\magnification=1200
\def\giorno{29/11/97}
\def\A{{\cal A}}
\def\Ga{\Gamma}
\def\sse{\subseteq}
\def\ss{\subset}
\def\om{\Omega}
\def\ga{\gamma}
\def\G{{\cal G}}
\def\adP{{\tt ad}P}
\def\a{\alpha}
\def\b{\beta}
\def\pa{\partial}
\def\grad{\nabla}
\def\d{{\rm d}}
\def\S{\Sigma}
\def\L{{\cal L}}
\def\ref#1{[#1]}
\parindent=0pt
{\nopagenumbers
~
\vskip 2 truecm
\centerline{\bf REDUCIBLE CONNECTIONS IN GAUGE
THEORIES}
\bigskip\bigskip\bigskip
\centerline{Giuseppe Gaeta$^*$}
\centerline{\it Dipartimento di Fisica,
Universit\`a di Roma I }
\centerline{\it I--00185 Roma (Italy)}
\centerline{\tt giuseppe.gaeta@roma1.infn.it}
\bigskip
\centerline{Paola Morando}
\centerline{\it Dipartimento di Matematica,
Politecnico di Torino}
\centerline{\it I--10139 Torino
(Italy)}
\centerline{\tt morando@polito.it}
\bigskip\bigskip\bigskip
\vfill
\footnote{}{$^*$ Present
address: Department of Mathematics, Loughborough
University, Loughborough LE11 3TU (England)}
\footnote{}{{\tt Last modified \giorno}}
\footnote{}{{\tt P.A.C.S. numbers: 11.15.Ex ~;~
02.20.Tw , 02.30.Wd , 02.40.Vh}}
{\bf Abstract.} When studying gauge theories
(e.g. with finite energy conditions), attention is
traditionally reserved to the subset of
irreducible connections, which is open and dense
in the full space of connections. We point out
that in general the residual set of reducible
connections contains critical points of the gauge
functionals, which moreover are in general the
only ones common to all theories with a given
symmetry, i.e. those determined by the symmetry
and the geometry of the problem alone, and not by
the specific choice of the functional.
\vfill\eject}
\pageno=1
\parskip=10pt
\parindent=0pt
{\bf 1. Introduction.}
When we study a pure gauge theory (e.g., but not
necessarily, of Yang-Mills type), the central
object is a (gauge-invariant) functional $L$
defined on a space of connections $\A$ of the
associated bundle $\adP$ to the principal
$G$-bundle $P$ over the base space $B$ (usually
the four-dimensional spacetime or a
compactification of it) which defines the gauge
invariance of the theory.
This space of connections $\A$ is not the full set
of connections, but a subset of it; this is
typically a Sobolev space (which ensures a number
of technically relevant properties) chosen in
agreement with physical requirements, usually a
finite energy condition appropriate for $L$ and
the physical nature of the problem.
However, it is by now customary to consider not
the full Sobolev space $\A$, but a subset of it,
which is open and dense in $\A$ and which is
technically more manageable to study. This subset
$\A^*$ corresponds to the so-called {\it
irreducible connections} (relevant definitions
will be recalled below).
Reducing one's attention to $\A^*$ rather than to
the full $\A$ is justified on the one side by the
fact that this corresponds to generic connections
-- we have already mentioned that $\A^*$ is open
and dense in $\A$ -- and on the other by the fact
that in this way the analysis is greatly
simplified, as the connections in $\A_0 \equiv \A
\backslash \A^*$ are indeed singular in many
senses, and one only overlooks a set of measure
zero. Moreover, the topology of $\A^*$ conveys
information on the structure of the ``erased
part'' $\A_0$ as well.
However, to the eyes of a reader familiar with
the theory of symmetry breaking in finite
dimensions -- e.g. in theories defined by
minimization of a finite dimensional
$G$-invariant potential $V(x)$, with $G$ a compact
Lie group and $x \in M \sse R^n$ -- this reduction
could cause some perplexity: in this setting,
it is quite well known that there is a ``generic
stratum'' $M^*$, open and dense in $M$ (or
similarly when considering the set $\Omega
\simeq M/G$ of $G$-orbits), but it is also known
that generically the critical point of $V$ lie
precisely in $M \backslash M^*$, and more
precisely in the ``most singular'' (in a precise
sense to be defined below) strata.
The simplest example of this situation is
trivial: it suffices to consider potentials $V$
defined on $R$ and which are invariant under $G =
Z_2$ acting as $g(x) = - x$: the set $M^*$ is
given by $R \backslash \{ 0 \}$, but the origin
is a critical point for all such potentials, and
indeed the only generic critical point for this
class of $Z_2$-invariant potentials.
As a less trivial application of the general
finite-dimensional theory \ref{1}, one can
consider the well known -- and celebrated --
works of Michel and Radicati on the breaking of
$SU(3)$ symmetry in hadronic interactions
\ref{2}, in which the physical particles are
shown to be in correspondence precisely with
points in the most singular strata, and thus in
particular with orbits which lie in $M \backslash
M^*$.
The Michel's geometric theory of symmetry
breaking has been recently extended \ref{3} to
the full setting of gauge theories (both pure ones
and with matter fields, and not necessarily of
Yang-Mills type and/or defined by first order
functionals), leading to a complete analogy with
the original results of Michel in the finite
dimensional setting.
This extensions shows in particular that gauge
orbits which are isolated in their stratum
(again, relevant definitions will be introduced
below) are necessarily critical points of any
gauge invariant functional. But such gauge orbits
are necessarily {\it not} in the generic set
$\A^*$ -- they are actually the ``most singular''
among the singular orbits -- and thus by
restricting to $\A^*$ one is overlooking a whole
class of critical points, which moreover are
the only critical points which are common to all
the gauge theories with the same gauge invariance,
i.e. which are fixed by the symmetry (and
geometry) of the problem itself.
The purpose of this note is to illustrate the
above statement, avoiding to go into excessive
mathematical detail (a mathematically complete,
detailed and rigorous proof is published
elsewhere \ref{3}) and focusing instead on the
physical side of the question. In particular, we
would like to attract the attention of the reader
on the fact that -- except for quite special groups
such as $SU(2)$ -- one cannot limit his/her
attention to $\A^*$, as in doing this one is
necessarily forgetting critical points of the
theory.
It should be stressed that our analysis is only
at the classical level; when one considers the
quantum theory -- which we do {\it not} do,
neither here nor elsewhere -- one has to
integrate over the full space of connections, and
by considering only $M^*$ one is indeed leaving
out of the integration only a set of measure zero
(corresponding to singular orbits).
However, if one is approaching the quantum theory
by a WKB or related approximations, it becomes
specially relevant even in this case to know the
critical points -- and, we stress, not only the
stable ones -- of the classical theory, which are
used to start the perturbative expansion, so that
again it is quite dangerous to sistematically
leave out of consideration the set $\A \backslash
\A^*$, which is now known to contain generic
critical points.
\bigskip
{\bf 2. Irreducible connections.}
We consider a finite dimensional connected
riemannian manifold $B$ (e.g. $R^4$ or $S^4$)
and a compact semisimple Lie group $G$ [e.g.
$SU(n)$] with Lie algebra $\G$; and a
principal bundle $P$ over $B$, with projection
$\pi : P \to B$ and fiber $\pi^{-1} (x) = G$. We
also consider the corresponding adjoint bundle
$\adP$, with fiber $\G$ and $G$-action on it
defined by the adjoint action of $G$ on its Lie
algebra $\G$.
In the affine space ${\cal C}$ of
connections on $P$ we can select a reference
point (i.e. connection) $A^0$; with this we can work in the vector space of
connection forms, i.e. in the space $\A$ of
one-forms on $B$ with values in $\G$: to a
one-form $\a = A_\mu \d x^\mu$ corresponds the
connection with components $A_\mu$, and the
covariant derivative $\grad_\mu = \pa_\mu +
A_\mu$. We will, for ease of notation, speak
improperly of ``the connection $\a$''.
We consider the associated bundle $\adP$ on
$B$, and the space of differentiable sections
$\Ga$ on it; this is the gauge group modelled on
$G$. An element $\ga \in \Ga$ acts naturally on
connections by conjugation: if $\ga$ is, in local
coordinates, given by $g(x)$, then $\ga : \a \to
\ga (\a ) = \b$ given in local coordinates by $\b =
B_\mu \d x^\mu$ with $$ B_\mu (x) = g(x) \cdot
A_\mu (x) \cdot g^{-1} (x) - (\pa_\mu g)(x) \cdot
g^{-1} (x) \ . \eqno(1) $$
Thus the gauge isotropy subgroup $\Ga_\a$ of the
connection $\a$, which is obviously defined as
$$ \Ga_\a = \{ \ga \in \Ga \ : \ \ga (\a ) = \a
\} \ , \eqno(2) $$
is given in local coordinates by the $g(x)$ such
that $(\pa_\mu g) (x) = [ g(x) , A_\mu (x) ]$, as
follows by requiring that $B_\mu = A_\mu$ in (1).
As mentioned in the introduction, one prefers --
for reasons of both mathematical and physical
nature, the latter ones being usually finite
energy conditions -- to consider Sobolev classes of
connections and gauge sections; by a suitable
choice of the Sobolev norms and of the Sobolev
class to consider (depending in particular on the
dimension $k$ of the base space $B$), one can work
properly in this setting; in this note we will not
discuss this analytical aspect of the question,
referring to \ref{3} for detail, and focus on the
main geometrical issue: here we only want to
reassure the reader that the analytical side is not
forgotten, and that $\A$ and $\Ga$ should more
precisely be thought of as subsets of the full
sets of connections and gauge sections, which are
Sobolev spaces of class, respectively, $k$ and
$k-1$ in suitable norm \ref{4,5}
It is a classical and well known result \ref{6}
that $\Ga_\a$ is easily characterized in terms of
any reference point $p_0 \in P$, with $\pi (p_0 )
= x_0$, as the set of sections in $\G$ which take
in $x_0$ values in $C_G [H_\a (p_0 ) ]$, the
centralizer in $G$ of the holonomy group $H_\a
(p_0 )$ of the connection $\a$ at $p_0$ -- that
is, such that $g (x_0) \in C_G [H_\a (p_0 ) ] \sse
G$ -- and which are covariantly constant along the
connection $\a$ itself.
This means that for any $\a$ the gauge isotropy
subgroup $\Ga_\a$ is isomorphic to a compact
group, but it should be noticed that the
isomorphism depends on $\a$ itself.
A dichotomy immediately arises in $\A$: the
connections for which $\Ga_\a$ is nontrivial --
i.e. $C_G [H_\a (p_0 ) ]$ does not reduce to $\{
e \}$ for all $p_0 \in P$ -- are called {\it
reducible} (we denote their set by $\A_0 \ss \A$),
and one can obtain a reduction of $P$ under them
(this means that $P$ can be decomposed into the
disjoint union of subbundles, invariant under the
connection); the others are called {\it
irreducible}, and we denote their set by $\A^* \ss
\A$.
It is a general result that, under the above
assumptions on $B$, $G$, $P$ and $\Ga$, the set
$\A^*$ is open and dense in $\A$ (e.g., for
$G=SU(2)$ any nonzero connection is in $\A^*$
\ref{4}); we also say that $\A^*$ constitutes the
generic {\it stratum} in $\A$.
The set $\A^*$ is obviously easier to study than
the full set $\A$; actually, one can prove
\ref{6} that it is a smooth Hilbert manifold,
and a principal $G$-bundle \ref{7}. Moreover, as
it is open and dense in $\A$, it seems quite
reasonable to confine attention to its study;
indeed, this is what is done in a vaste body of
literature on gauge theories.
\bigskip
{\bf 3. Reducible connections.}
We want now to briefly present a recent result
providing an identification of points which are
critical for any gauge functional (once $P$ --
and thus $B$ and $G$ -- is given); these turn out
to correspond to reducible connections.
We will not give proofs, referring again to
\ref{3}, but we need however some work to
introduce the notation suitable to formulate the
result.
We consider functionals $L : \A \to R$ which are
invariant under $\Ga$, i.e. such that $L [\a ] =
L [ \ga (\a ) ] $ for all $\ga \in \Ga$ and all
$\a \in \A$; we assume these functionals are
defined by a local density $\L [A;x]$, function of
$A (x)$ and its derivatives (up to any finite
order) at the point $x \in B$, and maybe
explicitely dependent on $x$ itself, so that
$$ L [A] \ = \ \int_B \L [A;x] \, \d^k x \ .
\eqno(3)$$
The $\Ga$-invariance of $L$ guarantees that
critical points of $L$ come in $\Ga$-orbits. We
say that a $\Ga$-orbit $\om$ is {\it
$\Ga$-critical} if it is an orbit of critical
points for any $\Ga$-invariant functional $L$.
To any $\a \in \A$ we associate the isotropy
subgroup $\Ga_\a$ defined above; through this,
$\A$ can be partitioned into {\it isotropy
types}, or {\it strata}, defined as sets of
connections having isotropy subgroups belonging to
the same conjugacy class under conjugation in
$\Ga$: thus the stratum $\S (\a )$ of $\a$ is
defined as
$$ \S (\a ) \ = \ \{ \b \in \A \ : \ \Ga_\b = \ga
\, \Ga_\a \, \ga^{-1} \ , \ \ga \in \Ga \} \ .
\eqno(4)$$
Obviously, $\Ga (\a ) \sse \S (\a )$;
notice that in the present gauge setting the set of
strata is countable. The strata are smooth
manifolds, and principal bundles (see \ref{7}
for detail and some subtleties involved in the
bundle structure; further detail on the geometry of
gauge orbits is given in \ref{8}).
The topology defined in $\A$ induces a topology
in the strata (as $G$ is compact and thus acts
regularly, in particular on its Lie algebra);
thus it makes sense to speak of an orbit $\om$
which is {\it isolated in its stratum}; this
intuitive concept can be defined more precisely
using slices and tubular neighbourhoods around
$\Ga$-orbits \ref{3}, and amounts to saying that
no other orbit belonging to the same stratum is
near (in the natural topology) to $\om$.
We have then the following result, proved in full
detail in \ref{3}, which is an immediate
extension -- including the method of proof -- of
a classical result obtained by Louis Michel
\ref{1} in 1971, motivated by the Physics of
hadronic interactions \ref{2}:
{\bf Theorem.} {\it A $\Ga$ gauge orbit $\om \ss
\A$ is $\Ga$-critical if and only if it is
isolated in its stratum.}
As the stratum of irreducible connections is open
and dense, it is obvious that $\Ga$-critical
orbits, if they exist, do necessarily correspond
to reducible connections.
It should be mentioned that, as also discussed in
detail in \ref{3}, the same result extends to
theories with matter fields (and actually also to
the case, maybe less relevant physically, of
gauge-equivariant equations which do not come
from a variational principle \ref{9}). Similarly,
the theorem given above can be combined with
bifurcation theory to describe symmetry breaking
arising in second-order phase transitions
\ref{3}.
\bigskip
{\bf 4. Relevance for Physics.}
We have thus, after introducing the (minimal)
needed notation, stated our result, and noticed
that it points out to critical connections -- and
in particular $\Ga$-critical $\Ga$-orbits of
connections -- which are reducible, i.e. which
would be overlooked if restricting attention to
$\A^*$ rather than considering the full $\A$.
Actually, we are in the opposite situation: these
are necessarily in $\A_0 = \A \backslash \A^*$, so
that our approach does naturally point in a
direction which is ``orthogonal'' to the standard
one.
In the present section we want to argue that such
a result is not a mathematical curiosity, or even
a mathematically relevant fact, but it is also
relevant from the point of view of Physics.
First of all, we stress that, as already
mentioned, there are gauge orbits which are
critical for {\it any } smooth functional with the
required gauge invariance: thus these critical
points are directly related to the symmetry
rather than to the detailed choice of the
functional. In this sense, i.e. from a symmetry
point of view, these $\Ga$-critical orbits (which
correspond necessarily to reducible connections,
see above) are more fundamental than any other
solution; in our opinion, this would already be by
itself a sufficient reason to study them in detail.
In this respect, it should be recalled what is
the situation in the finite-dimensional
approximation for strong interactions: here the
theory is described by a $G$-invariant potential
on the Lie algebra $\G$; we have $G = SU(3)$
acting on its algebra by the adjoint
representation, and we should fix the
normalization condition $|x|=1$ (for $x \in R^8
\approx \G$). When we look at the $G$-orbit
space, we deal with a subset of $R^2$
(corresponding to a Cartan subalgebra of $\G$; the
$G$-invariants can be chosen e.g. as the two
independent eigenvalues of the $su(3)$ matrices),
and again by normalization we restrict to the unit
circle. On this there are points which correspond
to orbits isolated in their stratum, i.e. to
$G$-critical orbits: these, together with the one
corresponding to the origin, give precisely the
$SU(3)$ octet of hadronic interactions
\ref{2,10}.
It should be stressed that the same situation is
encountered if we abandon the finite-dimensional
(i.e. potential) approximation, and consider the
gauge theory in its full (functional) setting: in
this case one obtain, with an analysis which is
only slightly more complicated than the finite
dimensional one -- and resorting also to the
``Symmetric Criticality Principle'' of Palais
\ref{11} -- again that the $\Ga$-critical orbits
corresponds to covariantly constant gauge fields
representing the physical particles of the
$SU(3)$ octet \ref{3}.
In a sense, it is surprising that the many
works adopting ``automatically'' the
restriction to irreducible connections avoid to
mention the fact that these -- relevant and very
simple -- solutions lie outside of such a domain.
On the other side, the technical complication met
in attempting a detailed analysis of the space of
reducible connections \ref{7,12} can explain
why this is so often avoided.
In a way, the approach proposed here suggest to
start from the most singular strata rather than
approaching them from the generic one; this is so
much more reasonable as the theorem reported
above focuses our attention not just on reducible
connections, but precisely on the ``most
reducible'' ones. These are the ones which are more
difficult to study in the standard way, i.e. as
the border of the generic stratum -- and of any
other one -- but being at the other extremum of
the range they can be profitably tackled in their
own sake.
A strength and at the same time a weakness of the
approach followed here is that our result holds
for {\it any} gauge invariant functional, i.e.
not only for gauge functionals of Yang-Mills
type. This is a strength because of the greater
generality, but a weakness in that we fail to
make use of the specificity of the YM, physically
relevant, case.
Similarly, one is not restricted to functionals of
first order (i.e. depending on the fields and
derivatives of order up to one alone), but could
consider functionals of any finite order: again,
this is a more general setting, but -- precisely
for this -- it does not take full advantage of the
specific YM structure.
It should also be mentioned that in this way we
have no limitation to minima and/or selfdual or
anti-selfdual critical points, but can obtain any
kind of critical point.
In this respect, it should maybe be recalled that
critical points which are not minima can be quite
relevant physically, e.g. in the WKB or related
approximations when dealing with the quantum case.
Also, the theorems of Bourguignon and Lawson
\ref{13} stating that all the weakly stable
critical points of YM functionals are selfdual or
anti-selfdual, apply under precise conditions,
which on the one side include the physically
relevant case but on the other are not generic:
thus, they only hold in dimension four and for a
restricted class of gauge groups [including the
physically relevant ones, $SU(2)$ and $SU(3)$];
and do not deal with unstable critical points.
In principle, the present approach could also be
useful in determining the metric in the moduli
space, as it localizes the critical points in
strata and thus could help in determining a
parametrization (maybe using considerations on
the Hilbert basis of invariants for $G$).
Finally, it should be mentioned that the approach
proposed here, as the whole Michel theory of
which this is just a small extension, relies
simply on the combination of group-invariant
geometry and of topology, provided there is a
suitable ``agreement'' between the symmetry and
the topology, i.e. essentially that the
projection to (gauge) orbit space ``agrees'' with
the topology of the space in which the theory is
defined. This is a quite general situation, and
thus it is not surprising that the range of
applications of Michel theory is correspondingly
ample. This same consideration also suggests that
Michel theory could maybe be formulated in a more
``intrinsic'' way, i.e. (roughly speaking) in
terms of Atyiah-Bott rather than Bredon
equivariant cohomology (this remark is due to L.A.
Ibort).
It is obvious, but maybe equally worth
mentioning, that all our discussion is at a
purely classical level, i.e. we have only looked
at critical points of the classical gauge
functionals, and not at the corresponding
functional integral; obviously, as already
recalled, the classical critical points can be
used to start a perturbative expansion (WKB
method), which in some cases (as in the
Duistermaat-Heckmann theory) can even become
exact.
\bigskip
{\bf 5. Conclusions.}
In the early days of gauge theories
it was noticed that the subspace $\A^*$ of
irreducible connections is on the one side much
easier to study than the full space $\A$ of
connections (in the chosen Sobolev class), and on
the other side that $\A^*$ is open and dense in
$\A$. Also, the topology of $\A^*$ conveys
information on the ``erased part'' $\A \backslash
\A^*$, i.e. on the set of reducible connections.
Thus, it was suggested that many relevant
informations -- as many as possible -- should be
extracted directly from $\A^*$, without embarking
in tackling directly the complex geometry of $\A /
\Ga$.
We have pointed out here that many relevant
features, in particular $\Ga$-critical orbits,
not only lie in the set of reducible connections,
but actually in the most singular strata.
This suggest that, rather than starting from the
study of the generic stratum (irreducible
connections), one could instead study directly
the most singular strata, and maybe the less and
less singular strata in order to analyze the
successive symmetry breakings.
This is, of course, in agreement with the
philosophy underlining Michel's symmetry breaking
theory, according to which -- in Louis Michel's
words -- ``quite often Physics is in the
most singular strata''.
\vfill
{\bf Acknowledgements.}
The work described here was performed at -- and
benefited from support by -- the M.F.O. in
Oberwolfach (under the {\it Research in Pairs}
program, supported by Volkswagen Stiftung) and the
I.H.E.S. in Bures sur Yvette: we warmly thank these
Institutions and their Directors.
%In particular, G.G. warmly thanks the I.H.E.S. and
%its Director for hospitality as well as for
%moral and material support while on unpaid leave
%from Loughborough University, in the key stage of
%the present work.
We also acknowledge financial support from
CNR-GNFM (Italy).
We would like to thank some friends for useful
discussions, and in particular C. Bachas and A.
Chakrabarti in the key stage of this work, and
C. Hayat-Legrand and L.A. Ibort when (we
wrongly thought) it was completed. The
encouragement of J.P. Bourguignon and of L. Michel
was essential at several stages.
\vfill\eject
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\def\ref#1{\medskip \item{[#1]} }
\def\ni{\item{} }
\parindent=30pt
\parskip=0pt
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\bye