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\begin{document}
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\begin{titlepage}
\begin{flushright}
DFF 289/9/97 \\
hep-th/9710071
\end{flushright}
\vspace{1.cm}
\begin{center}
{\Large {\bf Solving the Frustrated Spherical Model  \\
with $q$-Polynomials}}\\
\vspace{1.5cm}
{\bf Andrea Cappelli }\ \ \ {\em and} \ \ \ {\bf Filippo Colomo}\\
\vspace{0.8cm}
{\em I.N.F.N., Sezione di Firenze \\
and Dipartimento di Fisica, Universit\`a di Firenze, \\
Largo E. Fermi 2, 50125 Firenze, Italy}\\
\end{center}
\vspace{2.cm}

\begin{abstract}
We analyse the Spherical Model with frustration induced by an external
gauge field. In infinite dimensions, this  has been recently
mapped onto a problem of $q$-deformed oscillators, whose real parameter $q$
measures the frustration.
We find the analytic solution of this model by suitably
representing  the $q$-oscillator algebra with
$q$-Hermite polynomials.
We also present a related Matrix Model which possesses the same diagrammatic
expansion in the planar approximation. Its interaction potential
is oscillating at infinity with period $\log(q)$, and  may lead to
interesting metastability phenomena beyond the planar approximation.
The  Spherical Model is similarly $q$-periodic, but does not exhibit
such phenomena: actually its low-temperature phase is not glassy and
depends smoothly on $q$.
\end{abstract}
\vfill
\vspace{5mm}
{\hfill September 1997}
\end{titlepage}

\section{Introduction}

In two recent papers  \cite{parisi1,parisi2}, Parisi {\em et al.}
have introduced and analysed the Spherical and $XY$ Spin Models
with frustration, but in the absence of quenched disorder.
Their aim was to test the conjecture that the frustrated deterministic
systems at low temperature behave as some suitably chosen
spin-glass models with quenched disorder \cite{spinglass}.
They considered the frustrated models in the limit of large
dimensionality $D$ of the lattice,
where the saddle-point approximation becomes exact.
In their analysis of these models, they showed that
the high temperature expansion can be nicely rewritten by using the
$q$-oscillators algebra  \cite{parisi1,parisi2}. Here
$q$ measures the frustration per plaquette and
varies continuously  between the fully-frustrated case
($q=-1$, fermionic algebra)
and the ferromagnetic case ($q=1$, bosonic algebra).
Similar $q$-deformed algebraic relations have also appeared in
the Hofstadter problem of quantum particles hopping on a
two-dimensional lattice in a magnetic field  \cite{wiegmann}.
This problem is closely related to the frustrated spin models,
which can be considered as simplified models of hopping in the large
$D$ and classical limits.
This relation provides another motivation for our analysis.
Finally, we note that the frustrated XY Model can also describe
Josephson junctions  arrays in a magnetic field.

In this paper, we solve exactly the frustrated  Spherical Model
in the large $D$ limit, both in the high and low temperature phases.
We use the ``coordinate'' representation of the $q$-oscillators,
which is given by the $q$-Hermite polynomials \cite{maassen,gasper}.
We find that the spectrum of the lattice Laplacian is essentially given by a
Jacobi theta function; thus, it is periodic along the imaginary axis,
with  period $\log(q)$.
This $q$-periodicity does not affect
the low-temperature phase of the Spherical Model, which is rather standard and
non-glassy, the effect of frustration being quantitative only.
On the other hand, we show that the Spherical Model is associated to a
Matrix Model  \cite{brezin,matmod}, which has the same diagrammatic
expansion in the planar approximation.
The potential of this Matrix Model is oscillating at infinity,
where it has infinitely many minima at approximate distance
$\log(q)$ (see Fig.(2)).
Once the corrections to the planar limit are considered,
its states become metastable by tunnelling to these
minima: therefore, we argue that this model could have a band spectrum and
possibly behave as a spin glass.
Similar phenomena could also occur in the frustrated $XY$ Model,
whose analysis is however left to future investigations.

\bigskip

The models we consider are described by the Hamiltonian
\begin{equation}
H_0=-{1\over\sqrt{2D}}\ \sum_{\langle j k \rangle} \phi^\dagger_j U_{jk}\phi_k \
+ \  {\rm h.c.} \ .
\label{ho}
\end{equation}
The complex field $\phi_j\in {\bf C} $ is
defined on the sites (labelled by $j$) of a $D$-dimensional hyper-cubic lattice,
and there are nearest-neighbour interactions $U_{ij}$.
Three different models can be  obtained by constraining the field as
follows:
\begin{eqnarray}
\beta H_G&=&\beta H_0 +\sum_{j=1}^N|\phi_j|^2 \ ,\label{hgauss}\\
\beta H_S&=&\beta H_0 +\mu\left(\sum_{j=1}^N|\phi_j|^2 \  -N\right) \ ,
\label{hsfer}\\
\beta H_{XY}&=&\beta H_0 + \sum_{j=1}^N\ \mu_j\left(|\phi_j|^2 \  -1\right) \ .
\label{hxy}
\end{eqnarray}
The first is the Gaussian Model, which only exists in the
high-tempe\-rature phase, where the mass term  dominates the
kinetic term (\ref{ho}). The second is the Spherical Model
\cite{berlin}, which uses the
Lagrange multiplier $\mu$ to
enforce the condition $\sum |\phi_j|^2=N$,
$N$ being the total number of sites.
This Lagrange multiplier is promoted to a field $\mu_j$ in
the $XY$ Model (\ref{hxy}), whose variables are constrained to  live on
the unit circle $|\phi_j|=1$, $\ \forall j$.

The couplings $U_{jk}$ are complex numbers of modulus one and satisfy the
relation $U_{jk}=U^*_{kj}$; they are the link variables of an Abelian
lattice gauge field, without kinetic term, which produces a static external
magnetic field. This is chosen in such a way as to give
the same magnetic flux $\pm B$ for
any plaquette of the lattice (the product of the four $U$'s
around the plaquette is  $e^{\pm i B}$). Therefore,
the magnetic field should have the same
projection on all the axes of the lattice, modulo the sign.
In order to avoid the choice of a preferred direction in the lattice,
these signs are chosen randomly\footnote{
This is a small amount of randomness, of order
$D(D-1)/2$, to be compared to the $\sim L^D$ randomness present
in  systems with quenched disorder.}  \cite{parisi1}.

The ferromagnetic spin interaction is obtained for $B=0$, i.e.
$ U_{jk} =1$. Non-vanishing values of $B$ induce a frustration
around each plaquette, which is maximal for $B=\pi$,
the fully-frustrated case. Of course all the intermediate, partially
frustrated cases, with $0<B<\pi$, are  also interesting.
As explained in Ref.  \cite{parisi1}, the $B=0$ theory should be treated
with care, because the large $D$ limit is different in this case,
and the normalisation factor $1/\sqrt{2D}$ in (\ref{ho}) should be
replaced by  $1/(2D)$.

In order to study the previous models, the main difficulty consists
in finding the spectrum of the lattice Laplacian in the presence of  the
magnetic field, which is defined as:
\begin{equation}
(\Delta f)_j=\sum_{k=1}^{2D} \  U_{jk} \  f_k \ .
\end{equation}
Similarly to
the Hofstadter problem  \cite{wiegmann}, there is a competition between the
periodicity due to the lattice and those induced by commensurate magnetic
fields of the form $B=2\pi r/s$, with $r,s$ integers. Thus, we could
expect a complex band structure in the spectrum.
However, there are simplifications due to the large $D$ limit.
The authors of Refs.  \cite{parisi1,parisi2} have approached the
problem via the high-temperature expansion.
In the case of the free energy of the Gaussian Model
(eq. (\ref{hgauss})), this is given by a sum over all closed loops,
as follows:
\begin{equation}
\beta F=\sum_{n=0,\ {\rm even}}^\infty \ \ \ {1\over n}\
\left(\frac{\beta}{\sqrt{2D}}\right)^n \  {\cal N}(n)\  \langle W ({\cal C}) \rangle_n
=\sum_{k=0}^\infty\ {\beta^{2k}\over 2k}\ G_k \ .
\label{htexp}
\end{equation}
In this expression, the loops ${\cal C}$ are arranged according to their
length $n=2k$, and their number is ${\cal N}(n)$.
For each loop, the magnetic field yields a weight, which is given by
the Wilson loop $W({\cal C})$, the path-ordered product
of the couplings $U$ along the loop.
The brackets $\langle \ \rangle_n $ represent the average of this weight
over all the ${\cal N}(n)$ circuits of length $n$.
In the second expression of eq. (\ref{htexp}), we introduce the
notation $G_k$ for the product of the multiplicity
and the Wilson loop average.

Each loop encloses a number of plaquettes and receives a weight
proportional to $\exp{(i B A)}$, where $A$ is the sum of plaquettes
with signs depending on the orientations  \cite{selfcit}.
Due to the average over orientations and loops, the quantity
$G_k$ is a polynomial in the variable
\begin{equation}
q= \cos B \ ,
\label{qdef}
\end{equation}
of order ${k(k-1)/2}$, which is given by the maximal area
enclosed by the loop.
Parisi  \cite{parisi1} enumerated these
diagrams in the large $D$ limit.
First, he showed that the same counting is given by
the Feynman diagrams with $2k$ external points, which are joined pairwise
by lines (propagators) intersecting $I$ times, and have assigned the
weight $q^I$.
These diagrams also occur in the topological (large $N$) expansion of
Matrix Models \cite{brezin}, where  the planar limit corresponds to no
intersections, i.e. to the $q=0$ case.
Secondly, Parisi found a recursion
relation for the coefficients of the polynomial $G_k(q)$
-- a sort of Wick theorem -- which
can be nicely  expressed by the algebra of the $q$-oscillators
$a_q, a_q^\dagger$:
\begin{equation}
\aqa_q^\dagger- q a_q^\daggera_q=1 \ .
\label{qalg}
\end{equation}
These operators \cite{kulish} act on the Hilbert space spanned by the vectors:
$|m\rangle$, $m=0,\ 1,\ ... $, as follows,
\begin{eqnarray}
a_q^\dagger |m\rangle  &=& \sqrt{[m+1]_q}\ |m+1\rangle \ ,\nonumber\\
a_q |m\rangle  &=& \sqrt{[m]_q}\ \quad |m-1\rangle \ ,\qquad a_q |0\rangle= 0 \ ,
\end{eqnarray}
where
\begin{equation}
[m]_q=\frac{\  1-q^m}{\  1-q\ } \ .
\end{equation}
Using the recursion relation, the weighted multiplicities of the diagrams
of eq. (\ref{htexp}) were neatly written as an expectation value over
the ground state of
the $q$-oscillators  \cite{parisi1,parisi2}:

\begin{equation}
G_k(q) \ =\ \langle 0|(a_q^\dagger+a_q)^{2k}|0\rangle \ .
\label{qvev}
\end{equation}

\section{Spectrum of the Laplacian and High-Temperature Expansion}

We shall now make  expression (\ref{qvev}) more explicit by introducing
the coordinate representation for the $q$-oscillators.
First note that in the Gaussian Model (\ref{hgauss})
the quantity $G_k$ is nothing else then the trace of $(2k)$-th power of
the frustrated Laplacian, such that we can write the general relation
\begin{equation}
 {\rm Tr} \ \left[ f\left(\Delta\right)\right] \equiv \langle 0 |  f(x_q) |0\rangle\ ,
 \qquad x_q\equiv a_q^\dagger + a_q \ .
 \label{qveve}
\end{equation}

The $x_q$ coordinate representation, $x_q|x\rangle = x|x\rangle$,
is given by the so-called continuous $q$-Hermite polynomials  \cite{gasper,szego}.
These are defined by:
\begin{equation}
H_n(x) =  \langle x | n \rangle\ {\cal C}_n\ ,
\qquad {\cal C}_n = \left(\left[n\right]_q!\right)^{1/2}\ {\cal C}_0\ ,
\end{equation}
where the normalization constant ${\cal C}_0$ is fixed by
$H_0(x) =1 $ and the $q$-factorial is
\begin{equation}
\left[n\right]_q! =[n]_q \ [n-1]_q \dots [1]_q\ , \qquad
[1]_q=[0]_q=1 \ .
\end{equation}
These polynomials satisfy, of course, a three-term recursion relation
in the index $n$:
\begin{equation}
x H_n(x) = H_{n+1}(x) +[n]_q\ H_{n-1}(x)\ , \qquad n\ge 1 \ .
\end{equation}
Moreover, they obey a $q$-difference equation \cite{wiegmann}\footnote{
This $q$-periodicity will be better discussed later.}
in their coordinate $x$, which ranges over the interval
$x\in\left[-2/\sqrt{1-q},2/\sqrt{1-q}\right]$.
A convenient parametrisation is
\begin{equation}
x=\frac{2}{\sqrt{1-q}}\cos\theta \  ,\qquad \theta\in [0,\pi ].
\label{variasm}
\end{equation}
More properties of these $q$-Hermite polynomials can be found
in  Ref.  \cite{gasper}, where they are defined as
${\cal H}_n(\cos\theta)=(1-q)^{n/2} H_n (x)$.
The most important property for us is the orthogonalizing measure
$\nu_q(x)\ \ $  \cite{maassen,gasper,szego}:
\begin{equation}
\int_{-2/\sqrt{1-q}}^{2/\sqrt{1-q}}\ \nu_q(x)\ dx\
H_n(x)\ H_m(x) \ = \  \delta_{n,m}\ [n]_q! \ ,
\end{equation}
\begin{eqnarray}
\nu_q(x) &=&\frac{\sqrt{1-q}}{2\pi}\  q^{-1/8}\
\Theta_1\left({\theta\over\pi},q\right)\  \nonumber\\
&=&\frac{\sqrt{1-q}}{\pi}\ \sum_{n=0}^{\infty}\ (-1)^n q^{n(n+1)/2}
\ \sin[(2n+1)\theta] \nonumber\\
&=&\frac{\sqrt{1-q}}{\pi}\ \sin\theta\
\prod_{n=1}^{\infty}\ \left(1-q^n \right)\
\left(1-q^n \rm e^{2i\theta} \right)\ \left(1-q^n \rm e^{-2i\theta} \right) \ ,
\label{qmeasure}
\end{eqnarray}
where $\Theta_1(z,q)$ is the first Jacobi theta function.

Using these results, we can rewrite the general ground-state
expectation value in eq. (\ref{qveve}) as follows:
\begin{equation}
{\rm Tr} \left[ f\left(\Delta\right)\right] \equiv
\int\ dx\ \nu_q(x)\ f(x) \ .
\end{equation}
Therefore, we have shown that the Laplacian is diagonal in the coordinate
representation of the $q$-oscillators: it has a continuous spectrum
over the interval $[-2/{\sqrt{1-q}}, 2/{\sqrt{1-q}}]$,
with eigenvalue density given by $\nu_q$ in eq. (\ref{qmeasure}).

Let us discuss this spectrum in some interesting limits.
For $q=0$, we find $\nu_0(x)=1/\pi\ \sqrt{1-x^2/4}$, which is
the Wigner semi-circle law for the Gaussian Hermitean Matrix
Model  \cite{brezin}.
This results confirms the previous correspondence between
the high-temperature expansion of the $q=0$ Gaussian Model and the
planar Feynman diagrams without interaction vertices of the Matrix Model.
Actually, this correspondence can be extended to all values of $q$
(see Section 4).
Note also that the $q=0$ frustrated
model is diagrammatically equivalent to the gauge spin glass where
the coupling $U_{jk}$ are random quenched variables  \cite{parisi1}.
Next, for $q=-1$, the measure becomes a
representation for $(\delta(x-1)+\delta(x+1))/2$, and we recover
 the two states of the fermionic algebra \cite{maassen}.
The limit $q\rightarrow 1$  is singular, owing to the
$D\to\infty$ peculiarities said before; nevertheless,
$\nu_1(x)$ becomes the Gaussian distribution, after multiplicative
renormalization  \cite{maassen}.

Let us now check that our expression for the Laplacian reproduces
the high temperature expansion computed in
Refs.  \cite{parisi1,parisi2}.
The free energy of the Gaussian Model (\ref{hgauss}) can be written in
terms of the Laplacian as $\beta F=-\log Z= {\rm Tr} [\ln (1-\beta\Delta)]$.
Therefore, the internal energy $U(\beta)$ is:
\begin{equation}
1-\beta U(\beta)=R(\beta)\ ,\qquad
R(z)\equiv {\rm Tr}\left[\frac{1}{1-z\ \Delta}\right] \ .
\label{resolvent}
\end{equation}
In the last equation, we introduced
the resolvent $R(z)$, which will play a major role in the subsequent
study of the Spherical Model.
$R(z)$ is originally well-defined for real $z$ in the interval
$|z|<z_c=(\sqrt{1-q})/2$ and is then  analytically extended to the
whole complex plane of $z$.
By using the results of the previous section, we find:
\begin{eqnarray}
R(z)&=&\int_{-2/\sqrt{1-q}}^{2/\sqrt{1-q}}
\  dx \ \frac{\nu_q(x)}{1-z\ x} \nonumber\\
&=&{\sqrt{1-q}\over z}\ \sum_{n=0}^{\infty} \ (-1)^n\ q^{n(n+1)/2} \
\left[K\left(\frac{2z}{\sqrt{1-q}}\right)\right]^{2n+1} \ ,
\label{erre}
\end{eqnarray}
where we introduced
\begin{equation}
K(\alpha) = \frac{1 - \sqrt{1-\alpha^2}}{\alpha}\ .
\label{gi}
\end{equation}
The singularities of $R(z)$ in the complex plane
are completely determined by those of the simpler function
$K(2z/\sqrt{1-q})$.
Actually, the series is very well convergent for $|q|<1$ as any Jacobi theta
function. Moreover, for $|q|=1$, it is a geometric series,
which is still convergent because $|K| < 1$ for $|z|<z_c$.

The high-temperature expansion of the internal energy of the Gaussian
Model can be obtained by expanding $R(\beta)$ in series;
there only appear even powers of $\beta$ with $q$-dependent coefficients.
We have computed the series to order $O(\beta^{18})$
with Mathematica  \cite{math}, and obtained the polynomials
$G_k(q)$ , $k=1,\dots,9$ of eq. (\ref{htexp}).
We have verified that they indeed match the results of
Refs.  \cite{parisi1,parisi2}, which were found by direct enumeration of the
graphs on a computer.

For $q=0$, we have $U(\beta)=\left(\beta - K(2\beta) \right)/\beta^2$ and the
high-temperature expansion is singular at $\beta_c=1/2$, which is
the critical
temperature of the Gaussian Model. The specific heat diverges
as $(\beta_c-\beta)^{-1/2}$ at the transition, $i.e.$ the
value of critical exponent $\alpha$ is $1/2$.
This result still holds for generic $-1<q<1$ as a consequence of the
relation between the singularities of $R(z)$ and $K(z)$.

\section{Solution of the Spherical Model}

We are now ready to solve the Spherical Model. Our approach
essentially follows the original Berlin-Kac solution of the ferromagnetic
model in
$D=1,2,3$  \cite{berlin}. Indeed we shall see that the mechanism of the phase
transition is the same, only the form of the Laplacian is different.
The partition function is
\begin{eqnarray}
Z&=&\int_{\mu_0-i\infty}^{\mu_0+i\infty}d\mu\
\int{\cal D}\phi \ \exp\left[ \beta \sum\ \phi^\dagger\Delta\phi
                        -\mu\left(\sum |\phi|^2-N\right)\right] \nonumber\\
&=&\int_{\mu_0-i\infty}^{\mu_0+i\infty} d\mu\ \
\exp\left[-{\rm Tr}[\ln(\mu-\beta\Delta)]+\mu N \right] \ .
\label{partsfer}
\end{eqnarray}
The integration path over $\mu$ is along a straight line which runs
in the complex plane parallel to the imaginary axis; a positive real
part has been added in order to make the integration over $\phi$
convergent.
The trace in the exponent is proportional to $N$ ($\Delta$ is an
$N$-dimensional operator): in the large $N$ limit, we can apply the
saddle point method to evaluate the integral over $\mu$.
The saddle point equation is:
\begin{equation}
{\rm Tr}\left[\frac{1}{\mu-\beta\Delta} \right]=1\ ,
\label{saddl}
\end{equation}
and must be solved for real, positive values of $\mu$. Introducing the
variable $z=\beta/\mu$, this equation can be
written in terms of the resolvent (\ref{resolvent}), as follows:
\begin{equation}
\frac{\beta}{z} = R(z) \ ,\qquad z\equiv \frac{\beta}{\mu} \ .
\label{saddle}
\end{equation}

In the case of $q=0$, this equation can be easily solved:
\begin{equation}
\frac{1}{2z^2}(1-\sqrt{1-4z^2})=\frac{\beta}{z} \qquad \longrightarrow \qquad
z(\beta)=\frac{\beta}{1+\beta^2} \ ,
\label{spq0}
\end{equation}
which corresponds to $\mu=1+\beta^2$.
This solution is valid for $z<z_c=1/2$, namely
it describes the high-temperature phase $\beta<\beta_c=1$.
Indeed, let us follow the saddle point in the $z$ plane,
starting from $\beta=0$, where $z=0$ too. Upon increasing
$\beta$, $z$ moves toward the critical value $z_c=1/2$;
for $\beta >1$, $z$ cannot increase anymore because it finds
the square-root branch cut of $R(z)$,
and therefore ``it sticks to the singularity''  \cite{berlin}.
Hence, for $\beta>1$, the saddle point equation has no longer an
acceptable solution; nevertheless, the leading contribution to the
$\mu$ integral in (\ref{partsfer}) still comes from the neighbourhood of
$z=1/2$.
Therefore, the low-temperature solution is $z(\beta)=1/2$ independent of $\beta$,
$i.e.$ $\mu=2\beta$.
The free and internal energies are therefore given by
\begin{eqnarray}
\beta F&=&{\rm Tr}[\ln(\mu-\beta\Delta)]-\mu \ ,\\
U(\beta)&=&(\partial_{\beta}+(\partial_{\beta}\mu)\partial_{\mu})(\beta F)
\nonumber\\
&=&\frac{1}{\beta}\left(1-{\rm Tr}\left[\frac{1}{1-z\Delta}\right]\right)+
\left(\frac{1}{\mu}{\rm Tr}\left[\frac{1}{1-z\Delta}\right]-1\right)
{\partial\mu \over\partial\beta}\ .
\end{eqnarray}
The last term in this equation vanishes in  the high-temperature phase,
because the saddle point condition is satisfied; however, it should
be included for the low temperature regime.
As a matter of fact, the result can be written in both phases as follows:
\begin{equation}
U(\beta)=\frac{1}{\beta}-\frac{1}{z(\beta)} \ .
\label{usfer}
\end{equation}
More explicitly,
\begin{equation}
U(T)=\left\{
\begin{array}{lll}
-\frac{1}{T}  & & T>1 \\
T-2           & & T<1
\end{array} \right.
\qquad \left( q=0 \right ) \ .
\end{equation}
The internal energy is therefore continuous at the transition,
together with its first derivative. But the specific heat presents  a
discontinuity in its first derivative. These results are qualitatively
very similar to those of the Berlin-Kac analysis  \cite{berlin};
ne\-ver\-the\-less, their model is quite different, in the sense that
frustration is there completely absent.

The solution for any $-1< q < 1 $ is a straightforward generalization
of the $q=0$ case. The discussion of the saddle-point equation (\ref{saddle})
is analogous, because the singularities of $R(z)$
are still given by a square-root branch cut. Although we cannot find
an explicit expression for $z(\beta)$, we can follow its behaviour:
starting from low $\beta$, where $z\sim \beta$, the
saddle point moves towards higher values of $z$ until it hits
the cut of $R(z)$ at $z=z_c=\left(\sqrt{1-q}\right)/2$.
In the low-temperature phase, the value of $z$ sticks to $z_c$
and thus $\mu=\beta/z_c$ for $\beta>\beta_c$.
The critical temperature is given by
\begin{equation}
\beta_c=z_c R(z_c)=\sqrt{1-q}\ \sum_{n=0}^{\infty} (-1)^n q^{n(n+1)/2} \ .
\end{equation}

The internal energy is still given by the general formula (\ref{usfer}),
which now reads
\begin{equation}
U(T)=\left\{
\begin{array}{lll}
T - \frac{2}{\sqrt{1-q}} & & T<T_c \\
- \frac{1}{T} -q\frac{1}{T^3} -\left(q^3+3q^2\right)\frac{1}{T^5}
+ O\left( \frac{1}{T^7} \right)          & & T \gg T_c
\end{array} \right.
\qquad \left( -1< q<1 \right ) \ .
\end{equation}
\begin{figure}
\epsfxsize=15cm \epsfbox{q-o.fig1.eps}
\caption{Internal energy of the Spherical Model  versus
temperature for the three values: $(a)$ $q=-0.707$; $(b)$ $q= 0$;
$(c)$ $q=0.707$.}
\label{fig1}
\end{figure}
A plot of the internal energy is shown in Fig.({\ref{fig1})
for the three values $q=(-1/\sqrt{2}, \ 0, \ 1/\sqrt{2})$.
This can be easily obtained without solving the saddle-point equation,
by using $z\in [0,z_c]$ as a parameter
for both $U(\beta)$  in eq. (\ref{usfer}) and $\beta$ in eq. (\ref{saddle}).
Note that for $q\neq 0$, the first derivative of the internal energy is not
continuous at the transition. The critical temperature goes to
infinity for $q\to 1$ and to zero for $q\to -1$, respectively.
In these limiting cases, we can sum the series in $R(z)$ (\ref{erre})
and obtain the explicit form of $z(\beta)$ and the internal energy.
For $q=1$, we find the meaningless results $z=\beta$ and $\ U(T)=0$
in the high-temperature phase, which collapses to a point ($T_c=\infty$):
this case has a pathologic $D\to\infty$ limit,
as discussed in Ref.  \cite{parisi1}, and a sensible theory should include
$1/D$ corrections.

The limit $q=-1$ is well defined and
corresponds to the fully frustrated model at $D=\infty$\footnote{Note
that the limits $D\rightarrow\infty$ and $q\rightarrow -1$ do not commute,
as discussed in Ref.  \cite{parisi1}.}.
Here, the high-temperature phase extends down to $T=T_c=0$, i.e. there is
no transition. Actually, we find that the resummed $R(z)$ no longer
has the square-root branch cut of $K(z)$ and that the mapping
$\beta=\beta(z)$ is invertible on the whole positive $\beta$ axis:
\begin{eqnarray}
\beta &=&  \frac{z}{1-z^2} \ , \qquad\qquad \qquad \qquad
\left( q=-1\right) \nonumber\\
U &=& -z =  \frac{1}{2\beta}\left(1- \sqrt{1+4\beta^2} \right) \ .
\end{eqnarray}
Let us remark that the absence of a phase transition
has also been observed in the fully-frustrated
long-range $XY$ model  \cite{fermi}.

In conclusion, the qualitative behaviour of the (infinite-dimensional)
frustrated Spherical Model in the low-temperature phase is rather smooth and
standard: indeed, this model does not present the potential features of:
(i) glassy behaviour (many ground states), which
is usually found in  systems with quenched disorder;
(ii) commensurability in the spectrum for $B=2\pi r/s$,
which was observed in the $D=2$ hopping model in a magnetic field.
Regarding point (i), we would like to remark  that the corresponding
model with quenched disorder, $i.e.$ the Spherical Gauge Glass,
does not exhibit a glassy phase either. Actually, this model corresponds
diagrammatically to the $q=0$ frustrated model. In intuitive terms, the
spin variables are only loosely constrained by the spherical condition,
$\sum_{i} |\phi_i|^2 = N$, and thus can globally adapt themselves to any
complex coupling configuration. As far as point (ii) is concerned, the
commensurability effects are probably washed out  by the $D\rightarrow\infty$ limit,
which oversimplifies the geometry of hopping on the lattice.

On the other hand, we expect a glassy phase in the frustrated XY Model
\cite{parisi2}.
In this model, the Lagrange multiplier $\mu$ becomes a local
field $\mu_j$ and the saddle-point equation (\ref{saddl}) is functional.
Nothing changes in the high-temperature phase,
where the solution $\mu_j=\mu=$ const. is correct.
However, the solution(s) in the low-temperature phase can be rather different
from the one of the Spherical Model, and will not be discussed here.
Rather, we shall approach this problem from a different
perspective, by establishing a relation with the well developed
subject of Matrix Models.

\section{Analogy with the Matrix Models}

Let us first recall the solution of the Hermitean Matrix models
in the the large $N$ approximation, which corresponds to the
planar Feynman diagrams  \cite{brezin}.
These models are characterized by the $(N\times N)$  matrix variable
$M=M^\dagger$ and by the Hamiltonian
$\beta H={\rm Tr} V(M)= {\rm Tr} \left( M^2 /2 + \cdots \right)$.
After diagonalization of the matrix, one is left with the partition
function over the eigenvalues $\lambda_i$, $i=1,\dots , N$ :
\begin{equation}
Z_M =\int \prod_{i=1}^N \ d\lambda_i\ {\rm e}^{-\sum_i V\left(\lambda_i \right)}\
\prod_{i< j} \left(\lambda_i -\lambda_j \right)^2\ .
\end{equation}
This can be thought of as being the statistical mechanics of $N$
charges with coordinates $\lambda_i$ in one dimension,
which repel each other logarithmically and are kept together
by the external potential $V(\lambda)$.
This one-dimensional gas of charges usually has a unique phase.
In the large $N$ limit,
the partition sum is dominated by the contribution of the saddle point,
which corresponds to the equilibrium configuration of the charges,
neglecting fluctuations.
Moreover, the charges become a continuum with density $\nu(\lambda)$,
which is normalised to one by a convenient rescaling  \cite{brezin}.
The saddle point equation is:
\begin{equation}
\frac{1}{2}\ V^\prime (\lambda) =
P \int _{-a}^{a} dx\ {\nu(x) \over \lambda -x} \ ,\qquad
\lambda \in (-a,a) \ , \qquad \int _{-a}^{a} dx\ \nu(x) = 1 \ ,
\label{saddlemat}
\end{equation}
where $P$ stands for the principal value of the integral.
This is an equation for the unknown $\nu(x)$, as a function of
the given potential $V(\lambda)$.
Following Ref.  \cite{brezin}, we introduce the function
\begin{equation}
F(\lambda) =\int_{-a}^a \ dx\ {\nu(x) \over \lambda -x} \ ,
\label{effe}
\end{equation}
with $\lambda$ taking values in the complex plane. This
function is analytic outside
the segment of the real axis $(-a,a)$ corresponding to
the spectrum; moreover, it goes to zero
at infinity as $1/\lambda$, due to the normalisation of $\nu$.
For $\lambda$ inside the spectrum, we have
\begin{equation}
{\rm Re}\  F(\lambda) =\frac{1}{2} V^\prime (\lambda)\ ,\qquad
{\rm Im}\  F(\lambda) = -\pi \nu(\lambda) \ ,\qquad \lambda\in (-a,a) \ .
\label{reimeffe}
\end{equation}
The saddle point equation is thus equivalent to these relations for
the function $F(\lambda)$: they can be usually solved by analyticity arguments,
and determine the density $\nu(\lambda)$.

Here we would like to remark that
these formulas are rather similar to those encountered
in the Spherical model. Actually, we can identify the two saddle-point
equations (\ref{saddle}) and (\ref{saddlemat}) as follows:
\begin{equation}
F(\lambda)\equiv \frac{1}{\lambda}\  R\left( \frac{1}{\lambda} \right) \ ,
\qquad z\equiv \frac{1}{\lambda} \ .
\label{relation}
\end{equation}
More precisely, the saddle point equation for the Spherical Model is
discussed for $\lambda$ outside the spectrum, $\nu(\lambda)$ is given and
$\beta=\beta(z)$ is the unknown.
On the other hand, in the Matrix Model $\lambda$ is inside the
spectrum, $V(\lambda)$ is given and $\nu(\lambda)$ is the unknown.

\begin{figure}
\epsfxsize=15cm \epsfbox{q-o.fig2.eps}
\caption{The first derivative of the Matrix Model potential
in eq. (39) as a function of $\chi$, for $q=-0.707$.}
\label{fig2}
\end{figure}

We can now define the Matrix Model corresponding to
the frustrated Spherical Model, as the one which possesses
the same eigenvalue density $\nu_q(\lambda)$, eq.  (\ref{qmeasure}),
in the planar limit.
By using  eqs. (\ref{effe}) and (\ref{reimeffe}), we can determine
its potential:
\begin{equation}
V^\prime \left( \lambda \right) = 2 {\rm Re}\  F(\lambda) =
2 \sqrt{1-q} \ \sum_{n=0}^\infty (-1)^n q^{n(n+1)/2}
\cosh \left[(2n+1)\chi\right] \ ,
\label{vprime}\end{equation}
where we introduced the convenient parametrisation
\begin{equation}
\lambda = \frac{2}{\sqrt{1-q}} \cosh\chi \ ,
\label{variamat}
\end{equation}
for $\lambda$ outside the cut $(-2/\sqrt{1-q},2/\sqrt{1-q})$.
The Gaussian Matrix Model with $V(\lambda)=\lambda^2/2$ is indeed
recovered for $q=0$.

The relation (\ref{relation}) between the Spherical and Matrix Models can also
be understood at the level of diagrammatic expansions.
In the former model, $R(z)$ generates the high-temperature expansion,
whose diagrams were cast into the form of $n$ lines joining $2n$ points
with weight $q$ for each intersections  \cite{parisi1}.
In the latter model, $F(1/z)/z$ is the generating function for the observables
$\langle {\rm Tr} M^{2n} \rangle= \langle \sum_i\lambda_i^{2n} \rangle$  \cite{brezin}.
Their diagrams also have $2n$ external points: in the Gaussian Model ($q=0)$,
they are joined by propagator lines with no intersections, due to
the planar limit. For $q\neq 0$, the Spherical Model diagrams
have non-planar intersections, which are reproduced by interaction
vertices in the planar Matrix Model diagrams; actually, the potential
$V(\lambda)$ contains interactions of any order.

The Spherical Model in the high-temperature phase and the Matrix Model in
its planar limit have corresponding saddle point equations but
rather different free energies, so they are not quite the same
physical problem.
The analogy with Matrix Models is, nevertheless, interesting because it
could provide some useful technology  \cite{matmod} for solving the
$XY$ Model.
Furthermore, the Matrix Model itself could exhibit
interesting physics beyond the planar approximation:
in the following, we shall put forward some educated guesses
which are based on the well-known physical picture of the gas of charges.

The properties of the Matrix Model potential $V(\lambda)$ can be found by
analysing equation (\ref{vprime}). A plot of $V^\prime(\chi)$ is shown
in Fig. (\ref{fig2}), for $q=-1/\sqrt{2}$:
away from the origin, an oscillating behaviour sets in
with period\footnote{
The period is $|\log q|/2$ for $0<q<1$.}
$\chi \to\chi -(\log q)$, and amplitude growing to infinity.
This {\it q-periodicity} of the potential can be found by inspection
of (\ref{vprime}), and reads:
\begin{equation}
V^\prime\left( q^{-1/2}\rm e^\chi\right) =
2 \left(\frac{1-q}{q}\right)^{1/2} \rm e^{\chi} -
q^{-1/2}\rm e^{2\chi}\ V^\prime\left(\rm e^\chi\right) \ .
\end{equation}
This is not an exact periodicity, due to the presence of
the additive term. However, this term becomes negligible for $\chi$ far away
from the origin, i.e. $\lambda\gg 1$, because $V^\prime$ grows
more than exponentially (the amplitude of fluctuations is of order
$O\left(\exp(2\chi^2/|\log q|)\right)$).
The $q$-periodicity of the potential corresponds to a true
periodicity of the eigenvalue density
$\nu_q(\theta)\propto\Theta_1(\theta/\pi,q)$,
because these two quantities are the real and imaginary parts of
the function $F$ in (\ref{effe}), respectively (their variables
$(x,\theta)$ in (\ref{variasm}) and $(\lambda, \chi)$ in (\ref{variamat})
are also related by analytic continuation).
Actually, the eigenvalue density is periodic in the direction of
the imaginary $\theta$-axis:
$\nu_q(\theta - (i/2)\log q)\sim\nu_q(\theta)$, for $0<q<1$,
and $\nu_q(\theta +\pi/2 -(i/2)\log q)\sim\nu_q(\theta)$, for $-1<q<0$.

This periodicity can be interpreted as the
potentiality for metastable states, which, however, are not realised in
the Spherical Model, owing to its simplified dynamics.
Indeed, its saddle point equation involves the function $F(\lambda)\ $,
which is also $q$-periodic, as  its real part $V^\prime$;
however, $F(\lambda)$ never grows sufficiently high to develop the
oscillating behaviour
and, in fact, goes monotonically to zero at infinity.

On the other hand, the states might become metastable in the Matrix Model
beyond  the planar approximation. In this approximation, the charges form
an equilibrium configuration determined by the minimum of
$V(\lambda)$ at the origin, and the tunnelling of charges to lower nearby
minima is suppressed \cite{matmod}:
thus, the $q$-periodicity of the potential is
not felt. Beyond this approximation, tunnelling switches on and the states in
the spectrum can become metastable.
Clearly, a detailed analysis is necessary to understand the effect
of tunnelling into a $q$-periodic  set of local minima separated
by ever-rising barriers.
This might lead to a complex pattern of metastability,
which is a characteristic of the glass phase.

In conclusion, there is the possibility that this Matrix Model might describe
some of the expected effects
of frustrated magnetic systems in finite dimension
$D$  \cite{spinglass,wiegmann}.
Moreover, it should be solvable by known techniques  \cite{matmod}.
The frustrated $XY$ model is another system which might develop
these effects in the low-temperature phase; it would be interesting to
pursue the relation between the $XY$ model and the Matrix Model beyond
the planar limit.

\newpage

{\large\bf Acknowledgements}

We would like to thank Philippe Di Francesco, Enzo Marinari,
Giorgio Parisi and Paul Wiegmann for useful discussions.
This work is supported in part by the European Community Program
``Training and Mobility of Researchers'' FMRX-CT96-0012.


text:
\section{Introduction}

In this note we shall study a class of integral manifolds for a
generalization of the classical contact distribution.
This distribution arises naturally in algebraic geometry as
the infinitesimal restriction satisfied by the period maps
of algebraic surfaces.  To be more precise, consider the
group $G = SO(2p,q)$, the compact subgroup $U(p)\times SO(q)$,
and the homogeneous space
\[
    D = G / V.
\]
The homogeneous space parametrizes Hodge decompositions
\[
  H_{{\bf C}} = H^{2,0} \oplus H^{1,1} \oplus H^{0,2}
\]
on a fixed complex vector space endowed
with a nondegenerate, symmetric bilinear form
defined over an underlying real vector space
$H_{\bf R}$ whose complexification is $H_{\bf C}$.  The models
for these spaces are the primitive second cohomology
of an algebraic surface with real or complex coordinates.
If one has a family of algebraic surfaces $S_t$ parametrized
by $t$, then one has a family of Hodge decompositions
which are determined by the subspaces $H^{2,0}(S_t)$.
As Griffiths showed in \cite{Griff}, these spaces satisfy
\[
  { d \over dt } H^{2,0}(S_t)
    \subset H^{2,0}(S_t) \oplus H^{1,1}(S_t).
\]
This condition, which asserts that a famly of subspaces
$H^{2,0}_t$ cannot vary arbitrarily, but rather like a Frenet
frame of a space curve, defines the
\emph{Griffiths distribution} on the homogeneous space $D$.
Thus families of algebraic surfaces define integral
manifolds of this distribution.

In \cite{CTAMS} it was shown that for $p > 1$ even and $q > 1$,
integral manifolds which are of maximal dimension are
quite special: as germs they are congruent under
the action of $G$; moreover, they correspond to the germ
of an imbedding $U(p,q/2) \longrightarrow SO(2p,q)$.  Thus
germs of integral manifolds of this kind are rigid and depend on
finitely many parameters.

Here we shall study certain integral manifolds which are maximal
with respect to inclusion: they do not lie in a larger dimensional
integral manifold.  There is natural class of these which we call
``generic,'' characterized by the existence of
a family of vectors $v(t)$ in $H^{2,0}_t$
such that the  partial derivatives
$\partial v /\partial t_i$
span $H^{1,1}_t$.
(See definition \ref{def:generic}). These
integral manifolds behave quite differently: they are not rigid, and
they depend on infinitely
many parameters.  Moreover, as we shall see in Theorem \ref{maintheorem}
and the  paragraph which follows it, generic integral
manifolds are given quite explicitly by a system of generating functions.
This is in strict analogy to the case of the contact distribution,
where maximal integral manifolds are flexible and determined
explicity by a generating function.  The function(s) constitute
the infinite-dimensional parameters.

\section{Generalized Contact Distributions}

Recall that the contact distribution is the annihilator $E$ in the
tangent bundle of
${\bf R}^{2n+1}$ or ${\bf C}^{2n+1}$ of the one-form
$$
    \omega = dz - y\cdot dx = dz - \sum_{i=1}^n y_idx_i ,
    \label{contactform}
$$
and that integral manifolds are by definition submanifolds
tangent to $E$. They are of dimension at
most $n$ and those of maximum dimension --- the Legendre manifolds
(see \cite{Arnold}) --- are given, up to certain admissible
changes of coordinates, by a generating function
$f$ through the prescription
\begin{eqnarray}
  & z = f(x) \\
  & y = \nabla f .
\end{eqnarray}
Thus maximal integral manifolds are graphs
of one-jets of functions, and so constitute an infinite-dimensional
family.

After a
suitable change of variables, any distribution of codimension
one can be reduced locally  to a product of the trivial
distribution and a contact distribution.  Consequently the local
nature of their maximal integral manifolds
is completely understood.  By ``maximal'' we mean ``maximal with
respect to inclusion.''  The situation for distributions of
codimension greater than one is completely different.  There is no
general theory which answers the basic questions and the nature of
the integral manifolds is in general very complicated.  There is
always a maximum dimension for integral manifolds and manifolds
of that dimension are obviously maximal with respect to inclusion.
However, maximal integral manifolds may not be of maximal dimension.
The distribution which we shall study here is of codimension greater
than one and exhibits the just-mentioned behavior.  It is a natural
one to study for a number of reasons. First, it is defined by a
matrix-valued analogue of the contact distribution.  Second, it
arises as a local model $U$ for the Griffiths distribution
on a period domain $D$ of weight two \cite{Griff} discussed in the
introduction.

We shall now describe this local model $U$, then give the main result
of the paper. To this end, consider the group
$G$ of matrices of the form
$$
  g = \left(\matrix{
                  1_p & 0 & 0 \cr
                  X & 1_q & 0 \cr
                  Z & Y & 1_p \cr
                    }\right)
$$
where $1_n$ denotes the $n\times n$ identity matrix.
A basis for the left-invariant one-forms on $G$ is given by the
Maurer-Cartan form
\begin{equation}
  \Omega \eqdef
         g^{-1}dg = \left(\matrix{
                  0 & 0 & 0 \cr
                  dX & 0 & 0 \cr
                  \omega & dY & 0 \cr
                    }\right)
 \label{MCform}
\end{equation}
where
\begin{equation}
  \omega = dZ - YdX
 \label{matrixcontactform}
\end{equation}
On $G$ define a distribution $E$ as the set of tangent vectors
which annihilate the entries of $\omega$.  When $p = 1$, the group is
the Heisenberg group and $\omega$ is the contact form.  Our local
model $U$ is the unipotent subgroup defined by the equations
\begin{equation}
    \quad Y  = \tr X
\label{pereq-a}
\end{equation}
and
\begin{equation}
    \quad Z + \tr Z = \tr X X .
 \label{pereq-b}
\end{equation}
A neighborhood of the identity in $U$ is
isomorphic to a neighborhood of a fixed but arbitrary point of $D$.
Under this identification the restriction of the Griffiths distribution
is the same as the restriction of the distribution $E$ to $U$.

Since $X$ determines $Y$ and the
symmetric part of $Z$, coordinates on $U$ are given by the entries of
$X$ and the skew-symmetric part of $Z$. Therefore $U$ has dimension
$pq + p(p-1)/2$.   From (\ref{matrixcontactform}) and
(\ref{pereq-a}), one finds that
\begin{equation}
    \omega = dZ  - \tr X dX,
    \label{Ucontacteq-a}
\end{equation}
and from the exterior derivative of (\ref{pereq-b}) one finds that
\begin{equation}
     \omega^+ = 0,
\label{Ucontacteq-b}
\end{equation}
where $\omega = \omega^+ + \omega^-$ is the decomposition into
symmetric and antisymmetric parts.  Thus $E$ has dimension $pq$ and
codimension $p(p-1)/2$.  Our main interest will be in the case $p >
2$, i.e., codimension greater than one.

Consider now a subspace $S$ of the tangent space to $U$ at some point.
If it is tangent to an integral manifold it annihilates not only
$\omega$ but also $d\omega$.  An arbitrary subspace satisfying these
two conditions --- a potential tangent space to an integral manifold
--- is called an {\sl integral element}.  For the contact distribution
all integral elements are integrable.  For the Griffiths distribution
``integrability'' holds for integral elements of maximal dimension
($pq/2$ when $q$ even, $p(q-1)/2 + 1$ when $q$ odd, \cite{CKT}).  For other integral
elements, e.g., those which are maximal with respect to inclusion
but not of maximal dimension, one could presumably answer the integrability
question using the  Cartan-K\"ahler theory.  What we do instead
is to solve the integrability problem explictly for generic integral
elements, which are easily shown to be maximal \cite{CTAMS}:

\begin{theorem} Let $S$ be a generic $q$-dimensional integral
element for a period domain with Hodge numbers $p = h^{2,0}$, $q = h^{1,1}$,
where $p > 1$. Then $S$ is tangent to an integral
manifold. Such integral manifolds are determined in a canonical way by holomorphic functions
$f_2 \,,\,\cdots\,,\, f_p$ of a complex variable $u = (u_1 \,,\,\cdots\,,\,
u_q)$ which satisfy the system of partial differential equations
\begin{equation}
   [H_{f_i}, H_{f_j}] = 0
    \label{Hcommutatoreq}
\end{equation}
where $H_f$ is the Hessian matrix of $f$.  The space of solutions to
this equation for fixed $S$ is infinite-dimensional.
\label{maintheorem}
\end{theorem}

\noindent
The set of generic integral elements is open in the set of all
$q$-dimensional integral elements.  We shall formulate
and prove this result in the next section.  To explain the canonical
construction, recall that the entries of the matrices $X$ and the
skew-symmetric part of $Z$ give coordinates on $U$.  Thus an integral
manifold will be specified by giving
these coordinates in terms of the functions $f_i$. To do
so, let $[a_1 \,,\,\cdots\,,\, a_p]$ denote the matrix with
column vectors
$a_i$, set
$$
  X(u) = [\, u, \nabla f_2 \,,\,\cdots\,,\, \nabla f_p \, ],
$$
and put
$$
  Z_{j1}(u) = f_j(u),
$$
where $j > 1$.  For the entries $Z_{jk}$ with $j>k$, choose arbitrary
solutions of the equations
$$
   dZ_{jk} = \sum_a X_{aj}dX_{ak}
$$
deduced from the $jk$ entry of
$$
  \omega = dZ - \tr X dX = 0.
$$
Use these to
determine the antisymmetric part of $Z$.  For the symmetric part use
the quadratic equation (\ref{pereq-b}).  The analogy with the
contact system --- both in the form of the equations and the form of
the solutions, is clear.  Indeed, when $p = 2$ the equation $\omega =
0$ is the same as the equation
$$
   dZ_{12} - \sum_a X_{a1}dX_{a2} = 0
$$
Thus, if we set $z = Z_{12}$, $x = (x_{11} \,,\,\cdots\,,\, x_{1q})$.
and $y = (x_{21} \,,\,\cdots\,,\, x_{2q})$, then both equations and
solutions coincide with those of the contact case.  Note, however,
that the relations (\ref{Hcommutatoreq}) are a new feature of the
case $p > 2$.

\section{Integral elements}

In order to give a precise definition of ``generic'' we describe in
some detail the tangent space of $D$ at a fixed point of reference
and the integral elements it contains.  To this end we choose
the local correspondence between
$D$ and $U$ so that identity matrix of $U$ is mapped to the reference
point.  Thus the Lie algebra ${\germ u}$ --- the tangent space at the
identity of $U$ --- corresponds to the tangent space of $D$ at the
reference point.  Now consider a curve $g(t)$ based at the identity
matrix with arbitrary initial tangent $\tau = g'(0)$.
It has the form
$$
  g(t) = \left(\matrix{
                  1_p & 0 & 0 \cr
                  a(t) & 1_q & 0 \cr
                  b(t) & \tr a(t) & 1_p \cr
                    }\right)
$$
where
$$
b = c(t) + {1 \over 2 } \tr a(t)a(t)
$$
with $c(t)$ is skew-symmetric, and where $a(0) = 0$, $c(0)
= 0$.  Differentiating, we find
$$
  \tau = \left(\matrix{
                  0 & 0 & 0 \cr
                  \phi & 0 & 0 \cr
                  \psi & \tr \phi & 0 \cr
                    }\right)
$$
where $a'(0) = \phi$ and $c'(0) = \psi$ are arbitrary matrices subject
to the condition that $\psi$ be skew-symmetric.  We can read this as
saying $\Omega(\tau) = \tau$, where $\Omega$ is the Maurer-Cartan form
(\ref{MCform}).  In more detail,
\begin{eqnarray}
  & dX(\tau) = \phi \\
  & dZ(\tau) = \psi .
\end{eqnarray}
Thus a matrix $\tau(\phi, \psi)$ is an element of $E$ if an only if
$\psi = 0$.  Consequently the map
$$
   \phi \mapsto \tau(\phi) \eqdef\tau(\phi,0)
$$
defines an isomorphism of $E$  at the identity with $q \times p$
matrices, i.e., with linear maps
$$
  \phi:{\bf C}^p \longrightarrow {\bf C}^q
$$

Now let ${\germ a} \subset {\germ u}$ be an integral element and let
$\tau_i = \tau(\phi_i)$ be vectors in ${\germ a}$.  By definition
${\germ a}$ annihilates
$$
   d\omega = -\tr dX \wedge dX.
$$
But
\begin{eqnarray}
   d\omega(\tau_1, \tau_2)
    &=
   - \tr  dX(\tau_1) dX(\tau_2) + \tr  dX(\tau_2) dX(\tau_1) \cr
    &=
   - \tr \phi_1 \phi_2 + \tr \phi_2 \phi_1 \cr
\end{eqnarray}
so that the commutator
\begin{equation}
  (\phi_1, \phi_2)
    \eqdef
  \tr \phi_1 \phi_2 - \tr \phi_2 \phi_1
  \label{newcommutator}
\end{equation}
vanishes.  Equivalently, the Lie bracket $[\tau_1, \tau_2]$ vanishes.
Thus we may regard ${\germ a}$, either as a subspace of ${\germ u}$ or of the linear
maps from ${\bf C}^p$ to ${\bf C}^q$, as an {\sl abelian subspace}.  To
summarize:

\begin{lemma} A subspace of ${\germ u}$ is an integral element if and only
if it is an abelian subspace.
\end{lemma}

\noindent
We can now define what me mean by generic:

\begin{definition} A $q$-dimensional abelian subspace ${\germ a}$ of ${\germ u}$
is {\sl generic} if there is a vector $v \in {\bf C}^p$ such that
${\germ a}(v) = {\bf C}^q$.
\label{def:generic}
\end{definition}

\noindent
By ${\germ a}(v)$ we mean the space $\sett{ \phi(v) }{\phi\in{\germ a} }$.  To
justify the terminology we claim that (a) there are such spaces and
(b) the condition that a space be generic is an open one.  For the
first point let $\set{ e_i }$ denote the standard basis for ${\bf C}^n$
and let $v \cdot w$ denote the complex dot product.  Then the
commutator as defined in (\ref{newcommutator}) of $q \times p$
matrices
$a = [a_1 \,,\,\cdots\,,\, a_p]$ and $b = [b_1 \,,\,\cdots\,,\, b_p]$,
is the skew-symmetric matrix with entries
$$
  (a,b)_{ij} = a_i \cdot b_j - b_i \cdot a_j .
$$
Set
$$
   M_i = [e_i, 0 \,,\,\cdots\,,\, 0 ]
$$
for $i = 1..q$ and let ${\germ a}_0$ be their span.  It is clear that
$(M_i,M_j) = 0$, so that ${\germ a}_0$ is abelian.  For the second point
consider the space $A$ of framed abelian subspaces, that is, abelian
subspaces endowed with a basis $\set{ M_i }$. Consider the function
$F_v$ on $A$ defined by
$$
   (M_1 \,,\,\cdots\,,\, M_q)
      \mapsto
    M_1(v)\wedge \cdots \wedge M_q(v)
$$
The union ${\cal G}$ of the sets ${\cal G}(v) = \sett{ (M_i) = \hbox{basis for
an abelian space}}{ F_v(M_i)\ne 0 }$ is open in $A$ and contains any
framing of ${\germ a}_0$, whence the claim.

Now consider the $f$ transformation defined by
\begin{eqnarray}
  & X \longrightarrow BXA \cr
  & Z \longrightarrow \tr AZA \cr
\end{eqnarray}
where $X$ is a $q\times p$ matrix, $Z$ is a $p\times p$ matrix,
where $A$ is invertible and where $B$ is complex orthogonal.
The set of such transformations constitutes a complex Lie group
$H \cong GL(p,{\bf C})\times O(p,{\bf C})$ which acts on the local model $U$.
This action fixes the identity and acts on the form $\omega$ by
$$
  \omega \longrightarrow \tr A \omega A
$$
Consequently it preserves the distribution $E$ and so maps integral
manfifolds to integral manifolds.  Therefore in studying integral
manifolds of $E$ we may do so up to the action of $H$. Note also that
for commutators,
$$
  (BX_1A, BX_2A) = \tr A(X_1,X_2)A,
$$
so that the transform of an abelian space is an abelian space.
From this one sees that the orbit of ${\cal G}(e_1)$ is
${\cal G}$;  consequently, we may, without loss of generality, reason about
${\cal G}(e_1)$ in place of ${\cal G}$.  But an element of ${\cal G}(e_1)$ is a
$q$-dimensional abelian space with a basis elements of the form
$$
  M_j = [\, e_j, * \,,\,\cdots\,,\, * \,],
$$
where ``$*$'' stands for a column vector.  We shall call
such bases ``distinguished.''

Distinguished bases for abelian spaces can be characterized as
follows.  Given a matrix $A$, let $(A)_k$ denote the $k$-th column.
Consider next a system of $q\times q$ matrices  $\set{A_j}$,
where $j = 2 \dots p$, and  construct a new system
of $q\times p$ matrices
$$
   M_k = [ e_k,\ (A_2)_k \,,\,\cdots\,,\, (A_p)_k  ],
$$
where $k = 1 \dots q$.
The correspondence $\set{ A_j } \longrightarrow \set{ M_k} $
is one-to-one, and a routine computation shows the
following:

\begin{proposition} The span of a distinguished basis $\set{M_k}$,
$k = 1 \dots q$,
is an abelian space if and only if $\set{ A_j }$, $j = 2 \dots p$,
is a commuting set of
symmetric matrices.  The span of the $M_k$ is then a maximal abelian space.
\label{distbasisprop}
\end{proposition}

\noindent
Moreover, it is not hard to show that the natural relation between
Hessians and tangent spaces holds:

\begin{proposition} Let $\set{ A_j }$ be a commuting set of symmetric
matrices and let $\set{ f_j }$ be a solution to (\ref{Hcommutatoreq})
such that $f_j(0) = 0$, $\nabla f_i(0) = 0$, and $H_{f_j}(0)  = A_j$.
Then the tangent space at the identity to the associated integral
manifold is the abelian space associated to $\set{ A_j }$ which
has basis $\set{ M_k }$.
\end{proposition}

\section{The canonical construction}

We will now show that an integral manifold of $E$
whose tangent space at the
identity is in ${\cal G}(e_1)$ is given locally by the canonical construction. To this
end, note that the genericity hypothesis (the condition ${\germ a}(e_1) = {\bf C}^q$) is
equivalent to the condition that the components of $(dX)_1$ are independent.
In this case
$$
   \eta = dX_{11} \wedge \cdots \wedge dX_{r1}
$$
is nonzero.  By shrinking $M$ we may assume that the product
$\eta$ is nonzero on all of
$M$, so that the functions $X_{a1}$ are independent on it.  Then
(\ref{Ucontacteq-a}) implies that
$$
  dZ_{ij} = \sum X_{ai}dX_{aj} .
$$
Consider in particular the case $j = 1$, $i > 1$, for which we obtain
the equation
$$
  dZ_{i1} = \sum X_{ai}dX_{a1}.
$$
Since no $dX_{ab}$ for $b \ne 1$ occur, we see that $Z_{i1}$ may be
viewed as function $f_i$ of the entries of $(X)_1$, the first column of
$X$. Moreover, the $X_{ai}$ are functions of these same entries,
namely,
$$
    X_{ai} = \pd{f_i}{X_{a1}}
$$
So far we have used just some of the equations $\omega = 0$ determined by
(\ref{Ucontacteq-a}).  These equations assert that there exist certain
functions
$Z_{ij}$ for $i > j > 1$ of  $(X)_j$ which are in turn
functions of $(X)_1$.  Consequently the already-determined forms
$\phi_{ij} = dZ_{ij}$ must be closed.  Now
$$
  \phi_{ij}
    =
     \sum_a
       \pd{f_i}{X_{a1}} d\left(\pd{f_j}{X_{a1}}\right)
    =
      \sum_{ab}
        \pd{f_i}{X_{a1}} \mmpd{f_j}{X_{a1}}{X_{b1}}{ dX_{b1}}
$$
and so
$$
  d\phi_{ij}
    =
       \sum_{abc} \left[
         \mmpd{f_i}{X_{a1}}{X_{c1}} \mmpd{f_j}{X_{a1}}{X_{b1}} dX_{c1}\wedge dX_{b1}
           +
          \pd{f_i}{X_{a1}} \mmmpd{f_j}{X_{a1}}{X_{b1}}{X_{c1}} dX_{c1}\wedge dX_{b1}
          \right]
$$
The third partial derivative is symmetric in $b$ and $c$ whereas the product
$dX_{c1}\wedge dX_{b1}$ is antisymmetic in these indices.  Consequently the
second sum vanishes.  Therefore the first sum must vanish.  Since the coefficient
of $dX_{c1}\wedge dX_{b1}$ is symmetric in $b$ and $c$ we conclude that
it must vanish.  But inspection reveals that coeffient to be the $bc$ entry
of the commutator
$$
  [H_{f_i},H_{f_j}],
$$
where $H$ denotes the Hessian matrix.  Consequently the consistency of our overdetermined
system is just the set of partial differential equations
(\ref{Hcommutatoreq}).

\section{Existence results}

Let us now consider the problem of the existence and nature of solutions
to the system of partial differential equations $[H_{f_i},H_{f_j}] = 0$.
The most basic question is whether enough there are enough solutions to
pass an integral manifold through an arbitrary integral element.  By Proposition
(\ref{distbasisprop}) this is
equivalent to the problem of constructing functions whose Hessians commute
and whose values at the origin are given commuting symmetric matrices $\set{A_\ell}$.
For these it is enough to take the quadratic functions
$$
   f_\ell = {1 \over 2 }\sum_{ij} (A_\ell)_{ij} u_iu_j
$$
We have therefore shown the following and with it part of (\ref{maintheorem}):

\begin{proposition} Any generic element is tangent to an integral manifold.
\label{integrabilityprop}
\end{proposition}

Let us consider now the problem of finding additional solutions to
the equations (\ref{Hcommutatoreq}). Note first that
if two functions $f$ and $g$ have diagonal Hessians then they
automatically satisfy $[H_f,H_g] = 0$.
The condition that the Hessians be diagonal is the
overdetermined system of equations
$$
   \mmpd g{x_i}{x_j} = 0 \qquad \hbox{for $i \ne j$},
   \eqno{(*)}
$$
In the case of two variables there is just one equation, a form of the wave equation,
which has solutions of the form $h_1(x_1) + h_2(x_2)$.  The solutions in the
general case have the same form,
$$
  g(x_1 \,,\,\cdots\,,\, x_n) = \sum_i h_i(x_i),
  \label{diagsolution}
$$
where the $h_i$ are arbitrary functions of one variable.
Therefore an integral manifold is specified by a set of functions
$\set{ h_{ij}(u) }$, where
$$
  f_i(x_1 \,,\,\cdots\,,\, x_s) = \sum_j h_{ij}(x_j) .
$$
This does not give a complete set of solutions, but it does give an infinite-dimensional
set.  Moreover, we can choose the $h_{ij}$ in such a way that so that the Hessian of
$f_i$ at the origin is an arbitrary diagonal quadratic form.  In fact, we can do somewhat
more.  Let $\set{ A_\ell }$ be a set of commuting complex symmetric matrices at least one
of which has distinct eigenvalues.  Then there is a set of common eigenvectors which form
a basis for ${\bf C}^q$ and which are orthogonal relative to the complex dot product.
Consequently there is a complex orthogonal matrix $C$ which simultaneously diagonalizes
the $A_\ell$.  Let $D_\ell$ be the diagonal matrix corresponding to $A_\ell$, where
$CA_\ell \tr C = D_\ell$.  There is an infinite dimensional family of solutions $(f'_\ell)$
to (\ref{Hcommutatoreq}) such that $H_{f'_\ell}(0) = D_\ell$.  Let $f_\ell(x) = f'_\ell(Cx)$.
Then
$$
   H_{f_\ell} = \tr C H_{f'_\ell} C
$$
and, since $C$ is complex orthogonal,
$$
   [H_{f_\ell},H_{f_m}]  = \tr C [ H_{f'_\ell}, H_{f'_m}] C
$$
Thefore the infinite-dimensional family of functions $(f_\ell)$ is also a set of
solutions to (\ref{Hcommutatoreq}), and each member has the specified initial Hessians
$(A_\ell)$. Consequently our previous integration result (\ref{diagsolution}) can be
strengthened:

\begin{theorem} Any generic abelian space of dimension $r$ is tangent to an integral
manifold.  Moroever,  the set of germs of integral manifolds tangent to this space is
infinite dimensional.
\end{theorem}

\subsection*{Remark.} Consider the case $q = 2$ with $p > 2$ arbitrary.
Fix $f_2$ arbitrarily but generically in the sense that the Hessian
generically has distinct eigenvalues.  Consider the equations
$[H_{f_2},H_{f_j}] = 0$, for $j > 2$.  For given $j$ one has a single
non-trivial partial differential equation which is linear of second
order in $f_2$.  By the Cauchy-Kowaleska Theorem \cite{FJohn} the solution
space is infinite-dimensional. Now let $V(u)$ be the matrix of eigenvectors
of $H_{f_2}(u)$.  Then the transformation $A \longrightarrow \tr V(u)AV(u)$ simultaneously
diagonalizes all of the matrices $H_{f_i}(u)$.  Consequently all of these
matrices commute with each other, i.e., the functions $f_i$ solve (\ref{Hcommutatoreq}).
The case $q > 2$ is more
complicated because the  partial differential equations constitute an overdetermined system, somewhat
like the system $\nabla f  = \xi$ for a given vector field $\xi$.

redpajama_set_name: RedPajamaArXiv