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{\nopagenumbers
~ \vskip 6 truecm
\parindent=0pt
{\bf REPLICA THEORY AND THE GEOMETRY OF SYMMETRY BREAKING}
\bigskip\bigskip\bigskip
{Giuseppe Gaeta\footnote{$^*$}{Address after 1/2/98:
Dip. di Fisica, Universit\`a di Roma I, 00185 Roma (Italy), {\tt
gaeta@roma1.infn.it}}}
\smallskip
%{\it Dipartimento di Fisica, Universit\`a di Roma I} \par
%{\it I--00185 Roma (Italy)} \par
%{\tt gaeta@roma1.infn.it} \par
\smallskip
{\it Department of Mathematics, Loughborough University} \par
{\it Loughborough LE11 3TU (England)} \par
{{\tt g.gaeta@lboro.ac.uk}} \par
\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip
{{\bf Abstract.} In the study of Replica Symmetry
Breaking (RSB), one is led to consider functions $f$ of {\it
pseudomatrices}, i.e. of matrices of order $\a$, with $\a$ a
real, rather than an integer, number. We propose a
mathematically rigorous definition of pseudomatrices, and show
that from this it follows that the minimum of $f$ over the space
of pseudomatrices $Q(x,y)$ which depend only on $(x-y)$ is also
a critical point for $f$; this corresponds to a property which
is usually assumed without proof in the study of RSB. We also find that
generic bifurcations from such a minimum lead to minima corresponding to
periodic quasimatrices. These results are obtained through use of Michel's
theory of symmetry breaking.}
{\tt P.A.C.S. numbers: 02.40.-k, 02.90.+p, 05.90.+m, 61.43.-j}
\bigskip\bigskip\bigskip
\parskip=0pt
\parindent=10pt
{\bf 1. Introduction.}
\bigskip
In the study of replica symmetry breaking \ref{1}, one is led to
look for minima of a function $f(Q)$ of a {\it pseudomatrix}
$Q$; in this physical situation, $f$ is typically an energy
functional depending on a partition function ${\cal Z} (Q)$,
defined on pseudomatrices.
These functions on pseudomatrices are usually studied in the
physical literature starting from corresponding functions on $N
\x N$ matrices $M = \{ M_{ij} \}$ (with $M_{ii}=0$) via a precise
albeit not so mathematically well defined procedure involving
several steps. Such steps (to be taken in a precise given order)
include in particular a limit for $N \to \infty$, analytical
continuation of $N$ to a continuous variable -- so to consider
derivatives with respect to $N$ of the partition function $Z_N
(M)$, defined on $N \x N$ matrices -- and finally the limit $N
\to \infty$ (see \ref{1,2} for more detail).
This seemingly self-contradictory extension of the usual concept
of functions of a matrix can be cast into a coherent procedure,
and provides significant results in the study of disordered
systems \ref{2}.
In this note, we propose a mathematically coherent way to define
pseudomatrices $Q$; this also has the advantage that we can use
the ``Symmetric Criticality Principle'' of Palais \ref{3} in the
search for minima of $f (Q)$. Actually, we will be able to
describe generically the location of minima of $f(Q)$, which
will be in a set of symmetric psudomatrices, and of
symmetry-breaking solutions.
The present paper is written in a quite mathematical style; this is quite
unavoidable, as its goal is to provide mathematical foundations (a
posteriori) to the work of physicists active in the field. We hope the
physicists readers will bear with such a style.
\bigskip\bigskip
{\bf 2. Matrices and line bundles.}
\bigskip
A real line bundle (RL bundle) is a fiber bundle with
fiber $F \simeq \R$; a complex line bundle, or line bundle {\it
tout court}, is a bundle with fiber $F \simeq {\bf C}$. In the
following we will consider RL bundles, for which it will be
understood without further mentioning that $F = \R^1$; notice
that if we had to consider complex (pseudo)matrices, our
considerations would have to be cast in terms of complex line
bundles, which would be immediate.
A real $N \x N$ matrix $M = \{ M_{ij} \, ; \, i,j = 1,...,N \}$
can be seen as a section $\mu $ of the RL bundle $(E_N , \pi_N ,
B_N )$ with base space $B_N = \Z_N \x \Z_N$. Points $(i,j) \in
\Z_N \x \Z_N = B_N$ corresponds to sites of the matrix, and the
values $ \mu (i,j)$ of the section $\mu$ at the points $(i,j) \in
B_N$ correspond to the entries of the matrix; that is, $\mu (i,j)
\equiv M_{ij}$. The projection $\pi_N : E_N \to B_N$ assigns to
the entry $M_{ij}$ its indices: $\pi_N (M_{ij} ) = (i,j)$.
In this setting, taking the limit $N \to \infty$ corresponds then
to consider the RL bundle $( E_* , \pi_* , B_* )$ with base space
$B_* = \Z \x \Z$ and projection $\pi_* : E_* \to B_*$ as before.
Notice that these bundles are trivial ones; notice also that $E_N
, E_*$ can be seen both as vector bundles and as principal fiber
bundles, since $\R$ is both a vector space and a Lie group.
We come now to a simple but useful remark: the space $\Z \x
\Z = B_*$ has a natural immersion in the space $\R \x \R := B_0$;
correspondingly, the RL bundle $(E_* , \pi_* , B_*) = B_* \x \R$
is naturally immersed in the RL bundle $(E_0 , \pi_0 , B_0 ) = B_0
\x \R$. Thus, sections $\s : B_0 \to E_0$ of this bundle are the
natural generalization of (infinite) matrices.
It is maybe worth stressing that the immersion of the base space
of the bundle into a continuous space can be performed also
without taking the limit $N \to \infty$. In this case the natural
immersion is $ \Z_N \ss I_N \simeq I$, where $I_N$ is the real
line interval $I_N = (0,N]$, $I = I_1$, and $\simeq $ denotes
isomorphism. In this way, any $E_N$ with finite $N$ is naturally
immersed into the RL bundle $(E_I , \pi_I , B_I )$, where $B_I = I
\x I$.
Up to now, we have not considered any special condition (such as
$M_{ii} = 0$) on matrices, nor we have considered base space
automorphisms, of relevance in the following; considering these
will be our next step.
Before proceeding to this, we would like to call the attention
of the reader to an important fact: the present interpretation of
a matrix as a section of the pertinent bundle requires that the
two spaces (we denote them generically by $S$) entering in the
product to give the base spaces of the bundles we consider here,
$B = S \x S$, should be seen as {\it two copies of the same
space} (mirrors of each other in the Web jargon), as it will be
discussed in a moment.
In the algebraic language \ref{4}, we should consider a ``doubling
functor'' $\de$ which assigns to a space $S$ a pair of copies of
the same space, $\de (S) = (S \x S) \equiv S_\de$; and to
morphisms $\rho : S \to S$ of a space $S$ a pair of copies of
the same morphism, each acting on a copy of the space. Thus, $\de
(\rho ) = (\rho \x \rho ) \equiv \rho_\de$, with $\rho_\de :
S_\de \to S_\de$ acting as $\rho_\de (S \x S) = \rho (S) \x \rho
(S)$.
\bigskip\bigskip
{\bf 3. Base space morphisms.}
\bigskip
As mentioned above, the interpretation of matrices as sections of
a bundle requires that the two spaces whose product is the base of
the bundle should be seen as mirror copies of the same space. This
means in particular that if we consider morphisms $\th : B \to B$
of the base space $B = S \x S$, they are acceptable only if they
are of the form $\th = \rho \x \rho$, with $\rho : S \to S$,
i.e. if $\th = \de (\rho )$ for $\de$ the doubling functor and
$\rho$ some morphism of $S$.
We will denote by $\M (\a )$ the set of morphisms of $\a$; thus
the above remark means that we should not consider the whole $\M
(B)$, but only $\de \[ \M (S) \] \ss \M (B)$.
In the case of $N \x N$ matrices, $S = \Z_N$, this means that if
$\rho : \Z_N \to \Z_N$ is a permutation, we should apply it to
both the row and the column indices (which is indeed a quite
natural requirement).
Let us now consider in more detail $\M (S)$ for the various kind
of $S$ we have considered so far, i.e. $S = \Z_N , \Z , I , \R$;
and let us wonder if all elements of $\M (S)$ are acceptable. In
the case of matrices, it is clear that only permutations are
acceptable, while a noninvertible map $\rho : \Z_N \to \Z_N$
should not be considered as legitimate. For later extension, we
characterize the allowed transformations (permutations) as {\it
invertible} and {\it volume preserving}. We denote the group of
invertible applications on $S$ as $\M_0 (S ) \ss \M (S)$.
Summarizing our discussion so far, an $N \x N$ matrix should be
seen as a section $\mu : B_N \to E_N$ of the fiber bundle $E_N$,
on which operates the group $\de \[ \M_0 (\Z_N ) \] \ss \de (B_N )$
of base space morphisms.
The generalization of this to the bundle $E_*$ is immediate, as
now it suffices to consider as $\M_0 ( \Z )$ the group of
invertible applications $\rho : \Z \to \Z$.
The situation is much less clear when we pass to a continuous
base space (i.e. for $S = I $ or $S = \R$); in this case the group
of invertible applications is still well defined, but very large.
Moreover, invertible applications can be not volume-preserving,
and/or not continuous (and even nowhere continuous, as for the
$\rho : \R \to \R$ defined as $\rho (x) = x$ for $x$ rational,
and $\rho (x) = 1/x$ for $x$ irrational) .
Two natural classes of applications to consider are indeed
(invertible) volume-preserving applications, and (invertible)
continuous ones; or the intersection of the two. We choose -- in
view of physical motivation -- to consider the latter case.
{\bf Definition 1.} Given a space $S$, the group of {\it proper
morphisms} of $S$, denoted by $\M_0 (S)$, is the group of
invertible, volume-preserving, continuous applications $\rho : S
\to S$.
For $S$ a discrete space, any map will be considered as
continuous. The following definitions formalize the
discussion conducted so far.
{\bf Definition 2.} Given a space $S$, the associated (real)
{\it pseudomatrix bundle} $E_S$ is the trivial (real) line bundle
$E_S = \de (S) \x \R$, with projection $\pi_S : E_S \to B_S = \de
(S)$.
{\bf Definition 3.} Given a space $S$, a (real) pseudomatrix over
$S$ is a section $\mu : B_S \to E_S$ of the associated (real)
pseudomatrix bundle.
In the standard matrix case, two matrices which can be
identified by a simultaneous permutation of rows and column
indices can be considered as equivalent; correspondingly,
{\bf Definition 4.} Given a space $S$, two pseudomatrices over
$S$ are equivalent if they are in the same equivalence class
under $\de [ \M_0 (S ) ] $; i.e. two pseudomatrices $\mu_1 ,
\mu_2$ are equivalent if exists a $\th \in \de [ \M_0 (S) ]$
such that, for all $z \in B_S = \de (S)$, they satisfy $\mu_1 (z)
= \mu_2 (\th z)$.
\bigskip\bigskip
{\bf 4. Boundary conditions on pseudomatrices.}
\bigskip
As mentioned in the introduction, in replica symmetry breaking
(RSB) one considers indeed (the generalization of) matrices with
diagonal element equal to zero, $M_{ii} = 0$. In the
present language, this means that we should impose the
{\bf Condition A.} $\mu (x,x) = 0$ for any $x \in S$.
It should be stressed that the set of pseudomatrices satisfying
condition A is invariant under $\de [ \M_0 (S )]$, and actually
under the whole $\de [ \M (S )]$. Thus, condition A is
compatible with the natural equivalence relation of definition 4.
In RSB, one does also sometimes consider ``periodic matrices''; this means
matrices with indices ``modulo $N$'' (i.e.
$j + N \approx j$). In this case, the continuous base space
generalization should be $S = S^1$ rather than $S = I$; the
corresponding pseudomatrix bundle will be denoted as $E_c = B_c
\x \R$ (notice that now a nontrivial structure of the bundle
would be in general possible), with $B_c = \de (S^1 )$; the
subscript ``c'' stands for ``circle''.
Notice that while $\M_0 (I) = \{ e \}$, in this case we have to
deal with $\M_0 (S^1 ) = \R$, corresponding to rotations of
$\S^1$ through an angle $\th \in \R$; when we take into account
equivalence of rotations through $\th$ and through $\th + 2
\pi$, we have $\M_0 (S^1 ) = S^1 = \R/\Z$.
\bigskip\bigskip
\vfill\eject
{\bf 5. Critical points of functions over pseudomatrices and
minimal RSB.}
\bigskip
Having obtained a proper mathematical setting for
pseudomatrices, we can pass to consider function on
pseudomatrices, and in particular the type of minimization
problems encountered in RSB.
Given a space $S$, we will denote by $\Q (S)$ the set of
pseudomatrices over $S$, i.e. of sections of the RL bundle over
$\de (S)$. With this notation, we have to consider critical
points of smooth functions $f : \Q (S) \to \R$.
In particular, we would like to consider functions which are
compatible with the natural equivalence relation considered in
definition 4, i.e. which take equal values on equivalent
pseudomatrices; hence $f$ should be invariant under $G = \de [
\M_0 (S)]$. We will refer to such an $f$ as the invariant
energy functional, and to $G$ as the symmetry (invariance) group.
In view of the physical problem one has to consider, it is
natural to introduce a {\it finite energy condition}, i.e. to
consider only the sections which lead to a finite value of the
energy functional $f$; once we define a scalar product in $\Q$,
this finite energy condition leads naturally to consider a
Sobolev space of sections in $\Q$. From now on, we assume this
has been selected and we have restricted to it, and just denote it
by $\Q$.
Notice that it is easy to introduce a (fibered) scalar product in
$\Q$, defined e.g. as $$ \langle \mu , \nu \rangle \ = \ \cases{
\sum_{x \in B} \( \mu (x) \, \nu (x) \) & for $B$
discrete \cr ~ & ~ \cr \int_B \, \mu (x) \,
\nu (x) \, \d x & for $B$ continuous. \cr} $$
A Sobolev product would be defined in the same way, involving also
derivatives (or finite differences) of sections; we will not dwell any
further on these technical points.
\bigskip
If we make abstraction from the RSB setting, we have a pretty
classical problem: namely, look for extremal points of a
functional $f$ defined on a space $\Q$ of sections of a fiber
bundle $E$, and invariant under a group $G$ which acts on $\Q$.
The work of previous sections providing a sound mathematical
setting for the ``objects $Q$''permits therefore to apply the
known results available in the literature for this classical
problem, even to the problem of RSB. We will now briefly mention
which ones of these classical results are specially useful to
deal with RSB, referring to literature for detail.
The key result in this field is Michel's theorem \ref{5} (see
also \ref{6}), dealing with a potential $V$ over a smooth finite
dimensional manifold $M$ and invariant under the action of a
compact Lie group $G$ (as for the $SU(3)$ theory of strong
interactions, which provided motivation for Michel's work). The
proper statement of Michel's theorem would require to introduce
the notion of {\it (isotropy) stratification of a $G$-manifold};
here we will just give a rough idea of the final result,
referring to \ref{6,7,8} for detail.
If $V : M \to \R$ is $G$-invariant, it can be thought as a
function $\Phi$ on the orbit space $\Om = M / G$. A point $\om_0
\in \Om$ -- i.e. a $G$-orbit in $M$ -- has an isotropy type $[G_0 ]$
corresponding to the conjugacy class of isotropy subgroups $G_x
\sse G$ for points $x$ on the $G$-orbit; orbits having the same
isotropy type are said to form a {\it stratum}.
Michel's theorem asserts that if $\om_0$ is isolated in its
stratum, then $\om_0$ is critical for any $\Phi$, i.e. $x \in
\om_0 \ss M$ is a critical point for {\bf any } $G$-invariant
potential $V$.
The simplest example is given by $G = \Z_2$ acting on $M = \R$
by $g(x) = - x$; here there is a stratum made of only one
orbit $\om_0$ (the origin), which is indeed a critical point for
any even potential. A less trivial example is provided by $G =
SU(3)$ acting via the adjoint action on its Lie algebra ${\cal G}
\simeq \R^8$, or more precisely on the unit sphere in this,
${\cal G}_0 \simeq S^7$. Here $M/G \simeq \R^2$, ${\cal G}_0 / G
= S^1$, and critical orbits correspond to physical particles in
the $SU(3)$ octet \ref{9}.
A simple result which, although obtained indipendently, can be
seen as a corollary of Michel's theorem \ref{6,10} and which has
quite far-reaching applications is the ``Equivariant Branching
Lemma'' (EBL) of Cicogna and Vanderbauwhede \ref{11} (extended by
Golubitsky and Stewart \ref{12} to Hopf bifurcations), which
together with the ``Reduction Lemma'' of Golubitsky and Stewart
\ref{12} is a cornerstone of equivariant bifurcation theory
\ref{13,8}, i.e. in the mathematical theory of phase transitions.
Roughly speaking, this ensures that if a subgroup $H \sse G$ has
a one-dimensional fixed space, then when a maximally symmetric
solution (invariant under the whole $G$) looses stability, the
existence of a stable symmetry breaking solution $x_0$ with
symmetry $H \sse G$ is generically guaranteed.
Michel's theorem can be extended to the present setting, i.e.
$G$-invariant functionals on a (Sobolev) space of sections of a
fiber bundle \ref{14}; this extension does essentially amount to
rephrase results by Palais, known as the ``Symmetric Criticality
Principle'' (SCP) \ref{3}. The extension carries over to the
bifurcation setting and to the EBL \ref{15}, i.e. to the symmetry
breaking case, and can therefore be applied to our problem of RSB.
Referring again to the literature for detail \ref{3}, let us
see what is the content of the SCP in the present notation. Given
the functional $f : \Q \to \R$, invariant under $G$, we can
consider the restriction $f_H$ of $f$ to the space $\Q_H$ of
$H$-invariant sections, where $H \sse G$. Suppose now that $\mu$
is a critical point for $f$: then, the SCP guarantees that $\mu$
is also a critical point for the unrestricted functional $f$.
We stress that this follows from the fact we managed to set our
problem so to satisfy the conditions of Palais' theorem (in
particular we have a Sobolev space of sections of a bundle, and
the $G$-action is linearizable around any symmetric point
\ref{3,14,15}), but the SCP is {\it not } valid, contrary to what
one could think at first, in more general situations; partial extensions
are however possible \ref{3,15}.
Making use of this, it is easy to see that an extension of the
EBL \ref{15} ensures that if $H \ss G$ is a maximal isotropy
subgroup of $G$ (this very notion can be quite delicate for
infinite groups, and in more general cases one has either to rely
heavily on geometry, or limit to a dense subset of the space of
sections, see \ref{16} for details) and the fully symmetric
solution looses stability, then generically there is a stable
solution, i.e. a minimum of $f$, given by some $\mu \in \Q_H$.
\bigskip\bigskip
{\bf 6. Application to RSB.}
\bigskip
Let us now apply the above mentioned results to the problem of
RSB. In the case $S = \R$ or $S = S^1$, the group $G = \M_0 (S)$
is just $G = \R$, acting as translations. The $G$-invariant
functionals $f: \Q \to \R$ are therefore invariant under
translations; $G$ acts explicitely as
$ g_a [\mu (x,y) ] = \mu (g_a x , g_a y ) \equiv \mu (x
+ a , y+a ) $.
It is clear from this that any $\mu \in \Q_G$ can be written as
$\mu (x,y) = \~\mu (x-y)$; the SCP guarantees that we can
consider $f_G$, the restriction of $f$ to $G$-invariant sections.
Considering pseudomatrices of this form is indeed what
is usually done in the study of RSB \ref{1,2}. We have therefore
provided a sound mathematical justification for the current
method of analysis for RSB. We summarize this as the
{\bf Proposition 1. }
{\it The pseudomatrix which
minimizes $f_G$ is also a critical point for $f$.}
As already mentioned, this is nothing else than a corollary of
the SCP of Palais; it should be stressed that the SCP ensures
only that $\mu_0$ is a critical point for $f$, but not
necessarily a minimum. The extension of the EBL mentioned above
\ref{15} gives the following corollary when $f$ depends on a real
parameter $\la$ belonging to an interval $\Lambda \sse \R$:
{\bf Proposition 2. }
{\it Suppose $f = f^\la : \Q \to \R$ depends smoothly on a real
parameter $\la$, and let $f_G = f^\la_G$ be
the restriction of $f^\la$ to $\Q_G \ss \Q$.
Let $\mu_0 \equiv \mu_0^\la \in \Q_G$ be the minimum of $f_G^\la$,
and suppose $\mu_G^\la$ is a minimum of $f^\la$ for $\la <
\la_0$, and looses stability at $\la = \la_0$.
Then there is an $\varepsilon_* > 0$ such that for each maximal
isotropy subgroup $H \sse G$ there is a stable critical point
$\mu_{[H]}^\la \in \Q_H$, for $\la \in (\la_0 , \la_0 +
\varepsilon )$.}
Notice that if $\mu_0 = 0$, this shows that when $\mu_0$ becomes
unstable we have a bifurcation of a branch of fully symmetric
(i.e. $G$-invariant) minima.
Notice also that these results are formulated for general $G$;
indeed, they would apply for any $G$-action on $\Q$ satisfying
the conditions required for the SCP to hold.
In our concrete case, we have $G = \R$ acting as simultaneous
translations in $x$ and $y$; and thus the maximal isotropy
subgroups of $G$ for this action on $\Q$ are isomorphic to $\Z$.
The sections over $\de (\R )$ invariant under $\Z$, i.e. such that
$\mu (x,y) = \mu (x + \a , y + \a)$ with $\a$ a fixed real
number, corresponds to ``periodic pseudomatrices'' (see sect.4),
and can be thought as sections over $\de (\R / \Z ) = \de
(S^1 )$, so that to a periodic $\mu \in \Q$ we can associate a
section $\~\mu_1 (\varphi , \vartheta )$ of a RL bundle over the
torus ${\bf T}^2$.
This means that if $\mu_0 (x,y) = \~\mu_0 (x-y)$ looses
stability, the symmetry breaking solution will be
generically given by a periodic pseudomatrix $\mu_1 (x,y)$, i.e.
a pseudomatrix for which $\mu_1 (x,y) = \mu_1 (x + \a , y +
\a)$, as discussed above.
\bigskip\bigskip
\vfill\eject
{\bf 7. Extensions, and conclusions.}
\bigskip
We would now like to briefly present some remarks concerning
possible extensions of the present work, and then summarize this
short note.
\bigskip
First of all, it should be mentioned that fiber bundles
corresponding to the product of a continuous and a discrete space
are also of use in Connes' {\it Non-Commutative Geometry}
\ref{17}, and in the approach to Field Theory of fundamental
interactions based on this \ref{18}; although in that case the
role of teh two spaces is interchanged with respect to the
present setting (i.e. one has a continuous base and a discrete
fiber), Connes' work shows that we can have a highly nontrivial
connection on a discrete space. An approach based on
Non-Commutative Geometry has also been used to address problems
arising in Condensed Matter Physics, where the discrete structure
is by all means essential for the Physics of the system\ref{19}.
In this respect, we mention that bundles with discrete base
space -- as those we are considering -- can be equipped with the
discrete equivalent of a connection, i.e. with a set of maps
$T_{\alpha \beta} : F_\beta \to F_\alpha$, where $F_x$ is the
fiber over $x$, and $\alpha , \beta$ are points in the base
space.
If, as in the present note, we only want to consider trivial
bundles, we can take $T_{\alpha \beta}$ to be the identiy for
any pair of points in the discrete base space $B$; notice that
when we are embedding $B$ in the contractible space $\R$, it
is entirely natural to consider maps such that $T_{xy} = T_{xz}
T_{zy}$ (no sum on $z$) for all $x,y$ and any $z$, and for the
open chain $B = \Z$ these are equivalent to choosing $T_{xy} =
I$.
On the other side, when we consider periodic matrices
and pseudomatrices, so that $B$ is a periodic chain -- and is
finally embedded in $S^1$ -- maps satisfying the transfer
property $T_{xy} = T_{xz} T_{zy}$ can however have a nontrivial
holoedry; in this case it is natural to embed the bundle over
$B$ into a nontrivial RL bundle over $S^1$, so that one is led
to considering nontrivial bundle structures.
In the present note we have implicitely assumed that $T_{ij}$ is
the identity for all $(i,j)$ -- i.e. just consider trivial RL
bundle -- as the Physics we are interested in does not require
more complicate structures. However, it would be interesting
to expand the present approach, especially in the case of complex
pseudomatrices and complex line bundles, to consider nontrivial
connections; in physical terms this would correspond to
introducing a ``geometric frustration'' \ref{20} in the system.
\bigskip
As already stressed in section 6, the theorems given there
apply not only for the case at hand -- i.e. $G = \R$ acting as
translations -- but also for any $G$-action on $\Q$ satisfying
the conditions required for the SCP to hold (discussed in \ref{3}
and e.g. in \ref{15}).
It should be stressed that in this general case the theorem does
not affirm that other bifurcations -- i.e. direct breaking to
smaller isotropy subgroups -- are not possible: indeed, this can
be possible depending on the stratified geometry of $\Q / G$; on
the other side, it affirms that certain solutions will however be
present and generic. In many cases, these will indeed be the
only ones. A discussion of these points would require to
introduce a substantial additional amount of notions from
Michel's theory of symmetry breaking and from Geometry, and is
thus not given here.
\bigskip
Finally, let us briefly summarize our discussion.
In the present note, we have defined (real) pseudomatrices in a
mathematically precise way, i.e. as sections of certain (real)
line bundles. We have then considered the situation encountered
in studying Replica Symmetry Breaking, i.e. the minimization of
a functional $f$ (free energy) which depends on a partition
function ${\cal Z} (Q)$ defined on pseudomatrices; by imposing a
finite energy condition, whose precise form depends on $f$, we
were led to selecting a Sobolev space of pseudomatrices $\Q$.
Such an abstract setting permits to utilize at once Michel's
geometric theory of symmetry breaking and the Symmetric
Criticality Principle of Palais.
A straightforward application of these powerful tools gave our
main results, and confirmed that -- as assumed in the physical
literature -- the minima of $f$ correspond to symmetric
pseudomatrices, i.e. to pseudomatrices $\mu (x,y) $ which depend
only on $(x-y)$, i.e. $\mu (x,y) = \~\mu (x-y)$. We were also
able to predict that generically a phase transition will lead to
new minima corresponding to periodic pseudomatrices i.e. to
pseudomatrices satisfying $\mu (x,y) = \mu (x + \a , y + \a )$
for some real number $\a$.
These results are neither surprising nor new from the point of
view of the Physics of disordered systems, but provide a solid
mathematical foundation to physicists' work.
\bigskip\bigskip
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