=\sigma_{-}$ .

The maps
$p_{+}:\mathcal{F}PDO(S^{1},\mathbb{C}^{k})\rightarrow\mathcal{F}PDO(S^{1},\mathbb{C}^{k})$
and
$p_{-}:\mathcal{F}PDO(S^{1},\mathbb{C}^{k})\rightarrow\mathcal{F}PDO(S^{1},\mathbb{C}^{k})$
are clearly algebra morphisms that leave the order invariant and are also
projections (since multiplication on formal symbols is expressed in terms of
pointwise multiplication of tensors).

######  Definition 6.17.

We define $\mathcal{F}PDO_{+}(S^{1},\mathbb{C}^{k})=Im(p_{+})=Ker(p_{-})$ and
$\mathcal{F}PDO_{-}(S^{1},\mathbb{C}^{k})=Im(p_{-})=Ker(p_{+}).$

Since $p_{+}$ is a projection, we have the splitting

$\mathcal{F}PDO(S^{1},\mathbb{C}^{k})=\mathcal{F}PDO_{+}(S^{1},\mathbb{C}^{k})\oplus\mathcal{F}PDO_{-}(S^{1},\mathbb{C}^{k}).$

Let us give another characterization of $p_{+}$ and $p_{-}$. Looking more
precisely at the formal symbols of $p_{E_{+}}$ and $p_{E_{-}}$ computed in
Lemma 6.15, we observe that

$\sigma(p_{E_{+}})=\left\\{\begin{array}[]{ll}1&\hbox{if }\xi>0\\\ 0&\hbox{if
}\xi<0\\\ \end{array}\right.\hbox{ and
}\sigma(p_{E_{-}})=\left\\{\begin{array}[]{ll}0&\hbox{if }\xi>0\\\ 1&\hbox{if
}\xi<0\\\ \end{array}\right..$

In particular, we have that $D^{\alpha}_{x}\sigma(p_{E_{+}}),$
$D^{\alpha}_{\xi}\sigma(p_{E_{+}}),$ $D^{\alpha}_{x}\sigma(p_{E_{-}}),$
$D^{\alpha}_{\xi}\sigma(p_{E_{-}})$ vanish for $\alpha>0$. From this, we have
the following result:

######  Proposition 6.18.

[22] Let $a\in\mathcal{F}PDO(S^{1},\mathbb{C}^{k}).$
$p_{+}(a)=\sigma(p_{E_{+}})\circ a=a\circ\sigma(p_{E_{+}})$ and
$p_{-}(a)=\sigma(p_{E_{-}})\circ a=a\circ\sigma(p_{E_{-}}).$

We then remind that if $V={\mathbb{C}}^{n}$ and use the notations

${DO}(S^{1},V)=\bigcup_{o\in\mathbb{N}}\left\\{\sum_{0\leq k\leq
o}a_{k}\partial^{k}\right\\},\quad{IO}(S^{1},V)=\left\\{\sum_{k\leq-1}a_{k}\partial^{k}\right\\}$

we get also the vector space decomposition

(6.3) $\displaystyle\Psi DO(S^{1},V)={DO}(S^{1},V)\oplus{IO}(S^{1},V).$

such that any (matrix) order $k$ pseudo-differential operator
$A=\sum_{i=-\infty}^{k}a_{i}\partial^{i}$ is splitted in two components
$A=A_{D}+A_{S}$ with $A_{D}=\sum_{i=0}^{k}a_{i}\partial^{i}$ and
$A_{S}=\sum_{i=-\infty}^{-1}a_{i}\partial^{i}.$ The Adler trace [Ad1979]
defined by

$Tr:A=\sum_{k\leq o}a_{k}\partial^{k}\mapsto\int_{S^{1}}tr(a_{-1})$

is the only non trivial trace on $\Psi DO(S^{1},V).$ Morover, see e.g.
[EKRRR1995] and [KZ1995],

######  Theorem 6.19.

$(\Psi DO(S^{1},V),{IO}(S^{1},V),{DO}(S^{1},V),Tr)$ is a Manin triple.

The Wodzicki residue ([Wod1984], see e.g. [Ka1989]) is usually known as an
“extension” of the Adler trace to $\mathcal{F}Cl(S^{1},V)$ and hence to
$Cl(S^{1},V).$ For the sake of deeper insight, we need to precise that the
space of traces on $\mathcal{F}Cl(S^{1},V)$ is 2-dimensional, generated by two
functionals:

$res_{+}:A\mapsto\int_{S^{1}}\sigma_{-1}(A)(x,1)|dx|$

and

$res_{-}:A\mapsto\int_{S^{1}}tr(\sigma_{-1}(A))(x,-1)|dx|.$

The functionals $res_{\pm}$ are the only non-vanishing traces on
$\mathcal{F}Cl_{\pm}(S^{1},V)$ (up to a scalar factor) and are vanishing on
$\mathcal{F}Cl_{\mp}(S^{1},V).$ The (classical) Wodzicki residue reads as
$res=res_{+}+res_{-}.$

####  6.2.1. Radul, Schwinger and index cocycle on $PDO(S^{1},E)$

Let $\pi:E\rightarrow S^{1}$ be a non trivial real vector bundle over $S^{1}$
of rank $k.$

######  Proposition 6.20.

[6] Let $\nabla$ be a Riemannian covariant derivative on the bundle
$E\rightarrow S^{1}$ and let $\nabla\over dt$ be the associated first order
differential operator, given by the covariant derivative evaluated at the unit
vector field over $S^{1}.$ We modify the operator $\nabla\over dt$ into an
injective operator $D={\nabla\over dt}+p_{ker{\nabla\over dt}}$, where
$p_{ker{\nabla\over dt}}$ is the $L^{2}$ orthogonal projection on
$ker{\nabla\over dt}\subset C^{\infty}(S^{1},E)\subset L^{2}(S^{1},E),$ and we
set

$\epsilon(\nabla)=D\circ\left|D\right|^{-1}.$

Then the formal symbol of $\epsilon(\nabla)$ is $i\xi\over|\xi|.$

######  Proposition 6.21.

[2, 3, 6] For each $A\in PDO(S^{1},E),$ $[A,\epsilon(\nabla)]\in
PDO^{-\infty}(S^{1};E).$

The fiber bundle $T^{*}S^{1}-S^{1}$ has two connected components and the phase
function is positively homogeneous, so that we can make the same procedure as
in the case of the symbols. The main results gradually discovered in [23, 3,
6] are now gathered. Here, $\epsilon(\nabla)$ is not a sign operator, but an
operator such that $\epsilon(\nabla)^{2}=Id$ up to a smoothing operator. :

######  Theorem 6.22.

[2, 3, 6] For any $A\in PDO(S^{1},E)$, $[A,\epsilon(\nabla)]\in
PDO^{-\infty}(S^{1},E).$ Consequently,

$c_{s}^{\nabla}:A,B\in PDO(S^{1},E)\mapsto{1\over 2}\hbox{\rm
tr}\left(\epsilon(\nabla)[\epsilon(\nabla),A][\epsilon(\nabla),B]\right)$

is a well-defined $\mathbb{R}$-valued 2-cocycle on $PDO(S^{1},E).$ Moreover,
$c_{s}^{\nabla}$ is non trivial on any Lie algebra $\mathcal{A}$ such that
$C^{\infty}(S^{1},\mathbb{R})\subset\mathcal{A}\subset PDO(S^{1},E).$

####  6.2.2. From $\Psi DO(S^{1},V)$ to $\mathcal{F}Cl(S^{1},V)$ based on [20]

There exists a decomposition
$\mathcal{F}Cl_{+}(S^{1},V)=\mathcal{F}Cl_{+,D}(S^{1},V)\oplus\mathcal{F}Cl_{+,S}(S^{1},V)$
and another
$\mathcal{F}Cl_{-}(S^{1},V)=\mathcal{F}Cl_{-,D}(S^{1},V)\oplus\mathcal{F}Cl_{-,S}(S^{1},V),$
and setting

$\mathcal{F}Cl_{D}(S^{1},V)=\mathcal{F}Cl_{+,D}(S^{1},V)\oplus\mathcal{F}Cl_{-,D}(S^{1},V),$
$\mathcal{F}Cl_{S}(S^{1},V)=\mathcal{F}Cl_{+,S}(S^{1},V)\oplus\mathcal{F}Cl_{-,S}(S^{1},V),$

we get the vector space decomposition analogous to (6.3):

$\mathcal{F}Cl(S^{1},V)=\mathcal{F}Cl_{D}(S^{1},V)\oplus\mathcal{F}Cl_{S}(S^{1},V),$

Let us consider the decomposition

$\mathcal{F}Cl(S^{1},V)=\mathcal{F}Cl_{odd}(S^{1},V)+\mathcal{F}Cl_{even}(S^{1},V),$

that we equip with the classical Lie bracket $[.,.]$ or with
$[.,.]_{\epsilon(D)}=\frac{1}{2}\left([\epsilon(D)\circ.,.]+[.,\epsilon(D)\circ.]\right)$
and with the bilinear form $(A,B)=res(AB).$

######  Theorem 6.23.

[20] res(AB) is a bilinear, non degenerate, symmetric and invariant form for
both brackets, and $\mathcal{F}Cl_{odd}(S^{1},V)$ as well as
$\mathcal{F}Cl_{eo}(S^{1},V)$ are isotropic vector spaces. Moreover,

  * •

for $[.,.],$ $\mathcal{F}Cl_{odd}(S^{1},V)$ is a Lie algebra

  * •

for $[.,.]_{\epsilon(D)},$ $\mathcal{F}Cl_{even}(S^{1},V)$ is a Lie algebra.

####  6.2.3. Extension of the classical Manin triple to
$\mathcal{F}Cl(S^{1},V)$ based on [20]

Because the partial symbol $\sigma_{-1}(A)$ of an operator $A\in\Psi
DO(S^{1},V)$ is skew-symmetric in the $\xi-$variable, $res$ is vanishing on
$\Psi DO(S^{1},V)=\mathcal{F}Cl_{odd}(S^{1},V),$ so that it is superficial to
state that the Wodzicki residue is “simply” the extension of the Adler trace.
However the two linear functionals already described, namely

$A\in\mathcal{F}Cl(S^{1},V)\mapsto\sum_{k\in\mathbb{Z}}\sigma_{k}(A)(x,1)\partial^{k}$

and

$A\in\mathcal{F}Cl(S^{1},V)\mapsto\sum_{k\in\mathbb{Z}}\sigma_{k}(A)(x,-1)\partial^{k},$

identity $res_{+}$ and $res_{-}$ respectively with $Tr.$ By the way, we can
state:

######  Theorem 6.24.

We have three Manin triples:

$(\mathcal{F}Cl_{+}(S^{1},V),\mathcal{F}Cl_{+,S}(S^{1},V),\mathcal{F}Cl_{+,D}(S^{1},V),res_{+}),$
$(\mathcal{F}Cl_{-}(S^{1},V),\mathcal{F}Cl_{-,S}(S^{1},V),\mathcal{F}Cl_{-,D}(S^{1},V),res_{-})$

and

$(\mathcal{F}Cl(S^{1},V),\mathcal{F}Cl_{S}(S^{1},V),\mathcal{F}Cl_{D}(S^{1},V),res).$

####  6.2.4. Injecting $\Psi DO(S^{1},\mathbb{K})$ in
$\mathcal{F}Cl(S^{1},\mathbb{K}).$

We already mentionned the identification of $\Psi DO(S^{1},\mathbb{K})$ with
$\mathcal{F}Cl_{odd}(S^{1},{\mathbb{C}}).$ We claim here that this
identification can be generalized to

$\Phi_{odd,\lambda}:\sum_{k\in\mathbb{Z}}a_{k}\left(\frac{d}{dx}\right)^{k}\in\Psi
DO(S^{1},\mathbb{K})\mapsto\sum_{k\in\mathbb{Z}}a_{k}\left(\lambda\frac{d}{dx}\right)^{k}\in\mathcal{F}Cl_{odd}(S^{1},{\mathbb{C}}).$

Similar to this identification, we have other injections for
$\lambda\in\mathbb{R}^{*}:$

$\Phi_{\epsilon(D),\lambda}:\sum_{k\in\mathbb{Z}}a_{k}\left(\frac{d}{dx}\right)^{k}\in\Psi
DO(S^{1},\mathbb{K})\mapsto\sum_{k\in\mathbb{Z}}a_{k}\left(\lambda\epsilon(D)\frac{d}{dx}\right)^{k}\in\mathcal{F}Cl(S^{1},\mathbb{K}),\hbox{
and}$
$\Phi_{\lambda,\mu}:\sum_{k\in\mathbb{Z}}a_{k}\left(\frac{d}{dx}\right)^{k}\in\Psi
DO(S^{1},\mathbb{K})\mapsto\sum_{k\in\mathbb{Z}}a_{k}\left(\lambda^{k}\left(\frac{d}{dx}\right)_{+}^{k}+\mu^{k}\left(\frac{d}{dx}\right)_{-}^{k}\right)\in\mathcal{F}Cl(S^{1},\mathbb{K})$

for $(\lambda,\mu)\in{\mathbb{C}}^{2}\backslash\\{(0;0)\\},$ with unusual
convention $0^{k}=0$ $\forall k\in\mathbb{Z}.$

######  Remark 6.25.

$\Phi_{1,1}=\Phi_{ee}$ and $\Phi_{1,-1}=\Phi_{\epsilon(D),1}.$

######  Remark 6.26.

$Im\Phi_{1,0}=\mathcal{F}Cl_{+}(S^{1},\mathbb{K})$ and $\Phi_{1,0}$ is a
isomorphism of algebras from $\Psi DO(S^{1},\mathbb{K})$ to
$\mathcal{F}Cl_{+}(S^{1},\mathbb{K}).$ The same way, $\Phi_{0,1}$ identifies
the algebras $\Psi DO(S^{1},\mathbb{K})$ and
$\mathcal{F}Cl_{-}(S^{1},\mathbb{K}).$

######  Remark 6.27.

Wa have also to say that the maps $\Phi_{\lambda,\mu}$ are not algebra
morphisms unless $(\lambda,\mu)\in\\{(1;0),(0;1),(1;1)\\}.$ For example, let
$\lambda\in\mathbb{C}-\\{0;1\\}.$ the map $\Phi_{\lambda,0}$ pushes forward
the multiplication on $\Psi DO(S^{1},\mathbb{K})$ to a deformed composition
$*_{k}$ on $\mathcal{F}Cl_{+}(S^{1},\mathbb{K})$ that reads as
$\sigma(A)*_{k}\sigma(B)=\sum_{\alpha\in\mathbb{N}}\frac{(-i)^{\alpha}}{\alpha!.k^{\alpha}}D^{\alpha}_{x}\sigma(A)D^{\alpha}_{\xi}\sigma(B).$

From our previous remarks, we get:

######  Theorem 6.28.

The map

$\Phi_{1,0}\times\Phi_{0,1}:\Psi
DO(S^{1},\mathbb{K})^{2}\rightarrow\mathcal{F}Cl_{+}(S^{1},\mathbb{K})\times\mathcal{F}Cl_{-}(S^{1},\mathbb{K})=\mathcal{F}Cl(S^{1},\mathbb{K})$

is an isomorphism of algebra.

We also remark a new subalgebra of $\mathcal{F}Cl(S^{1},\mathbb{K}):$

######  Definition 6.29.

Let $\mathcal{F}Cl_{\epsilon}(S^{1},\mathbb{K})$ be the image of
$\Phi_{\epsilon(D),1}$ in $\mathcal{F}Cl(S^{1},\mathbb{K}).$

We have the obvious identification
$\mathcal{F}Cl_{\epsilon}(S^{1},\mathbb{K})=C^{\infty}(S^{1},\mathbb{K})((i|D|^{-1}))$
as a vector space.

####  6.2.5. An integrable almost complex struxture on
$\mathcal{F}Cl(S^{1},{\mathbb{C}})$ based on [20]

Let

$i\epsilon(D)=\left(\frac{d}{dx}\right).|D|^{-1}=|D|^{-1}.\left(\frac{d}{dx}\right).$

We define the operator $J_{1}=i\epsilon(D)\circ(.)$ on
$\mathcal{F}Cl(S^{1},V).$

######  Theorem 6.30.

The operator $J_{1}$ defines an integrable almost complex structure on
$\mathcal{F}Cl(S^{1},V).$$\mathcal{F}Cl(S^{1},V)=\Psi
DO(S^{1},V)\otimes{\mathbb{C}}$ as a real algebra, identifying
$\mathcal{F}Cl_{odd}(S^{1},V)$ with $\Psi DO(S^{1},V)$ (real part) and
$\mathcal{F}Cl_{even}(S^{1},V)$ with $i\Psi DO(S^{1},V)$ (imaginary part).

Open problem: study the obstruction of extending the almost complex structure
$J_{1}$ from $\mathcal{F}Cl(S^{1},V)$ to $Cl(S^{1},V).$

###  6.3. Renormalized extension of the Hilbert-Schmidt Hermitian metric,
based on [19]

The vector space $Cl^{-1}(S^{1},V)$ is a space of Hilbert-Schmidt operators.
As a subspace, $Cl^{-1}(S^{1},V)$ inherits a Hermitian metric from the
classical Hilbert-Schmidt inner product. The renormalized trace $\hbox{\rm
tr}^{\Delta}$ extends the classical trace tr of trace class operators to a
smooth linear functional on $Cl(S^{1},V).$ We investigate here the possible
(maybe naive) extension of the classical Hilbert-Schmidt inner product to
$Cl(S^{1},V)$ via $\hbox{\rm tr}^{\Delta}.$

####  6.3.1. Extension of the Hilbert-Schmidt metric to $FCl.$

Let $(z^{k})_{k\in\mathbb{Z}}$ is the Fourier $L^{2}-$orthonormal basis. Let
us recall that there exists an ambiguity on $\epsilon(D)$ concerning its
action on $z^{0},$ which can be, or not, in the kernel of $p_{+},$ or in the
eigenspace of the eigenvalue $1$ or $-1.$ Depending on each of these three
possibilities respectively, we set $\epsilon(k)$ as the eigenvalue of
$\epsilon(D)$ at the eigenvector $z^{k}.$

######  Lemma 6.31.

Let $X=u\frac{d}{dx},Y=v\frac{d}{dx}$ be two vector fields over $S^{1},$ and
let $a,b\in C^{\infty}(S^{1},\mathbb{C}).$ Then

  1. (1)

$\hbox{\rm tr}^{\Delta}(a\bar{b})=0$

  2. (2)

$\hbox{\rm tr}^{\Delta}(XY^{*})=0$

  3. (3)

$\hbox{\rm tr}^{\Delta}(Xa)=\hbox{\rm tr}^{\Delta}(aX)=0$

######  Theorem 6.32.

[19] The Hilbert-Schmidt definite positive Hermitian product

$\left(A,B\right)_{HS}=\hbox{\rm tr}\left(AB^{*}\right)$

which is positive, definite metric on $Cl^{-1}(S^{1},V)$ extends:

  * •

to a Hermitian, non degenerate form on $Cl(S^{1},V)$ by
$(A,B)\mapsto(A,B)_{\Delta}=\hbox{\rm tr}^{\Delta}(AB^{*})$

  * •

to a Hermitian, non degenerate form on $Cl_{odd}(S^{1},V)$ by
$(A,B)\mapsto(A,B)_{\Delta}=\hbox{\rm tr}^{\Delta}(AB^{*})$

  * •

to a $(\mathbb{R}-)$ bilinear, symmetric non degenerate form on
$Cl(S^{1},V)\oplus Vect(S^{1})$ by
$(A,B)\mapsto\mathfrak{Re}(A,B)_{\Delta}=\mathfrak{Re}\left(\hbox{\rm
tr}^{\Delta}(AB^{*})\right)$ where $A=a+u,$ $B=b+v,$ with $(a,b)\in
Cl(S^{1},V)$ and $(u,v)\in Vect(S^{1}).$

######  Remark 6.33.

We remark that $(.,.)_{\Delta}$ is bilinear, non degenerate but not positive.
Indeed, from relation (1) of Lemma 6.31, $C^{\infty}(S^{1},M_{n}(\mathbb{C}))$
is an isotropic Lie subalgebra for $(.,.)_{\Delta}$ which proves that this
$\mathbb{R}-$bilinear symmetric form is not positive.

From the Lie algebra $Cl(S^{1},V)\oplus Vect(S^{1}),$ we then span by right-
invariant action of $FCl^{*}(S^{1},V)$ on $TFCl^{*}(S^{1},V)$ a right-
invariant pseudo-metric. For this goal, the Lie algebra elements are
identified as infinitesimal paths, and actions and Lie brackets are those
derived from the coadjoint action (and right-Lie bracket) of
$FCl^{*}(S^{1},V)$ on $Cl(S^{1},V)\oplus Vect(S^{1}),$ while we consider the
trivial mapping defined by the sum $Cl(S^{1},V)\oplus Vect(S^{1})\rightarrow
Cl(S^{1},V)=Cl(S^{1},V)+Vect(S^{1})$ in order to compute
$\mathfrak{Re}(.;.)_{\Delta}.$ The same constructions hold for the pseudo-
Hermitian metric $(.;.)_{\Delta}$ on $Cl^{*}(S^{1},V).$

######  Definition 6.34.

Let $A\in FCl^{0,*}(S^{1},V)$ and let $a\in Cl^{0}(S^{1},V)\oplus
Vect(S^{1}).$ We note by $R_{A}(a)$ the (right-)action by composition

$R_{A}(a)=a\circ A.$

Then, identifying $T_{A}FCl^{0,*}(S^{1},V)$ with
$R_{A}\left(Cl^{0}(S^{1},V)\oplus Vect(S^{1})\right)$ we set a smooth pseudo-
Riemannian metric on $TFCl^{0,*}(S^{1},V)$ by defining for

$(a,b)\in\left(Cl^{0}(S^{1},V)\oplus Vect(S^{1})\right)^{2},$

and hence for
$(R_{A}(a),R_{A}(b))\in\left(T_{A}FCl^{0,*}(S^{1},V)\right)^{2},$

$(R_{A}(a),R_{A}(b))_{\Delta,A}=(a,b)_{\Delta}.$

####  6.3.2. On bounded odd class $Diff(S^{1})-$pseudo-differential operators

Let us finish our remarks with the group of ($L^{2}-$)bounded even-even
$Diff(S^{1})-$pseudo-differential operators. Its Lie algebra

$Cl^{0}_{odd}(S^{1},V)\rtimes Vect(S^{1})$

also reads as

$Cl^{-1}_{odd}(S^{1},V)\oplus DO^{0}(S^{1},V)\oplus(Vect(S^{1})\otimes
Id_{V}).$

and the pseudo-Riemannian product $\mathfrak{Re}(.,.)_{\Delta}$ decomposes
blockwise as

$\left(\begin{array}[]{ccc}\mathfrak{Re}(.,.)_{HS}&*&*\\\ &0&0\\\
&0&0\end{array}\right),$

where $\mathfrak{Re}(.,.)_{HS}=\mathfrak{Re}\left((.,.)_{HS}\right)$ is the
scalar product derived from the Hilbert-Schmidt Hermitian product
$(.,.)_{HS}.$

###  6.4. In search of pseudo-Hermitian connections for $(.,.)_{\Delta}$ based
on [19]

There exists some difficulties in describing the whole space of connection
1-forms $\Omega^{1}(FCl_{Diff(S^{1})}(S^{1},V),Cl(S^{1},V)\rtimes
Vect(S^{1})).$ Indeed the space of smooth linear maps acting on $Cl(S^{1},V)$
is actually not well-understood to our knowledge. Finding an adjoint for $ad$
and for $(.,.)_{\Delta}$ fails apparently due to the non-traciality of
$\hbox{\rm tr}^{\Delta}$ (and surprisingly not due to the lack of the
classical arguments envolving strong metrics). We consider here a class of
connections where this smooth linear endomorphism is defined by composition by
a smoothing operator. The resulting technical simplifications enables us to
get pseudo-Hermitian connections for $(.,.)_{\Delta}.$ Most of them can be
easily adapted to get pseudo-Riemannian connections for
$\mathfrak{Re}(.,.)_{\Delta}.$

####  6.4.1. A class of connections

Let us define now, for $w\in Cl(S^{1},V)$ such that $\forall(a,b)\in
Cl(S^{1},V),$

$\Theta^{w}_{a}b=b[a,w].$

The curvature of $\Theta^{w}$ reads as

$\Omega_{\Theta^{w}}(a,b)c=[wbw,a]-[waw,b]+[wa,wb]-[wb,aw]-[[b,a]w]$

for $(a,b)\in\mathcal{F}Cl(S^{1},V).$ Let us analyze the connection
$\Theta^{w}$ with $w=i\epsilon(D).$

######  Theorem 6.35.

$\Theta^{i\epsilon(D)}$ is a $Cl^{-\infty}(S^{1},V)-$valued connection.

####  6.4.2. Pseudo-Hermitian connections associated with a skew-adjoint
pseudodifferential operator

Let $w\in Cl(S^{1},V)$ such that $w^{*}=-w.$ For example, one can consider the
example $w=i\epsilon(D).$

######  Lemma 6.36.

$\forall a\in Cl(S^{1},V),$ $\forall w\in Cl(S^{1},V)$ such that $w^{*}=-w,$
$\Theta^{w}_{a^{*}}$ is the adjoint of $\Theta^{s}_{a}$ for $(.,.)_{\Delta}$

Let us now analyze

$(a,b)\in
Cl(S^{1},{\mathbb{C}})^{2}\mapsto\theta^{w}_{a}b=b[a-a^{*},w]=(\Theta^{w}_{a}-\Theta^{w}_{a^{*}})(b).$

######  Theorem 6.37.

$\theta^{w}$ is the connection 1-form of a pseudo-Hermitian connection of
$(.,.)_{\Delta}.$ Moreover,

$\theta^{i\epsilon(D)}$ is $Cl^{-\infty}(S^{1},V)-$valued as
$\Theta^{i\epsilon(D)}$ is.

####  6.4.3. On another class $Cl^{-\infty}(S^{1},V)-$connections

Motivated by the previous example of $Cl^{-\infty}(S^{1},V),$ let us now give
families of $Cl^{-\infty}(S^{1},V)-$connections which a priori do not include
the connections $\theta^{i\epsilon(D)}$ and $\Theta^{i\epsilon(D)}$. Let us
define now, for $s\in Cl^{-\infty}$ and $\forall(a,b)\in Cl(S^{1},V),$

$\Theta^{s,l}_{a}b=sas^{*}b,$ $\Theta^{s,r}_{a}b=bsas^{*},$

and

$\Theta^{s,[]}_{a}b=\left[sas^{*},b\right].$

Let us describe here their associated class of pseudo-Riemannian connections
for $(.,.)_{\Delta}$ along the lines of the previous section. Let $s\in
Cl^{-\infty}(S^{1},V)$ be a smoothing operator. Let $a,b\in Cl(S^{1},V)^{2}$
and let

$\theta_{a}^{s,[]}b=\Theta_{a}^{s,[]}b-\Theta_{a^{*}}^{s,[]}b=\left[s(a-a^{*})s^{*},b\right],$
$\theta_{a}^{s,l}=\Theta_{a}^{s,l}b-\Theta_{a^{*}}^{s,l}b=s(a-a^{*})s^{*}b,$

and

$\theta_{a}^{s,r}=\Theta_{a}^{s,r}b-\Theta_{a^{*}}^{s,r}b=bs(a-a^{*})s^{*}.$

######  Lemma 6.38.

$\forall a\in Cl(S^{1},V),$ $\forall s\in Cl^{-\infty}(S^{1},V),$

  * •

$\Theta^{s,l}_{a^{*}}$ is the adjoint of $\Theta^{s,l}_{a}$ for
$(.,.)_{\Delta}$

  * •

$\Theta^{s,r}_{a^{*}}$ is the adjoint of $\Theta^{s,l}_{a}$ for
$(.,.)_{\Delta}$

  * •

$\Theta^{s,[]}_{a^{*}}$ is the adjoint of $\Theta^{s,l}_{a}$ for
$(.,.)_{\Delta}$

######  Theorem 6.39.

Then $\theta^{s,[]},$ $\theta^{s,l}$ and $\theta^{s,r},$ define three right-
invariant pseudo-Hermitian connections on $FCl(S^{1},V).$

Open problem: Determine the geodesic equations of these connections, determine
the corresponding dynamical systems that one can deduce, and determine
families of one parameter families of connections that link these systems.
These “homotopies” can exibit or not symmetry breakings.

####  6.4.4. The Schwinger cocycle and the connection $\Theta^{i\epsilon}$

Let us make the two following remarks

######  Proposition 6.40.

Let $(a,b)\in Cl(S^{1},V)\rtimes Vect(S^{1}).$ Then

$c_{s}(a,b)=-i\hbox{\rm tr}^{\Delta}(\Theta^{i\epsilon}_{a}b)=\hbox{\rm
tr}^{\Delta}(\Theta^{i\epsilon}_{a}\Theta^{i\epsilon}_{b}\epsilon(D)).$

######  Remark 6.41.

When defining a smoothing connection $\theta$ on $Cl(S^{1},V)\rtimes
Vect(S^{1}),$ we define a map with values on the first component of the
product $Cl(S^{1},V)\times Vect(S^{1}).$

######  Theorem 6.42.

The Schwinger cocycle $c_{s}$ has the same cohomology class as

$c_{1}^{i\epsilon}:(a,b)\in Cl(S^{1},V)^{2}\mapsto\frac{1}{2}\hbox{\rm
tr}^{\Delta}\left(\Omega^{i\epsilon}(a,b)\epsilon(D)\right)$

where $\Omega^{i\epsilon}$ is the curvature of $\Theta^{i\epsilon}.$

Open problem: Determine the cohomology classes that can be obtained this way,
by connections with $Cl^{-\infty}-$valued curvatures, and implement an
adequate holonomy bundle with suitable (maybe generalized) geometric
structures. A partial answer is given in the non-presented (because non-
published and only pre-published) work [35], that has to be generalized both
for other classes of connections, and for changing $S^{1}$ for a higher
dimensional manifold $M.$

####  6.4.5. On odd class $Diff(S^{1})-$pseudo-differential operators

Considering now

$Cl^{*}_{odd}(S^{1},V)\rtimes Diff(S^{1}),$

we remark that the renormalized trace $\hbox{\rm tr}^{\Delta}$ is tracial on
its Lie algebra $Cl_{odd}(S^{1},V)\rtimes Vect(S^{1}),$ i.e.

$\forall(a,b)\in Cl_{odd}(S^{1},V),\quad\hbox{\rm tr}^{\Delta}([a,b])=0$

(representing $Cl_{odd}(S^{1},V)\rtimes Vect(S^{1})$ in $Cl_{odd}(S^{1},V)$ as
in the rest of the text).This enables to state the following property:

######  Proposition 6.43.

$\forall a\in Cl_{odd}(S^{1},V),$ the adjoint map

$ad_{a}:b\mapsto ad_{a}b=[b,a]$

has an adjoint map for $(.,.)_{\Delta}$ given by

$ad_{a}^{*}=ad_{a^{*}}.$

As a consequence, applying the arguments of [Fr1988-2] and especially those
leading to [Fr1988-2, Proposition 1.7] to right-invariant vector fields on
$FCl_{odd,Diff(S^{1})}(S^{1},V)$, we get:

######  Theorem 6.44.

The pseudo-Riemannian metric $\mathfrak{Re}(.,.)_{\Delta}$ admits a unique
pseudo-Riemannian, torsion-free (i.e. Levi-Civita) connection
$\nabla^{\Delta}$ that reads as

$\nabla^{\Delta}_{a}b=\frac{1}{2}\left(ad_{a}b-ad_{a^{*}}b-ad_{b^{*}}a\right)$

Open problem: For a weak (pseudo-)metric in e.g. such a general framework,
determine, depending on the chosen structure group, if there exist a torsion
free (pseudo-)Riemannian or Hermitian connection.

###  6.5. Manifolds of mappings and embeddings

Let $M$ be a compact boundaryless manifold and let $N$ be a finite dimensional
Riemannian manifold. The basic structure of manifolds of maps
$C^{\infty}(M,N)$ is known since [Ee1966]. For $f\in C^{\infty}(M,N),$ the
tangent space $T_{f}C^{\infty}(M,N)=\gamma(M,f^{*}TN)$ and the $L^{2}-$
exponential map can be described pointwise, by the way, one can define a frame
bundle over $C^{\infty}(M,N)$ defined by $0-$order differential operators. The
space of embeddings $Emb(M,N)$ is an open submanifold of $C^{\infty}(M,N).$ On
these spaces, we have to mention works from us (presented in our PhD thesis
and hence not presented for as works for the habilitation evaluation) and
others, which started with Freed’s and Paycha’s works [Fr1988-1, Fr1988-2,
Pay2001] on the development of Chern-Weil forms on mapping spaces. These
construction were motivated by [RSF1985, BR1987-1, BR1987-2], where anomalies
are identified as kind of extension of a Chern form ”$\hbox{\rm tr}\Omega$”,
where tr needs to be understood only as a chosen summation over a. well-chosen
orthonormal basis. This ”well-chosen” summation has been interpreted,
following the ideas of [Pay2001, Mick1994], see e.g. [Ism1996], as a
renormalized trace $\hbox{\rm tr}^{Q}$ for $Q=\Delta.$ More precisely, re-
interpreting and extending some results [Fr1988-2],

######  Theorem 6.45.

[21] Let $G$ be a semi-simple compact Lie group, let $C^{\infty}_{b}(S^{1},G)$
be the based loop group equipped with the almost complex structure $J=D/|D|,$
and let $\omega$ the Kähler form of the based loop group. Then

$\hbox{\rm tr}^{\Delta}(\Omega^{1,0})=-i\omega,$

where $\Omega^{1,0}$ is the holomorphic part of the $H^{1/2}-$Levi-Civita
connection, associated to the $H^{1/2}$-metric $(\Delta^{1/2}.,.)_{L_{2}}.$

This theorem has been extended:

######  Theorem 6.46.

[23] With the same notations, if $\theta$ is a connection on
$C^{\infty}_{b}(S^{1},G)$ such that, read on left-invariant frames, $\forall
X\in C^{\infty}_{b}(S^{1},\mathfrak{g},)$ $\theta_{X}\in
ad_{X}+Cl^{-1}(S^{1},\mathfrak{g}),$ then the associated Chern form

$\hbox{\rm tr}(\Omega^{1,0})$

is closed and it has the same cohomology class as $-i\omega.$

Let us make two comments before going to other results:

  * •

Freed’s results in [Fr1988-1] only extend the contruction of the first Chern
form $\hbox{\rm tr}(\Omega)$ when $\Omega$ is trace-class, which is not the
case here since $\Omega^{1,0}$ lies in the Dixmier ideal $L^{1,\infty}$ in
Freed’s framework [Fr1988-2] or in $Cl^{-1}(S^{1},\mathfrak{g})$ in the
framework of [21].

  * •

With the infinite dimensional Ambrose-Singer theorem 4.34, and especially the
part stated in [1], we have a principal bundle over

$C^{\infty}_{b}(S^{1},G)_{\mathbb{C}},$

modelled on

$GL^{2}\cap Cl^{0}(S^{1},\mathfrak{g}),$

which is shown to be non trivial since its first Chern form has a non trivial
cohomology class. This is a deep contrast with Kuiper’s triviality results
[K1965] for $U(H)-$principal bundles, when $H$ is a Hilbert space.

Around this central example, natural questions raised:

  * •

What happens passing to $C^{\infty}(M,N)?$ This is mostly the aim of [23].
There are many cases when direct investigations on the properties of fields of
weights cannot conclude if the obtained Chern-Weil forms are closed or not,
and if this they are closed, when they belong to the same cohomology class.
For example, the first Chern form of the $H^{1}-$Levi-Civita connection on
$C^{\infty}_{b}(S^{1}\times S^{1},\mathfrak{u}(n))$ is shown to vanish, bu
with no comparison result with other $Cl^{-1,*}-$connections. The best result
that one can state is the following, adapted from [23]:

######  Proposition 6.47.

If $M$ is compact aand if $N$ is parallelizable, if $Q$ is a diagonal weight,
then the Chern-Weil forms $\hbox{\rm tr}^{Q}(\Omega^{k})$ lie in the same
cohomology class for $Cl^{-m,*}-$connections.

We have similar results with odd class operators:

######  Proposition 6.48.

In $M$ is compact and odd-dimensional, if $Q$ is an odd-class weight, then
then the Chern-Weil forms $\hbox{\rm tr}^{Q}(\Omega^{k})$ lie in the same
cohomology class for $Cl^{0,*}_{odd}-$connections.

But these two results only enables one to get vanishing characteristic
classes, or characteristic classes with undetermined cohomology class.

  * •

Can one choose other ”traces”? If one requires the traciality property
$\hbox{\rm tr}[A,B]=0,$ the only trace on $Cl(M,\mathbb{C})$ is the Wodzicki
residue [Wod1984] when $dimM>1,$ with a quite similar situation for $dimM=1$
with a Wodzicki residue derived from two Adler traces [Wod1984, Ka1989], as
suggested before with the splitting
$\mathcal{F}Cl(S^{1},\mathbb{C})=\mathcal{F}Cl_{+}(S^{1},\mathbb{C})\oplus\mathcal{F}Cl_{-}(S^{1},\mathbb{C}).$
An investigation of the situation with the Wodzicki residue, initiated in [23]
where we showed that the Chern-Weil forms are trivial for $C^{\infty}(M,N),$
and extended in [MRT2014] (without citing [23]) to Chern-Simons forms,
revealed potential applications. For bounded operators, the only traces on
$Cl^{0}$ are spanned by the Wodzicki residue and the leading symbol trace
[LP2007]. This means that there is no tracial extension of the classical trace
of trace-class operators on these algebras and all the traces vanish on trace-
class operators. However, the discussion of the previous point is far from
finished since some interesting traces can appear when restricting the
considered structure groups.

Another manifold of interest is the space of embeddings, see e.g. [BF1981].
The group of diffeomorphisms of $M$, $Diff(M)$, acts smoothly and on the right
on $Emb(M,N)$, by composition. Moreover,

$B(M,N)=Emb(M,N)/Diff(M)$

is a smooth manifold [BF1981, KM2000], and $\pi:Emb(M,N)\rightarrow B(M,N)$ is
a principal bundle with structure group $Diff(M)$ (see [KM2000, Mo2008]). Let
us now precise the vertical tangent space and a normal vector space of the
orbits of $Diff(M)$ on $Emb(M,N)$. $T_{f}PEmb(M,N)$, the tangent space at $f$,
is identified with the space of smooth sections of $f^{*}TN$, which is the
pull-back of $TN$ by $f$. Let $\mathcal{N}_{f}$ be the normal space to $f(M)$
with respect to the metric $(.,.)$ on $N$. For any $x\in M$,
$T_{f(x)}N=T_{f(x)}f(M)\oplus\mathcal{N}f(M)$. Hence, denoting
$f^{*}\mathcal{N}_{f}$ the pull back of $\mathcal{N}_{f}$ by $f$, we have that

$C^{\infty}(f^{*}TN)=C^{\infty}(TM)\oplus f^{*}\mathcal{N}_{f}.$

Moreover, for any volume form $dx$ on $M$, if

$<.,.>:X,Y\in C^{\infty}(f^{*}TN)\mapsto<X,Y>=\int_{M}(X(x),Y(x))dx$

is a $L^{2}$-inner product on $C^{\infty}(f^{*}TN)$, this splitting is
orthogonal for $<.,.>$. We get here a fundamental difference between the
inclusion $Emb(M,N)\subset C^{\infty}(M,N),$ where the model space of the type
$C^{\infty}(f^{*}TN),$ and $Emb(M,N)$ as a $Diff(M)-$ principal bundle:
sections of the vertical tangent vector bundle read as order 1 differential
operators, where as the operators acting on the normal vector bundle reads as
$0-$order differential operators, just like the structure group of
$TC^{\infty}(M,N).$

Now, let $f\in Emb(M,N)$ and let us consider the map

$\Phi^{U,f}:(f,v,X)\in TU\sim(1-p)TU\oplus
pTU\mapsto\Xi^{f}(v).exp_{Diff(M)}(X)\in Emb(M,N).$

This map gives a local (fiberwise) trivialization of the principal bundles
$Emb(M,N)\rightarrow B(M,N)$ following [HV2004, KM2000, Mo2008], and we see
that the changes of local trivializations have $Aut(\mathcal{N})$ as a
structure group.

If $M$ is oriented, we note by $Diff_{+}(M)$ the group of orientation
preserving diffeomorphisms and we have

$\frac{Diff(M)}{Diff_{+}(M)}=\mathbb{Z}_{2}.$

Then, defining

$B_{+}(M,N)=\frac{Emb(M,N)}{Diff^{+}(M)}$

we get that $B_{+}(M,N)$ is a 2-cover of $B(M,N).$

####  6.5.1. On the structures of spaces of embeddings, based on [6]

Forgotten in [GBV2014], one can consider also based embeddings. By taking
basepoints $x_{0}\in M$ and $y_{0}\in N,$ we define the principal bundle of
based embeddings.

######  Proposition 6.49.

[6] Let $Emb_{b}(M,N)=\\{f\in Emb(M,N)|f(x_{0})=y_{0}\\}.$ Let

$Diff_{b}(M)=\\{g\in Diff(M)|g(x_{0})=x_{0}\\}\hbox{ and
}Diff_{b,+}(M)=Diff_{b}(M)\cap Diff_{+}(M).$

Let

$B_{b}(M,N)=Emb_{b}(M,N)/Diff_{b}(M,N)\hbox{ and
}B_{b,+}(M,N)=Emb_{b}(M,N)/Diff_{b,+}(M,N).$

Then $Emb_{b}(M,N)$ is a principal bundle with base $B_{b}(M,N)$ (resp.
$B_{b,+}(M,N)$) and with structure group $Diff_{b}(M)$ (resp. $Diff_{b,+}(M)$)

This completes [GBV2014], where structure groups such as volume preserving
diffeomorphisms $Diff_{\mu}(M)$ were considered to build the quotient
$B_{\mu}(M,N)=Emb(M,N)/Diff_{\mu}(M).$

####  6.5.2. Chern forms in infinite dimensional geometry

##### Chern forms in infinite dimensional setting, based on [6]

Surprisingly, the development of Chern-Weil forms using the full space of
(adequate) polynomials, tstead of the ones derived from the standard
polynomials $tr(A^{k}),$ was not present in the literature till [6]. This
remark is fundamental because, followin a remark given by C. Roger, there can
exist, on infinite dimentional Lie algebras, other types of polynomials. Let
us then sketch the general description of Chern-Weil forms.

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
$((p,v),g)\in(P\times\mathfrak{g})\times
G\mapsto(p.g,Ad_{g^{-1}}(v))\in(P\times\mathfrak{g})$. 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 $Ad^{k}P=(AdP)^{\times
k}$ be the product bundle, of basis $M$ and of typical fiber
$\mathfrak{g}^{\times k}$.

######  Definition 6.50.

Let $k$ in $\mathbb{N}^{*}$. We define $\mathfrak{Pol}^{k}(P)$, the set of
smooth maps $Ad^{k}P\rightarrow\mathbb{C}$ that are $k$-linear and symmetric
on each fiber, equivalently as the set of smooth maps
$P\times\mathfrak{g}^{k}\rightarrow\mathbb{C}$ 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 $\mathfrak{Pol}(P)=\bigoplus_{k\in\mathbb{N}*}\mathfrak{Pol}(P)$.

Let $\mathcal{C}(P)$ be the set of connections on $P$. For any
$\theta\in\mathcal{C}(P)$, we denote, only for this section, by $F(\theta)$
its curvature and $\nabla^{\theta}$ (or $\nabla$ when it carries no ambiguity)
its covariant derivation. Given an algebra $A$, In this section, we study the
maps, for $k\in\mathbb{N}^{*}$,

$\displaystyle Ch\quad:\quad\mathcal{C}(P)\times\mathfrak{Pol}^{k}(P)$
$\displaystyle\rightarrow$ $\displaystyle\Omega^{2k}(M,\mathbb{C})$
$\displaystyle(\theta,f)$ $\displaystyle\mapsto$ $\displaystyle
Alt(f(F(\theta),...,F(\theta)))$

where $Alt$ denotes the skew-symmetric part of the form. Notice that, in the
case of the finite dimensional matrix groups $Gl_{n}$ with Lie algebra
$\mathfrak{gl}_{n}$, the set $\mathfrak{Pol}(P)$ is generated by the
polynomials $A\in\mathfrak{gl}_{n}\mapsto\hbox{tr}(A^{k}),$ for
$k\in{0,...,n}$. This leads to classical definition of Chern forms, see e.g.
[KN63-69]. 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 $\mathfrak{Pol}(P).$ Moreover, since the usual trace tr of
matrices is satisfies the ”trace property” $\hbox{\rm
tr}[A,B]=0\quad\forall(A,B)\in M_{n}(\mathbb{C}),$ the classical constructions
of Chern forms, and the related proofs, are deeply simplified compared to what
follows. We must also say that we have been surprised to find nowhere the
following material, proved in [6].

######  Theorem 6.51.

[6] Let $f\in\mathfrak{Pol}(P)$ for which there exists
$\theta\in\mathcal{C}(P)$ such that $[\nabla^{\theta},f]=0.$ We shall note
this set of polynomials by $\mathfrak{Pol}_{reg}(P).$ Then, the map

$Ch^{f}:\theta\in\mathcal{C}(P)\mapsto
Ch^{f}(\theta)=Ch(\theta,f)\in\Omega^{*}(P,\mathbb{C})$

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 $\theta$ on $P$.

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

######  Proposition 6.52.

[6] Let $\phi:\mathfrak{g}^{k}\rightarrow\mathbb{C}$ be a $k-$linear,
symmetric, $Ad-$invariant form. Let
$f:P\times\mathfrak{g}^{k}\rightarrow\mathbb{C}$ be the map induced by $\phi$
by the formula: $f(x,g)=\phi(g).$ Then $f\in\mathfrak{Pol}_{reg}.$

The problem of these infinite dimensional Chern forms was first raised by
Freed [Fr1988-1, Fr1988-2] where the connections considered where $Gl^{p}$
connections, i.e. with curvature valued in the Schatten ideal
$L^{p}=\left\\{A\in L(H)|\,|A|^{p}\hbox{ is trace class }\right\\}.$

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

Mimicking the approach of [23], the cohomology classes of Chern-Weil forms
should give rise to homotopy invariants. Applying Theorem 6.51, we get:

######  Theorem 6.53.

[6] The Chern-Weil forms $Ch^{f}$ is a $H^{*}(B(M,N))-$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. [Hir1976]. We now apply the material of the
previous section to manifolds of embeddings. For this, we can define invariant
polynomials of the type

$A\mapsto\lambda(A^{k})\in\mathfrak{Pol}^{k}_{reg},$

where $\lambda=\hbox{\rm tr}^{Q}$ for a well-chosen weight $Q.$ Let us give
the following example when $M$ is odd-dimensional: the Kontsevich and Vishik
trace [KV1994, KV1995] is a renormalized trace for which $tr^{Q}([A,B])=0$ 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
$\nabla$ is a connection induced on $\mathcal{N}_{f}$ by the Riemannian
metric, as described in [23]. It is an order 2 injective elliptic differential
operator (in the odd class), and the coadjoint action of
$Aut(\mathcal{N}_{f})$ will give rise to another order 2 injective elliptic
differential operator [Gil1984]. When $Q=Id+{\nabla}^{*}{\nabla},$ this only
changes $\nabla$ into another connection on $E.$ Thus,
$\phi(A,...,A)=tr^{Q}(A^{k})\in\mathfrak{Pol}_{reg}.$ Let us now consider a
connected component of $B(M,N),$ i.e. a homotopy class of an embedding among
the space of embeddings. Following [6],

######  Theorem 6.54.

Let $M$ be an odd dimensional boundaryless nanifold and let
$Q=Id+{\nabla}^{*}{\nabla}$ on $\mathcal{N}.$ The polynomial

$\phi:A\mapsto tr^{Q}(A^{k})$

is $Diff(M)-$invariant, and gives rise to an invariant of non oriented knots,
i.e. a Chern form on the base manifold

$B(S^{1},N)=Emb(M,N)/Diff(M)$

Open problem: Prove that these Chern-Weil forms give rise to non-vanishing
characteristic classes. This problem seems related with the work [MP2007]
which is more or less heuristically linked with $FCl^{0,*}-$ connections in
[Pay2013]. The full link between these two settings still needs to be
clarified, even if intuitionistic arguments given both in [Pay2013] and in [6]
show that there are big similitudes between the two independently devolopped
settings.

##  Chapter 7 Contributions to the theory of KP hierarchies

Let $R$ be an algebra of functions equipped with a derivation $\partial.$ For
us, $R=C^{\infty}(S^{1},\mathbb{K})$ with $\mathbb{K}=\mathbb{R},{\mathbb{C}}$
and $\mathbb{H},$ and $\partial=\frac{d}{dx}.$ Let
$T=\\{t_{n}\\}_{n\in\mathbb{N}^{*}}$ be an infinite set of formal (time)
variables and let us consider the algebra of formal series $\Psi
DO(S^{1},\mathbb{K})[[T]]$ with infinite set of formal variables
$t_{1},t_{2},\cdot$ with $T-$valuation $val$ defined by $val_{T}(t_{n})=n$
[Mu1984]. One can extend naturally on $\Psi DO(S^{1},\mathbb{K})[[T]]$ the
notion of smoothness from the same notion on $\Psi DO(S^{1},\mathbb{K}),$ see
[ER2013] for a more complete description. The Kadomtsev-Petviashvili (KP)
hierarchy reads

(7.1) $\frac{dL}{dt_{k}}=\left[(L^{k})_{D},L\right]\;,\quad\quad k\geq 1\;,$

with initial condition $L(0)=L_{0}\in\partial+\Psi^{-1}(R)$, where the
subscrit $(.)_{D}$ means the algebraic canonical projection

$\Psi DO(S^{1},\mathbb{K})[[T]]=(\Psi DO^{-1}(S^{1},\mathbb{K})\oplus
DO(S^{1},\mathbb{K}))[[T]]\rightarrow DO(S^{1},\mathbb{K})[[T]]$

. The dependent variable $L$ is chosen to be of the form
$L=\partial+\sum_{\alpha\leq-1}u_{\alpha}\partial^{\alpha}\in{\Psi}^{1}(S^{1},\mathbb{K})[[T]]\;.$
A standard reference on (7.1) is L.A. Dickey’s treatise [Dic2003], see also
[KZ1995, Mu1984, Mu1983, RS1981] as well as [Mick1989, PS1988, SW1985, Ue1985,
Wan1983, Wan1984] for a link with infinite dimensional geometric structures
(determinant bundles, restricted linear group $GL_{res}$) and with non linear
partial differential equations such as KdV, Boussinesq, KP equations. The KP
hierarchy produces a formal solution to all these systems, under mild
assumptions on the initial conditions and constraints. It can be derived from
the search of isospectral deformations of diferential operators, in link with
the so-called Gelfand-Dickey hierarchy, and it recently appeared in various
contexts such as Hodge theory and combinatorics.

In order to solve the KP hierarchy, we need the following groups (see e.g.
[16] for a latest adaptation of Mulase’s construction [Mu1984, Mu1983]):

$\bar{G}=1+\Psi DO^{-1}(S^{1},\mathbb{K})[[T]],$
$\widehat{\Psi}=\left\\{P=\sum_{\alpha\in{\mathbb{Z}}}a_{\alpha}\,\partial^{\alpha}\in{\Psi}(S^{1},\mathbb{K})[[T]]\;:\exists
N\in\mathbb{N},\,val_{T}(a_{\alpha})>_{\alpha\rightarrow+\infty}C\alpha-N\hbox{
and }P|_{t=0}\in 1+\Psi DO^{-1}(S^{1},\mathbb{K})\right\\}$

and

$\widehat{\mathcal{D}}=\left\\{P=\sum_{\alpha\in\mathbb{Z}}a_{\alpha}\,\partial^{\alpha}:P\in\widehat{\Psi}\mbox{
and }a_{\alpha}=0\mbox{ for }\alpha<0\right\\}\;.$

We have a matched pair $\widehat{\Psi}=\bar{G}\bowtie\widehat{\mathcal{D}}.$
The same constructions apply in various settings, see e.g. [Dem1995-1,
Dem1995-2, Ku2000, McI2011], and all are envolved with purely algebraic
arguments in contrast with our work.

###  7.1. Smoothness of Mulase decomposition and well-posedness of the KP
hierarchy, based on [10, 15, 16]

When algebras of functions $R=C^{\infty}(S^{1},\mathbb{K})$ are Fréchet
algebras, a natural notion of differentiability occurs, making addition,
multiplication and differentiation smooth. By the way, considering addition
and multiplication in $\Psi DO(S^{1},\mathbb{K}),$ one can say that addition
and multiplication in $\Psi DO(S^{1},\mathbb{K})$ by understanding, under this
terminology, that, if $A=\sum_{n\in\mathbb{Z}}a_{n}\partial^{n}$ and
$B=\sum_{n\in\mathbb{Z}}b_{n}\partial^{n},$ setting
$A+B=C=\sum_{n\in\mathbb{Z}}c_{n}\partial^{n}$ and
$AB=D=\sum_{n\in\mathbb{Z}}d_{n}\partial^{n}$ the map

$\left((a_{n})_{n\in\mathbb{Z}},(b_{n})_{n\in\mathbb{Z}}\right)\mapsto\left((c_{n})_{n\in\mathbb{Z}},(d_{n})_{n\in\mathbb{Z}}\right)$

is smooth in the relevant infinite product, encoded into diffeological
concepts in [10, 15, 16] where a fully rigorous framework for smoothness on
these objects is described and used. Again following [10], the same smoothness
properties can be described for any differential unital algebra
$(A,\partial),$ but in this general setting, diffeologies seem to be an
essential tool. Let us describe more precisely the setting. Let $A^{*}$ be the
group of invertible elements (or, units) of $A$. Let $\xi$ be a formal
variable not in $A$. The algebra of symbols over $A$ is the vector space

$\Psi_{\xi}(A)=\left\\{P_{\xi}=\sum_{\nu\in{\bf
Z}}a_{\nu}\,\xi^{\nu}\;|\;a_{\nu}\in A\,,\;a_{\nu}=0\mbox{ for }\nu\gg
0\right\\}\;$

equipped with the associative multiplication $\circ\,$ given by

(7.2) $P_{\xi}\circ Q_{\xi}=\sum_{k\geq
0}\frac{1}{k!}\,\frac{\partial^{k}P_{\xi}}{\partial\xi^{k}}\,\partial^{k}Q_{\xi}\;,$

with the prescription that multiplication on the right hand side of (7.2) is
standard multiplication of Laurent series in $\xi$ with coefficients in $A$,
see [Dic2003]. The algebra $A$ is included in $\Psi_{\xi}(A)$. The algebra of
formal pseudodifferential operators over $A$ is the vector space

$\Psi(A)=\left\\{P=\sum_{\nu\in\mathbb{Z}}a_{\nu}\,\partial^{\nu}\;|\;a_{\nu}\in
A\,,\;a_{\nu}=0\mbox{ for }\nu\gg 0\right\\}$

equipped with the unique multiplication which makes the map
$\sum_{\nu\in\mathbb{Z}}a_{\nu}\,\xi^{\nu}\;\mapsto\;\sum_{\nu\in\mathbb{Z}}a_{\nu}\,\partial^{\nu}$
an algebra homomorphism. The algebra $\Psi(A)$ is associative but not
commutative. It becomes a Lie algebra over $K$ if we define, as usual,

(7.3) $[P,Q]=P\,Q-Q\,P\;.$

The order of $P\neq 0\in\Psi(A)$,
$P=\sum_{\nu\in\mathbb{Z}}a_{\nu}\,\partial^{\nu}$, is $N$ if $a_{N}\neq 0$
and $a_{\nu}=0$ for all $\nu>N$. If $P$ is of order $N$, the coefficient
$a_{N}$ is called the leading term or principal symbol of $P$. We note by
$\Psi^{N}(A)$ the vector space of pseudodifferential operators $P$ as above
satisfying $k>N\Rightarrow a_{k}=0.$ We note

$\Psi^{N}(A)=\left\\{a\in\Psi(A)\;|\;\forall m>N,a_{m}=0\right\\}\;.$

The special case of $\Psi^{0}(A)$ is of particular interest, since it is an
algebra. Adapting the notations used in the previous subsection, we write
$\Psi^{0,*}(A)$ for its group of units, i.e. the group of invertible elements
of $\Psi^{0}(A).$ We assume now that the algebra $A$ is a Frölicher algebra,
and that therefore addition, scalar multiplication and multiplication are
smooth, and that inversion is a smooth operation on $A^{*}$, in which $A^{*}$
is equipped with the subset Frölicher structure. We also assume that the
derivation $\partial$ is smooth. Then, identifying a formal operator
$P\in\Psi(A)$ with its sequence of partial symbols, we conclude that
$\Psi(A),$ as a linear subspace of $A^{\mathbb{Z}},$ carries a natural
Frölicher structure. We obtain:

######  Proposition 7.1.

[10] $\Psi(A)$ is a Frölicher algebra.

Now let us assume that $A^{*}$ is a Frölicher Lie group with Lie algebra
$\mathfrak{g}_{A}$, and that $A$ is an integral Frölicher vector space. Our
Theorems 4.6 and 4.7 imply the following two results:

######  Lemma 7.2.

[10] $1+\Psi^{-1}(A)$ is a regular Frölicher Lie group with regular Lie
algebra $\Psi^{-1}(A).$

######  Theorem 7.3.

[10] There exists a short exact sequence of groups:

$1\longrightarrow 1+\Psi^{-1}(A)\longrightarrow\Psi^{0,*}(A)\longrightarrow
A^{*}\longrightarrow 1$

such that:

  1. (1)

The injection $1+\Psi^{-1}(A)\rightarrow\Psi^{0,*}(A)$ is smooth.

  2. (2)

The principal symbol map $\Psi^{0,*}(A)\rightarrow A^{*}$ is smooth and it has
a global section which is the restriction to $A^{*}$ of the canonical
inclusion $A\rightarrow\Psi^{0}(A).$

As a consequence, $A^{*}$ is a fully regular Frölicher Lie group if and only
if $\Psi^{0,*}(A)$ is a fully regular Frölicher Lie group with Lie algebra
$\mathfrak{g}_{A}\oplus\Psi^{-1}(A).$

Open problem: It is natural to wonder if we can extend Theorem 7.3 to the full
algebra $\Psi(A),$ and to the Frölicher Lie group $\Psi(A)^{*}$ separately.

  * •

On one hand, in [16], for $A=\mathbb{R},$ we get that
$\Psi(A)\sim\mathbb{R}((X))$ is the Lie algebra of a non-regular Lie group,
which shows that the problem of enlargibility of $\Psi(A)$ cannot be solved by
finding an underlying regular Frölicher Lie group.

  * •

The group $\Psi(A)^{*}$ has formally two candidates as Lie algebras. The first
one is a Lie subalgebra of $\Psi(A)$ which has to be determined depending
certainly on $\mathfrak{g}_{A}$ and the other one yields a formal
correspondence with the group by a formal exponential described in e.g.
[KW2009] for $A=C^{\infty}(\mathbb{R}).$ In each of these more or less formal
settings, the exact sequence envolving the group $1+\Psi^{-1}(A)$ does not
seem so straightforward to generalize.

####  7.1.1. Smoothness of Mulase decomposition and refinements

We can generalize our foregoing discussion. We hope this generalization will
be of use, for instance, in the development of $p$-adic KP theory. Our basic
idea is to replace algebras of power series by general algebras equipped with
non-archimidean valuations. We only present the main points of this
generalization and we refer to [10] for details. Let $A$ be an associative
(but not necessarily commutative!) $K$–algebra with unit $1$, in which $K$ is
an arbitrary field of characteristic zero. We assume that $A$ is a
diffeological algebra and that $A$ is equipped with a smooth derivation $D$.
We consider $\Psi(A)$ as in the previous subsection, but instead of assuming
that $A$ is an algebra of series, we equip it with a valuation, adapting an
idea from [Dem1995-1, Dem1995-2]. A valuation allows us to equip $A$ with a
topology. For $\alpha\in\mathbb{Z}$ and $x_{0}\in A$ we set

$V_{\alpha}(x_{0})=\\{x\in A:\sigma(x-x_{0})>\alpha\\}\;;$

we easily see that for each $x_{0}\in A$, the collection
$\\{V_{\alpha}(x_{0})\\}_{\alpha\in\mathbb{Z}}$ is a basis of neighborhoods
for a first countable Hausdorff topology on $A$. It is a classical fact that
this topology is metrizable: we define the absolute value classicaly and if we
set $d(x,y)=|x-y|$ we obtain a metric $d$ on $A$ and the metric topology
coincides with the topology introduced above. Very importantly, in this
topology the sets $V_{\alpha}(x_{0})$ coincide with the balls $\\{x\in
A:|x|<c^{\alpha}\\}$, and are both open and closed. $A$ becomes a topological
algebra and $D$ a continuous derivation. We also remark that compatibility of
derivation and valuation implies that $|D(a)|\leq|a|$ for all $a\in A$.

Now we let $\hat{A}$ be the completion of the metric space $(A,d)$, and we
extend $D$ to a continuous derivation on $\hat{A}$. We continue denoting the
extension of the absolute value on $A$ by $|\cdot|$, and the extension of the
continuous derivation on $A$ by $D$.

In the previous subsection we used Frölicher algebras and obtained Frölicher
structures on spaces of (regularized) formal pseudodifferential operators. It
turns out that, because of the presence of quotients, the use of diffeologies
is more natural than the use of Frölicher spaces, since the latter structure
does not pass so easily to quotients and carry some problems of non-
reflexivity (for a definition of reflexivity in diffeological spaces, see
[Wa2012]). At the present level of generality, we proceed as follows, after
[10]:

For all $p\in\mathbb{Z},$ the quotient vector space projection
$\pi_{p}:A\rightarrow A/A_{p}\,$, in which $A_{p}=\\{a\in A\,:\,\sigma(a)\geq
p\\},$ extends to $\hat{A}$ in the following way: for $\hat{a}\in\hat{A}$, we
set $\pi_{p}(\hat{a})=a+A_{p}$ if and only if
$\sigma(a)=\hat{\sigma}(\hat{a})$ and $\hat{\sigma}(a-\hat{a})\geq p$. It is
possible to find such an $a\in A$ because of standard properties of
valuations, as explained in [10].

######  Definition 7.4.

We equip the quotients $A/A_{p}$ with their quotient diffeology. The
completion $\hat{A}$ is equipped with the pull-back diffeology with respect to
the family of maps $\\{\pi_{p};p\in\mathbb{Z}\\}$ (see [IZ2013, p. 32]). The
valuation $\sigma$ of $A$ is called a diffeological valuation if and only if
the diffeology of $A$ is the pull-back of the diffeology of $\hat{A}$ and all
plots are continuous in the valuation topologies of $A$ and $\hat{A}$.

We assume that $A$ and $\hat{A}$ are equipped with diffeological valuations
$\sigma$ and $\hat{\sigma}$, and that $D$ is smooth with respect to the
product diffeology on $\hat{A},$ which implies that its restriction to $A$ is
also smooth. The spaces of formal pseudodifferential and differential
operators of infinite order now become:

######  Definition 7.5.

[10] The space of formal pseudodifferential and differential operators of
infinite order are, respectively, $\widehat{\Psi}(\hat{A})$ and
$\widehat{\mathcal{D}}_{\hat{A}}\,$, in which

(7.4)
$\begin{split}\widehat{\Psi}(\hat{A})=\left\\{P=\sum_{\alpha\in{\mathbb{Z}}}a_{\alpha}\,D^{\alpha}\;|\;a_{\alpha}\in\hat{A}\mbox{
and }\exists A_{P},B_{P}\in\mathbb{R}^{+}\mbox{ and
}M_{P},N_{P},L_{P}\in\mathbb{Z}^{+}\mbox{ so that }\right.\\\ \left.M_{P}\geq
N_{P}\,,\;|a_{\alpha}|<\frac{A_{P}}{\alpha-N_{P}}\ \forall\
\alpha>M_{P}\,,\mbox{ and }|a_{\alpha}|<B_{P}\ \forall\
\alpha<-L_{P}\right\\}\end{split}$

and

(7.5)
$\widehat{\mathcal{D}}_{\hat{A}}=\left\\{P=\sum_{\alpha\in\mathbb{Z}}a_{\alpha}\,D^{\alpha}\;|\;P\in\widehat{\Psi}(\hat{A})\mbox{
and }a_{\alpha}=0\mbox{ for }\alpha<0\right\\}\;.$

The definition of the absolute value $|\cdot|$ implies that $\hat{A}$ is
contained in $\widehat{\Psi}(\hat{A})$ and, as a by-product, we note that our
assumptions on $A$ and $\hat{A}$ imply that $\widehat{\Psi}(\hat{A})$ and
$\widehat{\mathcal{D}}(\hat{A})$ are diffeological spaces.

######  Lemma 7.6.

[10] The space $\widehat{\Psi}(\hat{A})$ has an algebra structure and
$\widehat{\mathcal{D}}_{\hat{A}}$ is a subalgebra of
$\widehat{\Psi}(\hat{A})$. Moreover, if $\hat{A}$ is a diffeological
K-algebra, $\widehat{\Psi}(\hat{A})$ and $\widehat{\mathcal{D}}_{\hat{A}}$ are
diffeological K-algebras.

Now we construct groups. There exist two standard structures associated to the
valuation $\hat{\sigma}$ on $\hat{A}$. The subring
$\mathcal{O}_{\hat{A}}=\\{a\in\hat{A}:\hat{\sigma}(a)\geq 0\\}$, and the two-
sided ideal
$\mathcal{P}_{\hat{A}}=\\{a\in\mathcal{O}_{\hat{A}}:\hat{\sigma}(a)>0\\}$. If
$A$ is a diffeological algebra, these algebraic constructions carry natural
underlying diffeologies. Since we can check that the derivation $D$ on
$\hat{A}$ is compatible with $\hat{\sigma}$, we have
$D(\mathcal{P}_{\hat{A}})\subset\mathcal{P}_{\hat{A}}\,$; it follows that the
derivation $D$ is well-defined on the quotient ring
$\mathcal{O}_{\hat{A}}/\mathcal{P}_{\hat{A}}\,$. We let
$\pi:\mathcal{O}_{\hat{A}}\rightarrow\mathcal{O}_{\hat{A}}/\mathcal{P}_{\hat{A}}$
be the canonical projection. Since $A$ is a diffeological algebra, the map
$\pi$ is smooth.

The set
$G_{\hat{A}}=1+\mathcal{I}_{\mathcal{O}_{\hat{A}}/\mathcal{P}_{\hat{A}}}$ is a
multiplicative group (see [Dem1995-2]), and a diffeological group according to
Theorem 4.6. For
$P=\sum_{\nu\in\mathbb{Z}}a_{\nu}D^{\nu}\in\widehat{\Psi}(\mathcal{O}_{\hat{A}})$
we set $\pi(P)=\sum_{\nu\in\mathbb{Z}}\pi(a_{\nu})D^{\nu}$.

######  Definition 7.7.

We define the spaces

(7.6)
$\widehat{\Psi}(\hat{A})^{\times}=\\{P\in\widehat{\Psi}(\mathcal{O}_{\hat{A}})\;|\;\pi(P)\in
G_{\hat{A}}\\}$

and

(7.7)
$\widehat{\mathcal{D}}_{\hat{A}}^{\times}=\\{P\in\widehat{\mathcal{D}}_{\mathcal{O}_{\hat{A}}}\;|\;\pi(P)=1\\}\;.$

######  Proposition 7.8.

The space $\widehat{\Psi}(\hat{A})^{\times}$ is a group: each element $P$ in
$\widehat{\Psi}(\hat{A})^{\times}$ has an inverse of the form

$P^{-1}=\sum_{n\geq 0}(1-P)^{n}\;.$

In addition, the space $\widehat{\mathcal{D}}_{\hat{A}}^{\times}$ is a
subgroup of $\widehat{\Psi}(\hat{A})^{\times}$.

We can prove the following result on smoothness, see [10].

######  Proposition 7.9.

The group $G(\Psi(\hat{A})):=\widehat{\Psi}(\hat{A})^{\times}$ is a
diffeological Lie group with Lie algebra $\Psi(\mathcal{O}_{\hat{A}})$; the
group $G_{+}(\mathcal{D}_{\hat{A}}):=\widehat{\mathcal{D}}_{\hat{A}}^{\times}$
is a diffeological Lie group with Lie algebra
$\mathcal{D}_{\mathcal{O}_{\hat{A}}}$; the group
$G_{-}(\mathcal{I}_{\hat{A}}):=1+{\mathcal{I}}_{\mathcal{O}_{\hat{A}}}$ is a
diffeological Lie group with Lie algebra
${\mathcal{I}}_{\mathcal{O}_{\hat{A}}}$. Moreover, the exponential map

$\exp:P\in{\mathcal{I}}_{\mathcal{O}_{\hat{A}}}\mapsto\sum_{n\in\mathbb{N}}\frac{(sP)^{n}}{n!}\in
G_{-}(\mathcal{I}_{\hat{A}})$

is one-to-one and onto, with inverse the classical logarithmic series $\log$,
and both $\exp$ and $\log$ are smooth. As a consequence, the inversion in
smooth in $G_{-}(\mathcal{I}_{\hat{A}})$.

Let us now specialize to $A=C^{\infty}(S^{1},\mathbb{R})$ as in [15, 16],

######  Theorem 7.10.

The following algebras are Frölicher algebras:

(1) $A_{t}\,$; (2) $\Psi(A_{t})\,$; (3) $\widehat{\Psi}(A_{t})\,$; (4)
$\widehat{\mathcal{D}}_{A_{t}}\,$.

######  Theorem 7.11.

  1. (1)

$\widehat{\mathcal{D}}_{A_{t}}^{\times}$ is a Frölicher group.

  2. (2)

$G_{R}=1+\Psi^{-1}(R)$ and $G_{A_{t}}=1+\Psi^{-1}(A_{t})$ are Frölicher Lie
groups.

  3. (3)

$G(\widehat{\Psi}(A_{t}))$ is a Frölicher group.

We stress the fact that we have not stated the existence of an exponential
map. In fact, it seems difficult to show the existence of the exponential map
on $\widehat{\Psi}(A_{t})$, and very difficult to determine the tangent space
$T_{1}G(\widehat{\Psi}(A_{t}));$ even more, it is difficult to differentiate
the possible adjoint action of $G(\widehat{\Psi}(A_{t}))$ on it. Most of the
difficulties come from the very general definition of $\widehat{\Psi}(A_{t})$.
This is why we construct a Frölicher subalgebra
$\overline{\Psi}(A_{t})\subset\widehat{\Psi}(A_{t})$ as follows:

######  Definition 7.12.

The regularized space of formal pseudo-differential and differential operators
of infinite order are, respectively, $\overline{\Psi}(A_{t})$ and
$\overline{\mathcal{D}}_{A_{t}}$, in which

(7.8)
$\overline{\Psi}(A_{t})=\left\\{\sum_{\alpha\in{\mathbb{Z}}}a_{\alpha}\,\partial^{\alpha}\in\widehat{\Psi}(A_{t})\;|\;val_{t}(a_{\alpha})\geq\alpha\right\\}$

and

(7.9)
$\overline{\mathcal{D}}_{A_{t}}=\left\\{P=\sum_{\alpha\in\mathbb{Z}}a_{\alpha}\,\partial^{\alpha}\;|\;P\in\overline{\Psi}(A_{t})\mbox{
and }a_{\alpha}=0\mbox{ for }\alpha<0\right\\}\;.$

In addition, we define

(7.10)
$G(\overline{\Psi}(A_{t}))=\\{P\in\overline{\Psi}(A_{t})\;|\;P|_{t=0}\in
G_{A_{t}}\\}$

and

(7.11)
$\overline{\mathcal{D}}_{A_{t}}^{\times}=\\{P\in\overline{\mathcal{D}}_{A}\;|\;P|_{t=0}=1\\}\;,$

and we can prove, see [16], that $G(\overline{\Psi}(A_{t}))$ and
$\overline{\mathcal{D}}_{A_{t}}^{\times}$ are fully regular Frölicher Lie
groups.

####  7.1.2. Well-posedness of the KP hierarchy

The following result gives a synthesied statement of the main result of [16]
and of [10] both on the KP hierarchy (7.1), by considering two approaches of
“smoothness”

  1. (1)

In the general setting of [10], smoothness is only with respect to
$A-$coefficients of pseudodifferential operators, and the field of scalars is
assumed to be equipped with the discrete topology (and discrete diffeology).
this is classically what is needed to take safe limits in the ultrametric
completions,

  2. (2)

In the classical setting of [16], when $A=C^{\infty}(S^{1},\mathbb{R})$
principally and extended to $A=C^{\infty}(S^{1},\mathbb{R})^{\mathbb{N}}$ in
the same reference, the smoothness is understood with respect to $A$ and with
respect to the variable
$(t_{1},t_{2},...)\in\mathbb{R}^{\infty}=\bigcup_{n\in\mathbb{N}}\mathbb{R}^{n}.$

It states smooth dependence on the initial conditions and on the values
$t_{1},t_{2},...$ which justifies the terminology of well-posedness.

######  Theorem 7.13.

[16] Consider the KP hierarchy 7.1 with initial condition $L(0)=L_{0}$. Then,

  1. (1)

There exists a pair $(S,Y)\in\widehat{G}\times\widehat{\mathcal{D}}$ (resp.
$\in\bar{G}\times\overline{\mathcal{D}}$ ) such that the unique solution to
Equation $(\ref{eq:KP})$ with $L|_{t=0}=L_{0}$ is
$L(t_{1},t_{2},\cdots)=Y\,L_{0}\,Y^{-1}=SL_{0}S^{-1}$.

  2. (2)

The pair $(S,Y)$ is uniquely determined by the smooth decomposition problem

$exp\left(\sum_{k\in\mathbb{N}}\tau_{k}L_{0}^{k}\right)=S^{-1}Y$

and the solution $L$ depends smoothly on the initial condition $L_{0}$.

  3. (3)

The solution operator $L$ is smoothly dependent on the initial value $L_{0}.$

######  Remark 7.14.

One may notice in [16] two proofs of Theorem 7.13. In the first proof of the
third item of this Theorem, and more generally the first proof of this
theorem; is inspired by Reyman and Semenov-Tian-Shansky approach to
integrability via R-matrices and factorization theorems, see for instance
[Per1990, Section 1.12, Theorem 7]. However, our result is not exactly an
instance of the Reyman–Semenov-Tian-Shansky theory since in this paper we are
not considering the hamiltonian content of Equation $(\ref{jph})$. What we are
observing here is that techniques appropriated for the study of integrability
of Hamiltonian systems can be adapted to prove well-posedness of the
interesting equation $(\ref{eq:KP})$. As in Mulase’s papers [Mu1984, Mu1983],
the crucial point of the proof is the existence of a factorization of an
infinite-dimensional Lie group, and not the possible hamiltonian character of
the equation being investigated.

Open question: sequences of approximation, pseudo-differential operators,
Hamiltonnians and unstable solutions.

Initiated in [16], the procedure that consists in replacing non smooth
functions by approximating sequences of smooth functions enable to enlarge the
theory of non-smooth symbols initiated in [BR1984], see e.g. [Marsch1998], to
symbols which approximation sequence converge in function spaces which are not
embedded in $C^{0}.$ This embedding is the technical limitation of the actual
theory. The classical results on pseuo-differential operators, such as
boundedness, norm estimates, kernel analysis, spectral theory, then need to be
analyzed in this generalized context, with in mind the main application of
this theory which is the analysis of PDEs. Among these PDEs, the hamiltonnian
equations can generalize straight way, again along the lines of the ideas
announced in [16], replacing the usual $\mathbb{C}-$valued non degenerate
pairing on regular polynomials on $DO(M)$ by a $\mathbb{C}[[z]]-$valued
pairing for hamiltonnians of the equation extended to approximation sequences.
Thus first integrals of the (generalized) motion are $\mathbb{C}[[z]]$-valued
functionals, which convergence may intuitively depend on the stability of the
solutions.

###  7.2. A scaling for the KP hierarchy and the $h-$KP hierarchy, based on
[4]

We make the following definition, along the lines of the theory developed in
[4] for formal pseudo-differential operators:

######  Definition 7.15.

Let $h$ be a formal parameter. The set of odd formal class $h-$pseudo-
differential operators is the set of formal series

$\Psi DO_{h}(S^{1},V)=\left\\{\sum_{n\in\mathbb{N}}a_{n}h^{n}\,|\,a_{n}\in\Psi
DO^{n}(S^{1},V)\right\\}\;.$

We state the following result on the structure of $\Psi DO_{h}(S^{1},V)$:

######  Theorem 7.16.

The set $\Psi DO_{h}(S^{1},V)$ is a Fréchet algebra, and its group of units
given by

$\Psi
DO_{h}^{*}(S^{1},V)=\left\\{\sum_{n\in\mathbb{N}}a_{n}h^{n}\,|\,a_{n}\in\Psi
DO^{n}(S^{1},V),a_{0}\in\Psi DO^{0,*}(S^{1},V)\right\\}\;,$

is a regular Fréchet Lie group.

This result is mostly an application of Theorem 4.6. Decomposition
$L=L_{S}+L_{D}$, $L_{S}\in\Psi DO^{-1}(S^{1},V)$, $L_{D}\in DO^{1}(S^{1},V)$
valid on $\Psi DO(S^{1},V)$ extends straightforwardly to the algebra $\Psi
DO_{h}(S^{1},V)$. We now introduce the $h-$KP hierarchy with non-formal
pseudo-differential operators. Let us assume that
$t_{1},t_{2},\cdots,t_{n},\cdots,$ are an infinite number of different formal
variables. Then, again adapting work carried out in [4], we make the following
definition:

######  Definition 7.17.

Let $S_{0}\in Cl^{-1,*}_{odd}(S^{1},V)$ and let
$L_{0}=S_{0}(h\frac{d}{dx})S_{0}^{-1}.$ We say that an operator

$L(t_{1},t_{2},\cdots)\in\Psi DO_{h}(S^{1},V)[[ht_{1},...,h^{n}t_{n}...]]$

satisfies the $h-$deformed KP hierarchy if and only if

(7.12) $\left\\{\begin{array}[]{cl}L(0)=&L_{0}\\\
\frac{d}{dt_{n}}L=&\left[(L^{n})_{D},L\right]\;.\end{array}\right.$

We recall from [4] that the $h-$KP hierarchy is obtained from the classical KP
hierarchy by means of the $h-$scaling

$\left\\{\begin{array}[]{ccc}t_{n}&\mapsto&h^{n}t_{n}\\\
\frac{d}{dx}&\mapsto&h\frac{d}{dx}\end{array}\right.\;,$

and we also recall that formal series in $t_{1},\cdots,t_{n},\cdots$ can be
also understood as smooth functions on the algebraic sum

$T=\bigoplus_{n\in\mathbb{N}^{*}}(\mathbb{R}t_{n})$

for the product topology and product Frölicher structure, see [4]. Now we
solve the initial value problem for (7.12).

######  Theorem 7.18.

Let $U_{h}=\exp\left(\sum_{n\in N^{*}}h^{n}t_{n}(L_{0})^{n}\right)\in
Cl_{h}(S^{1},V).$ Then:

  * •

There exists a unique pair $(S,Y)$ such that

    1. (1)

$U_{h}=S^{-1}Y,$

    2. (2)

$Y\in\Psi DO_{h}^{*}(S^{1},V)_{D}$

    3. (3)

$S\in\Psi DO_{h}^{*}(S^{1},V)$ and $S-1\in Cl_{h,odd}(S^{1},V)_{S}.$

Moreover, the map

$(S_{0},t_{1},...,t_{n},...)\in Cl^{0,*}_{odd}(S^{1},V)\times
T\mapsto(U_{h},Y)\in(\Psi DO_{h}^{*}(S^{1},V))^{2}$

is smooth.

  * •

The operator $L\in\Psi DO_{h}(S^{1},V)[[ht_{1},...,h^{n}t_{n}...]]$ given by
$L=SL_{0}S^{-1}=YL_{0}Y^{-1}$, is the unique solution to the hierarchy of
equations

(7.13)
$\left\\{\begin{array}[]{ccl}\frac{d}{dt_{n}}L&=&\left[(L^{n})_{D}(t),L(t)\right]=-\left[(L^{n})_{S}(t),L(t)\right]\\\
L(0)&=&L_{0}\\\ \end{array}\right.\;,$

in which the operators in this infinite system are understood as formal
operators and $A\mapsto(A)_{D}$ means projection into the space of
differential operators, obtained by cutting the part of negative order along
the lines of e.g. [Mu1984], and which corresponds to the projection $A\mapsto
A_{D}$ already defined on non formal, odd class pseudo-differential operators.

######  Definition 7.19.

In what follows, $val_{h}$ denotes standard valuation of $h$ series. Let

$\Psi_{h}(R)=\left\\{\sum_{\alpha\in\mathbb{Z}}a_{\alpha}\partial^{\alpha}\in\Psi(R)[[h]]\;|\;val_{h}(a_{\alpha})\geq\alpha\right\\}\;,$
$G\Psi_{h}(R)=\left\\{\sum_{\alpha\in\mathbb{Z}}a_{\alpha}\partial^{\alpha}\in\Psi_{h}(R)\;|\;a_{0}=1+b_{0},\quad
val_{h}(b_{0})\geq 1\right\\},$ $G_{R,h}=\left\\{A\in
G\Psi_{h}(R)\;|\;A=1+B,\quad B\in\Psi^{-1}(R)[[h]]\right\\}\;,$
$\mathcal{D}_{h}(R)=\left\\{\sum_{\alpha\in\mathbb{Z}}a_{\alpha}\partial^{\alpha}\in\Psi_{h}(R)\;|\;a_{\alpha}=0\hbox{
if }\alpha<0\hbox{ and }a_{0}=1+b_{0},\quad val_{h}(b_{0})\geq 1\right\\}\;.$

The next Lemma is proved in [4]; a shorter proof appears in [16].

######  Lemma 7.20.

$\Psi_{h}(R)$ is a Frölicher algebra, an integral Lie algebra, and the groups
$G\Psi_{h}(R),$ $G_{R,h}$ and $\mathcal{D}_{h}(R)$ are regular Frölicher Lie
groups with Lie algebras given respectively by:

$\mathfrak{g}\Psi_{h}(R)=\left\\{\sum_{\alpha\in\mathbb{Z}}a_{\alpha}\partial^{\alpha}\in\Psi_{h}(R)\;|\;val_{h}(a_{0})\geq
1\right\\}\;,$ $\mathfrak{g}_{R,h}=\Psi^{-1}(R)[[h]]\;,$

and

$\mathfrak{d}_{h}(R)=\left\\{\sum_{\alpha\in\mathbb{Z}}a_{\alpha}\partial^{\alpha}\in
D_{h}(R)\;|\;a_{0}=0\hbox{ if }\alpha<0\quad val_{h}(a_{0})\geq 1\right\\}\;.$

######  Theorem 7.21.

[16] Mulase decomposition holds for $W\in G\Psi_{h}(R),$ $S\in G_{R,h}$ and
$Y\in\mathcal{D}_{h}(R)$ and, in particular, with
$T=\cup_{n\in\mathbb{N}}\mathbb{R}^{n},$
$R=C^{\infty}(T,C^{\infty}(S^{1},\mathbb{R}))=C^{\infty}(T\times
S^{1},\mathbb{R})$ and $\partial=\frac{d}{dx}$ on $S^{1},$ recovering [4]. The
map $U\mapsto(S,Y)$ is smooth.

###  7.3. The KP hierarchy on an extended class of formal Pseudo-differential
operators based on [20]

The algebra of operators that we intend to use in this section is the algebra
of formal classical pseudo-differential operators
$\mathcal{F}Cl(S^{1},\mathbb{K}^{n})$ that are obtained from classical pseudo-
differential operators acting on smooth sections of the trivial vector bundle
$S^{1}\times\mathbb{K}^{n}$ over $S^{1},$ for $K=\mathcal{C}$ or $\mathbb{H},$
see e.g. [Gil1984, Pay2012]. The key properties of
$\epsilon(D)=D|D|^{-1}=|D|^{-1}D$ that we use in our constructions are:

  * •

the formal operator $\epsilon(D)\in\mathcal{F}Cl(S^{1},\mathbb{K}^{n})$
commutes with any formal operator $A\in\mathcal{F}Cl(S^{1},\mathbb{K}^{n}),$

  * •

$\epsilon(D)^{2}=Id$

  * •

the composition on the left $A\mapsto\epsilon(D)\circ A$ is an endomorphism of
the algebra $\mathcal{F}Cl(S^{1},\mathbb{K}^{n}),$ which restricts to a
bijective map from $\Psi
DO(S^{1},\mathbb{K}^{n})=\mathcal{F}Cl_{odd}(S^{1},\mathbb{K}^{n})$ to an
algebraic complement in $\mathcal{F}Cl(S^{1},\mathbb{K}^{n})$ noted as
$\mathcal{F}Cl_{even}(S^{1},\mathbb{K}^{n})$ following the terminology of
[Sco2010]

  * •

the restriction of the Wodzicki residue to $\Psi
DO(S^{1},\mathbb{K}^{n})=\mathcal{F}Cl_{odd}(S^{1},\mathbb{K}^{n})$, which is
similar to but not equal to the Adler functional, is vanishing.

We have already stated the following:

  * •

The space $\mathcal{F}Cl(S^{1},\mathbb{K}^{n})$ splits in various ways: one is
derived from the splitting of $T^{*}S^{1}-S^{1}$ into two connected components
, the splitting with respect to $\Psi DO(S^{1},\mathbb{K}^{n})$ as a
subalgebra , and the extension of the splitting related to the classical Manin
triple on $\Psi DO(S^{1},\mathbb{K}^{n})$ to
$\mathcal{F}Cl(S^{1},\mathbb{K}^{n})$.

  * •

The operator $\epsilon(D)$ is in the center of
$\mathcal{F}Cl(S^{1},\mathbb{K}^{n}).$ It generates then a polarized Lie
bracket using it as a $\mathbf{r}-$matrix and an integrable almost complex
structure on $\mathcal{F}Cl(S^{1},\mathbb{K}^{n}).$

These technical features enables us to state the announced main results of
this section: existence and uniqueness of solutions of the KP hierarchy with
various initial conditions (section 7.3.1) and KP hierarchy with complex
powers (section 7.3.2).

####  7.3.1. Multiple classical KP hierarchies on
$\mathcal{F}Cl(S^{1},\mathbb{K})$

The (classical) KP hierarchy on $\Psi DO(S^{1},\mathbb{K})$ can then push-
forward on $\mathcal{F}Cl$-classes of operators by various ways.

###### Push-Forward via $\Phi_{\lambda,\mu}$ maps

Let $\mathbb{K}={\mathbb{C}}$ or $\mathbb{H}.$ For each choice of
$(\lambda,\mu)\in\mathcal{C}^{2}\backslash\\{0;0\\}$ identifies
$\frac{d}{dx}\in\Psi DO(S^{1},\mathbb{K})$ with an operator in
$\mathcal{F}Cl(S^{1},\mathbb{K})$ with the same algebraic properties.
Notation: $\partial_{\lambda,\mu}=\Phi_{\lambda,\mu}\left(\frac{d}{dx}\right)$
and $\mathcal{F}Cl_{\lambda,\mu}(S^{1},\mathbb{K})=Im\Phi_{\lambda,\mu}.$ Then
we can develop the KP hierarchy on
$\mathcal{F}Cl_{\lambda,\mu}(S^{1},\mathbb{K}).$ We first remark that, since
each map $\Phi_{\lambda,\mu}$ is a degree $0$ morphism of filtered algebras,
each push-forward of the unique solotion $L$ of the KP hierachy (7.1)
generates a solution of the corresponding equation in
$\mathcal{F}Cl(S^{1},\mathbb{K})$ which reads the same way:

$\frac{dL}{dt_{k}}=\left[(L^{k})_{D},L\right]\;,\quad\quad k\geq 1\;,$

where solutions operators now belong to
$\mathcal{F}Cl^{1}(S^{1},\mathbb{K})[[T]]$ and where each initial value
$\Phi_{\lambda,\mu}(L_{0})\in\partial_{\lambda,\mu}+\mathcal{F}Cl_{\lambda,\mu}^{-1}(S^{1},\mathbb{K})$
with obvious extension of notations. Therefore, for any initial value
$L_{0}\in\Psi DO(S^{1},\mathbb{K}),$ we get a family of operators

$L_{\lambda,\mu}\in\mathcal{F}Cl_{\lambda,\mu}^{1}(S^{1},\mathbb{K})[[T]]\subset\mathcal{F}Cl^{1}(S^{1},\mathbb{K})[[T]]$

parametrized by the complex parameters $\lambda$ and $\mu$ chosen as before,
which satisfies the KP hierarchy in $\mathcal{F}Cl(S^{1},\mathbb{K})$ and with
initial values $\Phi_{\lambda,\mu}(L_{0}).$

###### Existence, uniqueness and well-posedness of the KP system in
$\mathcal{F}Cl(S^{1},\mathbb{K}).$

We adapt here the $r-$matrix approach for the construction of the solutions,
along the lines of [10] with the following specific choices:

  * •

The algebra of smooth coefficients for formal pseudo-differential operators is
$\mathcal{R}=C^{\infty}(S^{1},M_{n}(\mathbb{K}))\oplus\epsilon(D)C^{\infty}(S^{1},M_{n}(\mathbb{K}))$
with multiplication rules inherited from $Cl(S^{1},\mathbb{K}^{n}).$

  * •

The differential operator is $\partial=\frac{d}{dx}.$

######  Proposition 7.22.

$\Psi DO(\mathcal{R})=\mathcal{F}Cl(S^{1},\mathbb{K}^{n})$ and there is an
identification of the Manin triples $(\Psi
DO(\mathcal{R}),DO(\mathcal{R}),IO(\mathcal{R}))$ with
$(\mathcal{F}Cl(S^{1},\mathbb{K}^{n}),\mathcal{F}Cl_{D}(S^{1},\mathbb{K}^{n}),\mathcal{F}Cl_{S}(S^{1},\mathbb{K}^{n})).$

Hence, applying the main result of [RS1981] completed, for well-posedness, by
[10, Theorem 4.1] or by [16, Theorem 4.1] when
$\mathcal{R}=C^{\infty}(S^{1},\mathbb{K})=M_{1}(C^{\infty}(S^{1},\mathbb{K}))$
is a commutative algebra, we can state the following:

######  Proposition 7.23.

The Kadomtsev-Petviashvili (KP) hierarchy (7.1) on $\Psi DO(\mathcal{R})$
(resp. $\mathcal{F}Cl(S^{1},\mathbb{K}^{n})$) with initial condition
$L(0)=L_{0}\in\partial+\Psi DO^{-1}(\mathcal{R})$ (resp.
$\in\partial+\mathcal{F}Cl^{-1}(S^{1},\mathbb{K}^{n})$) satisfies Theorem
7.13.

######  Remark 7.24.

We have used here, intrinsically, the integrable almost complex structure
$J_{1}.$ Indeed,
$\mathcal{R}=C^{\infty}(S^{1},M_{n}(\mathbb{K}))+J_{1}C^{\infty}(S^{1},M_{n}(\mathbb{K}))$
is an algebra.

######  Remark 7.25.

There exists another way to justify Proposition 7.23. One can use
alternatively the splitting

$\mathcal{F}Cl(S^{1},\mathbb{K}^{n})=\mathcal{F}Cl_{+}(S^{1},\mathbb{K}^{n})\oplus\mathcal{F}Cl_{-}(S^{1},\mathbb{K}^{n}).$

Then Equation (7.1) on $\mathcal{F}Cl(S^{1},\mathbb{K}^{n})$ splits into two
independent equations, similar to Equation (7.1) on
$\mathcal{F}Cl_{\pm}(S^{1},\mathbb{K}^{n}).$ Through the identification maps
$\Phi_{1,0}$ and $\Phi_{0,1}$ of $\mathcal{F}Cl_{\pm}(S^{1},\mathbb{K}^{n})$
with $\Psi DO(S^{1},\mathbb{K}^{n}),$ we get existence, uniqueness and well-
posedness for Equation (7.1) on $\mathcal{F}Cl(S^{1},\mathbb{K}^{n})$ with
initaial value $L_{0}\in\partial+\mathcal{F}Cl^{-1}(S^{1},\mathbb{K}^{n}).$

From this last remark, we can generalize the identification procedure,
changing the maps $\Phi_{ee},$ $\Phi_{1,0}$ and $\Phi_{0,1}$ by the family of
maps $\Phi_{\lambda,\mu}.$

######  Theorem 7.26.

Let $(\lambda,\mu)\in({\mathbb{C}}^{*})^{2}.$ Then the KP equation (7.1) in
$\mathcal{F}Cl(S^{1},\mathbb{K}^{n})$ with initial value
$L_{0}\in\partial_{\lambda,\mu}+\mathcal{F}Cl^{-1}(S^{1},\mathbb{K}^{n})$ has
an unique solution $L$ in
$\partial_{\lambda,\mu}+\mathcal{F}Cl^{-1}(S^{1},\mathbb{K}^{n})[[T]]$ and the
problem is well-posed: the solution $L$ depends smoothly on $L_{0}.$

###### Twisted KP hierarchy

Let us now change the standard multiplication on
$\mathcal{F}Cl(S^{1},\mathbb{K})$ by $(A,B)\mapsto\epsilon AB$ where
$\epsilon=\epsilon(D)$ or $a\epsilon(D)$ for any $a\in\mathcal{C}^{*}.$ Since
$\epsilon(D)$ commutes with any element of $\mathcal{F}Cl(S^{1},\mathbb{K})$
for the standard multiplication, this new multiplication defines a new algebra
structure on $\mathcal{F}Cl(S^{1},\mathbb{K}).$ When necessary we note by
$\circ$ the standard multiplication, and by $\circ_{\epsilon}$ the twisted
one. Associated to this multiplication, we get the deformed Lie bracket
$[.,.]_{\epsilon}.$ Then we get again and equation similar to (7.1)

(7.14)
$\frac{dL}{dt_{k}}=\left[\epsilon^{k-1}(L^{k})_{D},L\right]_{\epsilon}=\epsilon^{k}\left[(L^{k})_{D},L\right]\;,\quad\quad
k\geq 1\;,$

where powers in this equation are taken with respect to $\circ.$

######  Theorem 7.27.

The Let $L_{0}$ such that
$L_{0}\in\partial_{\lambda,\mu}+\mathcal{F}Cl^{-1}(S^{1},\mathbb{K}^{n}),$
with $(\lambda,\mu)\in(\mathcal{C}^{*})^{2}.$ Then the $\epsilon-$KP hierarchy
(7.14) with initial value $L_{0}$ has an unique solution. Moreover, the
problem is well-posed.

####  7.3.2. KP hierarchies with complex powers

We finally extend all the constructions of the last section to complex powers,
along the lines of [EKRRR1995]. Let $\mathbb{K}=\mathcal{C}$ or $\mathbb{H}.$
We consider an operator $L_{0}$ of complex order $\alpha$ such that

(7.15)
$L_{0}\in\left(\frac{d}{dx}\right)^{\alpha}+\mathcal{F}Cl^{\alpha-1}(S^{1},\mathbb{K})$

or

(7.16)
$L_{0}\in\left|D\right|^{\alpha}+\mathcal{F}Cl^{\alpha-1}(S^{1},\mathbb{K})$

For each setting (7.15) and (7.16), we define the complex KP hierarchy on
$\mathcal{F}Cl^{\alpha}(S^{1},\mathbb{K})$ by

(7.17)
$\frac{dL}{dt_{k}}=\left[(L^{k/\alpha})_{D},L\right]_{\epsilon}=-\left[(L^{k})_{S},L\right]\;,\quad\quad
k\geq 1\;,$

where $L^{k/\alpha}=\exp\left(\frac{k}{\alpha}\log L\right)$ and the solution
$L\in\mathcal{F}Cl^{\alpha}(S^{1},\mathbb{K})[[T]].$

######  Theorem 7.28.

The KP hierarchy (7.17) with initial value $L_{0}$ defined along the lines of
(7.15) or (7.16) has an unique solution in
$\mathcal{F}Cl^{\alpha}(S^{1},\mathbb{K})[[T]].$ Moreover, the prblem is well-
posed.

Open question: The extension of the KP hierarchy from
$\mathcal{F}Cl(S^{1},{\mathbb{C}})$ to $Cl(S^{1},{\mathbb{C}})$ is a non
trivial open problem, due to the lack of local slice
$\mathcal{F}Cl(S^{1},{\mathbb{C}})\rightarrow Cl(S^{1},{\mathbb{C}})$ in the
short exact sequence

$0\rightarrow Cl^{-\infty}(S^{1},{\mathbb{C}})\rightarrow
Cl(S^{1},{\mathbb{C}})\rightarrow\mathcal{F}Cl(S^{1},{\mathbb{C}})\rightarrow
0.$

## Bibliography

  * [-] Presented works
  * [1] Magnot,J-P.; Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 no6, 513-529 (2004)
  * [2] Magnot, J-P.; Renormalized traces and cocycles on the algebra of $S^{1}$-pseudo-differential operators; Lett. Math. Phys. 75 no2, 111-127 (2006)
  * [3] Magnot, J-P.; The Schwinger cocycle for algebras with unbounded operators; Bull. Sci. Math. 132 no. 2, 112-127 (2008).
  * [4] Magnot, J-P.; Ambrose-Singer theorem on diffeological bundles and complete integrability of the KP equation; Int. J. Geom. Meth. Mod. Phys. 10, No. 9, Article ID 1350043, 31 p. (2013). (2013)
  * [5] Magnot, J-P.; Infinite dimensional integrals beyond Monte Carlo methods: yet another approach to normalized infinite dimensional integrals; IC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences, 3-7 September 2012, Budapest, Hungary; J. Phys.: Conf. Ser. 410, 012003 (2013)
  * [6] Magnot, J-P.; On $Diff(M)-$pseudo-differential operators and the geomery of non-linear grassmannians; Mathematics 4 no 1; articleID 1 (2016)
  * [7] Magnot, J-P.; Differentiation on spaces of triangulations and optimized triangulations.  J. Phys.: Conf. Ser., 738, No 1 articleID 012088 (2016)
  * [8] Magnot, J-P.; The mean value for infinite volume measures, infinite products and heuristic infinite dimensional Lebesgue measures; J. Math. Vol. 2017 (2017), Article ID 9853672, 14 pages
  * [9] W.W. Koczkodaj, J.-P. Magnot, J. Mazurek, J.F. Peters, H. Rakhshani, M. Soltys, D. Strzałka, J. Szybowski, A. Tozzi; On normalization of inconsistency indicators in pairwise comparisons Int. J. Approx. Reasonning 86 73 - 79 (2017)
  * [10] Eslami-Rad, A.; Magnot, J-P.; Reyes, E. G.; The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra J. Nonlinear Math. Phys. 24 Supp. issue no 1, 103-120 (2017)
  * [11] Magnot, J-P.; A Mathematical Bridge between Discretized Gauge Theories in Quantum Physics and Approximate Reasoning in Pairwise Comparisons Adv. Math. Phys. 2018 articleID 7496762, 5 pages (2018)
  * [12] Magnot, J-P.; The group of diffeomorphisms of a non compact manifold is not regular. Demonstr. Math. 51 8-16 (2018)
  * [13] Magnot, J-P.; Remarks on a New Possible Discretization Scheme for Gauge Theories Int. J. Theoret. Phys. 57 2093–2102 (2018)
  * [14] Magnot, J-P.;On mathematical structures on pairwise comparisons matrices with coefficients in an abstract group arising from quantum gravity Heliyon 5 e01821 (2019)
  * [15] Magnot, J-P. Reyes, E.G.; The Cauchy problem of the Kadomtsev-Petviashvili hierarchy and infinite-dimensional groups. in  Nonlinear Systems and Their Remarkable Mathematical Structures, Volume 2; Norbert Euler and Maria Clara Nucci Editors, CRC press (2019) section B6
  * [16] Magnot, J-P.; Reyes, E.G.; Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups Annales Henri Poincaré 21, 1893-1945 (2020)
  * [17] Magnot, J-P.; On the domain of implicit functions without extra norm estimates Demonstr. Math. 53 no 1 112-120 (2020)
  * [18] Magnot, J-P.; On the differential geometry of numerical schemes and weak solutions of functional equations. Nonlinearity 33, no. 12, 6835-6867 (2020) .
  * [19] Magnot, J-P.; On the geometry of $Diff(S^{1})-$pseudodifferential operators based on renormalized traces Proceedings of the International Geometry Center 14 No. 1, 19-48 (2021)
  * [20] Magnot, J-P.; Rubtsov, V.; On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators Theoret. Math. Phys. 207, No. 3, 799-826 (2021)
  * [-] Non presented works: works from the PhD thesis
  * [21] Cardona, A.; Ducourtioux, C.; Magnot, J-P.; Paycha, S.; Weighted traces on pseudo-differential operators and geometry on loop groups; Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 no4 503-541 (2002)
  * [22] Magnot, J-P.; The Kähler form on the loop group and the Radul cocycle on Pseudo-differential Operators; GROUP’24: Physical and Mathematical aspects of symmetries, Proceedings of the 24th International Colloquium on Group Theorical Methods in Physics, Paris, France, 15-20 July 2002; Institut of Physic conferences Publishing 173, 671-675, IOP Bristol and Philadelphia (2003)
  * [23] Magnot, J-P.; Chern forms on mapping spaces, Acta Appl. Math. 91, no. 1, 67-95 (2006).
  * [-] Other non presented works
  * [24] Magnot, J-P.; Difféologie sur le fibré d’holonomie d’une connexion en dimension infinie C. R. Math. Acad. Sci., Soc. R. Can. 28, no. 4, 121-128 (2006)
  * [25] Magnot, J-P., q-deformed Lax equations and their differential geometric background, Lambert University Publishing, Saarbruken, Germany, 2015.
  * [26] Magnot, J-P; From infinitesimal symmetries to deformed symmetries of Lax-type equations; J. Of Physics: Conference series 633 (2015) 012013
  * [27] Kakiashvili, T.; Koczkodaj, W.W; Magnot, J-P.; Approximate reasoning by pairwise comparisons: “Topodynamics of metastable brains” by Arturo Tozzi, et al. Phys. of Life reviews 21 July 2017, Pages 37-39 (2017)
  * [28] Magnot, J-P. ; From configurations to branched configurations and beyond Res. Rep. Math. 1 no1 art. ID 1000105 (2017)
  * [29] Magnot, J-P.; Watts, J.; The diffeology of Milnor’s classifying space Top. Appl. 232 189-213 (2017)
  * [30] Magnot, J-P.; On multidisciplinary applications of gauge theories Biostat. Biomet. Open access J. 7 no1, art. ID.555701 (2018)
  * [31] Magnot, J-P.; Remarks on the geometry and the topology of the loop spaces $H^{s}(S^{1},N),$ for $s\leq 1/2.$ ; International Journal of Maps in Mathematics 2 no 1 14-37 (2019)
  * [32] Singh, N.; Son, L.H.; Chiclana, F.; Magnot, J-P.; A new fusion of salp swarm with sine cosine for optimization of non-linear functions. Engineering with Computers 36 185–212 (2020)
  * [33] Magnot, J-P.; On algebras and groups of formal series over a groupoïd and application to some spaces of cobordism; ArXiv:1507.00932
  * [34] Magnot, J-P. ; Reyes, E. ; On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms arXiv:1808.03719
  * [35] Magnot, J-P.; On a class of closed cocycles for algebras of non-formal, possibly unbounded, pseudodifferential operators ArXiv:2012.06941
  * [-] Other references
  * [ACMM1989] Abbati, M. C.; Cirelli, R.; Mania, A.; Michor, P.; The Lie group of automorphisms of a principal bundle. J. Geom. Phys. 6, 215-235 (1989)
  * [AM2007] Abbati, M. C.; Mania, A. A geometrical setting for geometric phases on complex Grassmann manifolds. J. Geom. Phys. 57 (2007), no. 3, 777–797
  * [ARS1986-1] M. Adams, T.Ratiu, R. Schmid A Lie group structure for pseudodifferential operators Math. Ann. 273 no4 (1986), 529-551
  * [ARS1986-2] Adams, M.; Ratiu, T.; Schmidt, R.; A Lie group structure for Fourier integral operators; Math. Annalen 276, no 1, 19-41 (1986)
  * [Ad1979] Adler, M.; On a trace functionnal for formal pseudo-differential operators and the symplectic structure of Korteweg-de Vries type equations Inventiones Math. 50 219-248 (1979)
  * [AFKHKL1986] Albeverio, S.; Hoegh-Krohn, R.; Fenstad, J. E.; Lindstrom, T.; Nonstandard methods in stochastic analysis and mathematical physics. Reprint of the 1986 original ed. Dover Publications 514 p. (2009).
  * [AB1994] Albeverio, S; Brzeźniak, Z.; Feynman path integrals as infinite dimensional oscillatory integrals: some new developpements; Acta Appl. Math. 35; 5-27 (1994)
  * [AHKM2005] Albeverio, S.; Hoegh-Krohn, R.; Mazzuchi, S.; Mathematical theory of Feynman Path Integrals; an introduction 2nd edition; Lecture Notes in Mathematics 523 , Springer (2005)
  * [AKLU2000] Albeverio, S.; Kondratiev, Y., Lytvynov, E.; g. F. Us, G.F.; Analysis and geometry on marked configuration spaces; Infinite Dimensional Harmonic Analysis (Kyoto, September 20-24, 1999), H. Heyer et al., eds, Gräbner, Altendorf, 1-39 (2000)
  * [AM2005] Albeverio, S.; Mazzucchi, S.; Generalized Fresnel integrals Bull. Sci. Math. 129, no1, 1-23 (2005)
  * [AM2016] Albeverio, S.; Mazzucchi, S.; A unified approach to infinite-dimensional integration. Rev. Math. Phys. 28, No. 2, Article ID 1650005, 43 p. (2016)
  * [AM2018] Albeverio, S.; Mazzucchi, S.; A unified approach to infinite dimensional integrals of probabilistic and oscillatory type with applications to Feynman path integrals in: Gesztesy, Fritz (ed.) et al., Non-linear partial differential equations, mathematical physics, and stochastic analysis. The Helge Holden anniversary volume on the occasion of his 60th birthday. Based on the presentations at the conference ‘Non-linear PDEs, mathematical physics and stochastic analysis’, NTNU, Trondheim, Norway, July 4–7, 2016. (2018)
  * [AZ1990] Albeverio, S.; Zegarlinski, B.; Construction of convergent simplicial approximations of quantum fields on Riemannian manifolds Comm. Mat. Phys. 132 39-71 (1990)
  * [ADL2001] Albeverio, S.; Daletskii, A.; Lytvynov, E.; De Rham cohomology of configuration spaces with Poisson measure J. Funct. Anal. 185, No.1, 240-273 (2001).
  * [AS1953] Ambrose, W.; Singer, I.M.; A theorem on holonomy Trans. Amer. Math. Soc.; 75 (1953) 428-443
  * [Arn1966] Arnold, V.I.; Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16, 316-361 (1966).
  * [Asa2000] Asada, A; regularization of differential operators on a Hilbert space and geometric meaning of Zeta-regularization. Steps in differential geometry, proceding of the colloquium on differential geometry 25-30 July 2000, debrecen, Hungary
  * [BH2011] Baez, J. C.; Hoffnung, A.E.; Convenient categories of smooth spaces. _Trans. Amer. Math. Soc._ , 363, 5789–5825 (2011).
  * [Bak1991] Baker, R. L.; Lebesgue measure on $\mathbb{R}^{\infty}$. Proc. Amer. Math. Soc. 113 no4, 1023-1029 (1991)
  * [Bak2004] Baker, R. L.; Lebesgue measure on $\mathbb{R}^{\infty}$. II. Proc. Amer. Math. Soc. 132 no9, 2577-2591 (2004)
  * [BBHM2012] Bauer,M.; Bruveris, M.; Harms, P.; Michor, P.; Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Global Anal. Geom. 41 no 4 (2012) 461–472
  * [BR1984] Beals, M.; Reed, M.; Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems; Trans. Amer. Math. Soc. 285 (1984), 159-184
  * [BFR2018] Beltran, J.; Farinati, M.; Reyes, E. G.; Central extensions of the algebra of formal pseudo-differential symbols via Hochschild (co)homology and quadratic symplectic Lie algebras. J. Pure Appl. Algebra 222 no. 8, 2006-2021 (2018).
  * [BGV1992] Berline, N.; Getzler, E.; Vergne, M.; Heat kernels and Dirac operators; Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 298. Springer-Verlag, Berlin, (1992)
  * [BDIP1996] Bertin, J.; Demailly, J-P.; Illusie, L.; Peters, C.; Introduction à la théorie de Hodge SMF Panoramas et Synthèses 3 (1996)
  * [BF1981] E. Binz, H.R. Fischer: The manifold of embeddings of a closed manifold. Proceedings of Differential geometric methods in theoretical physics, Clausthal 1978, Lecture Notes in Physics 139, Springer-Verlag 1981, pp. 310–329
  * [Bog2010] Bogachev, V.I.; Differentiable measures and the Malliavin calculus AMS Math. Surveys and monographs 164 (2010)
  * [BS2017] Bogachev, V. I.; Smolyanov, O. G.; Topological vector spaces and their applications. Expanded and revised translation, originally published 2012. Springer Monographs in Mathematics, 456 p. (2017).
  * [BK1969] Bokobza-Haggiag, J.; Opérateurs pseudo-différentiels sur une variété différentiable; Ann. Inst. Fourier, Grenoble 19,1 125-177 (1969)
  * [Bou] Bourbaki, N.; Eléments de mathématiques; Masson, Paris (1981)
  * [BR1987-1] Bowick, M.; Rajeev, S.; String theory as the Kähler geometry of the loop space Phys. Rev. Lett. 58, 535 (1987)
  * [BR1987-2] Bowick, M.; Rajeev, S.; Anomalies as curvature in complex geometry Nucl. Phys. B 296 1007 (1987)
  * [BS1994] Brenner, S. C.; Scott, L. R. The mathematical theory of finite element methods.  Texts in Applied Mathematics. 15. Springer-Verlag. 294 p. (1994).
  * [Bry2008] Brylinski, J-L.; Loop spaces, characteristic classes and geometric quantization. Reprint of the 1993 edition. Modern Birkhäuser Classics. Basel: Birkhäuser (2008).
  * [CS1983] Cameron, R.H.; Storvick, D.A.; A simple definition of the Feynman integral, with applications Mem. of the AMS 46 no 288 (1983)
  * [Can2015] Canarutto, D.; Frölicher-smooth geometries, quantum jet bundles and BRST symmetry. J. Geom. Phys. 88 113-128 (2015).
  * [Can2020] Canarutto, D.; Gauge field theory in natural geometric language. A revisitation of mathematical notions of quantum physics. Oxford University Press 368 p. (2020)
  * [CdAS2010] Cavallo, B.; D’Apuzzo, L.; Squilllante,M.; Pairwise Comparison Matrices over abelian Linearly Ordered Groups: A Consistency Measure and Weights for the Alternatives; in : ”Multicriteria and Multiagent Decision Making with Applications to Economics and Social Sciences” Studies in Fuzziness and Soft Computing 305 (2010) 49-64
  * [CW2014-2] Christensen, J. D.; Wu, E.; The homotopy theory of diffeological spaces. New York J. Math. 20, 1269-1303 (2014)
  * [CvS2021] Connes, A.; van Suilekom, W.D.; Tolerance relations and operator systems arXiv:2111.02903
  * [CrW1985] Crawford, G.; Williams, C.; The analysis of subjective judgment matrices; A project AIR FORCE report prepared for united states air force report number R-2572-1-AF (may 1985)
  * [Dem1995-1] Demidov, E.E.; On the Kadomtsev-Petviashvili hierarchy with a noncommutative timespace. Funct. Anal. Appl. 29, no. 2, (1995), 131–133.
  * [Dem1995-2] Demidov, E.E.; Noncommutative deformation of the Kadomtsev-Petviashvili hierarchy. In “Algebra. 5, Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI)”, Moscow, 1995. (Russian). J. Math. Sci. (New York) 88, no. 4, (1998), 520–536 (English).
  * [Dic2003] L.A. Dickey, Soliton equations and Hamiltonian systems, second edition (2003). Advanced Series in Mathematical Physics $12$, World Scientific Publ. Co., Singapore.
  * [Di1968] Dieudonné, J.; Éléments d’analyse I,II,III; Gauthier-Villars (1968)
  * [Dui1974] Duistermaat, J.J.; Oscillatory integrals, Lagrange immersions and unfoldings of singularities; Comm. Pure Appl. Math. 27 207-281 (1974)
  * [DGV2015] Dodson, C.; Galanis, G.; Vassiliou, E.; Geometry in the Fréchet context: a projective limit approach London Mathematical Society Lecture Notes Series 428, Cambridge University Press (2015)
  * [Duc2001] Ducourtioux, C.; Weighted traces on pseudo-differential operators and associated determinants; Ph.D thesis, Clermont-Ferrand, France (2001)
  * [Dup1978] Dupont, J. L.; Curvature and characteristic classes. Lecture Notes in Mathematics. 640 (1978).
  * [Ee1966] Eells, J.; A setting for global analysis Bull. Amer. Math. Soc. 72 751-807 (1966)
  * [ET1984] Elworthy, D.; Truman, , A.; Feynman maps, Cameron-Martin formulae and anharmonic oscillators; Ann. Inst. H. Poincaré Phys. Theo. 41 (2) 115-142 (1984)
  * [EKRRR1995] Enriquez, B.; Khoroshkin, S.; Radul, A.; Rosly, A.; Rubtsov, V.; Poisson-Lie aspects of classical W-algebras; in: The interplay between Differential geometry and differential equations Amer. Math. Soc. Transl. (2) 167, 37-59 (1995)
  * [ER2013] Eslami Rad, A.; Reyes, E. G.; The Kadomtsev-Petviashvili hierarchy and the Mulase factorization of formal Lie groups J. Geom. Mech. 5, no 3 (2013) 345–363.
  * [vEK1964] van Est, W. T.; Korthagen, Th. J.; Non-enlargible Lie algebras. Nederl. Akad. Wet., Proc., Ser. A 67, 15-31 (1964).
  * [FH2001] Fadell, E.R.; Husseini, S.Y.; Geometry and topology of configuration spaces Springer, Berlin (2001)
  * [Fed1969] Federer, H.; Geometric measure theory, Springer (1969)
  * [Fr1988-1] Freed, D.; An index theorem for families of Fredholm operators parametrized by a groupTopology 27, No.3, 279-300 (1988)
  * [Fr1988-2] Freed, D.; The geometry of loop groups, J. Diff. Geom. 28 (1988) 223-276
  * [FK1988] Frölicher, A; Kriegl, A; Linear spaces and differentiation theory Wiley series in Pure and Applied Mathematics, Wiley Interscience (1988)
  * [Fu2017] Fujiwara, D.; Rigorous time slicing approach to Feynman path integrals.  Mathematical Physics Studies. Springer, 333 p. (2017).
  * [GV1997] Galanis, G. and Vassiliou, E.; A generalized frame bundle for certain Fréchet vector bundles and linear connections. Tokyo J. Math. 20 no.1 (1997), 129–137
  * [GBV2014] Gay-Balmaz, F; Vizman, C.; Principal bundles of embeddings and non linear grassmannians; Annals of Global Analysis and Geometry,46, 293–312 (2015)
  * [Gil1984] Gilkey, P; Invariance theory, the heat equation and the Atiyah-Singer index theorem Publish or Perish (1984)
  * [Gl2002] Glöckner, H; Algebras whose groups of the units are Lie groups Studia Math.  153, no2 (2002), 147-177
  * [Gl2007] Glöckner, H; Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories. J. Funct. Anal. 245 19-61 (2007)
  * [GN2012] Glöckner,, H.; Neeb, K-H.; When unit groups of continuous inverse algebras are regular Lie groups Studia Math. 211 no2, 95-109 (2012)
  * [GN2017] Glöckner,, H.; Neeb, K-H.; Diffeomorphism groups of compact convex sets. Indag. Math. (N.S.) 28 no. 4, 760–783 (2017).
  * [GK1971] Gelfand, I.; Kazhdan, D.; Certain questions of differential geometry and the computation of the cohomologies of the Lie algebras of vector fields. Soviet Math. Doklady 12, 1367–1370 (1971)
  * [Gr1997] Gromov, M., Metric structures in Riemannian and non-Riemannian spaces, 2nd ed. Birkauser, 1997.
  * [GR2007] Guieu, L. and Roger, C.; L’ Algèbre et le groupe de Virasoro: Aspects géométriques et algébraiques, generalisations. Centre de Recherches Mathematiques, Université de Montreal, (2007).
  * [Hah2004] Hahn, A.; The Wilson loop observables of Chern-Simons theory on $\mathbb{R}^{3}$ in axial gauge , Comm. Math. Phys. 248 , no. 3, 467?499 (2004)
  * [HV2004] Haller, S.; Vizman, C.; Non-linear grassmannian as coadjoint orbits Math. Ann. 139, no4, 771-785 (2004)
  * [Ham1984] Hamilton, R.S.; The inverse function theorem of Nash and Moser; Bull. Amer. Math Soc. (NS) 7 (1984) 65-222
  * [HL2020] Herzog, R.; Loayza-Romero, E.; A manifold of planar triangular meshes with complete Riemannian metric arXiv:2012.05624