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{\nopagenumbers \parindent=0pt
\centerline{\bf Michel theory of symmetry breaking}
\medskip
\centerline{\bf and pure gauge theories}
\footnote{}{{\tt \giorno }}
\footnote{}{Research supported by the Volkswagen
Stiftung under the RiP program at Oberwolfach.}
\bigskip\bigskip\bigskip
\vfill
\centerline{Giuseppe Gaeta}
\medskip
\centerline{\it Department of Mathematical Sciences, Loughborough University,}
\centerline{\it Loughborough LE11 3TU (Great Britain)}
\centerline{\tt G.Gaeta@lboro.ac.uk}
\bigskip%\centerline{and {\it C.P.Th., Ecole Polytechnique, 91128 Palaiseau
(France)}}
\bigskip
\centerline{Paola Morando}
\medskip
\centerline{\it Dipartimento di Matematica, Politecnico di Torino,}
\centerline{\it Corso Duca degli Abruzzi 24, 10129 Torino (Italy)}
\centerline{\tt Morando@polito.it}
\vfill
{\bf Summary.} {\petitrm We extend Michel's theorem on the geometry of symmetry
breaking \ref{1} to the case of pure gauge theories, i.e. of gauge-invariant
functionals defined on the space $\A$ of connections of a principal fiber
bundle. Our
proof follows closely the original one by Michel, using several known
results on the
geometry of $\A$.}
\bigskip\bigskip
\vfill \eject}
\pageno=1
\parskip=10pt
\parindent=0pt
The Michel theorem on critical orbits of $G$-invariant functions on a
$G$-manifold \ref{1} allows to identify $G$-orbits which are critical for {\it
any} $G$-invariant function; thus, it permits to {\it study spontaneous symmetry
breaking in a model-independent way} \ref{2}.
The purpose of the present short note is to show that Michel's theory can be
extended to the study of (pure) gauge theories, such as those defined by a
Yang-Mills functional.
It is quite remarkable (and, to us, surprising) that one can extend not only
Michel's results, but also the essence of Michel's construction and proof, from
its original setting ($M$ a finite dimensional manifold, $G$ a compact Lie
group) to the case of gauge theories (infinite dimensional, non-compact manifold
and group).
It is also remarkable that a large part of the background results we need
have been
available for over fifteen years \ref{3,4}; it seems that lack of communication
between different mathematical communities (and between mathematicians and
physicists) prevented such an extension to be obtained earlier.
In the present note, we limit ourselves to the main result, i.e. the direct
extension of Michel's theorem; a more complete discussion of the topic will be
given elsewhere.
\medskip
Let $B$ be an $n$-dimensional ($n$ finite) riemannian manifold (e.g. $B= R^4$ or
$S^4$); let $G$ be a compact, connected and semisimple Lie group (e.g. $G =
SU(N)$)
acting through a given matrix representation; let $P$ be a principal fiber
bundle over
$B$ with projection $\pi : P \to B$ and fiber $\pi^{-1}(x) \approx G$. We
denote by
$\G$ the Lie algebra of $G$.
Let $\C$ be the set of connections on $P$. It is well known that $\C$ is an
affine
space modelled on the vector space $\A := \Lambda^1 \( \T^*B,\G \)$ of the
one-forms
on $B$ taking value in $\G$. With any connection $A \in \C$ we can associate a
covariant derivative $\D^\a$ and a one-form $\a \in \A$ that are given, in local
coordinates $(x^1 , ... , x^n )$ on $B$ and with $A_\mu : B \to \G$, by
$$ \a \ = \ A_\mu (x) \ \d x^\mu \ ; \eqno(1) $$
$$ \D^\a_\mu \ = \ \pa_\mu \ + \ A_\mu \ . \eqno(2) $$
Choosing as reference point in $\C$ the null connection $A^0$ such that
$A^0_\mu = 0$
($\mu = 1,..., n$), from now on we will say ``the connection $\a$'' to mean
``the
connection $A=A^0+\a$ whose associated connection form is $\a$''.
Let $\Ga$ denote the space of differentiable sections of $P$; $\Ga$ has the
structure of a Hilbert-Lie group, and a section $\ga \in \Ga$ will be
written in local coordinates as
$$ \ga \ = \ g(x) \ , \eqno(3) $$
with $g : B \to G$.
We say that $\Ga$ is a gauge group modelled on $G$.
The gauge group $\Ga$ acts naturally on the operator of covariant derivative
associated to a connection $A \in \C$ by conjugation, i.e. $\ga : \D^\a \to
\ga \D^\a \ga^{-1}$; the action of $\Ga$ on $\A$ is given in local
coordinates by
$$ \ga \ : \ A_\mu \ \Longrightarrow \ g(x) \cdot A_\mu (x) \cdot g^{-1} (x) \ -
\ \( \pa_\mu g \) (x) \cdot g^{-1} (x) \ \ . \eqno(4) $$
The action of $\ga \in \Ga$ on the connection form $\a \in \A$ will be
denoted by
$\ga (\a )$.
With any connection form $\a$ (i.e. with any connection $A$) we can
associate a {\it
gauge isotropy subgroup} $\Ga_\a$,
$$ \Ga_\a \ = \ \{ \ga \in \Ga \ : \ \ga (\a ) = \a \} \ \ . \eqno(5) $$
It is well known \ref{3} that, with $p_0 \in P$ any reference point in $P$
and $x_0
= \pi (p_0 )$,
$$ \Ga_\a \ = \ \{ \ga \in \Ga \ : \ \D^\a (\ga ) = 0 \ , \ g (x_0 ) \in C_G \[
H_\a (p_0 ) \] \ \} \ \ , \eqno(6) $$
where $H_\a (p_0 )$ is the holonomy group of the
connection $\D^\a$ at $p_0$, and $C_G [H]$ the centralizer of $H$ in $G$, i.e.
$$ C_G [H] \ = \ \{ g \in G \ : \ [g,h] = 0 \ \ \forall h \in H \} \ \ .
\eqno(7)
$$ Thus, $\Ga_\a$ is isomorphic to a subgroup of the compact group $G$; notice
that the isomorphism, given by $\D^\a \ga = 0$, depends on $A$.
Given a connection $\a$ with isotropy group $\Ga_\a$ we can consider: {\tt (i)}
the {\it fixed space} of $\Ga_\a$, i.e. the space of connections
(associated with
connection forms) which are left invariant by $\Ga_\a$,
$$ \F (\a ) \ = \ \{ \b \in \A \ : \ \ga (\b ) = \b \ \ \forall \ga \in
\Ga_\a \}
\ = \ \{ \b \ : \ \Ga_\a \sse \Ga_\b \} \ \ ; \eqno(8) $$ and {\tt (ii)} the
{\it isotropy type} of $\a$, i.e. the space of connections (associated with
connection forms) having isotropy subgroups which are $\Ga$-conjugated to that
of $\a$,
$$ \S (\a ) \ = \ \{ \b \in \A \ : \ \exists \ga \in \Ga: \ \Ga_\b = \ga \Ga_\a
\ga^{-1} \ \} \ \ . \eqno(9) $$
The orbit of $\a$ under $\Ga$ will be denoted by $\Ga (\a )$ (or, for ease of
notation, by $\om_\a$). If we consider the gauge transformed of $\a$, we
have easily
that $\Ga_{\ga (\a )} = \ga \Ga_\a \ga^{-1}$, or in other words
$$ \Ga (\a ) \ \sse \ \S (\a ) \ \ ; \eqno(10) $$
thus, the equivalence classes under the relation of belonging to the same
isotropy
type consist of (necessarily, disjoint) unions of gauge orbits.
Actually, this equivalence relation leads to a {\it stratification} of $\A$
\ref{3} (see \ref{5} for more detail), pretty much as in the classical case of
compact group action on finite dimensional manifolds; however, we have now a
countable -- rather than finite -- set of strata \ref{4,5}.
The set $\S (\a )$ will be called the {\it stratum} of $\a$, and it can be shown
to be a smooth manifold, and actually a principal bundle \ref{5}.
\medskip
Let us now consider $\A$ in some more detail. We have already noticed that it is
a linear space; using the $G$-invariant scalar product in $\G$, denoted by
$\<.,.\> $, we can define a scalar product in $\A$ by
$$ \( \a , \b \) \ = \ \int_B \[ \sum_{\mu=1}^n \< A_\mu (x) , B_\mu (x) \> \]
\d^n x \eqno(11) $$
where $\a = A_\mu \d x^\mu$, $\b = B_\mu \d x^\mu$. Let $|\a | = \( \a , \a
\)^{1/2}$
denote the corresponding norm. Fixing a connection $C^0 \in \C$ and using the
induced covariant derivative $\D^0$ we define a Sobolev scalar product and
a Sobolev
norm of class $k$ on $\A$ by
$$ \( \a, \b \)_k \ = \ \sum_{j=0}^k \ \[ \ \int_B \( \(\D^0 \)^j\a,
\(\D^0 \)^j \b
\) d^n x \ \] \eqno(12) $$ and
$$ || \a ||_k \ = \ \( \sum_{j=0}^k \int_B | (\D^0 )^j ) \a |^2 d^n x
\)^{1/ 2} \ .
\eqno(13) $$
We will call $\A_k$ the completion of $\A$ with respect to this norm. Then
$\A_k$ is an Hilbert space, and different choices of the connection $C^0
\in \C$
give rise to equivalent norms \ref{6,7}. If we consider also the Sobolev
completion
$\Ga_k$ of the gauge group $\Ga$, we have that, for $k > \[ ({\rm dim}(B)
+1 )/2\]$,
$\Ga_k$ is an infinite dimensional Hilbert-Lie group modelled on a
separable Hilbert
space. The action of $\Ga$ on $\A$ can be extended to a smooth action of
$\Ga_k$ on
$\A_{k-1}$ and the $\Ga_k$ orbits are closed in $\A_{k-1}$ \ref{8-10}
>From now on we will assume that all objects requiring Sobolev completion
have been
completed in appropriate norms and we will write again $\A$ and $\Ga$ instead of
$\A_k$ and $\Ga_{k+1}$, and $\( \a, \b \)$ instead of $\( \a, \b \)_k$ .
We remark that the norm (13) induce a $\Ga$-invariant distance $d$ on $\A$
defined
in the usual way, i.e. as $d\( \a, \b \) = || \a-\b ||$ .
It is known \ref{3,4,7,10} that the action of $\Ga$ on $\A$ admits a slice
$S_\a$
at any point $\a \in \A$. The existence of a slice guarantees the existence of a
tubular neighbourhood $\U_\a$ of the $\Ga$ orbit $\om_\a$; this can be
obtained by
$\Ga$-transporting $S_\a$, i.e. $\U_\a = \Ga \( S_\a \)$.
We introduce the following
{\bf Definition:} {A gauge orbit $\om_\a$ is
{\it isolated in its stratum} if and only if $\U_\a \cap \Sigma(\a) = \om_\a$.}
We recall that a slice at $\a$ is a submanifold $S_\a \sse \A$
such that $\a \in S_\a$ and:
\parskip=0pt\parindent=0pt
{\tt (i)} $S_\a$ is transversal and complementary to the orbit
$\om_\a$ in $\A$ at $\a$;
{\tt (ii)} $S_\a$ is trasversal to all the $\Ga$ orbits which meet $S_\a$;
{\tt (iii)} $S_\a$ is (globally) invariant under $\Ga_\a$;
{\tt (iv)} For $\b \in S_\a$
and $\ga \in \Ga$, $\ga ( \b ) \in S_\a$ implies $ \ga \in \Ga_\a$,
i.e. $\Ga_\a$ is the maximal subgroup which leaves $S_\a$ globally
invariant; this also implies $\Ga_\b \sse \Ga_\a$. \parskip=10pt\parindent=0pt
For later reference, we rewrite {\tt (i)} as
$$ \T_\a \A = \T_\a S_\a \oplus \T_\a \om_\a \ ; \eqno(14) $$
we also denote, again for later reference,
$$ S_\a^0 \ = \ S_\a \cap \F (\a ) \ . \eqno(15) $$
It follows from properties {\tt (iii)} and {\tt (iv)} of $S_\a$, and the
compactness
of $\Ga_\a$, that
$$ S^0_\a \ = \ \Sigma(\a) \cap S_\a \ , \eqno(16) $$
$$ \Sigma(\a) \cap \U_\a \ = \ \bigcup_{\b \in \om_\a } S^0_\b \ ; \eqno(17) $$
hence, from (10),(14) and (17), we have that
$$ \T_\a \Sigma(\a) \ = \ \T_\a \omega_\a \oplus \T_\a
S^0_\a \ . \eqno(18) $$
Let us now consider a $\Ga$-invariant functional $f: \A \to R$ of class $C^1$
(a special case of this is the Yang-Mills functional), i.e. a
functional such that
$$ f \( \ga (\a ) \) \ = \ f ( \a ) \qquad \forall \ga \in \Ga \ , \ \forall \a
\in \A \ ; \eqno(19) $$
its differential at $\a$, $\d f_\a : \T_\a \A \to R$, will be a linear and
continuous
operator; as $\A$ is a Hilbert space, this will correspond to an element
$\phi_\a
\in \T_\a \A \simeq \A$, such that $\d f_\a (\b ) = (\phi_\a , \b )$.
It is easy to see that the invariance of $f$ implies
$$ \phi_{\ga (\a )} \ = \ \ga (\phi_\a) \ . \eqno(20) $$
This also implies that if $\ga \in \Ga_\a$, then $\d f_\a $ is invariant under
$\ga$; that is,
$$ \phi_\a \ \in \ \T_\a \F (\a ) \ \ . \eqno(21) $$
On the other side, it is clear from (19) that $(\phi_\a , \xi ) = 0$ for all
$\xi \in \T_\a \om_\a$; thus we conclude [see (14),(18)] that
$$ \phi_\a \ \in \ \T_\a S_\a^0 \ \ . \eqno(22) $$
It is clear from (19) that if $\a \in \A$ is a critical point for $f$, all the
points $\b \in \Ga (\a )$ are also critical for $f$\footnote{$^1$}{It is then
natural to consider as index of $\a$ its index as a critical point of the
restriction of $df_\a$ to the slice $S_\a$, or equivalently to $S_\a^0$.}; we
introduce the
{\bf Definition:} {The $\Ga$-orbit $\om \ss \A$ is {\it critical}
if points on $\om$ are critical for any $\Ga$-invariant functional.}
The above discussion shows that if an orbit is isolated in its stratum,
then it is
critical.
We could also prove the converse, i.e. that if an orbit is critical, then it is
isolated in its stratum; the proof of this would be just the same as the
one given
by Michel \ref{1} for compact groups and finite dimensional manifolds, and
thus is
omitted.
>From the above discussion we conclude that for functionals defined on the
space of
connections of a principal bundle (in Physics' language, pure gauge
theories) we have
-- in the framework, with the definitions, and under the conditions
introduced so far
-- the following extension of Michel's theorem:
{\bf Theorem.} {\it A gauge orbit $\om$ is critical if and only if it is
isolated in
its stratum.}
\medskip
In the classical case (compact group acting on a finite dimensional
manifold), it is
well known that the Michel's theory and its symmetry-based approach can be
extended
to consider general equivariant dynamics rather than just variational
problems; in
this way one can obtain\footnote{$^2$}{These results were
originally obtained with no use of Michel's theory.} \ref{11} in particular the
Equivariant Branching Lemma (EBL) \ref{12} (and its extension to the Hopf
case) and the
Reduction Lemma (RL) \ref{13}; most of the results in equivariant
bifurcation theory
are based on these lemmata \ref{13}.
The same holds here, i.e. one could obtain the corresponding results for
equivariant
vector fields on $\A$. Indeed, in this case equivariance means that
$$ X \( \ga (\a ) \) \ = \ \ga^* \[ X \( \a \) \] \qquad \forall \ga \in \Ga ,~
\forall \a \in \A \ , \eqno(23) $$
where $\ga^*$ denote the action of $\Ga$ on $\T \A$ induced by the action
of $\Ga$
on $\A$. Thus, for $\ga \in \Ga_\a$ and $X$ an equivariant vector field on
$\A$, we
have
$$ \ga^* \[ X(\a ) \] = X \( \ga ( \a ) \) = X (\a ) \ . \eqno(24) $$
We have thus proven the lemma below, from which the EBL and the RL can be
obtained in
the same way as for the finite dimensional case:
{\bf Lemma.} {\it Let $X : \A \to \T \A$ be a vector field on $\A$, equivariant
under the gauge group $\Ga$; then, $X \( \a \) \in \T_\a \F (\a )$.}
\bigskip\bigskip
{\bf Acknowledgements.~} {\petitrm We would like to thank the Mathematische
Forschunginstitut Oberwolfach and its Director M. Kreck for hospitality
during the
essential part of this work; this stay was supported by the Volkswagen
Stiftung under
the {\petitit Research in Pairs} program. We also acknowledge a PAS
Research Funds
grant, which made possible our joint work in Loughborough; the work of P.M.
was also
supported by GNFM-CNR and MURST.}
\vfill\eject
{\bf References} \parindent=10pt\parskip=4pt
\bigskip\bigskip
\def\CMP{{\petitit Comm. Math. Phys.}}
\def\ref#1{\item{[#1]} }
\font\petitrm = cmr8
\font\petitit = cmsl8
\font\petitbf = cmbx8
\baselineskip=10pt
\def\tit#1{{\petitrm ``#1''}}
\petitrm
\ref{1} L. Michel, \tit{Points critiques de fonctions invariantes sur une
G-vari\'et\'e}, {\petitit Comptes Rendus Acad. Sci. Paris} {\petitbf 272-A}
(1971),
433-436
\ref{2} L. Michel, \tit{Symmetry defects and broken symmetry.
Configurations. Hidden
symmetry}, {\petitit Rev. Mod. Phys.} {\petitbf 52} (1980), 617; \tit{Les
brisure
spontan\'ees de sym\'etrie en physique}, {\petitit J. Phys. (Paris)}
{\petitbf 36}
(1975), C7-41; L. Michel and L. Radicati, \tit{Properties of the breaking of
hadronic internal symmetry}, {\petitit Ann. Phys. (N.Y.)} {\petitbf 66}
(1971), 758;
\tit{The geometry of the octet}, {\petitit Ann. I.H.P.} {\petitbf 18}
(1973), 185. For
a very recent application to solid state theory, see: L. Michel,
\tit{Extrema des
fonctions sur la zone de Brillouin, invariantes par le groupe de sym\'etrie
du cristal
et le renversement du temps}, {\petitit C. R. Acad. Sci. Paris} {\petitbf B-322}
(1996), 223-230
\ref{3} I.M. Singer, \tit{Some remarks on the Gribov ambiguity}, \CMP
{\petitbf 60}
(1978), 7-12
\ref{4} J.P. Bourguignon, \tit{Une stratification de l'espace des structures
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\ref{5} A. Heil, A. Kersch, N. Papadopolous, B. Reifenhauser and F. Scheck,
\tit{Structure of the space of reducible connexions for Yang-Mills theories},
{\petitit J. Geom. Phys.} {\petitbf 7} (1990), 489-505
\ref{6} K.B. Marathe and G. Martucci, \tit{The geometry of gauge fields},
{\petitit J.
Geom. Phys.} {\petitbf 6} (1989), 1-106
\ref{7} H.B. Lawson , {\petitit The theory of gauge fields in four
dimensions}, A.M.S.
(Providence) 1985. See also J.P. Bourguignon and H.B. Lawson, \tit{Stability and
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189-230; and
J.P. Bourguignon, H.B. Lawson and J. Simons, \tit{Stability and gap
phenomena for
Yang-Mills fields}, {\petitit Proc. Natl. Acad. Sci. USA} {\petitbf 76} (1979),
1550-1553, and
\ref{8} P.K. Mitter and C.M. Viallet, \tit{On the bundle of connections and the
gauge orbit manifold in Yang-Mills theory}, \CMP {\petitbf 79} (1981), 457-472
\ref{9} G. Dell'Antonio and D. Zwanziger, \tit{Every gauge orbit passes
inside the
Gribov horizon}, \CMP {\petitbf 138} (1981), 291-299
\ref{10} J. Sniatycki, G. Schwarz and L. Bates, \tit{Yang-Mills and Dirac fields
in a bag, constraints and reduction}, \CMP {\petitbf 176} (1996), 95-115
\ref{11} G.Gaeta, \tit{A splitting lemma for equivariant dynamics},
{\petitit Lett.
Math. Phys.} {\petitbf 33} (1995), 313-320; \tit{Splitting equivariant
dynamics},
{\petitit Nuovo Cimento B} {\petitbf 110} (1995), 1213-1226
\ref{12} G. Cicogna, \tit{Symmetry breakdown from bifurcation}, {\petitit
Lett. Nuovo
Cimento} {\petitbf 31} (1981), 600-602; and \tit{A nonlinear version of the
equivariant
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Vanderbauwhede, {\petitit Local bifurcation and symmetry}, Pitman (Boston) 1982
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Golubitsky, D. Schaeffer and I. Stewart, {\petitit Singularities and groups in
bifurcation theory - vol. II}, Springer (New York) 1988
\bye