% This paper appeared in "Helvetica Physica Acta"
% vol. 65 (1992), p. 622
\def\Example{{\sl Example}}
\def\MIB{MIB }
\def\all{\forall}
\def\ss{\subset}
\def\sse{\subseteq}
\def\Lra{\Leftrightarrow}
\def\pa{\partial}
\def\pmu{\pa_\mu}
\def\c{\cdot}
\def\cd{\nabla}
\def\cdm{\cd_\mu}
\def\lapl{\triangle}
\def\x{\times}
\def\o+{\oplus}
\def\l{\ell}
\def\a{\alpha}
\def\A{{\cal A}}
\def\b{\beta}
\def\S{\Sigma}
\def\s{\sigma}
\def\G{{\cal G}}
\def\Ga{\Gamma}
\def\g{\gamma}
\def\de{\delta}
\def\de{\delta}
\def\L{{\cal L}}
\def\La{\Lambda}
\def\om{\omega}
\def\Om{\Omega}
\def\phi{\varphi}
\def\th{\vartheta}
\def\tha{\theta}
\def\Th{\Theta}
\def\Vscr{{\cal V}}
\def\Diff{{\rm Diff}}
\def\GDiff{{\rm GDiff}}
\def\CGDiff{{\rm CGDiff}}
\def\1{{(1)}}
\def\({\left(}
\def\){\right)}
\def\dim{{\rm dim}}
\def\Fix{{\rm Fix}}
\def\for{{\rm ~for~}}
\def\E{ E \oplus \^E }
\def\xd{{\dot x}}
\def\~#1{{\tilde #1 }}
\def\^#1{{\widehat #1 }}
\def\=#1{{\bar #1 }}
\magnification=1200
{\nopagenumbers
\parskip=0pt
\parindent=0pt
\vfill
\centerline{\bf Michel's theorem and critical sections of gauge functionals}
\bigskip \bigskip
\centerline{Giuseppe Gaeta}
\centerline{\sl Centre de Physique Theorique}
\centerline{\sl Ecole Polytechnique, F-91128 Palaiseau (France)}
\vfill
{\tt ~~ I - Critical orbits of invariant potentials}
1. Introduction \dotfill 1
2. Strata in $R^N$ \dotfill 3
3. Michel's theorem \dotfill 4
\vskip 10pt
{\tt~~II - Critical orbits of gauge functionals}
4. Zero-th order gauge functionals \dotfill 7
5. Discussion \dotfill 10
6. First order gauge functionals \dotfill 12
\vskip 10pt
{\tt~~III - The stratification of $\Theta = \Phi / \Gamma$}
7. Geometry and stratification of $\Om$ \dotfill 17
8. Stratification of gauge orbit space \dotfill 21
9. Maximal strata in gauge orbit space \dotfill 25
\vskip 10pt
{\tt~~IV - Reduction and critical orbits}
10. The equivariant branching lemma \dotfill 28
11. A reduction lemma for gauge invariant potentials \dotfill 31
12. Some examples of reduction \dotfill 32
\vskip 10pt
{\tt ~~V - Further developements}
13. Base space symmetries \dotfill 35
14. A scenario for pattern formation \dotfill 37
15. A scenario for phase coexistence \dotfill 39
\vskip10pt
~~~~~~ References \dotfill 40
\vfill
\eject}
\pageno=1
\parskip=10pt
\parindent=0pt
\bigskip \bigskip {\bf 1. Introduction} \bigskip
In many physical problems, one has to minimize a G-invariant
potential, i.e. a potential invariant under the action of a symmetry
group. In this case a powerful tool is offered by a theorem of L. Michel
[1] and generalizations thereof, see e.g. [2-11]. The purpose of this note
is to discuss just another such generalization, to gauge potentials.
After introducing the appropriate notation, we will briefly recall
the content of Michel theorem and the ingredients of its proof; we will
then pass to discuss the infinite dimensional case, and see that for gauge
symmetries one can still use tools not too different from the original
Michel's ones.
Since results as strong as in the finite dimensional case hold only
restricting to a subset of sections (although a structurally stable one)
and can be difficult to use to obtain directly the required minima of a
gauge functional, we will also look at Michel's theorem as an heurhystical
tool, allowing one to restrict to a simpler problem whose
solutions are, generically, in correspondence with solution of the
original problem (this will be done by means of the "reduction lemma").
\bigskip
The scheme of the paper is as follows: the whole paper is divided into
five parts, each one composed of three sections (in studying gauge
symmetry breaking, we felt obliged to give some formal symmetry to the
paper).
Part I recalls Michel's theorem for critical points of invariant
potentials $V : M \sse R^n \to R$ and the appropriate mathematical tools
and concepts (section 2), needed not only for the proof but also for the
statement of the theorem; a short proof of Michel's theorem is also
reported in section 3.
In part II we first extend Michel's theorem to sections and gauge orbits
of sections of a bundle on which a zero-th order gauge invariant
functional is defined (section 4); this is essentially accomplished by
introducing an appropriate (and natural) metric in the space of sections.
We then discuss (section 5) in some detail this extension, and how this
can be used in the case of higher order gauge functionals (section 6).
Part III deals with the difficulties arising in the stratification of
gauge orbit space: we first recall some results for the stratification of
orbit space for the finite-dimensional action of a compact Lie group
(section 7), then use these to study the gauge orbit space (section 8),
and point out that some restriction is needed in order for a
stratification of this to make sense. We propose such a restriction, to
the set of transverse sections (defined there); this is structurally
stable and dense in the set of sections. Even with this, the
stratification of gauge orbit space is by far too complex to be completely
described; we will limit to study the strata Michel's theorem is concerned
with, i.e. maximal ones (section 9).
In Part IV (sections 10 - 12) we discuss how, similarly to the finite
dimensional case, Michel's theorem opens the way to a reduction of the
problem of finding critical orbits, connecting to results of Palais; and
if a control parameter appears in the theory, to an equivariant branching
lemma, connecting to recent results of Cicogna.
Part V is different in spirit (which breaks the symmetry of the paper,
after all). In facts we first show how our approach can be extended to
take into account base-space, or in physical terms space-time, symmetries
(section 13). At this point we renounce to obtain rigorous results, and
shortly discuss how this could provide a scenario for spontaneous pattern
formation and describe situations of phase coexistence.
\bigskip
Parts I-IV constitute a self contained paper, containing rigorous (and, at
least for the author, interesting) results, which are illustrated by
repeatedly analyzing in detail a number of examples, first introduced in
section 7.
The original motivation of the paper was an attempt to understand Blue
Phases; although this is not pursued here, we believe this paper sets
mathematical bases for an attempt to study Blue Phases [12-14] in a rigorous
and seemingly original way, along the lines of Part V, and plan to pursue
such an approach in the near future.
\bigskip
As recalled above, the paper connects to recent results of Cicogna [15] for
the bifurcation case; a discussion of equivariant bifurcation theorems
general enough to accomodate the case of gauge symmetries is given in
[16,17]; in a related paper [18] we discuss at lenght reduction and
equivariant bifurcation lemma for such a general class of symmetries, and
in particular gauge functionals and nonlinear evolution PDEs.
As also recalled above, the paper re-obtains some results of Palais
[19,20]. These were unknown to the author, who thanks prof. Bourguignon for
pointing them out to him (unfortunately after they were re-obtained). The
author believes anyway that the present treatment of the connection between
them and Michel's theorem is original.
\bigskip
Finally, I would like to thank a number of persons for interesting
discussions, first of all proff. Michel and Bourguignon; prof. Palais was so
kind to give me some of his time to discuss the present work during one of his
visits in Paris; the mathematical
part of the paper was also discussed with proff. Gallot and Lascoux, while
proff. Chakrabarti, Collet, Doelman, B. Pansu, Peliti and Testa had the
patience to listen to, and discuss, my ideas about physical applications of
it. I would also like to thank an unknown referee for suggesting the remark in
section 7 concerning reference [54].
\vfill \eject
\bigskip \bigskip {\bf 2. Strata in $R^N$} \bigskip
Let us first consider a (smooth) potential $V$ defined on an $N$-
dimensional
real space,
$$ V : R^N \to R^N \eqno(1) $$
which is invariant under a representation $\La= \{\La_g ~/~ g \in G \}$
of a Lie
group $G$ acting in $R^N$
$$ \La_g : R^N \to R^N \eqno(2) $$
$$ V( \La_g x ) = V (x) ~~~~~~ \all x \in R^N ~,~ \all g \in G \eqno(3) $$
We will be interested in the critical points of $V$,
$$ \nabla V( x_c ) = 0 \eqno(4) $$
Under the representation $\La$, each point $x \in R^N$ has an
isotropy
subgroup $G_x$,
$$ G_x = \{ g \in G ~/~ \La_g x = x \} \eqno(5) $$
It is immediate to see that points on the same $G$-orbit, i.e. points
$y,x$ such that for some $g \in G$ it is $y = \La_g x$, have isotropy
subgroups conjugated in $G$:
$$ y = \La_g x \Rightarrow G_y = g G_x g^{-1} \eqno(6) $$
The set of points of $R^N$ having conjugated (in $G$) isotropy
subgroup is called a stratum [1,2,5] and will be denoted by $\S_x$,
with $x$ any of its points,
$$ \S_x = \{ y \in R^N ~/~ G_y = g G_x g^{-1} ~,~ g \in G \} \eqno(7) $$
We will denote by $\om (x)$ the $G$-orbit through $x$,
$$ \om (x) = \{ y \in R^N ~/~ y = \La_g x ~,~ g \in G \} \eqno(8) $$
It follows from (6) that
$$ \om (x) \sse \S_x \eqno(9) $$
so that the stratification of $R^N$ also induces a stratification of the
orbit space $ \Om = R^N / G$ (the orbit $\om (x)$ will be denoted as
$\om_x$ when thought as a point of $\Om$), satisfying
$$ \S_\om = \{ \om ' \in \Om ~/~ G_y = g G_x g^{-1} ~,~ g \in G ,~ x \in \om
,~ y \in \om' \} \eqno(10) $$
which actually does not depend on the choice of
representative points $x,y$ on $\om , \om '$.
It is immediate to check that belonging to the same stratum,
denoted as $x \sim y$ or $\om \sim \om '$, is an equivalence
relation.
In the space $\Xi$ of strata of $R^N$ one
can introduce a partial ordering by
$$ \S_x < \S_y \Lra G_y = g H g^{-1} ~;~ g \in G ,~ H \ss G_x \eqno(11) $$
where $H$ is a proper subgroup of $G_x$ and again the relation
does not depend on the choice of the representatives $x,y$.
Analogously, for the space $\Xi_\Om$ of strata of $\Om$,
$$ \S_\om < \S_{\om '} \Lra G_y = g H g^{-1} ~;~ g \in G ,~x\in \om ,~ y \in
\om ' ,~ H \ss G_x \eqno(12) $$
A stratum in $\Om$ is also called an orbit type.
It should be stressed that strata, both in $R^N$ and in $\Om$, are manifolds
[2,5], although $\Om$ is in general not a manifold.
All the above is still valid if instead of $R^N$ we consider a
manifold $M \ss R^N$ which is invariant under $\La$:
$$ \La_g : M \to M ~~~~~ \all g \in G \eqno(13) $$
For more details on the material of this section, see [2,5]; other results
concerning stratification will be recalled in sect.7.
\bigskip \bigskip {\bf 3. Michel's theorem} \bigskip
Let $\Vscr$ be the set of $C^\infty$ functions from $R^N$ to $R$
invariant under $\La$, i.e. such that $V ( \La_g x) = V(x)$ ~ $\all g \in G$,
$\all x \in R^N$.
If there is a point $x$ such that $d V(x ) =0$, then necessarily
$d V(y) = 0 ~ \all y \in \om (x)$, so that critical points of $V \in
\Vscr$ will came in $G$-orbits.
An orbit $\om = \om (x_c )$ such that $d V(x_c ) =0 ~ \all V \in \Vscr$ is
called a critical orbit for $G$ [1].
To see that these exist, just consider $N=1$ and $G=Z_2 = \{ e,g \}$, with
$\La_e : x \to x$, $\La_g : x \to -x$. In other words, every (smooth) even
potential has a critical point in the origin. Analogously, for $G=
SO(N)$ we have that any rotationally invariant smooth potential has
a critical point in the origin.
A less trivial example is obtained by considering $M = S^2 \ss R^3$
and $G = SO(2)$ acting in $R^3$ as rotations around the $z$-axis,
$$ \La_g = \pmatrix{ \cos \theta & - \sin \theta & 0 \cr \sin \theta &
\cos \theta & 0 \cr 0&0&1 \cr } \eqno(1) $$
%\midinsert
%\vskip 5truecm
%$$ fig. ~ 1 $$
%\endinsert
Any invariant potential will have critical points at North and
South poles of the sphere (i.e. at $(0,0, \pm 1 )$).
The orbit space is isomorphic to the segment $[-1 , 1 ]$ (this can be
thought as the $z$ coordinate of the orbit); all the points in the
interior of this belong to the same stratum ($G_x = \{ e\}$), and the
extrema $x= \pm 1$ form another stratum ($G_x = SO(2)$).
The theorem of L. Michel [1] tells that
{\it An orbit is critical for $G$ if and only if it is isolated in its stratum
}
An orbit is, roughly speaking, isolated in its stratum if one can take
a neighbourhood of $\om$ in $\Om$ which does not contain points
of $\S_\om$ other than $\om$ itself.
In order to talk of neighbourhoods in $\Om$, one has to provide it
with a topology, which will be taken to be the quotient topology:
the distance of two orbits will be defined by means of the distance
in $R^N$ to be
$$ d ( \om , \om ' ) = {\rm min}_{x \in \om ; y \in \om '} ~d ( x , y)
\eqno(2) $$
where $d (x,y)$ is the standard distance in $R^N$ (or the distance
corresponding to the metric defined on $M \ss R^N$ if we deal with
this case).
The minimum of (2), whose existence has to be proven, can be seen
also as
$$ d ( \om , \om ') = {\rm min}_{x \in \om } d (x, \om ' ) \eqno(3) $$
where we have introduced the distance of a point from an orbit,
$$ d (x, \om ' ) = {\rm min}_{y \in \om '} d (x,y) \eqno(4) $$
If the point $y \in \om '$ for which $d(x,y) = d (x, \om ')$ is unique
(locally), it is called the retraction of $x$ on $\om$ and denoted
$\rho_{\om '} (x)$.
The function $\rho_\om ( \cdot )$ is an equivariant one:
$$ \rho_\om (\La_g x) = \La_g \rho_\om (x) \eqno(5) $$
and is therefore also called the equivariant retraction [1,2,5,6,8,21].
Notice that if $G$ is compact, $\om , \om '$ are compact sets, and
the minima of (2),(3),(4) do surely exists, and therefore also the
equivariant retraction does exist.
We stress that for noncompact group orbits the equivariant
retraction could very well not exist, and therefore the concept of an
orbit "isolated in its stratum" be ill-defined (that is why we
discussed the concept at some lenght). Besides this, the very existence of a
stratification is not granted for noncompact groups, as it will be discussed
later.
Let us now sketch, without going to details [1,2,6,8], how Michel's
theorem is proved (for compact Lie groups).
At any point $x \in M$, one has a tangent and a normal space to
$\om (x)$, $T_x \om$ and $N_x \om$, with
$$ T_x \om \oplus N_x \om = T_x M \eqno(6) $$
These are linear spaces; in $N_x \om$ we can consider the invariant
subspace $N_x^0 \om \sse N_x \om$;
$$ N_x^0 \om = \{ \xi \in N_x \om ~/~ \La_g \xi = \xi ~~ \all g \in G_x \}
\eqno(7) $$
$N_x \om$ is also called the slice through $x$, and $N_x^0 \om$
the invariant slice through $x$ [5,21] (for nonlinear group actions the
slices are manifolds, tangent in $x$ to these linear spaces [21]).
It should be noted that the tangent space in $x \in \om$ to the stratum $\S_x$
is simply, by the definitions of $\S_x$ and $N^0_x \om$,
$$ T_x \S_x = T_x \om \oplus N_x^0 \om $$
Now, the gradient $dV$ of an invariant function $V$ has to be
perpendicular to the orbit $\om (x)$ at $x$
$$ d V(x) \in N_x \om ~~~;~~~ d V : x \to N_x \om \ss T_x M \eqno(8) $$
Moreover, $dV$ must be tangent to $\S_x$ in $x$.
To see this consider that for an equivariant function $f : M \to TM$,
$$ f (\La_g x ) = \La_g f(x) \eqno(9) $$
and
$$ G_x \sse G_{f(x)} \eqno(10) $$
which also means
$$\S_{f(x)} \le \S_x ~~~;~~~ f(x) \in \S_x \eqno(11) $$
(the equality holds in (10) and (11) if and only if $f$ is one to one
[1,2,5,6]), so that
$$ d V(x) \in T_x \S_x \eqno(12) $$
which gives immediately Michel's theorem.
By (8) and (12) we get [4,6]
$$ dV (x) \in N_x^0 \om \eqno(13) $$
and conversely one can prove that, given an integrity basis
[5,21,22,23] for $\La$, $N_x^0 \om$ is spanned by gradients of
elements of the integrity basis [6]. Relation (13) will also be fundamental in
establishing the "equivariant branching lemma", as discussed in the following
(sect.10); see also [15,18].
\bigskip \bigskip {\bf 4. Zero-th order gauge functionals} \bigskip
In gauge theories, one considers functions (gauge fields) defined on
a manifold $M \sse R^N$ with values in the Lie algebra $\G$ of a Lie
group $G$ (the gauge group); if the theory is not a pure gauge one,
one also considers functions (matter fields) from the same manifold
$M$ to a space $F$ in which a representation $\La$ of the group $G$
is defined [24-26]. One is then faced with the problem of
minimizing a functional $L$ (the lagrangian), usually expressed in
terms of a local density $\L$,
$$ L = \int_M \L (x) d^N x \eqno(1) $$
This kind of problem is better set in terms of fiber bundles [26,27].
We should in particular introduce a fiber bundle of total space $E$,
base $M$, projection $\pi : E \to M $, and fiber $\pi^{-1} (x) = F $, with
structural group $G$. The matter fields will then be (smooth) sections $\phi :
M \to E$, $\phi (x) \in \pi^{-1} (x)$, of this bundle; the space of smooth
sections of $E$ will be denoted by $\Phi$,
$$ \Phi = \{ \phi : M \to E ~/~ \pi \phi (x) = x ~~,~ \phi ~ {\rm smooth} \} $$
We should also consider another fiber bundle of total space $\~E$,
base $M$, projection $\~\pi : \~E \to M$, and fiber and structural
group $G$, $\~\pi^{-1} (x) = G$. The space of smooth
sections of $\~E$ will be denoted by $\Ga$,
$$ \Ga = \{ \g : M \to \~E ~/~ \~\pi \g (x) = x ~~,~ \g ~ {\rm smooth} \} $$
Let us first consider the unphysical case of a zero-th order density
$\L$, i.e. assume that the functional $\L$ depends only on the
matter fields $\phi (x)$,
$$ \phi : M \to F \sse R^s \eqno(2) $$
but not on its derivatives, which will be meant by the notation
$$ L = \int_M \L [\phi ] dx ~~~~~~~~ ; ~~ \L : F \to R \eqno(3) $$
and assume that $M$ is compact.
To say that $\L$ has a (local) gauge symmetry described by the
representation $ \La = \{ \La_g \}$ of the group $G$, $\La_g : F \to F$,
means that for any smooth function $\g : M \to G$,
$$ \L [ \La_{\g (x) } \phi (x) ] = \L [ \phi (x) ] \eqno(4) $$
In other words, we can consider the orbit space (under $\La$) $\Om =
F/G$. Then $\phi $ induces a $\phi_\Om : M \to \Om$, by $\om ( \phi
(x)) = \phi_\Om (x)$, and (4) tells that $L$ can be thought as a
functional on the space $\Phi_\Om$ of the $\phi_\Om$. (One should anyway pay
attention to smoothness problems, see the remark in section 5).
%\vskip 4truecm
This can also be seen as introducing a fiber bundle of total space $\=E$, with
base $M$, projection $\=\pi : \=E \to M$, and fiber $\=\pi^{-1} (x) = \Om
\equiv
F/G $ (notice that now $G$ acts on the fiber as the identity). Then $\=\Phi$ is
the space of sections of this bundle,
$$ \=\Phi = \{ \=\phi : M \to \=E ~/~ \=\pi \=\phi (x) = x \} $$
We should try to parallel the construction of sections 2-3 for the
space $\Phi$, in order to get an analogue of Michel's theorem in this case.
Notice that now the space $\Phi$ on which $G$ acts is infinite dimensional,
and therefore not compact.
The $\Ga$-orbit of a section $\s \in \Phi$, denoted $\th (\s)$, will be
defined
as
$$ \om (\s ) = \{ \s ' \in \Phi ~/~ \s ' (x) = \La_{\g (x)} \s (x) ~,~ \g (x)
\in \Ga \} \ss \Phi \eqno(5) $$
where $\Ga$ is the space of smooth sections $\g : M \to \~E$,
$\~\pi \cdot \g (x) =x $ of the principal fiber bundle $\~E$
introduced above.
The orbit space for sections will be denoted $\Th \equiv \Phi /
\Ga$; the orbit $\th (\s )$ will be denoted as $\th_\s$ when thought as a point
of $\Th$. Notice that $\Th$ corresponds to $\=\Phi$ defined above; an orbit
$\th$ can be seen as a section $\=\phi$ of the bundle $\=E$.
$\Ga$ is better seen as a subgroup of the group of fiber-preserving
(or gauge) diffeomorphisms of $E$,
$$ \GDiff (E ) = \{ f \in \Diff ( E ) ~/~f : \pi^{-1} (x) \to
\pi^{-1} (x) ~~ \all x \in M \} \eqno(6) $$
in particular when considering higher-order functionals (as in the
following section). This subgroup is simply given by
$$ \Ga \simeq \Ga_E = \{ f \in \GDiff (E) ~/~ f_x \equiv f\vert_{\pi^{-1} (x)}
=
\La_{\g (x)} ~,~ \g \in \Ga \} \ss \GDiff (E) $$
where $f_x$ is the restriction of $f$ to $\pi^{-1} (x)$.
In the same vein, $\Ga$ can be seen as coinciding with $\GDiff (\~E )$.
Given a section $\s \in \Phi$, we can define its isotropy subgroup
$\Ga_\s$ as
$$ \Ga_\s = \{ \g \in \Ga ~/ ~\g \cdot \s = \s \} \ss \GDiff (E ) \eqno(7) $$
where $\g \cdot \s$ has to be meant as
$$ \g \cdot \s \equiv \s ' (x) \equiv \La_{\g (x) } \cdot \s (x) \eqno(8) $$
If $\s ' = \g \cdot \s$, it is easy to see that
$$\Ga_{\s '} = \g \Ga_\s \g^{-1} \eqno(9) $$
We can therefore define as before a stratification of $\Phi$, at least
formally.
The reason for which this is only formal is that in the case of infinite
dimensional groups one can have a group conjugated to some of his proper
subgroups, so that the order relation could not be well defined. We will assume
for the moment that a stratification can be defined, and defer to a later
section (sect. 8) the issue of how to actually do it, and consideration of
the difficulties this can present.
We have seen before that the proof of Michel's theorem relies mainly on purely
geometrical concepts, which are transferred with no harm to the present
infinite dimensional setting. The only exception, i.e. obstacle to an infinite
dimensional extension, is represented by giving a topology to the orbit space.
In our case, anyway, we can take advantage of the fibered structure of the
problem, and define a distance between two sections $\s , \s' \in \Phi$ as
$$ d_\Phi ( \s , \s ' ) = { 1 \over |M| } \int_M d_F \left( \s (x) , \s '(x)
\right) dx \eqno(10) $$
where $|M| = \int_M 1 \cdot dx$, and $d_F (.,.)$ is a distance defined in $F$.
In order to define a distance in $\Th$, $\de : \Th \x \Th \to R_+$ (here
$R_+$ is the set of nonnegative reals), we can make use of the distance $d_\Om$
defined in $\Om$, see (3.2), i.e. of the equivariant retraction $\rho_\om$, by
$$ \de (\th_1 , \th_2 ) = { 1 \over |M| } \int_M d_\Om \left( \th_1 (x) , \th_2
(x) \right) dx $$
where $\th_i : M \to \Om $. In other words, we are defining a distance between
sections of $\=E$ along the lines of what we have done for sections of $E$,
i.e. by
$$ \de_\Om (\=\s_1 , \=\s_2 ) = { 1 \over |M| } \int_M d_\Om \left( \=\s_1 (x)
,
\=\s_2 (x) \right) dx \equiv { 1 \over |M| } \int_M d_\Om \left( \om ( \=\s_1
(x) ) , \om ( \=\s_2 (x) ) \right) dx $$
and use the isomorphism of $\Th$ with $\=\Phi$.
At this point, we can just repeat the proof of finite dimensional
Michel's theorem to obtain its extension to gauge functionals.
We will call a $\Ga$-orbit $\om (\s ) \ss \Phi$ a critical gauge orbit
for $\Ga$ if for every $\Ga$-invariant functional $L =\int_M \L [\s ]
dx$, $L :\Phi \to R$, $\om ( \s ) $ is a critical orbit for $L$. This
means that $\all \s \in \om (\s ), ~ \de L [ \s ] = 0$; or, $\L (\s + \epsilon
\de \s) = \L ( \s ) + O (\epsilon^2 ) $.
\bigskip \bigskip {\bf 5. Discussion} \bigskip
We think it can be useful to present some remarks.
The first is that to a
smooth section $\phi \in \Phi$ of $E$ can correspond a nonsmooth $\=\phi \in
\=\Phi$. To see an example, consider a trivial bundle of base $M= S^1$ and
fiber $F=R^1$; let $x$ be the coordinate on $M$ and $y$ the one on $F$; let the
group $G=Z_2 $ act on $F$ as $y \to -y$, so that $\Om = R_+$. The section $\phi
(x) = \cos (x)$ is smooth, but to it corresponds $\=\phi (x) \equiv \om ( \phi
(x)) = | \cos (x) |$ which is not such. If wishing to consider continuous
groups, one can instead e.g. consider $F = R^2$ and $G=SO(2)$.
A little thinking shows also that singular points of sections $\=\phi \in
\=\Phi$ lie in non-maximal (i.e. nongeneric, see [2,5] or sect.7) strata of
$\Om$ [5].
The second remark is quite closely related to this previous one: the careful
reader will have noticed that we defined a distance in the gauge orbit space
$\Th = \Phi / \Ga$ without defining an equivariant retraction in $\Phi$. This
is not only due to the fact that what we actually need is a distance in $\Th$,
but actually to an impossibility, as we now shortly discuss.
Given a section $\phi \in \Phi$ and a gauge orbit $\th \in \Th$, a distance of
$\phi$ from $\th$ can be defined using the analogous finite dimensional (i.e.
on $F$) concept, see eq. (3.4),
$$ \de (\phi , \th ) = {1 \over |M| } \int_M d ( \phi (x) , \th (x) ) dx $$
which also reads, in terms of the equivariant retraction on $F$,
$$ \de (\phi , \th ) = {1 \over |M| } \int_M d (\phi (x) , \rho_{\th (x) }
(\phi (x) ) ) dx $$
Now, an equivariant retraction of $\phi$ to $\th$ can be defined point-like at
any point $x \in M$, i.e. on any fiber $\pi^{-1} (x)$, as
$$ \tau_\th (\phi , x) = \rho_{\th (x)} ( \phi (x)) $$
but the section $\phi_{\tau \th}$ defined as
$$ \phi_{\tau \th} (x) = \tau_\th (\phi ,x) $$
can well fail to be smooth even if $\phi$ is.
In order to see an example, consider once again $M=S^1$, $F=R^1$, $G=Z_2$ as
before. The orbits in $F$ are made of two points, $\om (y) = \{ y,-y \}$,
except the singular orbit $\om (0) = \{ 0 \}$. Consider in $\Th$ the gauge
orbit $\th_1$ represented by the section $\s (x) = 1$. The retraction is simply
$$ \tau_{\th_1} ( \phi ,x) = \matrix{ ~1 & {\rm if} & \phi (x) > 0 \cr
-1 & {\rm if} & \phi (x) < 0 \cr} $$
(notice that $\tau_{\th_1} ( \phi ,x)$ is not defined if $\phi (x) \in \om (0)$
) so that for e.g. the section $\phi (x) = \cos (x)$, the retraction would be
$$ \phi_{\tau \th_1} (x) = \matrix{ ~1 & x < \pi/2 ~,~ x > 3 \pi /2 \cr -1 &
\pi /2 < x < 3 \pi /2 \cr} $$
No smooth section $\s \in \Phi$ exists such that
$$ \de (\phi , \s ) = \de (\phi , \th_1 ) = {\rm min}_{\s ' \in \th_1 } \de
(\phi
, \s ' ) $$
Let us recall the main result obtained: the basics facts valid for finite
dimensional compact group action, i.e.
{ $i)$ The gradients $dV$ of invariant functions $V(x)$ are in $T_x \S_x$ and
orthogonal to $\om (x)$ at $x$; they lie therefore in $N_x^0 \om \ss T_x \S_x
$.
$ii)$ One can define, by means of the distance defined in the $x$-space $X$, a
distance in the orbit space $\Om = X /G$; therefore the concept of
neighbourhood of a point $\om \in \Om$ in orbit space is well defined, as well
as that of an orbit isolated in its stratum.
$iii)$ A $G$-orbit is critical if and only if it is isolated in its stratum
(Michel's theorem; it follows from $i)$ and $ii)$ above). }
can be extended to gauge compact group action.
If $\Ga$ is the set (group) of smooth functions $\g : M \to G$ and $\Phi$ the
space of smooth sections $\phi : M \to E$, $\phi (x) \in \pi^{-1} (x)$, a
stratification of $\Phi$ under the action of $\Ga$ through the representation
$\La$ is well defined, and we have for $i) - iii)$ above, the corresponding:
{$i)$ The variations $\de \L$ of invariant functional densities $\L [ \phi ]$
are in $T_\phi \S_\phi$ and orthogonal to $\th (\phi )$ at $\phi$; they lie
therefore in $N_\phi^0 \th \ss T_\phi \S_\phi$.
$ii)$ One can define, by means of the distance defined on the finite
dimensional fiber $F$, a distance in the space of $G$-orbits in $F$, $\Om =
F/G$; by means of this one can define a distance in the gauge orbit space
($\Ga$-orbits) $\Th = \Phi / \Ga$. The concept of neighbourhood of a point $\th
\in \Th$ in gauge orbit space is well defined, as well as that of an orbit
isolated in its stratum.
$iii)$ A $\Ga$-orbit is critical if and only if it is isolated in its stratum
(Michel's theorem; it follows from $i)$ and $ii)$ above). }
It should be noticed that the deduction of $iii)$ from $i)$ and $ ii)$, once
the concept of critical orbit has been defined, is immediate.
Point $i)$ is of
geometrical nature, and once a stratification has been defined, it does not
make any difference if it refers to a finite or infinite dimensional space as
far as this point is concerned.
Therefore, as remarked earlier, the extension of Michel's theorem actually
consists only in defining a distance in gauge orbit space; it should be
stressed once again that we used in a crucial way the gauge (fibered) structure
of orbits.
\bigskip \bigskip {\bf 6. First order gauge functionals} \bigskip
In most physical cases, one is faced with a first order gauge theory
[25-27]; this means that the functional $L$ and its local density $\L$
depend not only on $\s (x)$, but on its first derivatives
$$ \pa_\mu \s (x) \equiv {\pa \s (x) \over \pa x^\mu } \eqno(1) $$
as well, which is what is meant by the notation
$$ L = \int_M \L [\phi , \pa \phi] dx \equiv \int_M \L [ \phi^\1 ] dx \eqno(2)
$$
Notice that under the transformation
$$ \s \to \g \cdot \s \equiv \La_{\g (x)} \s (x) \eqno(3) $$
the $\pa_\mu \s$ do not transform covariantly (i.e. in the same way as $\s$
does): in facts,
$$ \pmu \s \to \pmu (\g \c \s ) = (\pmu \g ) \c \s + \g \c (\pmu \s ) \equiv
(\pmu
\La_{\g (x)} ) \s (x) + \La_{\g (x) } (\pmu \s (x) ) \eqno(4) $$
One can introduce a covariant derivative $\cdm$ by
$$ \cdm = \pmu + \La_{A_\mu} \eqno(5) $$
$$ A_\mu : M \to \G \eqno(6) $$
where the $A_\mu$'s are called gauge fields (in physical notation) or
connection forms (in mathematical one). Here $\G$ is the Lie algebra of the Lie
group $G$, and $\La_\eta$ the infinitesimal generator
corresponding to the element $\eta$ of $\G$ in the representation $\La$.
One can check that if
$$ A \to \g A \g^{-1} + ( \pmu \g ) \g^{-1} \eqno(7) $$
(which means $\La_{A_\mu (x) } \to \La_{\g (x)} \La_{A_\mu (x)} \La_{\g
(x)}^{-1} + (\pmu
\La_{\g (x)} ) \La_{\g (x)}^{-1}$ ), then $\cd \s$ transforms as
$$ \cd \s \to \g \c (\cd \s ) \equiv \La_{\g (x)} ( \cd \s (x) ) \eqno(8) $$
i.e. covariantly.
Usually, in physical problems one starts from a density $\L (\s , \pmu \s )$
which is invariant under a global (i.e. rigid, $\g (x) = \g_0 = {\it const.}$)
gauge transformation $\La_{\g_0}$, and transform it into a local gauge one
(i.e.
a density invariant under local gauge transformations) [25] by considering
$\~\L (\s , \cd \s , A) = \L ( \s , \cd \s ) + \L_G (A)$, i.e. by
substituting $\cdm \s^i$ for $\pmu \s^i$ in $\L$ and adding the "pure
gauge" density $$ \L_G [A^\1 ] = F_{\mu \nu} F^{\mu \nu} \eqno(9) $$
$$ F_{\mu \nu} = \pa_\mu A_\nu - \pa_\nu A_\mu + [A_\mu , A_\nu ] \eqno(10) $$
As already remarked, a gauge theory is better seen, in mathematical language,
in terms of fiber bundles [26,27].
To this purpose, we introduce a new fiber bundle of total space $\^E$, with
base $M$, projection $\^\pi : \^E \to M$ and fiber $\^\pi^{-1} (x) = \G$ (this
is a vector bundle since a Lie algebra is a vector space). The gauge fields
will then be (smooth) sections $A_\mu : M \to \^E $, $A_\mu (x) \in \G$, of
this bundle; the space of smooth sections of $\^E$ will be denoted as
$$ \A = \{ A : M \to \^E ~/~ \^\pi A (x) = x ~,~ A ~ {\rm smooth} \} $$
$\Ga$ can be seen as a subgroup of $\GDiff (\^E )$,
$$ \Ga \simeq \Ga_{\^E } = \{ \^f \in \GDiff (\^E ) ~/~ \^f_x \c A(x) = \g (x)
A(x) \g^{-1} (x) + ( \pa \g (x) ) \g^{-1} (x) \} $$
where $\^f_x \equiv \^f |_{\^\pi^{-1} (x)}$ and we have used the notation of
(7).
We can also define the isotropy subgroup of $A \in \A$ as
$$ \Ga_A = \{ \g \in \Ga ~/~ \^g \c A = A \} $$
The setting discussed above requires then to consider (sections of) a sum
bundle $E_+ = E \oplus \^E$, with base $M$, fiber $F \o+ \G$ and projection
$\pi \o+ \^\pi$ (and structural group $G$).
For our purposes, it is actually more convenient to consider the bundle
$$ E_* = E^\1 \o+ \^E^\1 = E_+^\1 $$
with base $M$, fiber $F_* = JF \o+ J \G$ and projection $\pi_* = \pi^\1 \o+
\^\pi^\1 \equiv d \pi \o+ d \^\pi$. Here $JU$ is the (first) jet space of $U$
[28,33]; therefore the bundle $E_*$ is naturally equipped with a contact
structure [28-32].
The space of smooth sections $\chi_* : M \to E_*$, $\chi_* (x) \in \pi_*^{-1}
(x)$, of $E_*$ compatible with the contact structure of $E_*$ will be denoted
$\Phi_*$.
The sections $\chi = (\s , A) : M \to E_+$ of $E_+$ induce sections $\chi^\1 =
(\s^\1 , A^\1 )$ of $E_*$ obtained by prolongation [28,33]; conversely any
section $\chi_*$ of $E_*$ compatible with the contact structure defined in
$E_*$ is the prolongation of a section $\chi$ of $E_+$, $\chi_* = \chi^\1$, and
therefore the sections $\chi_* \in \Phi_*$ allow to recover the corresponding
sections $\chi \in \Phi \x \A$ [33].
The group $\GDiff (\E )$ extends (actually, is isomorphic) to the group
$\GDiff^\1 (\E ) \ss \GDiff (E_* )$ of prolongations [28,33] of
elements of $\GDiff (\E )$; this can also be seen as the subgroup of
$\GDiff (E_* )$ which preserves the contact structure of $E_*$, as it will be
discussed in the following.
In terms of explicit formulas, this setting gives back the previous ones: if
$\g \in \Ga \ss \GDiff (\E )$ transforms $\s$ into $\s ' = \g \s$,
then the prolongation $\g^\1 \in \Ga^\1 \ss \GDiff^\1 (\E )$
transforms $\s ,~ \cd \s ,~\pa \s , ~ A$ as
$$ \eqalign{ \s \to & ~ \s ' = \g \c \s \equiv \La_{\g (x)} \s (x) \cr
\cd \s \to & ~ ( \cd \s )' = \g \c \cd \s \equiv \La_{\g (x)} (\cd \s (x) ) \cr
A \to & ~ A' = \g A \g^{-1} + (d \g ) \g \cr
\pa \s \to & ~ \pa \s ' = \g (\pa \s ) + ( d \g ) \s \cr} \eqno(12) $$
The advantage of the present setting lies in that $L$ is a zero-th order
functional of sections of $E_*$, so that we recover the situation of the
previous section.
We would like to stress that when considering the variations $\de \L /
\de \chi $, only variations $\de \chi$ preserving the contact structure
should be allowed. This is satisfied if (and only if) $\de \chi = \( \de
\s^\1 , \de \g^\1 \)$, with $\de \s^\1 , \de \g^\1$ prolongations of $\de \s ,
\de \g$ [33,34] (see also [48]).
We can now define the isotropy subgroup of a section of $\^E$, and
therefore of one of $\E$. Let $\Ga$ be the group of smooth functions from $M$
into $G$:
$$ \Ga = \{ \g : M \to G \} \eqno(13) $$
which will be seen as a subgroup of $\GDiff (\E )$, by
$$ \Ga \simeq \Ga_{E_+} = \{ \g_+ : \s (x) \to \La_{\g (x)} \s ~~,~~ A \to \g A
\g^{-1} + (\pa \g ) \g^{-1} \} \ss \GDiff (\E ) \eqno(14) $$
(using again the notation of (7))
For a section $(\s , A )$ of $\E $, we define
$$ \Ga_{(\s ,A)} = \{ \g \in \Ga ~/~ \g \c \s = \s ~,~ \g A \g ^{-1} + (\pa \g
) \g^{-1} = A \} = \Ga_\s \cap \Ga_A \eqno(15) $$
(where the meaning of $\Ga_\s$ and $\Ga_A$ is obvious).
It is immediate to check that, if
$$ \s ' = \a \c \s ~~~;~~~ A' = \a A \a^{-1} + (\pa \a ) \a^{-1} \eqno(16) $$
then
$$ \Ga_{(\s ' , A')} = \a \Ga_{(\s , A)} \a^{-1} \eqno(17) $$
In facts, let $\Ga ' \equiv \Ga_{A'} ;~ \Ga \equiv \Ga_A$. By definition,
$$ \Ga A \Ga^{-1} + (\pa \Ga ) \Ga^{-1} = A \eqno(18) $$
and one has
$$ \eqalign{ ~ & (\a \Ga \a^{-1} ) A' \a \Ga^{-1} \a^{-1} + [ \pa (\a \Ga
\a^{-1}
)] \a \Ga^{-1} \a^{-1} = \cr
= & \a \Ga A \Ga^{-1} \a^{-1} + \a \Ga \a^{-1} (\pa \a )
\Ga^{-1} \a^{-1} + (\pa \a ) \a^{-1} + \a (\pa \Ga ) \Ga^{-1} \a^{-1} - \a \Ga
\a^{-1} (\pa \a ) \Ga^{-1} \a^{-1} = \cr
= & \a [ \Ga A \Ga^{-1} + (\pa \Ga ) \Ga^{-1}
] \a^{-1} + (\pa \a ) \a^{-1} = \a A \a^{-1} + (\pa \a ) \a^{-1} = A'
\cr} \eqno(19) $$
where we have used (16), (18) and the identity
$$ \pa \a^{-1} = - \a^{-1} (\pa \a ) \a^{-1} \eqno(20) $$
It is trivial that
$$ \Ga_{\s '} = \g \Ga_\s \g^{-1} \eqno(21) $$
so that (17) is satisfied.
One can therefore define a stratification in $(\Phi \x \A)$, and any stratum
will be the union of $\Ga$-orbits. By this, a stratification will also be
defined in the space of $\Ga$ orbits $ \Th_+ \equiv \( \Phi /\Ga \x \A / \Ga
\)
$.
It is also possible to define a distance in $\Th_+$ in the same way (and with
the same remarks) as done for $\Th$: we just have to define, given $\chi = (\s
,A)$, $\chi ' = (\s ' , A')$,
$$ \de (\chi , \chi ') = \de (\s , \s ') + \de (A,A') $$
and similarly for gauge orbits $\th_+ , \th_+ ' \in \Th_+$, $\th_+ = \th \x
\^\th$ (the $\^\th$ are orbits in the gauge orbit space $\A / \Ga$) we define
$$ \de (\th_+ , \th_+ ') = \de (\th , \th ') + \de (\^\th , \^\th ') $$
With this, a distance is defined in $\Th_+$. It is clear that we can proceed in
the same way for an $n$-sum vector bundle.
We have, anyway, to deal with sections $\chi_* \in \Phi_*$ of $E_*$; to this
purpose we will take advantage of the contact structure in $E_*$.
In order to deal with sections of $E_*$, we just use the fact that $JU = U \o+
U_{[1]}$ (where $U_{[1]}$ can be seen as the space of first derivatives of
functions $f(x) : M \to U$ [28,33]; if $M$ is one dimensional the jet space
$JU$ is nothing else than the familiar tangent bundle $TU$), so that $E_* $
is a 4-sum bundle equipped with an additional structure (the contact
structure).
Any section $\chi_* = \chi^\1 = (\s^\1 , A^\1 ) \in \Phi_*$ can be decomposed
as $\chi_* = ( \s , \cd \s ,A,\pa A)$, with $\s (x) : M \to F$, $(\cd \s ) (x)
: M \to F_{[1]}$, $A (x) : M \to \G$, $(\pa A) (x) : M \to \G_{[1]}$. Since
$F_{[1]}, \G_{[1]}$ are themselves vector spaces [28,33], we can proceed as
before in order to define the distance $\de ( \chi_* , \chi_* ')$ of two
sections in $\Phi_*$: with an obvious notation, this will be
$$ \de (\chi_* , \chi_* ') = \de (\s , \s ') + \de ( \cd \s , \cd \s ') + \de
(A,A') + \de (\pa A , \pa A') $$
When considering the action of $\Ga$ on $\Phi_*$, we should use the fact that
only sections of $E_*$ which are compatible with the contact structure are
allowed; this means that $\g \in \Ga$ acts on $\Phi_*$ by its (first)
prolongation $\g^\1$ [28,33].
The subgroup of $\GDiff (E_*)$ which preserves, in addition to the fibered
structure, the contact structure of $E_*$ will be denoted $\CGDiff (E_* )$; it
is clear that
$$ \CGDiff (E_* ) = \{ f^\1 ~/~ f \in \GDiff (E_+ ) \} = \GDiff^\1 (E_+ ) $$
(where as usual the superscript denotes first prolongation); we know by this
that [33]
$$ \CGDiff (E_* ) \simeq \GDiff (E_+ ) $$
$\Ga$ can be seen as a subgroup of $\CGDiff (E_* )$, by
$$ \Ga \simeq \Ga_* = \{ f_*^\1 \ss \GDiff (E_* ) ~/~ f_* = f \o+ \^f \in
\Ga_E \o+ \Ga_{\^E} \equiv \Ga_{E_+} \} $$
We can now define the isotropy subgroup of prolonged sections in the natural
way:
$$ \Ga_{\s^\1} = \{ \g \in \Ga ~/~ \g^\1 \c \s^\1 = \s^\1 \} ~~;~~ \Ga_{A^\1}
= \{ \g \in \Ga ~/~ \^\g^\1 \c A^\1 = A^\1 \} $$
$$ \Ga_{\chi^\1} = \Ga_{\s^\1} \cap \Ga_{A^\1} $$
It should be noticed that the action of $\g$ both on $\cd \s$ and $\pa A$
depends on the action of $\g$ on $A$.
Clearly, $\Ga_{\chi^\1} \sse \Ga_\chi$, as $\g^\1 \c \chi^\1 = \chi^\1$
requires in particular $\g \c \chi = \chi$ [33], but it is also true that if
a function is not changed, its derivatives are neither, so that
$\Ga_{\s^\1} = \Ga_\s$, $\Ga_{A^\1} = \Ga_A$ and
$$ \Ga_{\chi^\1} = \Ga_\chi $$
We stress that in general $\Ga_{\cd \s} \not= \Ga_{\s^\1}$, since $\cd \s = \pa
\s + A \s$, so that $\Ga_{\s^\1} \cap \Ga_\s = \Ga_\s \cap \Ga_A = \Ga_\chi$.
Finally, the above discussion allows to reconduce the stratification of
$\Phi_*$ under $\Ga$ (acting via $\La$) to that of $\Phi \x \A$. As usual, a
stratum in $\Phi_*$ will be the union of $\Ga$-orbits, so that a stratification
in the $\Ga$-orbit space $\Th_*$ is defined as well.
We still have to define a distance in $\Th_*$, but in order to do this we can
just proceed as for $\Th_+$; with obvious notation,
$$ \th_* (\chi_* ) = \th (\s ) \x \th^\1 (\cd \s ) \x \^\th (A) \x \^\th^\1
(\pa A) $$
and the distance is defined as
$$ \de (\th_* , \th_* ') = \de (\th , \th ') + \de (\th^\1 , {\th^\1 } ') +
\de (\^\th , \^\th ') + \de (\^\th^\1 , {{\^\th}^{\1}}' ) $$
With this, we have defined a distance in the $\Ga$-orbit space $\Th_*$ of
interest for first order gauge theories, which we see as zero order theories
with an assignement of a contact structure.
The discussion of sections 4 and 5 does therefore apply, and in particular
points $i) - iii)$ of section 5 continue to hold; quite clearly, we could in
the same way deal with gauge theories of any given finite order. This means
that Michel theorem for gauge orbits holds as well for first order gauge
theories, as those of physical interest.
We will consider some detailed examples in the following sections.
\bigskip \bigskip {\bf 7. Geometry and stratification of $\Om$} \bigskip
In order to understand the stratification of the space $\Phi$ of sections of
$E$ under the action of $\Ga$, one should first get a better knowledge of the
stratification of $F$ and $\Om \equiv F/G$ under the action of $G$.
This is actually a classical subject [2,5,21,22], but we will now recall some
relevant facts for the convenience of the reader, following [5]. We remind that
$F$ is a finite dimensional manifold, and $G$ a compact (actually, in physical
applications an orthogonal or unitary) Lie group acting on $F$ by a linear
representation $\La$. (It should actually be remarked that most of these
results could be extended to the case of smooth action of noncompact Lie
groups provided they have compact stabilizers, see [54]).
A theorem by Hilbert [5,22] states that there is a finite set of invariant
polynomials $\{ \tha_0 (x) , \tha_1 (x) , \dots , \tha_k (x) \} $ (here $x \in
F$, and we take $\tha_0 (x) \equiv 1$), $\tha_i (\La_g x ) = \tha_i (x)$ $\all
g
\in G$, such that any invariant polynomial $P (\La_g x ) = P (x) ~ \all g \in
G$
can be written as a polynomial in the $\tha$'s, $P(x) = \^P (\tha (x))$. The
$\{
\tha_i (x) \}$ form an integrity basis.
This theorem
was also extended to smooth functions [23,37]: any smooth invariant function
$f
(\La_g x) = f(x) ~\all g \in G$ can be written as a smooth function in the
$\tha$'s, $P(x) = \^P (\tha (x))$.
Now, invariant functions separate orbits, i.e. given two distinct orbits $\om ,
\om ' \in \Om$, there is at least one function $f$ (invariant, i.e. $f (\La_g
x)
= f (x)$ ) such that $f(x) \not= f(y)$ for $x \in \om,~ y \in \om '$.
Weierstrass approximation theorem tells that any $C^\infty$ function can be
locally written as the limit of a uniformly converging series of polynomials.
These two facts tell that the integrity basis separates orbits; in other words
we have that $\Om$ is a semialgebraic variety in $R^k$ (a subset of $R^k$
defined by equalities and inequalities of polynomials); $R^k$ can be thought as
the space of values assumed by the polynomials $\th_1 (x) , \dots , \th_k
(x)$.
A semialgebraic variety $\Om$ in $R^k$ has a natural primary stratification
[37],
i.e. can be seen as the disjoint union of open manifolds of dimensions from $k$
down to $0$ (as an example, a square $S$ is the union of interior points $S_i$
and of border $\pa S$; the latter is the union of points on edges $E_i$ and
border of edges $\pa E_i$, which are the vertices $V_i$), so that
$$ \Om = \cup_{\a , i} E^{(\a )}_i ~~~~~~~~~~~~~~~ {\rm dim} E^{(\a )}_i = \a
\eqno(1) $$
where $ \cup $ denotes disjoint union, and
$$ E_j^\b \in \pa E_i^\a ~~~~~~~~~~~ \b < \a \eqno(2) $$
The latter relation introduces a partial order in the set $\{ E^\a_i \}$ of
primary strata, i.e. bordering.
It can be proven that the stratification defined in sect. 2, i.e. based on
symmetry of orbits $\om \in \Om$ under $G$ (those strata will
sometimes be called isotropy strata) follows this primary stratification, in
the sense that connected components of isotropy strata correspond to union of
primary strata, i.e. $\S_\om = \cup_{\a \in A ; i \in I} E^\a_i$, for some
index sets $A,I$.
>From this it follow in particular two consequences: first, that only orbits in
primary strata of dimension $0$ can be isolated in their stratum; second, that
nearly all the $\om \in \Om$ belong to the maximal dimensional primary stratum,
and therefore to the principal stratum (some care should be taken if $\Om$ is
not
connected).
It can also be seen that more peripheral primary strata have higher symmetry;
i.e., the partial ordering given by the bordering relation coincides with the
partial ordering given by symmetry relations (see sect. 2).
Let us now consider some examples of stratification in orbit spaces (these same
examples will be considered again in the following to illustrate next steps).
\Example{ 1}: $F=R^1$, with coordinate $u \in R^1$; $G=Z_2 = \{ e,g \}$ acting
by $e: u \to u$ ; $g: u \to -u$. Then we have $\om (u) = \{ u,-u \}$, and
$$ \om
(u) \simeq Z_2 \for u \not= 0 ~~;~~ \om (u) = \{ 0 \} \simeq \{ e \} \for
u=0 $$
$$ G_u = \{ e \} \for u \not= 0 ~~;~~ G_u = Z_2 \for u=0 $$
(clearly, it must be $\om(u) = G / G_u$). The orbit space is $\Om = R_+ = \{ u
\ge 0 \}$ and we have two strata:
$$\Om_1 = \{ \th > 0 \} ~~~;~~~ \Om_0 = \{ 0 \} = \pa \Om_1$$
\Example{ 2}: $F=R^2 ,~ G=SO(2)$ acting by the standard representation. We have
$\om (u) = \{ u' ~/~ |u'|= |u| \}$, so that $$\om (u) \simeq SO(2) \for u \not=
0 ~~;~~ \om (0) = \{ 0 \} \simeq \{ e \}$$
$$G_u = \{ e \} \for u \not= 0 ~~;~~ G_0 = SO(2)$$
The orbit space is $\Om = R_+$, with strata
$$\Om_1 = \{ \th > 0 \} ~~;~~ \Om_0 = \{ 0 \} = \pa \Om_1 $$
\Example{ 3}: $F=R^3$, $G=SO(2)$ acting as rotations around the third axis,
generated by $\tau = \pmatrix{0&-1&0\cr1&0&0\cr0&0&0\cr}$. Then, with
coordinates $\{u_1 , u_2 , u_3 \}$ and $\rho (u) = u_1^2 + u_2^2$, we have $\om
(u) = \{ u' ~/~ \rho (u') = \rho (u) ~,~ u_3 ' = u_3 \}$. i.e. a minimal
integrity basis (MIB) is given by $\th_1 = u_1^2 + u_2^2~,~ \th_2 = u_3$. We
also
have
$$\om (u) \simeq SO(2) \for \th_1 (u) \not= 0 ~~;~~ \om (u) = \{ u \} \for
\th_1 (u) = 0$$
$$G_u = \{ e \} \for \th_1 (u) \not=0 ~~;~~ G_u = SO(2) \for
\th_1 (u) = 0$$
The orbit space is $\Om = \{ \th_1 , \th_2 ~/~ \th_1 \ge 0 \}
\ss R^2$; there are two strata,
$$\Om_1 = \{ \th_1 , \th_2 ~/~ \th_1 > 0 \} ~~;~~
\Om_0 = \{ \th_1 , \th_2 ~/~ \th_1 = 0 \} = \pa \Om_1$$
\Example{ 4}: $F=S^2 \ss R^3$ , $G=SO(2)$ acting as before. Then choose
coordinates $(\a , u_3 )$, $\a \in [0 , 2 \pi ]$, $u_3 \in [0,1]$, $(\a , \pm
1) = (0 , \pm 1) ~ \all \a$. Then $\om (u) = \{ u' ~/~ u_3 ' = u_3 \}$ (a
minimal integrity basis is given by $u_3$ alone);
$$\om (u) \simeq S^1 = SO(2)
\for u_3 \not= \pm 1 ~~;~~ \om (u) = \{ u \} \for u_3 = \pm 1 $$
The orbit space is $\Om = [-1,1] \ss R^1$; there are two strata,
$$\Om_1 = \{ \th
\not= \pm 1 \} ~~;~~ \Om_0 = \{ +1 , -1 \} = \pa \Om_1$$
Notice that here there are two orbits isolated in their stratum, while in
example 3 there are none.
\Example{ 5}: $F=R^3 ,~ G = SO(2) \x Z_2$, where $SO(2)$ acts as in examples 3
and 4, and $Z_2$ by $h = \pmatrix{1&0&0\cr0&1&0\cr0&0&1\cr}$; a \MIB is given
by $\{ \th_1 = u_1^2 + u_2^2 ~,~ \th_2 = u_3^2 \}$. The orbit space is $\Om =
\{
\th_1 , \th_2 ~/~ \th_1 \ge 0 ~,~ \th_2 \ge 0 \} = R_+ \x R_+$; $\om (u) = \{
u' ~/~ \th_1 (u' ) = \th_1 (u) ,~ \th_2 (u' ) = \th_2 (u) \}$, so that
$$ \eqalign{
\om(u) & \simeq S^1 \x Z_2 {\rm ~ for ~ } \th_1 \not=0 ;~ \th_2 \not= 0 \cr
\om(u) & \simeq Z_2 {\rm ~ for ~ } \th_1 =0 ;~ \th_2 \not= 0 \cr
\om(u) & \simeq SO(2) {\rm ~ for ~ } \th_1 \not=0 ;~ \th_2 = 0 \cr
\om(u) & \simeq \{ e \} {\rm ~ for ~ } \th_1 =0 ;~ \th_2 = 0 ~ (u=0) \cr } $$
Correspondingly, the isotropy subgroup is, in the four cases,
$$ G_u = \{ e \} ~~;~~ G_u = SO(2) ~~;~~ G_u = Z_2 ~~;~~ G_u = G $$
We have therefore four strata:
$$ \eqalign{
\Om_0 = \{ \th_1 \not=0 ;~ \th_2 \not= 0 \} \equiv E_1^2 \cr
\Om_1 = \{ \th_1 =0 ;~ \th_2 \not= 0 \} \equiv E_1^1 \cr
\Om_2 = \{ \th_1 \not=0 ;~ \th_2 = 0 \} \equiv E_1^1 \cr
\Om_3 = \{ \th_1 =0 ;~ \th_2 = 0 \} \equiv E_1^0 \cr } $$
One can check that actually
$$ \pa \Om_0 = \Om_1 \cup \Om_2 \cup \Om_3 ~~~;~~~ \pa \Om_1 = \pa \Om_2 =
\Om_3 $$
Notice that to these relations, also written
$$ \matrix{ & & \Om_1 & & \cr & \nearrow & & \searrow & \cr \Om_3 & & & & \Om_0
\cr & \searrow & & \nearrow & \cr & & \Om_2 & & \cr } $$
in terms of bordering relations, correspond the inclusion relations among
isotropy subgroups
$$ \matrix{ & & SO(2) & & \cr & \nearrow & & \searrow & \cr \{ e \} & & & &
SO(2) \x Z_2 \cr & \searrow & & \nearrow & \cr & & Z_2 & & \cr } $$
so that the diagram of bordering relations can also be seen as relative to
symmetry relations among strata.
\Example{ 6}: $F=S^2 \ss R^3$; $G=SO(2) \x Z_2$ acting as in example 5.
Choosing as coordinates $(\a , u_3 )$, $\a \in [0, 2 \pi ]$, $u_3 \in [-1, 1]$,
$(\a , \pm 1) = (0, \pm 1) ~ \all \a$, a \MIB is given by $\th_1 = u_3^2$, so
that $\Om = \{ \th_1 \} = [0,1] \ss R^1$. as for orbits, we have
$$ \matrix{
\om (u) \simeq & SO(2) \x Z_2 & {\rm ~ for ~} \th_1 \not=0 ,~ \th_1 \not= 1 \cr
\om (u) \simeq & SO(2) & {\rm ~ for ~} \th_1 =0 \cr
\om (u) \simeq & Z_2 & {\rm ~ for ~} \th_1 = 1 \cr } $$
and correspondingly for isotropy subgroups
$$ G_u = \{ e \} ~~~;~~~ G_u = Z_2 ~~~;~~~ G_u = SO(2) $$
There are three strata,
$$ \matrix{
\Om_0 = & \{ \th \not= 0 ,~ \th \not= 1 \} & \equiv E^2 \cr
\Om_1 = & \{ \th = 0 \} & \equiv E^1 \cr
\Om_2 = & \{ \th = 1 \} & \equiv E^0 \cr } $$
\bigskip \bigskip {\bf 8. Stratification of gauge orbit space} \bigskip
We can now discuss the stratification of the space $\Phi$ and of the
$\Ga$-orbit space $\Th = \Phi / \Ga$; from now on, subgroups of $G$ conjugated
in $G$ will simply be identified.
We will actually discuss a subclass $\Phi_T \ss \Phi$ of sections of $E$, that
of transversal ones (this name will be defined and explained in a moment).
Given a section $\s (x) : M \to F$, we can consider the set of values it takes
in $F$,
$$ F_\s = \{ \s (x) ~,~ x \in M \} \ss F \eqno(1) $$
and in an obvious way
$$ \Om_\s = \{ \om_{\s (x)} ~,~ x \in M \} \sse \Om \eqno(2) $$
Now, let us consider the primary stratification of $\Om$,
$$ \Om = \cup_\cdot E^\a_i ~~,~~ {\rm dim} E^\a = \a \eqno(3) $$
introduced in the previous section. The primary index $\a (\s )$ of the section
$\s$ is the greater $\a$ for which there is an $E^\a_i$ with
$$ \Om_\s \cap E^\a_i \not= \{ \oslash \} \eqno(4) $$
A section is transversal if it meets primary strata of dimension $\b$ strictly
less than its primary index transversally.
We do also define, for the sake of completeness, the set of accessible strata
for $\s$, $[E]_\s$:
$$ [E]_\s = \{ E^\a_i ~/~ \Om_\s \cap E^\a_i \not= \oslash \} \eqno(5) $$
Clearly,
$$ E_\s \equiv \cup_\cdot E^\a_i ~~,~~ E^\a_i \in [E]_\s \eqno(6) $$
is a semialgebraic variety; the locus of points $\om_{\s (x)} ,~ x \in M$
determines a curve $e(\s )$ in $E_\s$; it is immediate to see that for a smooth
section $\s$, $e(\s )$ is smooth at points belonging to strata of dimension $\a
(\s )$; it is also immediate to see that for transversal sections, $e(\s )$ is
singular ($C^0$ but not $C^1$) at points on strata of dimension strictly less
than $\a (\s )$.
It should be remarked that transversality is a structurally stable property,
and that transversal sections are dense in $\Phi$; we stress that this follows
from the results recalled in the previous section about the geometry of $\Om$.
This restriction to transverse sections could seem quite misterious, so let us
discuss it.
As briefly remarked in sect. 4 above, due to the fact that the gauge group is
infinite dimensional, the very concept of stratification is quite delicate. Let
us show by a simple example the kind of troubles one is faced with.
In the setting of example 2 of section 7, let the base space be $I = [0,1]$,
and consider two sections given by, with $v_0$ a given unit vector in $F$,
$$ \s_i (x) = e^{-1/(x-x_i )^2 } v_0 ~~,~~ x \le x_i ~~;~~ \s_i (x) =0 ~~,~~ x
\ge x_i ~~ ; ~~ i=1,2 $$
Clearly, the isotropy subgroups $\Ga_i \ss \Ga$ of these are
$$ \Ga_i = \{ \g \in \Ga ~/~ \g (x) = \{ e \} ~~ x \ge x_i \} \equiv
\{ \g : [0 , x_i ] \to SO(2) ~/~ \g (x_i )=0 \} $$
Now, $\Ga_1$ and $\Ga_2$ are isomorphic but if, say, $x_2 > x_1$, then $\Ga_1$
is a proper subgroup of $\Ga_2$.
It is quite clear that this "patology" does not occurr when restricting to
transversal sections. It {\it seems} to us that this restriction is sufficent
for a stratification to be properly defined, but we stress that this is not
being proved. We will anyway see that in our examples this restriction does
actually suffice to properly define a stratification.
Actually, to the best of author's knowledge, very little is known about the
geometry of orbit space for infinite dimensional, e.g. gauge, groups (see e.g.
the problem of Gribov ambiguity; even the proof that the Gribov region
contains representatives of every gauge orbit is very recent [49]). We will
formalize our reasonable guess into the
{\sl Assumption:} In the space $\Phi_T \ss \Phi$ of transversal sections, and
therefore in $\Th_T = \Phi_T / \Ga$, a stratification is properly defined.
Actually, we will mainly use only the
{\sl Weaker assumption:} For the action of $\Ga$ on $\Phi_T$, there is a set
(possibly infinite or even continuous) of subgroups $\Ga_\mu$ which are: $i)$
isotropy subgroups for some $\s \in \Phi_T$; $ii)$ not contained in any other
subgroup satisfying i); $iii)$ such that no proper subgroup $\Ga_\mu^{(\a)} \ss
\Ga_\mu$ is conjugated to any $\Ga_{\mu '}$ . In other words, the concept of
maximal isotropy subgroup of $\Ga$ is properly defined.
In the following we will consider only $\Phi_T$; since no confusion will be
possible, we will write - for ease of notation - $\Phi$ for $\Phi_T$ and $\Th$
for $\Th_T$. Should our assumptions be wrong, our discussion would be purely
formal; we believe that anyway it would remain of eurysthical value, as will be
seen in the following. Moreover, as already remarked, they apply at least to
the examples considered here.
Let us first consider a section $\s$ such that
$$[E]_\s =E^\a_i \eqno(7) $$
i.e. such that $\s (x)$ belongs to one and the same stratum for all $x \in M$;
then, if $G^\a_i $ is the isotropy subgroup corresponding to this stratum
$E^\a_i$, $\Ga_\s$ is the set of functions
$$ \Ga_\s = \{ \g (x) : ~ M \to G_\a \} \sse \Ga \eqno(8) $$
(we recall this is a group by $(\g_1 \cdot \g_2 ) (x) = \g_1 (x) \cdot \g_2
(x)$
).
Let us now consider a $\s$ such that
$$ \s (x) \in E^\a_i ~~~x \in D_1 ~~;~~ \s (x) \in E^\a_j ~~~ x \in D_2
\eqno(9) $$
for $D_1$ and $D_2$ domains of $M$, $\dim (D_1 )= \dim (D_2 )= \dim
(M)$, with border $ \pa D_1 = \pa D_2 = B$, $\dim B < \dim D_i$. Then, called
$G_1$ and $G_2$ the isotropy subgroups relative to the two strata $E^\a_i$ and
$E^\a_j$, we have
$$\Ga_\s \ss \Ga_\s^* = \{ \g : M \to G ~/~\g : D_1 \to G_1 ~;~\g : D_2 \to G_2
\} \ss \Ga \eqno(10) $$
One still has to impose boundary conditions on $B$; smoothness of $\g$ requires
that
$$ \g : B \to ( G_1 \cap G_2 ) \eqno(11) $$
(the above intersection is not empty since it contains at least $\{ e \} $).
Notice that, a priori, the section $\s (x)$ maps $B$ to more peripherical
strata,
so that one could think to have $\g (x) \in G^\b_k$ for $x \in B$, $\s (x) \in
E^\b_k$, since $E^\b_k \in \pa E^\a_i $, $ E^\b_k \in \pa E^\a_j$ and given
the discussion of the previous section; on the other side, $\dim B < \dim D_i $
amounts to the transversality condition, and this together with continuity of
$\g$ gives (11). Finally, for such a $\s$ we get
$$\Ga_\s = \{ \g : M \to G ~/~\g : D_1 \to G_1 ~;~\g : D_2 \to G_2 ~;~ \g : B
\to G_1 \cap G_2 \} \eqno(12) $$
Remark that for $G_1 = G_2 = G_\a$, this amounts to (8); in other words, the
transversality condition implies that "only higher dimensional strata in
$[E]_\s$
matter", in the sense that the isotropy subgroup $\Ga_\s$ does not depend on
the
strata of lower dimension, i.e. is insensible to the values taken by $\s (x)$
on
the lower dimensional set $B \ss M$.
With the previous discussion in mind, it is easy to understand the general
situation for transversal sections. We will discuss the case of compact
connected
$M$ and connected orbit space $\Om$, of interest here.
Given a transversal section $\s (x)$ of primary index $\a (\s ) \equiv \a$,
let
$\{ E^\a_i ~,~i=1, \dots , k \}$ be the set of $\a$-dimensional strata in
$[E]_\s$, that is, such that
$$\exists x \in M ~/~ \s (x) \in E^\a_i \eqno(13) $$
Then, let $D_i$ be the anti-image of $E^\a_i$ by $\s$,
$$ D_i = \{ x \in M ~/~\s (x) \in E^\a_i \} \eqno (14) $$
The transversality condition implies that
$$ B \equiv M \backslash \{ D_1 \cup \dots \cup D_k \} = \pa D_1 \cup \dots
\cup \pa D_k
\eqno(15) $$
and therefore
$$ \dim B < \dim D_i = \dim M \eqno (16) $$
Then, if $G_i$ is the isotropy subgroup of $E^\a_i$, we have
$$ \Ga_\s = \{ \g : M \to G ~/~ \g : D_i \to G_i \} \ss \Ga \eqno(17) $$
where the smoothness of $\g$ implies
$$ \g : (\pa D_i ) \cap ( \pa D_j ) \to G_i \cap G_j \eqno(18) $$
It should be stressed that $\Ga_\s$ does depend in a crucial way on the
geometry
of the $D_i$'s, as this does fix the "boundary conditions" (i.e. the conditions
on $B$) which the $\g \in \Ga_\s$ have to satisfy.
\bigskip
As an example of this, consider the setting of example 5 in the previous
section.
Let $M=S^1$, and $\s_1 (x) = (0,0,\sin (x) )$. Then $\a (\s) = 1$, $[E]_s = \{
E^1_1 , E^0_1 \}$; as for $D_1$, this is $D_1 = \{ (0, \pi ) \cup (\pi , 2 \pi
)
\},~B=\{ 0 , 2 \pi \}$. We have therefore
$$\Ga_1 \equiv \Ga_{\s_1} = \{ \g : S^1 \to SO(2) ~/~\g (0) = \g (\pi ) = e \}
\eqno(19) $$
(since $SO(2) \cap Z_2 = \{ e \}$ in this setting), so that $G_1$ is
parametrized by pairs of functions $(\g_1 , \g_2 )$ from the interval $I$ to
$SO(2)$ which satisfy boundary conditions $\g_i (\pa I ) = e$.
Let us now consider the section $\s_2 = (0,0, \sin (2x) )$. Now $\a (\s )$ and
$[E]_\s$ are as before, but
$ D_1 = \{ (0, \pi / 2 ) \cup ( \pi / 2 , \pi ) \cup ( \pi , 3 \pi /2) \cup (3
\pi /2, 2 \pi ) \}$, $B= \{ 0, \pi /2 , \pi , 3 \pi /2 \}$, and
$$ \Ga_2 \equiv \Ga_{\s_2} = \{ \g : S^1 \to SO(2) ~/~ \g (0) = \g ( \pi /2 ) =
\g ( \pi ) = \g ( 3 \pi /2 ) = e \} \eqno(20) $$
so that $\Ga_2$ is parametrized by quadruples $(\g_1 , \g_2
, \g_3 , \g_4 )$ of functions from the interval $I$ to $SO(2)$ which satisfy
boundary conditions $\g_i (\pa I ) = e$. Therefore, a priori (i.e. without
considering the requirement of smoothness of $\g$), we have $\Ga_1 \sse
\Ga_2$.
Notice also that there is a "basis space" $S^1$ symmetry associated to this
problem; this is broken to $Z_2$ for $\s_1$ and to $Z_4$ for $\s_2$; basis
space symmetry will be shortly discussed in the last part of this paper.
\bigskip
By looking at this setting, with $M$ an higher dimensional manifold, one gets
easily convinced that an explicit stratification of $\Th = \Phi / \Ga $ is
extremely complicate and difficult to describe. We will not attempt such a
description here, but will instead concentrate on the description of most
singular strata, i.e. those corresponding to maximal isotropy subgroups. These
are also the strata on which the extension of Michel's theorem given above can
be
applied.
It should be stressed that maximal isotropy subgroups are not always the only
ones corresponding to most singular strata: the hypotheses that they indeed are
is known as the maximal isotropy subgroup conjecture, and is now known to be in
general not true. A complete discussion of it, including identification of the
cases (i.e. of the groups) in which it holds true, has been given recently by
Field and Richardson for compact Lie groups [50-53].
\bigskip \bigskip {\bf 9. Maximal strata in gauge orbit space} \bigskip
Let us consider again the action of $G$ on $F$ by the representation $\La = \{
\La_g
subgroups (MIS) $G_\mu ,~ \mu = 1 , \dots , s$, i.e. of subgroups $G_\mu \sse
G$ such
that $\exists z \in F ~/~ \La_g z = z ~ \all g \in G_\mu$, and there is no
subgroup
$G_\mu^* \sse G$ such that $G_\mu \ss G_\mu^*$ and $G_\mu^*$ is an isotropy
subgroup. (We stress that the concept of MIS depends on both $F$ and $\La$, for
given $G$).
The set $\Om_\mu = \{ \om \in \Om ~/~ \La_g z = z ~\all z \in \om ,~ \all g \in
G_\mu
\}$ will correspond to a maximal (i.e. minimal dimensional) stratum.
A section $\s$ such that $\s (x) \in \Om_\mu ~\all x \in M$ (i.e. $\Om_\s
\sse \Om_\mu$) will admit as symmetry group
$$\Ga_\mu = \{ \g : M \to G_\mu \} \eqno(1) $$
The groups $\Ga_\mu \ss \Ga$, for $G_\mu$ a MIS of $G$, are MIS of $\Ga$. In
facts, to
have $\Ga_\mu \ss \Ga '$, $\Ga '$ must contain $\g$'s such that for some $x
\in M$, $\g (x) \in G \backslash G_\mu$. But we know that every $g$ such that
$\exists
z ~/~ \La_g z = z$ must belong to some $G_\mu ,~ \mu = 1 , \dots ,s$.
Therefore
$\g \in \Ga$ can belong to the isotropy group of some section only if
$$ \g (x) \in \cup_{\mu = 1,...,s} G_\mu \eqno(2) $$
Suppose now that for $x \in M$ $\g (x )$ belongs to at least two different
$G_\mu$'s, $\g (x_i ) \in G_i $, $G_i \not= G_j$ for $i \not= j$, and
let $M_i = \{ x \in M ~/~ \g (x) \in G_i \}$. Then necessarily there are
points $x \in M_i \cap M_j$; due to smoothness of $\g$, in these $\g (x) \in
G_i \cap G_j$.
Now, the functions $\g : M \to G ~/~ \g : M_i \to G_i$ can be seen as $n$-ples
of
functions $\g_i$ defined on $M_i$ with vales in $G_i$, each of them subject to
appropriate boundary conditions: on $M_{ij} \equiv M_i \cap M_j \in \pa M_i$,
$\g :
M_{ij} \to G_i \cap G_j$. Clearly for no distincts $i,j$ one can have $G_i \ss
G_i \cap G_j$, for the $G_\mu$ are MIS. This also means that it is not
possible
to find an isotropy group $\Ga ' \ss \Ga$ such that $\Ga_\mu \ss \Ga '$.
We stress that the above argument shows that all the $\Ga_\mu$ of the form (1),
with
$G_\mu$ a MIS of $G$, are maximal isotropy subgroups of $\Ga$, but in general
not all
the MIS (even on the set of transverse sections) need to be of the form (1).
In the same way, one can see that given a stratum $\S_\om \ss \Om$, with
isotropy
subgroup $G_0$, the sections $\s$ such that
$$ \om_{\s (x) } \in \S_\om ~~ \all x \in M \eqno(3) $$
have isotropy subgroup
$$ \Ga_0 = \{ \g \in \Ga ~/~ \g : M \to G_0 \} \eqno(4) $$
Let us go back to MIS: if in the stratification of $\Om$ by $G$ $\S_\om^{(\mu)}
\equiv
\Om_\mu$ is a maximal stratum (i.e. minimal dimensional, corresponding to a
MIS), we have just seen that
$$ \Phi_\mu = \{ \s \in \Phi ~/~ \s : M \to \Om_\mu \} \eqno(5) $$
form a maximal stratum with isotropy $\Ga_\mu$. In other words,
$$\Th_\mu = \Phi_\mu / \Ga \eqno(6) $$
is a maximal stratum in the stratification of $\Th$ by $\Ga$. (Remark that
actually
$\Th_\mu = \Phi_\mu / ( \Ga / \Ga_\mu )$, as $\Ga_\mu$ is the identity on
$\Phi_\mu$).
Now, let us assume that $\om \in \Om$ is isolated in its stratum; it is
immediate that
$\th \in \Th$, where $\th$ is the gauge orbit of sections $\s ~/~ \s (x) \in
\om ~ \all x
\in M$, is isolated in its stratum (recall that $\Th$ is equipped with the
topology
induced by the topology of $\Om$). This gives a constructive way for
determining
some (not all, in general !) of the critical gauge orbits.
We will summarize our discussion as follows:
\bigskip
\item{ } {\bf Theorem:} Let $G_1 \sse G$ be a MIS for the action of $G$ on $F$
by the representation $\La$, and let $\Om = F/G$. Then
\item{i)} $\Ga_1 = \{\g : M \to G_1 \}$ is a MIS of $\Ga$;
\item{ii)} The set $\Th_1 = \{ \th ~/~ \om \cdot \s (x) : M \to \Om_1 ~\all x
\in M ~\all
\s \in \th \}$, where $\Om_1 = \{ \om ~/~ \La_g z = z ~ \all z \in \om ~ \all g
\in G_1
\}$, is a maximal stratum of $\Th$;
\item{iii)} If $\om_0$ is isolated in its stratum $\Om_1$, then $\th_0$, the
gauge orbit
such that any section $\s (x) \in \om_0 ~\all x \in M$ belongs to $\th_0$, is
isolated in
its stratum $\Th_1$;
\item{iv)} As a consequence of iii), for every critical orbit $\om_0 \in \Om$
there is a
critical gauge orbit $\th_0 \in \Th$.
\bigskip
Let us now consider some examples, following (also in the numerotation) those
given in sect. 4.
\Example{ 1}: The section $\s_0 (x) =0$ constitutes a stratum $\S_0 \ss \Th$,
with isotropy subgroup $\Ga_0 = \{ \g : M \to G \} = \Ga$ (notice that, due to
the discreteness of $G$, these are actually constant functions). All the other
transverse sections form the principal stratum $\S_1 \ss \Th$, with isotropy
$\Ga_1 = \{ \g (x) \equiv e \}$.
\Example{ 2}: As in example 1, $\s_0 (x) = 0$ is the only section in the
stratum
$\S_0$, with isotropy $\Ga_0 = \Ga$; all the other transverse sections are in
the principal stratum $\S_1$ with isotropy $\Ga_1 = \{ \g (x) \equiv e \}$.
\Example{ 3}: The section $\s (x) : M \to \Om_0$ have isotropy $\Ga_0 = \Ga =
\{
\g : M \to SO(2) \}$; these form a stratum $\S_0 \ss \Phi$, to which
corresponds $\S_0^\th \ss \Th$, isomorphic to $\{ \th : M \to R \} \simeq
\S_0$. The other transverse section have $\a (\s ) =2$ and form the generic
stratum $\S_1$ with isotropy $\Ga_1 = \{ \g (x) \equiv e \}$. Notice that for
$\s \in \S_0$, $\a (\s ) = 1$ and in facts there are no sections isolated in
their stratum.
\Example{ 4}: The sections $\s_\pm (x) = (0,0,\pm 1)$ form a stratum $\S_0$
with
isotropy $\Ga_0 = \Ga = \{ \g : M \to G \}$. All the other transverse sections
belong to the principal stratum $\S_1$ with isotropy $\Ga_1 = \{ \g (x) \equiv
e
\}$. Notice that now $\s_\pm$ have $\a ( \s )=0$, and are indeed isolated in
their stratum.
\Example{ 5}: The section $\s_0 (x) = (0,0,0)$ form a stratum $\S_0$ with
isotropy $\Ga_0 = \Ga = \{ \g : M \to G = SO(2) \x Z_2 \}$; notice that $\a
(\s_0 )=0$ and $\s_0$ is in facts isolated in its stratum.
The transverse sections with $\a (\s )=2$, i.e. such that $E^2_1 \in [E]_\s$,
have isotropy $\Ga_2 = \{ \g (x) \equiv e \}$.
There are two one-dimensional strata, $E^1_1$ and $E^1_2$, in $\Om$; therefore
we have several possibilities for $\s$'s with $\a (\s )=1$. Two strata $\S_1^1
, ~\S_2^1$ are made of sections such that only one of the $E^1_i$ is in
$[E]_\s$; they have isotropy $\Ga_1^1 = \{ \g (x) : M \to Z_2 \}$ and $\Ga_1^2
=
\{ \g (x) : M \to SO(2) \}$.
As for the sections such that both $E^1_1$ and $E^1_2$ are in $[E]_\s$, notice
that since $SO(2) \cap Z_2 = \{ e \}$, necessarily on the domain $D_1 \ss M$,
$D_1 = \{x ~/~ \om ( \s (x)) \in E^1_1 \}$ we have $\g (x) = e ~ \all \g \in
\Ga_\s$, so that if $D_2 = \{ x ~/~ \om ( \s (x)) \in E^1_2 \} \ss M$ is the
disjoint union of subdomains $D_2^k ,~ k = 1 , \dots , K$, $\Ga_\s$ is made up
of $K$-ples of functions $\g_k : D_2^k \to SO(2)$ with boundary conditions
$\g_k : \pa D_2^k \to \{ e \}$.
\Example{ 6}: Here, the sections $\s_\pm = (0,0,\pm 1)$ form a stratum $\S^0_1$
with isotropy $\Ga^1_0 = \{ \g : M \to SO(2) \}$. The sections $\s (x) =
(z(x),w(x),0)$ form a stratum $\S^0_2$ with isotropy $\Ga^2_0 = \{ \g : M \to
Z_2 \}$; the other transverse sections form a stratum $\S_1$ with isotropy
$\Ga_1 = \{ \g (x) = e \}$.
\bigskip \bigskip {\bf 10. The equivariant branching lemma} \bigskip
One of the most fruitful applications of Michel's theorem in finite dimensional
case is to symmetric bifurcation theory [38,8,10,11]; this is done by means of
the so called {\sl equivariant branching lemma} (EBL), first proved for
bifurcation of stationary solutions [9,39] and extended later to other types of
bifurcations [40,41,11,15,17]. We will consider it only in the stationary and
variational settings. A related full discussion is given in [18]
We have seen before (sect.3) that
$$ \cd V(x) \in N_x^0 \om (x) \eqno(1) $$
If we denote by $W_x$ the space on which $G_x$ acts trivially,
$$ W_x = \{ y \in M ~/~ \La_g y = y ~~ \all g \in G_x \} = {\rm Ker}~G_x \sse M
\eqno(2) $$
we have immediately
$$ \cd V(x) \in T_x W_x ~~~\all x \in M \eqno(3) $$
Notice also that for linear representations the linearity of
$\La$ implies that if $M$ is embedded in $R^m$, $W_x$ is the intersection with
$M$ of a linear subspace om $R^m$, $W_x = R^{d(x)} \cap M$.
When considering gradient dynamical systems, i.e. of the form ($V$ an invariant
potential)
$$ \xd^i = F^i (x) = { \pa V(x) \over \pa x^i } ~~~x \in R^N \eqno(4) $$
$$ \xd = F(x) = \cd V(x) $$
equation (3) means that $W_x$ is invariant under the flow of (4), that is
$$ x(t) \in W_{x_0} ~~\all t \ge t_0 ~~ , ~~ x(t_0 ) = x_0 \eqno(5) $$
This is therefore called the reduction lemma (RL):
{\sl Reduction lemma:} The solutions of (4) with initial datum $x_0$ are also
solutions of the restriction of (4) to the linear subspace $W_0 \equiv
W_{x_0}$.
In other words, one is authorized to study the reduced system $\xd = \^F (x) ,
In the variational setting, the reduction lemma is most concisely expressed by
eq. (3); in other words, one has the
{\sl Reduction lemma (variational case):} The point $x_0 \in M$ is critical for
the invariant potential $V(x)$ if and only if the projection of $\cd V (x_0 )$
on $W_0$ vanishes; equivalently, if and only if $(y, \cd V(x) )=0 ~ \all y \in
T_x W_0$, where $(.,.)$ is the scalar product in $R^m$.
>From the RL, it follows at once the EBL; in this case, the variational
formulation is - in a sense which will be clear in the following - much more
powerful than the general evolution one.
In the generic case, we will consider
$$ \xd = F( \l , x) ~~~x \in M = R^m \eqno(6) $$
where $\l$ is a parameter, $\l \in R$, and $F: R \x M \to TM$ is an equivariant
vector field (i.e. $F(\l , \La_g x ) = \La_g F (\l , x) ~~\all g \in G $),
smooth in both arguments. $F$ will also assumed to be confining, i.e. it exists
a compact $B$ of the same dimension as $M$ such that, if $n(x) \in T_x M ~, ~x
\in \pa B$ is the unit tangent vector pointing outward of $B$, then $(n(x) ,
F(\l , x) ) \le 0 ~ \all x \in \pa B ~, ~ \all \l$. We have then
{\sl Equivariant bifurcation lemma:} Let G admit an isotropy subgroup $G_0$
with $\dim \Fix (G_0 ) = 1$ , where $\Fix (G_0 ) \equiv W_0 = \{ x \in M
stationary points $x_0 (\l ) \in W_0 $ under (6).
Remark that the assumption about $(x,F(\l , x) ) $ on $\pa B$ could be replaced
by an analogous assumption on points $x_- , x_+$ on $\pa I$, with $I$ a
nonempty
interval in $W_0$.
Remark also that we assumed $M=R^m$ in order to conform to the usual
bifurcation-theoretic setting: there (6) represents the bifurcation equation,
living by construction in (a neighbourhood of the origin in) $R^m$. One could
as well consider $M$ a general manifold; in this case, anyway, a
one-dimensional $W_0$ could be diffeomorphic to $S^1$ rather than to $R^1$, and
no stationary solution exist. In this case, it is opportune to drop the
confining condition, and use instead an analogue of the above mentioned one:
i.e. we ask that the one dimensional submanifold $W_0 \ss M$ contains an
invariant (under $\xd = F( \l ,x)$) interval $I_0 \ss W_0$.
Let us now discuss the variational case; i.e., we look for critical points of
the invariant potential $V$:
$$ \cd_x V (\l , x) = 0 ~~~~ x \in M \eqno(7) $$
or, equivalently, stationary solutions of
$$\xd = F (\l ,x) \equiv \cd_x V (\l ,x) \eqno(8) $$
Here again $\l \in R^1$ is a real parameter, $V$ is assumed to be smooth in
both arguments, and invariance means
$$ V (\l , \La_g x ) = V (\l , x ) ~~~~\all g \in G \eqno(9) $$
We can now allow $M$ to be of infinite (but numerable) dimension. We have then
that:
{\sl Equivariant bifurcation lemma (variational case):} Let G admit an isotropy
subgroup $G_0$ such that $\dim W_0 = d < \infty$ , where $W_0 \equiv \Fix (G_0
)
= \{ x \in M ~/~ \La_g x = x ~ \all x \in G_0 \}$. Then, if it exists a
$d$-dimensional compact set $B \ss W_0$ (topologically, a ball), such that
$\cd V (\l , x)$ points outward of $B$ for all $x \in \pa B$, $\all \l$, then
it exists a smooth family of critical $G$-orbits $\om (\l ) \ss B \ss W_0$ for
the potential $V$; each point $x \in \om$ is a minimum for $V$.
Obviously, we could as well consider $\cd V$ pointing inward on $\pa B$, and
grant the existence of a maximum.
The proof of the lemma is immediate: the reduction lemma allows to reduce to
$W_0$, but now we can further reduce to $B$; this is a compact set in a finite
dimensional space, and $V$ is a confining potential for $B$ (due to the
assumption $\cd V$ points outward of $B$ on $\pa B$). Therefore $V$ has a
minimum in $B$.
Remark that the restriction $\^V$ of $V$ to $W_0$ will exhibit some
invariance:
in particular, it will be symmetric under $N (G_0 ) = \{ g \in G ~/~ g G_0
g^{-1} =G_0 \} \equiv \{ g \in G ~/~ \La_g : W_0 \to W_0 \} $ (see e.g. [11]),
the normalizer of $G_0$ in $G$. Actually, $G_0$ acts trivially in $W_0$, and
is normal by definition in $N (G_0 )$, so that the symmetry of $\^V$ inherited
by the $G$-invariance of $V$ corresponds to the group
$$ D_0 \equiv D (G_0 ) = N(G_0 ) / G_0 \eqno(10) $$
Therefore, critical points of $\^V$ appear in $D_0$-orbits, and that is why in
the statement of the lemma we have a critical set - which is actually a
$D_0$-orbit - rather than a critical point.
Remark also that $\^V$ could possess other symmetries beside those described by
$D_0$, as a result of the reduction process. For a discussion of this matter,
we refer the reader to [10,11,17,42].
\bigskip \bigskip {\bf 11. A reduction lemma for gauge invariant functionals}
\bigskip
The EBL has recently been generalized by Cicogna [15] (see also [43]) to the
case
of symmetry under general (i.e. not necessarily linear) diffeomorphism groups;
his approach can be readily applied to the case ay hand, i.e. to gauge
symmetries.
We stress that the gauge group $G$ is assumed to be a compact finite
dimensional Lie group, the fiber $F$ to be finite dimensional and the action
of $G$ on $F$ to be linear; we will not repeat this caveat in our statements,
but
these statements are false if we drop these conditions, i.e. if extended to
more
general cases (see also [19,20]).
Let $\{ u_1 , ... , u_N \}$ be a basis of $F$, and $\{ x_1 , ... , x_m \}$
coordinates in $M \sse R^m$. One can consider the Lie algebra $\G$ of $G$; let
$\{ L_1 , ... , L_k \}$ be a basis of $\G$: the $L_i$ can be
represented as linear differential operators of the form
$$ L_i = A^{(i)}_j (u) \pa_j \eqno(1) $$
where $\pa_j \equiv \pa / \pa u_j$ and $A^{(i)}_j (u)$ is linear in the $u$;
i.e. $A^{(i)}$ can be represented by a matrix $A^{(i)}_{jk}$, and
$$ L_i = \( A^{(i)}_{jk} x_k \) \pa_j \eqno(2) $$
An arbitrary element $\eta$ of $\G$ can be written as a linear combination of
the $L_i$:
$$\eta = a_k L_k ~~~~~ a_k \in R \eqno(3) $$
Now, an arbitrary function $f:M \to \G$ can also be written as a vector
function $\phi : M \to R^k$, by
$$ f(x) = \sum_{j=1}^k \phi_j (x) L_j \equiv \phi (x) \c L \eqno(4) $$
The above representation shows that $\Ga$ is a module [44] (over the ring of
functions on $M$), and is generated by the constant functions $f_i (x) = L_i$,
$i=1,...,k$.
In facts, we have just seen that $\a (x) \eta_1 + \b (x) \eta_2 \in \Ga$ if
$\eta_1 , \eta_2 \in \Ga$, and that any $\g \in \Ga$ can be written as $\g =
\phi^i \c L_i$. Moreover,
$$ [\g_1 , \g_2 ] = [ \phi^i L_i , \psi^j L_j ] = ( \phi^i (x) \psi^j (x) )
[L_i , L_j ] = \phi^i (x) \psi^j (x) c^k_{ij} L_k \equiv \chi^k (x) L_k
\eqno(5) $$
where the $c^k_{ij}$ are the structure constants of the algebra $\G$.
Now we will consider a function $\L (x,u) ,~ \L : M \x F \to R$, such that
$$ \g \c \L = 0 ~~~ \all \g \in \Ga \eqno(6) $$
This means that at fixed $x$, $\L (x_0 , u)$ is invariant under $G$, i.e. $\L
(x_0 , \La_g u ) = \L (x_0 , u)$ $\all g \in G ,~ \all x \in M$.
We recognize the setting of the previous section: $x$ can be thought as a
multidimensional parameter and (6) just reflects the $G$-invariance of the
"potential" $\L$. One just has to remark that the discussion of the previous
section survives the extension from one-dimensional to $m$-dimensional
parameter space; then we conclude it holds the
{\sl Reduction lemma (variational gauge case):} Let $L[\phi ] = \int_M \L [\phi
(x) ] dx$ be a gauge invariant functional with gauge group $G$. Let the group
$G$ admit a Lie subgroup $G_0$ with $W_0 = \Fix (G_0 ) = \{ u \in F ~/~ \La_g u
= u ~\all g \in G_0 \}$. Let $\Phi_0 \ss \Phi$ be the set of sections such that
$\s (x) \in W_0 \sse F ~~ \all x \in M$, and let $L_0 [\phi ] = \int_M \L_0
[\phi (x) ] dx$ be the restriction of $L : \Phi \to R$ to $\Phi_0$. Then, a
section $\s \in \Phi_0$ is critical for $\L_0$ if and only if it is critical
for $L$.
It also follows immediately the generalization of the EBL:
{\sl Corollary (existence lemma):} Let $L,~\L ,~ G , ~G_0 ,~ W_0 ,~F$ be as
above, $\dim W_0 = d < \infty$, and let $W_0$ contain a nonempty compact subset
$B \ss W_0$ of dimension $d$ such that the vector $\cd_u \L (x,u)$ points
outward of $B$ on $\pa B$ $\all x \in M$. Then there is a local section $\s : M
\to E$, $\s (x) \in F_x$ ($F_x$ is the fiber through $x$) which is entirely
contained in $W_0$, i.e. $\s (x) \in W_0 \ss F_x$ $\all x \in M$, and which is
critical for the functional $L:\Phi \to R$.
We remark that the section whose existence is ensured by this corollary could
happen to be trivial, $\s (x) = 0$. If the trivial section is a local maximum
for $L$, the same reasoning leads to affirm the existence of a local section
$\s_0 (x) \in W_0$ which is a minimum.
We stress that the existence of global nonzero sections (and a fortiori of
global
sections if the fiber is not a linear space, e.g. if the fiber is the group
itself) cannot, quite obviously, be granted on purely algebraic terms and would
require a topological discussion, out of the scope of this paper.
We remark also that one could introduce in $\L (x,u) $ a dependence on a
control parameter $\l$ and consider bifurcations of critical sections of $L_\l
[\phi ] = \int_M \L_\l (x, \phi (x) ) dx$, but this is beyond the scope of the
present paper; the problem will be dealt with in [18].
\bigskip \bigskip {\bf 12. Some examples of reduction} \bigskip
We would now like to briefly see some examples of applications of the above
reduction lemma for some of the group actions already considered in sections 7
and 9; we will follow the notation and numerotation employed there.
For physical reasons, one is mainly interested in the case $\L = \L_0 + \L_1$,
where
$$\L_1 [\phi ] = {1 \over 2} ( \cd \phi )^2 = {1 \over 2} \sum_i (\cd \phi_i )
(\cd \phi_i ) \eqno(1) $$
and $\L_0$ is a polynomial of degree $\le 4$ in the $\phi_i$'s, and independent
of $x \in M$; we also denote $\L_0 [\phi ] $ by $V(\phi )$. We will always
tacitely assume that $\L_0 [\phi ] \to \infty $ for $\vert \phi \vert \to
\infty$.
\Example{ 1}: Here $\phi (x) \in F = R^1$; the invariance of $\L$ under $G=Z_2
$ implies $V=V( \phi^2 )$. The space $W_0$ invariant under $G_0 =G$ is $W_0 =
\{ 0 \}$, so that the RL and its corollary just say that $\phi (x) = 0$ is a
critical section.
\Example{ 2}: Here $\phi (x) \in F = R^2$; the invariance of $\L$
under $G=SO(2) $ implies $V=V( \phi^2 )$. The space $W_0$ for the MIS $G_0
=G$ is again $W_0 = \{ 0 \}$, and $\phi (x) = 0$ is a critical section.
\Example{ 3}: Here $\phi (x) \in F = R^3$; invariance
under $G=SO(2) $ implies that, with obvious notation, $V=V( \rho^2 , \phi_3 )$,
where $\rho^2 = \phi_1^2 + \phi_2^2$. The space $W_0$ for $G_0 =G$
is $W_0 = \{ (0,0,\phi_3 ) \}$, and it is easy to see that on $W_0$ indeed $\cd
V (x) \in T_x W_0$. We can therefore consider the restrictions $V_0 (\phi_3 ) =
V (0, \phi_3 )$, $\^\L = (1/2) (\cd \phi_3 )^2 + V_0 (\phi_3 )$ and $L_0
[\phi_3
] = \int_M \^\L [\phi_3 ] dx$, whose critical points are also critical points
of $L$; conversely, critical sections $\phi $ of $\L$ such that $\phi (x) \in
W_0 $ $\all x \in M$ are clearly also critical sections of $\^\L$.
\Example{ 4}: Here $\phi (x) \in F = R^3$; now there is a relation among
$\phi_3$ and $\rho^2$, i.e. $\phi_3^2 + \rho^2 =1$. Now for $G_0 =G =SO(2)$, we
have $W_0 = \{ (0,0,\pm 1) \}$, and the RL tells that the sections $\s_\pm (x)
= (0,0,\pm 1)$ are critical.
\Example{ 5}: Here $\phi (x) \in F = R^3$; invariance under $G = SO(2) \x Z_2$
implies that $V (\phi ) = V ( \tha_1 , \tha_2 )$ (see sect. 9); apart from $G_0
=G$ with $W_0 = \{ (0,0,0) \}$, the MIS are $G_1 = SO(2)$ and $G_2 = Z_2$.
These have $W_1 = \{ (0,0,\phi_3 ) \}$, to which it corresponds $\tha_1 = 0$,
and $W_2 = \{ (\phi_1 , \phi_2 , 0 )\}$, to which it corresponds $\tha_2 = 0$.
We stress that, a priori, the critical sections $\s_i (x) \in W_i ~ \all x
\in M$ whose existence is ensured by the RL could be trivial, $\s_i (x) =0$,
since $W_1 \ss W_0 ,~ W_2 \ss W_0$.
We will consider this example in some lenght. Let us consider the general
density $\L$ satisfying the constraints stated in the beginning of this
section, i.e.
$$ \L [\phi ] = \L_1 [\phi ] + \L_0 [\phi ] \eqno(2) $$
with
$$\L_1 [\phi ] = {1 \over 2} \sum_{i=1}^3 (\cd \phi_i )^2 \eqno(3) $$
$$\L_0 [\phi ] = -{1 \over 2} (\l \tha_1 + \mu \tha_2 ) + {1 \over 4} (a
\tha_1^2 + b \tha_2^2 + c \tha_1 \tha_2 ) \eqno(4) $$
where $a > 0 ,~ b>0 ,~c^2 < 4ab$.
A section $\phi \in W_1 $ satisfies $\phi_3 = \cd \phi_3 = 0$, and we get
$$\^\L_1 = {1 \over 2} \sum_{i=1}^2 (\cd \phi_i )^2 - {1 \over 2} \l \tha_1 +
{1 \over
4} a \tha_1^2 \eqno(4) $$
while for $\phi \in W_2$, $\phi_1 = \phi_2 =0$ and
$$ \^\L_2 = {1 \over 2} (\cd \phi_3 )^2 - {1 \over 2} \mu \phi_3^2 + {1 \over
4} b \phi_3^4 \eqno(5) $$
The Euler-Lagrange equations
$$ { \pa \L \over \pa \phi_i } = \cd_\a { \pa \L \over \pa (\cd_\a \phi_i ) }
\eqno(6) $$
give then in the two cases
$$\eqalign{ \lapl \phi_1 =& - \l \phi_1 + a (\phi_1^2 + \phi_2^2 ) \phi_1 \cr
\lapl \phi_2 =& - \l \phi_2 + a (\phi_1^2 + \phi_2^2 ) \phi_2 \cr} \eqno(7) $$
and
$$ \lapl \phi_3 = - \mu \phi_3 + b \phi_3^3 \eqno(8) $$
where $\lapl = ( \cd_\a \c \cd_\a )$ is the Laplacian.
The existence of nontrivial solutions to (7), (8) depends on the base space $M$
and on boundary conditions (i.e. on the global structure of the bundle $E$ of
base $M$ and fiber $F$), but it is easy to imagine $M$'s such that these exist
(e.g. $M= S^1 ,~ S^2 ,~ T^2 , ... $).
\Example{ 6}: Here $\phi (x) \in F = R^3$; now there is a relation among
$\tha_1$ and $\tha_2$, i.e. $\tha_1 + \tha_2 =1$. With the notations of the
previous example, $W_0$ is now empty (i.e. $G_0 = G$ is not an isotropy
subgroup), $W_1 = \{ (0,0,\pm 1 ) \}$, $W_2 = \{ (\phi_1 , \phi_2 , 0) ~;~
\phi_1^2 + \phi_2^2 =1 \}$, so that when applied to $G_1 = SO(2)$ the RL just
tells that the sections $\phi_\pm = (0,0,\pm 1)$ are critical; applied to $G_2
= Z_2$, the RL ensures the existence of a local critical section lying entirely
on the $S^1$ circle $W_2$.
We stress once again that what the RL tells is just that if a critical section
exists for the restriction $\^\L_i$ of $\L$ to $W_i$, then it is also a
critical section for $\L$. The global existence of sections depends on the
global structure of the fiber bundle, and infacts the corollary of the RL just
ensures the existence of local critical sections. When the space $W_i$ has the
structure of a linear space the local existence can be extended to global one
(but the section will not be nonzero, in general), but this depends on this
additional structure.
\vfill \eject
\bigskip \bigskip {\bf 13. Base space symmetries} \bigskip
Up to now, we have considered pure gauge symmetries; anyway, other kind of
symmetries can be present for the functional $L[\phi] = \int_M \L [\phi (x) ]
dx$, i.e. {\sl base space symmetries}. In physical gauge theories, these are
typically space-time symmetries. Also, motivated by physical considerations, we
will limit to consider the semidirect product of base space symmetries by gauge
ones (i.e., the transformation of space time does not depend on the values of
the fields $\phi$).
We have briefly dealt with the representation of the Lie algebra of the gauge
group as an algebra of differential operators ; in this language, if $x$ are
coordinates on the base space $M$ and $u$ are coordinates along the fiber $F$,
and
$$ L_i = \l^{(i)}_j (u) {\pa \over \pa u_j } \equiv \l^{(i)}_j (u) \pa_j
\eqno(1) $$
are the generators of the gauge group $G$, then the pure gauge transformations
can be written as
$$ \g = \sum_i \a_i (x) L_i = \~\a_j (x,u) \pa_j \eqno(2) $$
The infinitesimal base space symmetries can be written as
$$ \chi = \b_a (x) {\pa \over \pa x_\a } \eqno(3) $$
and if the Lie algebra of (the group $R$ of) base space symmetries of our
functional is $X$ with generators
$$ X_i = \nu^i_j (x) {\pa \over \pa x_j } \eqno(4) $$
we can rewrite (3) as
$$\chi = \sum_i \b_i X_i = \b_i \nu^i_j {\pa \over \pa x_j } = \~\b_j (x)
{\pa \over \pa x_j } \eqno(5) $$
The semidirect sum $X \oplus_\to L$ will be made of vectors fields of the form
$$\eta = \xi (x) \pa_x + \zeta (x,u) \pa_u \eqno(6) $$
where we have used the shortcut notation
$$ \xi \pa_x \equiv \sum_i \xi^i {\pa \over \pa x^i } ~~;~~ \zeta \pa_u \equiv
\sum_j \zeta^j {\pa \over \pa u^j } \eqno(7) $$
Vector fields of the form (6), and the corresponding algebras, are for obvious
reasons also called projectable [33].
A typical case of occurrence of base space symmetries is that of a density $\L$
of the functional
$$ L[\phi] = \int_M \L [\phi (x) , x] dx \equiv \int_M \L [\phi (x)] dx
\eqno(8)
$$
which does not depend explicitely on $x$ (we also say that the base
manifold $M$ is homogeneous: all its points are equivalent). In this case, the
group of rigid transformations of $M$ leaves $L$ invariant; notice that a
general - i.e. not rigid - diffeomorphism of $M$ does not leave $L$ invariant:
even in the case $\L$ depends only on $\phi$ and not on its derivatives $\cd
\phi$, such a transformation would modify the integration measure.
In this paper we always assume that $\L$ does not depend explicitely on $x$
(this is also expressed by saying that $\L$ is autonomous).
When $M=R^m$, the group of corresponding rigid transformations is the group
$E(m) = R^m \x O(m)$ of euclidean transformations, with Lie algebra generated
by
the $m$ traslations $ \pa / \pa x_1 , ... , \pa / \pa x_m $ and by the Lie
algebra of rotations in $R^m$, $so(m)$ (whose generators are of the form $x_i
\pa / \pa x_j - x_j \pa / \pa x_i$ ); for $M = S^{(m-1)} \ss R^m$, we just have
the group $O(m)$ (with Lie algebra $so(m)$) of rotations in $R^m$.
If now we consider also these base space symmetries, the full symmetry group of
the functional $L$ is
$$ \~\Ga = B \otimes_\to \Ga \eqno(9) $$
where $\Ga$ is, as before, the group of gauge diffeomorphism, $\Ga = \{ \g : M
\to G \}$. Clearly, $B$ acts on $\Ga$ but $\Ga$ does not act on $B$.
In the analysis of symmetries of sections $\s \in \Phi$, one should now
consider $\~\Ga$ rather than $\Ga$. On one side, this leads to consider the
orbit space $\~\Th = \Phi / \~\Ga$ rather than $\Th = \Phi / \Ga$, which means
that sections differing only by a rotation and/or a change in the origin of the
coordinate system in $M$ are identified; this is indeed quite a reasonable and
desirable feature.
On the other side, base space symmetries will also enter in the isotropy group
of the sections $\s \in \Phi$ and of the gauge orbits $\~\th \in \~\Th$; this
feature is capable, as we will readily discuss, to favour pattern formation or,
in other words, the selection of sections with some spatial (in the sense of
the base space $M$; it can well be a spatiotemporal one in physical terms)
regularity.
We would like to stress that the (structurally stable) set of transversal
sections is also invariant under rigid transformations of $M$.
We will now specifically discuss the case of $M=S^m \ss R^{m+1}$ or $M=R^m$; in
the last case we will also consider two compactifications (we recall that in
sects. 4-9 we needed to assume $M$ compact): one is the one-point
compactification to $S^m$, which is the natural one if we ask our sections to
vanish sufficently quickly at infinity and which comes back to the previous
case; the other amounts to ask periodicity to the sections, $\s (x) = \s ( x +
K \c \La_0 )$, where $K \in Z^m$ and $\La_0 \in R^m$ represents the basis
vectors of a lattice in $R^m$, so that $R^m$ is compactified to the (standard)
$m$-torus $T^m = \( S^1 \)^m$; in this case we can still discuss in the full
$R^m$ setting, but integration will be over a compact region $M$ corresponding
to a (simple, or finite) covering of $T^m$ or, in physical terminology, to a
unit cell of the lattice generated by $\La_0$.
For $M$ compact, also the group $B$ of rigid diffeomorphisms of $M$ is a
compact Lie group; we stress anyway that its action on the infinite dimensional
space $\Phi$ (or even $\Th$) gives an infinite dimensional representation.
In the case $M = S^m$, $B$ is actually the rotation group in $(m+1)$
dimensions, $R = O(m+1)$, $X=so(m+1)$, and the situation is well known,
amounting to an analysis in spherical harmonics.
In the case $M=R^m$, i.e. $M = T^m$; $B$ is the euclidean group $E(m)$ (modulo
the compactification $R^m \to T^m$); this is generated by rotations, i.e.
$O(m)$ again, and by $m$ traslations, i.e. $R^m$. With the torus
compactification, the latter becomes just rotations along the $m$ fundamental
cycles of the torus, i.e. $\( SO (2) \)^m $; we are therefore in a familiar
situation, corresponding to multidimensional Fourier analysis.
We stress that we could have chosen different compact manifolds $M$, giving
again familiar situations: e.g., if $M$ is (flat) space-time, $M=R^d \x R$, we
can pretend sections to go fast enough to zero at spatial infinity and consider
only time periodic sections. This amount to one-point compactification of
$R^d$, and torus compactification of $R$, i.e. to considering $M= S^d \x S^1$;
this is again a well known situation.
\bigskip \bigskip {\bf 14. A scenario for pattern formation} \bigskip
If we now consider the full symmetry algebra of the functional
$$ L [ \phi ] = \int_M \L ( \phi ) dx \eqno(1) $$
i.e., as discussed in the previous section,
$$ \~\Ga = X \oplus_\to \Ga \eqno(2) $$
(the symbol $\oplus_\to$ denotes semidirect sum)
the maximal subalgebras of $\~\Ga$ will be of the form
$$ \~\Ga_0 = X_0 \oplus_\to \Ga ~~~{\rm or}~~~ \~\Ga_0 = X_0 \Ga_0 \eqno(3) $$
with $X_0$ a maximal subalgebra of $X$, $\Ga_0$ a maximal subalgebra of $\Ga$.
Our aim would be to obtain informations about the existence of solutions (or
bifurcating solutions) with maximal symmetry. Clearly, the section $\s \equiv
0$ has the full $\~\Ga_0$ symmetry, so we will consider only nontrivial
solutions (we notice explicitely that these could vanish at some points;
anyway, they will not be identically zero).
Notice that, since $X$ generates the rigid transformations of $M$ (and we want
to consider $M= S^n$ or $M=T^n$) the only sections having the full $X$ symmetry
are constant ones, $\s (x) = \phi_0$. In this case we have
$$ L [ \s ] = \int_M \L ( \phi_0 ) dx = \L ( \phi_0 ) \c \mu \eqno(4) $$
where $\mu=\int_M dx$ is the volume of the manifold $M$, and we reduce to the
finite dimensional problem of finding minima of $\L : F \to R$, i.e. to the
problem solved by Michel's theorem. In particular, there will be solutions
corresponding to maximal $\Ga_0$, as discussed in sect.9.
Let us now consider non constant sections. It follows from the above discussion
that for these, the symmetry algebra is of the form
$$ \~\Ga_\s = X_0 \oplus_\to \Ga_0 \kern 2cm X_0 \ss X ~;~ \Ga_0 \sse \Ga
\eqno(5) $$
Now we have to distinguish two cases: either there are nonzero $\phi \in F$
such that $\G_\phi = \G$, either not. In the first case, one can have sections
$\s$ such that
$$ \s (x) \in W_0 ~~~, ~~~ \~\Ga_\s = X_0 \x \Ga \eqno(6) $$
where $W_0 = \{ \phi \in F ~/~ \eta \phi =0 ~~ \all \eta \in \G \} \simeq R^s
\sse F$, $s \ge 1$. In the second case this is impossible, since the only
section having the whole $\G$ and therefore $\Ga$ symmetry is the trivial one.
Therefore, in this case
$$ \s \not\equiv 0 \Longrightarrow \~\Ga_0 = X_0 \oplus_\to \Ga_0 ~~;~~~~~ X_0
\ss X ,~ \Ga_0 \ss \Ga \eqno(7) $$
In both cases, what we want to stress is that the symmetry $X$ is broken to
$X_0$ (or, at group level, $B$ is broken to $B_0$); the discussion given in
previous parts of this paper shows that, under some additional conditions in
order to precise the setting, maximal isotropy subgroups (MIS) are favoured if
the trivial solution $\s_0$ becomes unstable and bifurcates to a new solution
$\s_b (x)$.
The facts that MISs of $R^n$ (except those containing traslations) correspond
to
regular lattices (with the same holding for $T^n$, $S^n$) suggests that this
could be a mechanism implied in {\it pattern formation} [45,46].
Notice that, if $\Fix (G) = \{ 0 \}$, in such a bifurcation both the $X$ and
$\Ga$ symmetries must break, and the maximal symmetries of nontrivial solutions
correspond to both a {\it pattern formation}, i.e. a breaking of the symmetry
$B$ to a regular lattice, and a {\it gauge-symmetry breaking}.
More precisely, given a MIS $\~\Ga_0 = B_0 \x_\to \Ga_0 \ss B \x \Ga_0 \equiv
\~\Ga$, we restrict $L : \Phi \to R$ (recall $\Phi$ is the space of sections
of $M \x F$) to the subspace $\Phi_0$ of sections invariant under $\~\Ga_0$,
and get the functional $L_0 : \Phi_0 \to R$. As we have proved before, sections
$\s_1 \in \Phi_0$ which are critical for $L_0$ are also critical for $L$.
Consider now the case that $\L$, and through it $L$, depend (smoothly) on a
real
parameter $\l$, and for $\l < \l_0$ the trivial section $\s_0$ is a minimum for
$L$, while at $\l = \l_0$ it loses stability (for precise conditions of this
losing of stability, see e.g. [10,11,47]), and let us look for minima of $L$
for
$\l > \l_0$.
We can go over the set of MISs $B_i \ss B$, $\Ga_\a \sse \Ga$, and study
critical sections for $L_{i,\a} : \Phi_{i,\a} \to R$, where $\Phi_{i,\a} \ss
\Phi$ is the set of sections admitting $\~\Ga_{i,\a} = B_i \x_\to \Ga_\a$ as
symmetry. The minima for $L_{i,\a}$ are also critical sections (not necessarily
minima !) for $L$. This gives a constructive algorithm: we first determine
sections $\s_{i,\a} \in \Phi_{i,\a}$ which are minima for $L_{i,\a}$, and
then check their stability against general perturbations.
\bigskip \bigskip {\bf 15. A scenario for phase coexistence} \bigskip
In previous parts of this paper we pointed out that our results concerning
stratification and critical gauge orbits need, to be rigorous, to restrict to
the set of transverse sections, defined in sect.8. It was also remarked that
in some special cases a transverse section could connect different strata of
the $G$-orbit space $\Om$. In physical terms, this suggests that such solutions
correspond to {\it phase coexistence}
Again, we could consider a bifurcation from the trivial solution; it was
pointed out in sect.9 that the isotropy group $\Ga_0$ of the
solution connecting strata corresponding to non-conjugated subgroups $G_1 , G_2
\ss G$ is not a subgroup of $\Ga_1$, neither of $\Ga_2$, where $\Ga_i = \{ g :
X \to G_i \}$. This means in particular that for $G_1 , G_2$ MISs of $G$, and
therefore $\Ga_1 , \Ga_2$ MISs of $\Ga$ (for transverse sections), also $\Ga_0$
would be a MIS of $\Ga$. By the same argument as in the case of "pattern
formation" dealt with in the previous section, one can explain the appearance
of these "phase-coexistence" solutions.
We remark that in our qualitative discussion pattern formation is an
independent phenomenon, but phase coexistence seems to be intimately tied with
pattern formation, as it is indeed the case in a number of experimental
observations.
\vfill \eject
\parskip=5pt
\parindent=10pt
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\bye