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\centerline{\bf Fundamental solutions of
gauge theories}
\bigskip
\centerline{Giuseppe
Gaeta}
\centerline{\it Dipartimento di Fisica,
Universit\`a di Roma, 00185 Roma (Italy)}
\centerline{\it Department of Mathematics,
Loughborough University, Loughborough LE11
3TU (U.K.)}
\bigskip\bigskip
{\bf Summary.} {We discuss the physical
meaning of a recent differential
geometric result, also presented here,
stating that gauge fields whose symmetry
is essentially different from that of
nearby (non gauge-equivalent) fields are
critical points for {\it any } gauge
invariant action functional, and are the
only fields with this property.}
\section{Introduction.}
A gauge theory is defined by a
gauge-invariant lagrangian $L$,
identified by a density $\L$ to be
integrated over a spacetime domain $D$,
$$ L \ = \ \int_D \L [A;x] \, \d^n x \
; \eqno(1) $$
this is a functional defined over the
space $\A$ of gauge fields $A \equiv
A_\mu$ ($\mu = 1,...,n$), depending
also on derivatives of the $A_\mu$.
The $A_\mu (x)$ are fields taking values
in the Lie algebra $\G$ of a Lie group
$G$. We denote by $\Ga$ the
corresponding gauge group.
In mathematical terms, the gauge fields
corresponds to the components of a
connection form $\a = A_\mu \, \d
x^\mu$, and are sections of a bundle
associated to a principal $G$-fiber
bundle $P$ over $D$. The gauge group
$\Ga$ corresponds to the space of
sections of $P$ \ref{1-3}.
The space $\A$ is not the space of all
possible gauge fields, but should be
taken to be that corresponding to
fields of {\it finite energy} (this
restriction is analogous e.g. to that to
the space of square-integrable functions in
quantum mechanics). In mathematical
terms, this means selecting a Sobolev
space of connections \ref{4,5}. The
same applies to $\Ga$, i.e. this as
well will be a Sobolev space of gauge
transformations (sections).
Thus, the functional $L$ is
defined on such a space $\A$. However,
it is by now customary to consider not
the full space $\A$ (of finite energy
gauge fields), but only an open and
dense set in it, corresponding to what in
mathematical terms are {\it irreducible
connections}; in physical terms, we can
think of these as follows. After a gauge
fixing, a generic gauge field will not
have any residual invariance; however,
special fields can still be invariant
under a compact group $\Ga_0 \approx G_0
\sse G$: these
correspond to reducible connections,
while the generic ones correspond to
irreducible connections. We will
correspondingly speak of reducible and
irreducible gauge fields. Notice that
these fields should be thought of
without any gauge fixing.
As the space $\A^*$ of irreducible gauge
fields is open and dense in $\A$, by
restricting to it one overlooks a set
of measure zero; moreover, the topology
of $\A^*$ conveys information on the
space $\A_0 = \A \backslash \A^*$ (of
reducible gauge fields). The space
$\A^*$ also has a number of properties --
essentially due to the fact that all
gauge fields in it are equivalent under
symmetry considerations -- that make
its study technically easier.
All these reasons account for the usual
choice to focus attention on
irreducible gauge fields. The purpose
of this note is to point out, on the
basis of recent relatively abstract and
general mathematical results \ref{6},
that this is not necessarily the best
way of attacking gauge theories and
Yang-Mills equations, as relevant
Physics lies precisely in the ``singular
part'' $\A_0$ of the space $\A$, and
actually in the ``most singular'' (in a
way to be explained below) part of
$\A_0$.
The abstract mathematical results
mentioned above have actually a
physical origin, as they represents a
generalization to gauge theories of a
classical result by L. Michel \ref{7}
on spontaneous symmetry breaking in
theories described by a $G$-invariant
potential [$G$ a semisimple compact Lie
group, e.g. $SU(n)$] defined on a finite
dimensional manifold $M$ [e.g. $R^m$,
or the Lie algebra $\G$ of $G$ itself, $M =
su(n)$, with the adjoint action]. This
theorem, in turns, was motivated by the
study of the
$SU(3)$ symmetry in hadronic interaction and
provided an explanation in general term
for the $SU(3)$ octet \ref{8}.
However, these results are so far available
only in a mathematical language, as
they rely heavily on the differential
geometry of fiber bundles and on the
(geometric) Michel theory; such an heavily
mathematical language is probably an
hindrance for the understanding of their
physical content and for physical
applications.
We want here to give an exposition of such
results in physical terms, and to discuss
its relevance for Physics.
\section{Reducible and irreducible
gauge fields.}
A gauge transformation $ \ga \in \Ga$
can be written in local coordinates as
$\ga = g(x) \in G$. If we apply this on
the gauge fields $A(x)$, we obtain new
gauge fields $B(x) = (\ga A) (x)$,
given explicitely by
$$ B_\mu (x) \, = \, g(x) \, A_\mu (x)
\, g^{-1} (x) \, - \, (\pa_\mu g) (x)
\, g^{-1} (x) \ . \eqno(2) $$
We denote the gauge isotropy subgroup of
$A(x)$ as $\Ga_A$; it is known \ref{9}
that for any $A$ this is equivalent to
a compact subgroup of $G$, but the
equivalence depends on $A$ itself. In
slightly more precise terms, $ \Ga_A$ is
given by the $\ga$ which are constant
under the covariant derivative $\grad$,
where $$ \grad_\mu \ = \ \pa_\mu \, + \,
A_\mu \ , \eqno(3) $$
and such that at a reference point $x_0
\in D$,
$$ g (x_0 ) \in G_A := C_G [ H_A (x_0) ]
\ ; \eqno(4) $$
here $C_G [H]$ is the centralizer of $H
\subseteq G$ in $G$, and $H_A (x_0) $ is
the {\it holonomy group} under $\grad$
computed in $x_0$ \ref{1-3}.
As already mentioned, if $G_A = \{ e
\}$ we say that the gauge field $A$ is
irreducible, otherways $A$ will be
reducible. Generic fields are
irreducible, so that reducible ones are
``special'' and correspond to a
degeneracy.
\section{Critical points.}
The physical fields corresponds to
critical points for the functional $L$.
We want to argue here that there will
be fields which are critical for {\it
any } $L$ with the prescribed gauge
invariance under $\Ga$, and that these
will actually be the ``most degenerate''
fields.
We will not attempt to give a proof of
the result, referring instead to
\ref{6} for a complete, detailed and
rigorous proof. A discussion of the
significance of this result in
geometric terms is given in \ref{10}.
First of all, as $L$ is gauge
invariant, critical fields will come in
gauge orbits; thus we will speak of a
critical orbit for $L$ if it is a gauge
orbit of critical fields. If a gauge
orbit $\om = \Ga (A)$ is critical for
{\it any } functional $L$ which is gauge
invariant under $\Ga$, we say that it is
{\it $\Ga$-critical}.
Now, to any $A$ we associate an
isotropy subgroup $\Ga_A$; fields which
have isotropy subgroups conjugated in
$\Ga$ (i.e. such that $\Ga_B = \ga
\Ga_A \ga^{-1}$ for some $\ga \in
\Ga$) should be seen as having an
equivalent symmetry; thus we denote the
conjugacy class of $\Ga_A$ in $\Ga$ as
$[A]$, and call it the {\it isotropy
type } of $A$.
The set of all the gauge fields having
the same isotropy type as $A$ will be
denoted by $\S (A)$; notice that
obviously any fields $B$ obtained from
$A$ via a gauge transformation ($B =
\ga A$) has obviously the same isotropy
type as $A$. Thus we can speak of the
isotropy type $[ \om ]$ of a gauge
orbit $\om$; obviously, if $A \in \om$
we have $[A] = [\om ]$.
The set $\S(A)$ -- which is a smooth
manifold, and actually a fiber bundle
\ref{11} -- is also called the {\it
stratum} of $A$.
The topology (essentially, the distance)
in $\A$ induces a topology (essentially,
a distance) in the $\S (A)$ \ref{6},
and we can thus speak of an orbit being
{\it isolated in its stratum}. This
means that if we take a sufficiently
small neighbourhood ${\cal U}$ of the
gauge orbit of $A$ in $\A$, all the
fields $B\not=A$ in ${\cal U}$ have
isotropy type different from that of $A$,
$[B] \not= [A]$.
We have then the following result, which
generalizes the theorem of L. Michel
\ref{7,12}:
{\bf Theorem. } \ref{6}
{\it A $\Ga$ gauge orbit
$\om$ is $\Ga$-critical if and only if
it is isolated in its stratum.}
Thus the fields which are critical for
all gauge invariant functionals are
those which are ``maximally degenerate'':
all nearby fields which are not
obtained simply by a gauge
transformation, have a substantially
different (actually, smaller \ref{6})
symmetry.
This is in a way natural: it is clear
that the property of being critical for
any $L$ -- i.e. a solution for any $\Gamma$
gauge theory -- is extremely special,
and thus should correspond to very
special fields; this theorem tells
precisely in what this ``special''
character lies.
Let us add some comments, also to avoid
possible misunderstandings. First of
all, when we speak of ``any'' gauge
invariant, it should be understood that
the gauge group $\Ga$ (and thus $G$) is
fixed, as well as the space domain $D$
(and thus its dimension $n$). We should
also stress that here we have not
restricted consideration to Yang-Mills
theories, i.e. we have not fixed the
form nor the order (order of the
maximal derivative of $A$ on which it
depends) of the functional $L$.
It should also be stressed that the
theorem only deals with critical
points, but is not able to distinguish
between stable (minima) and unstable
(maxima or saddle points) ones.
This result can be extended
to include dependence on parameters to
describe successive symmetry breakings
and phase transitions \ref{6,10}.
It can also be extended to include matter
fields $\phi^a$ (in mathematical terms,
these are sections of a vector bundle over
the spacetime $D$); the way to obtain such
an extension goes through the fact that for
a gauge-matter field configuration
$(A,\phi)$, the gauge isotropy subgroup is
just
$$ \Ga_{(A,\phi)} \ = \ \Ga_A \, \cap \,
\Ga_\phi \ , \eqno(5) $$
so that we can first use this result on the
pure gauge sector, and once a $\Ga$-critical
gauge orbit $\om$ is determined, we can
consider matter fields with isotropy
$\Ga_\phi \sse \Ga_A$ for $A \in \om$; see
\ref{6,10} for detail.
\section{Physical significance.}
After describing this recent abstract
result, we should now discuss its
meaning and significance for Physics.
First of all, the reader could wonder
if there really are gauge fields
(orbits) like those considered in
the theorem, i.e. which are critical
for any gauge invariant $L$.
As discussed in detail in \ref{6}, one
can easily find explicit (and physically
meaningful) examples of them e.g. by
considering $L[A,\phi]$ in the form
$$ L \ = \ L_g + L_p + L_{ym} \eqno(6) $$
and looking for fields which are
critical for each of the three parts
separately.
In (5) the gradient, potential, and
Yang-Mills part of the lagrangian $L$
are, as usual, given by
$$ \matrix{
L_g = & \int_D \vert \grad_\mu
\phi^a (x) \vert^2 \,
\d x^n \cr & \cr
L_p = & \int_D V [ \phi (x) ] \, \d
x^n \cr & \cr
L_{ym} = & \int_D F^{\mu \nu} F_{\mu
\nu} \, \d
x^n \ ; \cr} \eqno(7) $$
here $\phi (x)
\in C^k$ represents matter fields, $V :
C^k \to R$ is a potential, $\grad_\mu$ is
as in (3), $F_{\mu \nu} = [\grad_\mu ,
\grad_\nu ]$, indices are raised and
lowered with the $g_{\mu \nu}$ metric on
$D$, and the summation convention is
implicit.
For $G = SU(3)$ this class of examples
contains, in particular, the case of
$A=0$ and constant matter fields $\phi
(x)$ corresponding to the critical
directions discussed by Michel and
Radicati, i.e. to the particles of the
octet. This shows that
the theorem is far from being empty,
and encompasses physical relevant
cases.
Thus we see that there can be fields
which are solutions to any gauge theory
(with given gauge group); it is natural
to see these as {\it the most fundamental
solutions of a gauge theory}, as they are
implied by the very symmetry properties
of the theory alone, and are there
idependently of the details of $L$, i.e.
of the specific model under study.
Now, gauge fields whose gauge orbits is
isolated in its stratum cannot belong
to the set of irreducible ones (as this
set is open and dense). Thus, these
fundamental solutions will necessarily
correspond to reducible fields.
This does not mean, of course, that the
usual approach based on the study of
the set $\A^*$ (of irreducible
connections) is wrong, nor that it cannot
cope with the problems considered here:
indeed, as mentioned in the introduction,
one can obtain informations on the
structure of the set $\A_0$ of reducible
fields from the structure of $\A^*$.
On the other side, such an approach is
probably not the most convenient one if
we are looking for the fundamental
solutions identified above, as these are
in a sense the most different from the
generic ones, i.e. from those in
$\A^*$: it seems much more convenient
to start from consideration of
maximally symmetric nontrivial fields,
i.e. fields $A$ such that $\Ga_A \approx G$,
and then -- if these do not exist or
correspond to unstable solutions --
pass to consider $A$ such that $\Ga_A
\approx G_1$ is a maximal isotropy subgroup
\ref{12} of $G$, and so on, reducing at
each step the symmetry.
The proposal of such a procedure
requires maybe some further
explanation. Let us first look for
maximally symmetric fields $A(x)$ and
gauge orbits $\om$; suppose we have
found say only one maximally symmetric
gauge orbit $\om$, which hence is
isolated in its stratum, but that we
have determined this is an unstable
critical point, and thus not interesting
for our purposes. We can then eliminate
it from $\A$, and consider $\A
\backslash \om$. As there are no other
orbits with $[\om ] = G$, we can then
pass to consider gauge fields and gauge
orbits with $[\om ] = G_i$, where $G_i$
are the possible maximal isotropy
subgroups of $G$, and repeat the
procedure.
It should be stressed that the approach
based on highly symmetric {\it ansatzes} has
always been used to determine solutions for
gauge theories; however, we pointed out
here that these highly symmetric solutions
are not only the easiest to determine, but
also have a fundamental physical (and
mathematical) meaning, as they are
determined by the symmetry requirements
alone.
Similarly, the fact that sufficiently
symmetric (e.g. fully symmetric) points are
critical for any invariant potential has
been known since the work of Michel
\ref{7}, but the fact that this applies
also to full gauge theories has only been
proved in \ref{6}
The same theory developed to obtain the
theorem quoted above, also leads to
consider the restriction $L_0$ of $L$ to
gauge fields of given symmetry $\Ga_0
\subseteq \Ga$; the fact that critical
points of $L_0$ will also be critical
for the full $L$ is not so obvious
(indeed it could fail for general --
but luckily unphysical, at least in
this context -- cases \ref{13,14}),
but can be proved to be true under
suitable conditions \ref{13,6}. Thus,
in looking for critical gauge fields
with given symmetry, we can actually
use a simplified analysis, as we can
use a functional defined on a much
smaller space of possible gauge field
configurations; this should be seen as
an additional and not irrelevant
advantage of the approach proposed here.
We would also like to point out some
other features of the result quoted
here and of the approach based on it we
propose for the analysis of gauge
theories. Our theorem, and in general
the ``Michel-gauge'' theory of
symmetry breaking, does not postulate
the specific form (5) of the gauge
invariant functional, nor requires it
to be first-order; i.e. is valid beyond
the realm of proper Yang-Mills
theories. This is a strength of the
theory, as it is more general; but it is
also a weakness, as it fails to use
specific features of the Yang-Mills
functional.
Also, in this way we can determine
critical points which are not
necessarily stable (for physically
relevant cases, general theorems
\ref{15,16} ensure these will correspond
to self-dual or anti-self-dual fields).
Again, this is a strength as it can deal
with more general critical points, but
a weakness as it does not use the
special features of stable critical
points, which are most relevant
physically; notice however that this
approach can also be applied directly to
self-dual or anti-self-dual Yang-Mills
functionals.
However, in this context it should be
recalled that saddle points are also
relevant physically, e.g. when using a
WKB (semiclassical) approximation or
thanks to localization-like properties.
Our results, and all the considerations
so far, only dealt with classical gauge
theories; from a physical point of
view, however, we are compelled to
think of their quantized version. In
the quantum theory, we should perform a
functional integration
$$ S \ = \ \int_\A e^{i \L [A]} \, {\cal D}
A \eqno(8) $$
over all field configurations, and we
know that the set $\A_0$ of reducible
fields has measure zero; thus, it could
naively seem that restricting the
functional integral to the set of
irreducible fields $\A_* \subset \A$
will not cause essential harm. This is
of course not true, because of the
nontrivial topology of $\A_*$; taking it
into account can be a highly nontrivial
task.
Our result shows that there is also
another reason for which restriction to
$\A^*$ is an unconvenient way of
dealing with the functional integration
required here: generically (i.e. for
generic gauge invariant functionals)
the critical points of $L$ will lie
precisely in the part $A_0$ which lies
outside $A^*$; thus, the stationary
points for $e^{i L[A]}$ are in $A_0 =
\pa \A^*$.
This means that in looking for
approximate versions of the functional
integral (7), we should rather
integrate over a neighbourhood of the
set $\A_0$; the simplest domain to be
considered for a generic (in the same
sense as above) gauge theory would thus
be a neighbourhood ${\cal U}_0$ of the
gauge orbits which are isolated in
their stratum (i.e. of the fundamental
critical points).
\section{Ackowledgements}
This work is based on common mathematical
work with Paola Morando, performed in
Oberwolfach and in IHES. I would like to
thank some friends for useful discussions,
and in particular C. Bachas, A.
Chakrabarti, C. Hayat-Legrand, L.A. Ibort
and T. Torgut. The encouragement of J.P.
Bourguignon and of L. Michel was essential
at several stages.
\def\tit#1{{``{#1}''}}
\def\CMP{{\it Comm. Math. Phys. }}
\def\bibitem#1{\item{#1.}}
\vfill\eject
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\bye