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{\nopagenumbers
\titlea{Critical sections of gauge functionals: a symmetry approach}
\bigskip \bigskip
\centerline{Giuseppe Gaeta}
\centerline{ \it Centre de Physique Theorique, Ecole Polytechnique}
\centerline{\it F-91128 Palaiseau (France) }
\centerline{\it E-mail: gaeta@orphee.polytechnique.fr}
\vfill
{\bf Abstract:} We extend the classical theorem of L. Michel on the symmetry
breaking patterns in theories described by minimization of a symmetric
potential to the case of theories described by minimization of a
gauge-symmetric functional.
\vfill
\vfill
\eject}
\pageno=1
\titleb{Introduction}
Some time ago, L. Michel proved a remarkable theorem [1] on geometry of orbit
space, and therefore on symmetry patterns in theories described by minimization
of
an invariant potential (see also [2,3]); indeed, his work was partly motivated
by the SU(3) model of particle physics [4].
I aim here at an extension of Michel theorem to gauge theories, i.e. to the
case one deals with a
functional invariant under a gauge group and looks for critical points of this
in the space of
sections of a fiber bundle. In this setting one is faced with a number
of difficulties, due to non-compactness of the gauge group and infinite
dimensionality of the space of sections; indeed, I will not be able to extend
Michel
theorem in full generality, mainly due to poor understanding of the geometry of
gauge orbit space. By a judicious use of the fibered structure (and actually
also paying the price
of restricting to a dense subspace of the space of sections) one can anyway
obtain
at least a partial extension of Michel's results to the case of interest here.
It
turns out that in this way one makes contact with results of Palais [5].
As it already happens with the original work of Michel, which gave origin to
the "Equivariant
Branching Lemma" [6-8], of great utility in equivariant bifurcation theory
and its applications [9-12], the present extension is readily adapted to
the bifurcation setting, and fits nicely in the framework of bifurcation theory
in
the presence of Lie-Point symmetries [13,14] and of "nonlinear" reduction
and equivariant branching lemma [15,16] (for an introduction to Lie-Point
symmetries, see [17-20]).
Here I only sketch a few (key) steps of this work; a complete version, with
full detail,
examples, and suggestions of further developements and applications (notably to
pattern formation and
phase coexistence), will appear elsewhere [21]; related results appear in [16];
see also [20].
\titleb{Michel theorem}
Let us first shortly recall the setting and statements of Mishel's theorem; the
reader is supposed
to be familiar with the geometry of group action on manifolds and with the
concept of (symmetry)
strata; see e.g. [2,3,20,21] for an introduction to these.
Let $M$ be a finite dimensional riemannian manifold (say, embedded in $\R^n$
for definiteness); let
$G$ be a compact Lie group,
%with finite dimensional Lie algebra $\G$,
and let $\La = \{ \La_g ,~g \in G \} $ be a representation of $G$ acting on
$M$, $\La : M \to M$.
We will denote the space of $G$-orbits on $M$, or {\it orbit space} for short,
by $\Om$; with the
present hypotheses for $G$ and $M$ this can be proved to be a semialgebraic
manifold [2,3,22].
Let$\V_\La$ be the set of ${\cal C}^\infty$ functions $V: M \to \R$ invariant
under $\La$, i.e. such
that $V ( \La_g x) = V(x)$ $\all x \in M$, $ \all g \in G$.
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 $V \in
\V$ 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 {\it 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,
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 ($G_x = SO(2)$).
An even less trivial example is obtained by considering the adjoint
representation of SU(3) (which
provided indeed physical motivation for the work of Michel), see [4,9,11].
The theorem of L. Michel [1] tells that:
{\bf 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 $\S_\om$ (the stratum of $\om$ in $\Om$) 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
$d (x,y)$ defined in $M$, and will be
$$ d ( \om , \om ' ) = {\rm min} \{ ~d ( x , y) ~;~ {x \in \om , y \in \om '}
\}
\eqno(1) $$
The minimum of (1), whose existence has to be proven, can be seen
also as
$$ d ( \om , \om ') = {\rm min} \{ d (x, \om ' ) ~;~ {x \in \om } \} \eqno(2)
$$
where we have introduced the distance of a point from an orbit,
$$ d (x, \om ' ) = {\rm min} \{ d (x,y) ~;~ {y \in \om '} \} \eqno(3) $$
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:
$$ \rho_\om (\La_g x) = \La_g \rho_\om (x) \eqno(4) $$
and is therefore also called the equivariant retraction [2,3]
Notice that if $G$ is compact, then $\om , \om '$ are compact sets, and
the minima of (1),(2),(3) 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.
\titleb{Gauge theoretical setting}
Let us consider a fiber bundle $(E,B,\pi )$ of base space $B$, fiber $\pi^1 (x)
=
M$ and structure group a compact Lie group $G$ acting on $M$ through the
representation $\La$ (here we have the same hypotheses made earlier on $G$,
$M$, $\La$). The space of smooth sections $\phi (x)$ of this bundle, $\phi : B
\to
E$, $\pi \phi (x) = x$, will be denoted by $\Phi$.
Consider now the bundle $(\~E , B ,\~\pi )$ of base $B$ and fiber $\pi^{-1} (x)
= G$;
the space of smooth sections $\g (x)$ of this bundle, $\g : B \to G$, $\~\pi \g
(x)
= x$, will be denoted by $\Ga$
\footnote{$^{(1)}$}{$\Ga$ can also 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 B \}$.
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 $\pi^{-1} (x)$.}.
Elements of $\Ga$ act in a natural way in $\Phi$:
$$ (\g \circ \phi ) (x) = \La_{\g (x)} \phi (x) \eqno(5) $$
Consider now a local functional $L : \Phi \to \R$, with local homogeneous
density
$\L$:
$$ L (\phi ) = \int_B \L [ \phi (x) ] dx \eqno(6) $$
In particular, we want to consider the case of $L$ invariant under the action
of
the gauge group (we say that $L$ is a {\it gauge functional} for short), i.e.
$$ L (\g \circ \phi ) = L (\phi ) ~~~~~ \all \g \in \Ga ~,~ \all \phi \in \Phi
\eqno(7') $$
or equivalently
$$ \L [ \La_{\g (x) } \phi (x) ] = \L [ \phi (x) ] ~~~~~ \all \g \in \Ga ~,~
\all
\phi \in \Phi \eqno(7") $$
Here and in (6) the square brackets mean that $\L$ can depend on derivatives of
the
field $\phi$ as well. In this note, anyway, we will consider for the sake of
brevity only the case of zero-th order gauge functionals, i.e. $\L$ will depend
only on $\phi (x)$ and not on its derivatives. The general case can be readily
recovered by this special one: e.g. for first order theories one considers a
field $\Psi$ whose
components $ \Psi_{0i} = \phi^i $, $\Psi_{ij} = \partial \phi^j / \partial x^i$
obey the
constraints $ \partial \Psi_{0j} (x) / \partial x^i = \Psi_{ij} (x)$ (and
similarly
for higher derivatives). This amounts to introducing a {\it contact structure}
[23,24,20]; this construction is discussed in full detail in [21] and will
not be discussed here.
\titleb{A distance in gauge orbit space}
One could then try to parallel the standard proof of Michel theorem
[1,3,21] for the space $\Phi$, in order to get an
analogue of Michel's theorem in this case. A difficulty immediately encountered
is
that now the space $\Phi$ on which $\Ga$ acts is infinite dimensional, and
therefore
not compact; correspondingly, also $\Ga$ is
non-compact\footnote{$^{(2)}$}{Although $G$ is compact.}.
The $\Ga$-orbit of a section $\phi \in \Phi$, denoted $\th (\phi)$, will be
defined
as
$$ \th (\phi ) = \{ \phi ' \in \Phi ~/~ \phi ' (x) = \La_{\g (x)} \phi (x) ~,~
\g
(x) \in \Ga \} \ss \Phi \eqno(8) $$
The orbit space for sections will be denoted $\Th \equiv \Phi /
\Ga$; the orbit $\th (\phi )$ will be denoted as $\th_\phi$ when thought as a
point
of $\Th$.
Given a section $\phi \in \Phi$, we can define its isotropy subgroup
$\Ga_\phi$ as
$$ \Ga_\phi = \{ \g \in \Ga ~/ ~\g \cdot \phi = \phi \} \ss \GDiff (E )
\eqno(9) $$
If $\phi ' = \g \cdot \phi$, it is easy to see that
$\Ga_{\phi '} = \g \Ga_\phi \g^{-1}$.
We can therefore [2,3] define 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 time a discussion of how (and how far) one can actually do
it,
and of the difficulties it can present.
The standard proof of Michel's theorem [1,3,21] 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 convenient topology to the
orbit space.
In our case, anyway, we can take advantage of the fibered structure of the
theory, and define a distance between two sections $\phi , \phi' \in \Phi$ as
$$ d_\Phi ( \phi , \phi ' ) = { 1 \over |B| } \int_B d_M \left( \phi (x) ,
\phi
'(x) \right) dx \eqno(10) $$
where $|B| = \int_B 1 \cdot dx$, and $d_M (.,.)$ is a distance defined in $M$.
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 earlier in $\Om$, i.e. of the equivariant retraction $\rho_\om$, by
$$ \de (\th_1 , \th_2 ) = { 1 \over |B| } \int_B d_\Om \left( \th_1 (x) , \th_2
(x) \right) dx $$
where $\th_i : B \to \Om $.
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 $\om (\phi ) \ss \Phi$ a {\it critical gauge orbit}
for $\Ga$ if for every $\Ga$-invariant functional $L =\int_B \L (\phi )
dx$, $L :\Phi \to \R$, $\om ( \phi ) $ is a critical orbit for $L$. This
means that $\all \phi \in \om (\phi ), ~ \de L [ \phi ] = 0$; or, $\L (\phi +
\epsilon \de \phi) = \L ( \phi ) + O (\epsilon^2 ) $.
The reader will notice 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. We will not discuss this here, but just
refer, as
usual, to [21].
\titleb{Stratification of gauge orbit space}
Our understanding of the stratification of the gauge orbit space $\Th$ will be
based on the stratification of the orbit space $\Om = M/G$. This latter one is
a classical
subject, and we will not discuss it here; see [2,3,22] for a discussion.
We just recall 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 $P_1 (x) , \dots , P_k (x)$ of
the Hilbert integrity basis for the $G$-action on $M$.
A semialgebraic variety $\Om$ in $R^k$ has a natural primary stratification
[25],
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 $
where $ \cup $ denotes disjoint union, and
$ E_j^\b \in \pa E_i^\a $, $ \b < \a$
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 symmetry stratification, i.e. the one based on
symmetry of orbits $\om \in \Om$ under $G$ follows this primary stratification,
in
the sense that connected components of symmetry 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.
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 $\s (x) : B \to M$, we can consider the set of values it takes
in $M$,
$$ M_\s = \{ \s (x) ~,~ x \in B \} \ss M \eqno(11) $$
and in an obvious way
$$ \Om_\s = \{ \om_{\s (x)} ~,~ x \in B \} \sse \Om \eqno(12) $$
Now, let us consider the primary stratification of $\Om$,
$$ \Om = \cup_\cdot E^\a_i ~~,~~ {\rm dim} E^\a = \a \eqno(13) $$
introduced above. The primary index $\a (\s )$ of the section
$\s$ is the greater $\a$ for which there is an $E^\a_i$ with
$$ \Om_\s \cap E^\a_i \not= \{ \oslash \} \eqno(14) $$
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 $\s$, $[E]_\s$:
$$ [E]_\s = \{ E^\a_i ~/~ \Om_\s \cap E^\a_i \not= \oslash \} \eqno(15) $$
Clearly,
$$ E_\s \equiv \cup_\cdot E^\a_i ~~,~~ E^\a_i \in [E]_\s \eqno(16) $$
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, but it is
actually related to the above remark, that 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.
Let $B=[0,1]$, $M=\R^2$, $G=SO(2)$; consider two sections given by (with $v_0$
a
given unit vector in $M$)
$ \s_i (x) = e^{-1/(x-x_i )^2 } v_0 $ for $x \le x_i$; $ \s_i (x) =0 $ for $ 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 \}$, or
$\{ \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 also 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 remark anyway this restriction does
actually suffice to properly define a stratification in a number of meaningful
examples, see [21].
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, if we require only what we really need for our
purposes, it will suffice the
{\sl Weaker assumption:} For the action of $\Ga$ on the space of transversal
sections $\Phi_T$, 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 write - for ease of notation - $\Phi$ for $\Phi_T$ and
$\Th$
for $\Th_T$.
Actually, by looking at examples, one gets easily convinced that an explicit
stratification of $\Th$ is extremely complicate and difficult to
describe. We therefore
concentrate on the description of most singular strata, i.e. those
corresponding to
maximal isotropy subgroups\footnote{$^{(3)}$}{It should be stressed that
maximal isotropy subgroups
correspond to most singular strata, but 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 [26].}. These are also the strata
on which the extension of
Michel's theorem given above can be applied.
\titleb{Maximal isotropy groups in gauge orbit space}
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 $\exists z \in M ~/~ \La_g z = z ~ \all g \in G_\mu$, 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 $M$ 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 $\s (x) \in \Om_\mu ~\all x \in B$ (i.e. $\Om_\s
\sse \Om_\mu$) will admit as symmetry group
$$\Ga_\mu = \{ \g : B \to G_\mu \} \eqno(17) $$
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 $x
\in B$, $\g (x) \in G \backslash G_\mu$. 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
$$ \g (x) \in \cup_{\mu = 1,...,s} G_\mu \eqno(18) $$
Suppose now that for $x \in B$ $\g (x )$ 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 $B_i = \{ x \in B ~/~ \g (x) \in G_i \}$. Then necessarily there are
points $x \in B_i \cap B_j$; due to smoothness of $\g$, in these $\g (x) \in
G_i \cap G_j$.
Now, the functions $\g : B \to G ~/~ \g : B_i \to G_i$ can be seen as $n$-ples
of
functions $\g_i$ defined on $B_i$ with values in $G_i$, each of them subject to
appropriate boundary conditions: on $B_{ij} \equiv B_i \cap B_j \in \pa B_i$,
$\g :
B_{ij} \to G_i \cap G_j$. 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
$$ \om_{\s (x) } \in \S_\om ~~ \all x \in B \eqno(19) $$
have isotropy subgroup
$$ \Ga_0 = \{ \g \in \Ga ~/~ \g : B \to G_0 \} \eqno(20) $$
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
$$ \Phi_\mu = \{ \s \in \Phi ~/~ \s : B \to \Om_\mu \} \eqno(21) $$
form a maximal stratum with isotropy $\Ga_\mu$. In other words,
$$\Th_\mu = \Phi_\mu / \Ga \eqno(22) $$
is a maximal stratum in the stratification of $\Th$ by $\Ga$. (Remark that
actually
$\Th_\mu = \Phi_\mu / ( \Ga / \Ga_\mu )$, since $\Ga_\mu$ 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 $\s ~/~ \s (x) \in
\om ~ \all x
\in B$, 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:
{\parskip=0pt \parindent=20pt
\item{ } {\bf Theorem:} Let $G_1 \sse G$ be a MIS for the action of $G$ on $F$
by the representation $\La$, and let $\Om = M/G$. Then
\item{i)} $\Ga_1 = \{\g : B \to G_1 \}$ is a MIS of $\Ga$;
\item{ii)} The set $\Th_1 = \{ \th ~/~ \om \cdot \s (x) : B \to \Om_1 ~\all x
\in B ~\all
\s \in \th \}$, 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 $\s (x) \in \om_0 ~\all x \in B$ 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 $\th_0 \in \Th$. }
\titleb{Reduction and equivariant branching lemma}
A full discussion of reduction and equivariant branching lemma for gauge
theories would be out of
place here. We will just mention the main results one can
obtain in this direction; a detailed discussion is given in [16,21].
{\sl Reduction lemma (variational gauge case):} Let $L[\phi ] = \int_B \L [\phi
(x) ] dx$ be a gauge invariant functional with gauge group $\Ga = \{ \g : B \to
G$.
Let the group
$G$ admit a Lie subgroup $G_0$ with $W_0 = \Fix (G_0 ) = \{ u \in M ~/~ \La_g u
= u ~\all g \in G_0 \}$. Let $\Phi_0 \ss \Phi$ be the set of sections such that
$\s (x) \in W_0 \sse M ~~ \all x \in B$, and let $L_0 [\phi ] = \int_B \L_0
[\phi (x) ] dx$ 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 $L$.
This is indeed the {\it "symmetric criticality principle"} of Palais [5],
recast in the language
usually employed in equivariant bifurcation theory\footnote{$^{(4)}$}{Palais
does actually also
consider the case of $G$ acting nontrivially on the base manifold $X$; we have
excluded this case
from his statement as well as from our discussion just for simplicity, but it
does not present new difficulties.}.
{\sl Corollary (existence lemma):} Let $L,~\L ,~ G , ~G_0 ,~ W_0 ,~M$ be as
above, $\dim W_0 = d < \infty$, and let $W_0$ contain a nonempty compact subset
$D \ss W_0$ of dimension $d$ such that the vector $\cd_u \L (x,u)$ points
outward of $D$ on $\pa D$ $\all x \in B$. Then there is a local section $\s : B
\to E$, $ \s (x) \in M_x$ ($M_x$ is the fiber through $x$) which is entirely
contained in $W_0$, i.e. $\s (x) \in W_0 \ss M_x$ $\all x \in B$, 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 $L$, as it is natural to assume in a bifurcation setting, the same
reasoning leads to affirm the
existence of a local section $\s_0 (x) \in W_0$ which is nontrivial and a
minimum.
Indeed, one could introduce in $\L (x,u) $ a dependence on a
control parameter $\l$ and consider bifurcations of critical sections of $L_\l
[\phi ] = \int_M \L_\l (x, \phi (x) ) dx$; in this setting the above existence
lemma becomes an
equivariant bifurcation lemma for gauge sections. A complete discussion of this
is given in [16].
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.
\titleb{Acknowledgements}
I would like to warmly thank proff. L.Michel, R.Palais and J.P. Bourguignon for
interesting discussions on the work reported here. I also thank J.v.d.Boon and
M.v.Westen for warm hospitality in Amsterdam, where this note was written.
\vfill \eject
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\parskip=3pt
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}
\bye