Krzysztof Gawedzki and Pascal Tran-Ngoc-Bich

Self-duality of the SL_2 Hitchin integrable system at genus two

32 pages, latex, no figures, references and a discussion inspired by
  one of them added

Abstract:
  We revisit the Hitchin integrable system whose phase space is the bundle
cotangent to the moduli space $N$ of holomorphic $SL_2$-bundles over a smooth
complex curve of genus two. $N$ may be identified with the 3-dimensional
projective space of theta functions of the second order, We prove that the
Hitchin system on $T^*N$ possesses a remarkable symmetry: it is invariant under
the interchange of positions and momenta. This property allows to complete the
work of van Geemen-Previato which, basing on the classical results on geometry
of the Kummer quartic surfaces, specified the explicit form of the Hamiltonians
of the Hitchin system. The resulting integrable system resembles the classic
Neumann systems which are also self-dual. Its quantization produces a commuting
family of differential operators of the second order acting on homogeneous
polynomials in four complex variables. As recently shown by van Geemen-de Jong,
these operators realize the Knizhnik-Zamolodchikov-Bernard-Hitchin connection
for group SU(2) and genus 2 curves.

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\begin{document}
\title{\bf{Self-duality of the $SL_2$ Hitchin integrable\\
system at genus 2}}
\author{\ \\Krzysztof Gaw\c{e}dzki \\ I.H.E.S., C.N.R.S.,
F-91440  Bures-sur-Yvette, France\\ \\Pascal Tran-Ngoc-Bich \\
Universit\'{e} de Paris Sud, F-91405 Orsay, France}
\date{ }
\maketitle

\vskip 0.3cm
\vskip 1 cm

\begin{abstract}
\vskip 0.3cm

\noindent We revisit the Hitchin integrable system
\cite{Hitch}\cite{VGP} whose phase space is the bundle cotangent
to the moduli space $\cal N$ of holomorphic $SL_2$-bundles
over a smooth complex curve of
genus 2. As shown in \cite{NarRa}, $\hspace{0.025cm}\cal{N}$
may be identified with the 3-dimensional projective space of theta
functions of the 2$^{\rm nd}$ order, i.e. ${\cal N}\cong \NP^{3}$.
We prove that the Hitchin system on $T^*{\cal N}\cong T^*\NP^{3}$
possesses a remarkable symmetry: it is invariant under
the interchange of positions and momenta. This property allows
to complete the work of van Geemen-Previato \cite{VGP} which,
basing on the classical results on geometry of the Kummer quartic
surfaces, specified the explicit form of the Hamiltonians
of the Hitchin system. The resulting integrable system resembles
the classic Neumann systems which are also self-dual.
Its quantization produces a commuting family of differential operators
of the 2$^{\rm nd}$ order acting on homogeneous polynomials in four
complex variables. As recently shown by van Geemen-de Jong
\cite{VGDJ}, these operators realize
the Knizhnik-Zamolodchikov-Bernard-Hitchin
connection for group $SU(2)$ and genus 2 curves.
\end{abstract}
\vskip 2cm

\nsection{Introduction}
\setcounter{equation}{0}
\vskip 0.5cm

In \cite{Hitch}, Nigel Hitchin has discovered an interesting
family of classical integrable models related to modular
geometry of holomorphic vector bundles or to 2-dimensional
gauge fields. The input data for Hitchin's construction
are a complex Lie group $G$ and a complex  curve $\Sigma$
of genus $\gamma$. The configuration space of the integrable
system is the moduli space $\cal N$ of (semi)stable holomorphic
$G$-bundles over $\Sigma$. This is a finite-dimensional
complex variety and Hitchin's construction is done in the holomorphic
category. It exhibits a complete family of Poisson-commuting
Hamiltonians on the (complex) phase space $T^*\cal{N}$.
The Hitchin Hamiltonians have open subsets of abelian varieties
as generic level sets on which they induce additive
flows \cite{Hitch}. More recently, Hitchin's construction
was extended to the case of singular or punctured curves
\cite{Mark}\cite{Nekr}\cite{EnR} providing a unified construction
of a vast family of classical integrable systems. For $\Sigma=\NC P^1$
with punctures, one obtains this way the so called Gaudin chains
and for $G=SL_N$ and $\Sigma$ of genus 1 with one puncture,
the elliptic Calogero-Sutherland models which
found an unexpected application in the
supersymmetric 4-dimensional gauge theories \cite{DonaW}.
\vskip 0.4cm

In Section 2 of the present paper we briefly
recall the basic idea of Hitchin's construction. The main
aim of this contribution is to treat in detail the
case of $G=SL_2$ and $\Sigma$ of genus 2 (no punctures).
The genus 2 curves are hyperelliptic, i.e.\hspace{0.05cm}\s given
by the equation
\begin{eqnarray}
\zeta^2\hspace{0.05cm}=\hspace{0.05cm}\prod\limits_{s=1}^6(\lambda-\lambda_s)
\label{hellp}
\end{eqnarray}
where $\lambda_s$ are 6 different complex numbers.
The semistable moduli space ${\cal N}$ has a particularly simple form
for genus 2, \cite{NarRa}: it is the
projectivized space of theta functions of the 2$^{\rm nd}$
order:
\begin{eqnarray}
{\cal N}\ =\ \NP H^0(L_{\Theta}^2)
\label{be}
\end{eqnarray}
where $L_\Theta$ is the theta-bundle over the Jacobian $J^1$
of (the isomorphism classes of) degree
$\gamma-1=1$ line bundles\footnote{we use
the multiplicative notation for the tensor product of line
bundles} $l$ over $\Sigma$. \hspace{0.05cm}${dim}_{_{\bf C}}(H^0(L_\Theta^2))=4$
so that ${\cal N}\cong\NP^3$. This picture of ${\cal N}$ is related
to the realization of $SL_2$-bundles as extensions
of degree 1 line bundles. We review some
of the results in this direction in Section 3 using
a less sophisticated language than that of the original
work \cite{NarRa}. The relation between the extensions
and the theta functions is lifted to the level of the cotangent
bundle $T^*{\cal N}$ in Section 4. The language of extensions
proves suitable for a direct description of the Hitchin Hamiltonians
on $T^*{\cal N}$. The main aim is, however, to present the Hitchin system
as an explicit 3-dimensional family of integrable systems
on $T^*\NP^3$, parametrized by the moduli of the curve.
This was first attempted, and almost achieved, in reference
\cite{VGP}.
\vskip 0.4cm

Let us recall that the Hitchin Hamiltonians
are components of the map
\begin{eqnarray}
{\cal H}\hspace{0.025cm}:\ T^*{\cal N}\ \longrightarrow\ H^0(K^2)
\label{Hm}
\end{eqnarray}
with values in the (holomorphic) quadratic differentials
($K$ denotes the canonical bundle of $\Sigma$).
Due to relation (\ref{be}), the map ${\cal H}$ may be viewed as
a $H^0(K^2)$-valued function of pairs $(\theta,\phi)$ where
$\theta\in H^0(L_{\Theta}^2)$ and $\phi$ from the dual space
$H^0(L_\Theta^2)^*$ are s.t. ${\langle\hspace{0.025cm}}\theta,
\hspace{0.025cm}\phi{\hspace{0.025cm}\rangle}=0$. Fix a holomorphic trivialization
of $L_\Theta$ around $l\in J^1$ and denote by $\phi_l$
the linear form that computes the value of the theta function
at $l$. As was observed in \cite{VGP},
\begin{eqnarray}
{\cal H}(\theta,\hspace{0.025cm}\phi_l)\ =\ -\hspace{0.025cm}{_1\over^{16\pi^2}}
\hspace{0.05cm}(d\theta(l))^2
\label{onK}
\end{eqnarray}
(with appropriate normalizations). In the above
formula, $\theta$ is viewed as a function
on $J^1$ and $d\theta(l)$ as an element of $H^0(K)$.
Since $\theta(l)=0$, the equation is consistent with
changes of the trivialization of $L_\Theta$.
\vskip 0.4cm

The map $J^1\ni l\mapsto\phi_l$ induces
an embedding of the Kummer surface $J^1/\NZ_2$
with $l$ and $l^{-1}K$ identified into a quartic
${\cal K}^*$ in $\NP H^0(L_\Theta^2)^*$. The Kummer quartic
is a carrier of a rich but classical structure, a subject
of an intensive study of the nineteenth century geometers,
see \cite{Kummer} and also the last chapter of \cite{GH}.
The reference \cite{VGP} used the relation (\ref{onK}) and
a mixture of the classical results and of more modern
algebraic geometry to recover an explicit form of the components
of the Hitchin map ${\cal H}$ up to a multiplication by a function on
the configuration space. The authors of \cite{VGP} checked
that the simplest way to fix this ambiguity leads
to Poisson-commuting functions but they fell short of showing
that the latter coincide with the ones of the Hitchin
construction.
\vskip 0.4cm

Among the aims of the present paper is to fill the
gap left in \cite{VGP}. We observe that the proposal
of \cite{VGP} has a remarkable {\bf self-duality} property:
it is invariant under the interchange of the positions
and momenta in $T^*\NP^3$. We show that the Hitchin
construction leads to a system with the same symmetry. This limits
the ambiguity left by the analysis of \cite{VGP} to a multiplication
of the components of ${\cal H}$ by constants. A direct check based on
Eq.\hspace{0.05cm}\s(\ref{onK}) fixes the normalizations and results
in a formula for the Hitchin map which uses the hyperelliptic
description (\ref{hellp}) of the curve. Namely,
\begin{eqnarray}
{\cal H}\ =\ -\hspace{0.025cm}{_1\over^{128\hspace{0.025cm}\pi^2}}\hspace{0.05cm}\sum\limits_{1\leq s\not=t\leq 6}
{r_{st}\over(\lambda-\lambda_s)(\lambda-\lambda_t)}\,(d\lambda)^2
\label{GR}
\end{eqnarray}
where $r_{st}$ are explicit polynomials in $(\theta,\phi)$
given, upon representation of $(\theta,\phi)$ by
pairs $(q,p)\in\NC^4\times\NC^4$, by Eqs.\hspace{0.05cm}\s(\ref{rs})
below. The above expression for ${\cal H}$ has a similar form as that
for the Hitchin map on the Riemann sphere with
6 insertion points $\lambda_s$, see e.g. Sect. 4 of \cite{FalKG},
except for the structure of the terms $r_{st}$. This is not
an accident but is connected to the reduction of conformal field
theory on genus 2 surfaces to an orbifold theory in genus 0
\cite{Knizh}\cite{Zamo}. We plan to return to this relation
in a future publication.
\vskip 0.4cm

Let us discuss in more details how we establish the
self-duality of the Hitchin Hamiltonians. The main tool
here is an explicit expression for the values of the
Hitchin map off the Kummer quartic ${\cal K}^*$ which
we obtain in Section 5. Our formula
for ${\cal H}(\theta,\phi)$ requires a choice
of a pair of perpendicular 2-dimensional
subspaces $(\Pi,\Pi^\perp)$ where $\theta\in\Pi\subset
H^0(L_\Theta^2)$ and $\phi\in\Pi^\perp\subset H^0(L_\Theta^2)^*$
(there is a complex line of such choices).
The plane $\Pi^\perp$ corresponds to a line $\NP\Pi^\perp$
in $\NP H^0(L_\Theta^2)^*$ which intersects the Kummer quartic
${\cal K}^*$ in four points $\NC^*\phi_{l_j},\ j=1,2,3,4,$ (counting
with multiplicity). Whereas the analysis of \cite{VGP}
was mainly concerned with the geometry of bitangents
to ${\cal K}^*$ with two pairs of coincident $\phi_{l_j}$'s,
we concentrate on the generic situation with $\phi_{l_j}$'s
different. Then any two of them, say $\NC^*\phi_{l_1}$
and $\NC^*\phi_{l_2}$, span $\Pi^\perp$. $\Pi$
is composed of the 2$^{\rm nd}$ order theta functions vanishing
at $l_1$ and $l_2$. In particular,
\begin{eqnarray}
\phi\hspace{0.05cm}=\hspace{0.05cm} a_1\hspace{0.025cm}\phi_{l_1}\hspace{0.025cm}+\hspace{0.025cm} a_2\hspace{0.025cm}\phi_{l_2}\quad
{\rm and}\quad\theta(l_1)\hspace{0.05cm}=\s0\hspace{0.05cm}=\hspace{0.05cm}\theta(l_2)\hspace{0.05cm}.
\label{ss0}
\end{eqnarray}
Let $x_1+x_2$ and $x_3+x_4$ be the divisors of $l_1l_2$ and
of $l_1l_2^{-1}K$, respectively, where $x_i$ are four
points\footnote{the other two lines of intersection
of $\NP\Pi^\perp$ with ${\cal K}^*$ correspond
to $l_3$ and $l_4$ with $l_1l_3={\cal O}(x_1+x_3)$,
$l_1l_3^{-1}K={\cal O}(x_2+x_4)$, $l_1l_4={\cal O}(x_1+x_4)$,
$l_1l_4^{-1}K={\cal O}(x_2+x_3)$} in $\Sigma$.
If $l_1^2\not=K$, which holds in a general situation,
then the quadratic differential ${\cal H}(\theta,\hspace{0.025cm}\phi)$
is determined by its values at $x_i$ which, as we show
in Section 5, are
\begin{eqnarray}
{\cal H}(\theta,\hspace{0.025cm}\phi)(x_i)\ = \
-\hspace{0.025cm} {_1\over^{16\pi^2}}\hspace{0.05cm}\left(a_1\hspace{0.05cm} d\theta(l_1)\pm a_2\hspace{0.05cm}
d\theta(l_2)\right)^2(x_i)\hspace{0.05cm}.
\label{fr}
\end{eqnarray}
Sign plus is taken for $x_1$ and $x_2$
and sign minus for $x_3$ and $x_4$. Note that for $\phi=\phi_l$
with $\theta(l)=0$ the above equation reproduces
the result (\ref{onK}).
\vskip 0.4cm

As we recall at the end of Section 3, there exists an almost natural
linear isomorphism $\iota$ between $H^0(L_\Theta^2)^*$
and $H^0(L_\Theta^2)$. What follows is independent of the remaining
ambiguity in the choice of $\iota$. The identity
$\langle\hspace{0.025cm}\theta,\phi\hspace{0.025cm}\rangle=\langle\hspace{0.025cm}
\iota(\phi),\hspace{0.025cm}\iota^{-1}(\theta)\hspace{0.025cm}\rangle$ implies that
if $(\theta,\phi)$ is a perpendicular pair then so
is $(\theta',\hspace{0.025cm}\phi')$ where $\theta'=\iota(\phi)$ and
$\phi'=\iota^{-1}(\theta)$.  Thus $\iota$
interchanges the positions and momenta in $T^*{\cal N}$.
We may take $(\Pi',\hspace{0.025cm}{\Pi'}^\perp)
=(\iota(\Pi^\perp),\hspace{0.025cm}\iota^{-1}(\Pi))$ as a pair of perpendicular
subspaces containing $(\theta',\phi')$. The line
$\NP{\Pi'}^\perp$ meets ${\cal K}^*$ in four points
$\NC^*\phi_{l'_j}$. Equivalently,
$\NC^*\iota(\phi_{l'_j})$ are the points of intersection of $\NP\Pi$
with the Kummer quartic ${\cal K}=\iota({\cal K}^*)\subset\NP H^0(L_\Theta^2)$.
In general situation, ${\Pi'}^\perp$ is spanned by any pair
of $\phi_{l'_j}\hspace{0.025cm}$'s so that
\begin{eqnarray}
\phi'\hspace{0.05cm}=\hspace{0.05cm} a'_1\hspace{0.025cm}\phi_{l'_1}\hspace{0.025cm}+\hspace{0.025cm} a'_2\hspace{0.025cm}\phi_{l'_2}\quad{\rm and}
\quad\theta'(l'_1)\hspace{0.05cm}=\s0\hspace{0.05cm}=\hspace{0.05cm}\theta'(l'_2)
\label{ss2}
\end{eqnarray}
which is the dual version of relations (\ref{ss0}).
Equivalently,
\begin{eqnarray}
\theta\hspace{0.05cm}=\hspace{0.05cm} a'_1\hspace{0.05cm}\iota(\phi_{l'_1})\hspace{0.05cm}+\hspace{0.05cm} a'_2\hspace{0.05cm}\iota(\phi_{l'_2})
\quad{\rm and}\quad{\langle\hspace{0.025cm}}\iota(\phi_{l'_1}),
\hspace{0.05cm}\phi{\hspace{0.025cm}\rangle}\hspace{0.05cm}=\s0\hspace{0.05cm}=\hspace{0.05cm}
{\langle\hspace{0.025cm}}\iota(\phi_{l'_2}),\hspace{0.05cm}\phi{\hspace{0.025cm}\rangle}\hspace{0.05cm}.
\label{ss1}
\end{eqnarray}
Let $y_i$ be the points associated
to $l'_j$ the same way as the points $x_i$ were associated
to $l_j$.  \hspace{0.05cm}$l'_j$ may be chosen so that $y_i$ and $x_i$ coincide
modulo the natural involution of $\Sigma$ fixing the six
Weierstrass points. Formula (\ref{fr}) implies then that
\begin{eqnarray}
{\cal H}(\theta',\hspace{0.025cm}\phi')(y_i)\ = \
-\hspace{0.025cm} {_1\over^{16\pi^2}}\hspace{0.05cm}\left(a'_1\hspace{0.05cm} d\theta'(l'_1)\pm a'_2\hspace{0.05cm}
d\theta'(l'_2)\right)^2(y_i)\hspace{0.05cm}.
\label{frd}
\end{eqnarray}
Points $y_i$ in Eq.\hspace{0.05cm}\s(\ref{frd}) may be replaced by $x_i$
since the quadratic differentials are equal at point $x$
if and only if they are equal at the image of $x$ by the
involution of $\Sigma$. A direct calculation of
the coefficients $a_1,\hspace{0.05cm} a_2$ and $a'_1,\hspace{0.05cm} a'_2$ appearing
on the right hand sides of Eqs.\hspace{0.05cm}\s(\ref{fr})
and (\ref{frd}) shows then that both expressions coincide,
establishing the self-duality of ${\cal H}$. The verification
of this equality is the subject of Section 6.
\vskip 0.4cm

In Section 7, we recall the main result of reference \cite{VGP}
and show how the self-duality may be used to complete
the analysis performed there and to obtain the explicit form
(\ref{GR}) of the Hitchin map. We briefly discuss the relation
of that form to the classical Yang-Baxter equation.
\vskip 0.4cm

An appropriate quantization of Hitchin Hamiltonians leads to operators
acting on holomorphic sections of powers of the determinant line bundle
over ${\cal N}$ and defining the Knizhnik-Zamolodchikov-Bernard-Hitchin
\cite{KZ}\cite{Ber}\cite{Ber2}\cite{Hitch2} connection.
In our case, the sections of the powers
of the determinant bundle are simply homogeneous
polynomials on $H^0(L_\Theta^2)$. It is easy to quantize
the Hamiltonians corresponding to the components of
the Hitchin map (\ref{GR}) in such a way that one obtains
an explicit family of commuting 2$^{\rm nd}$ order differential
operators acting on such polynomials. The corresponding connection
coincides with the explicit form of the (projective) KZBH connection
worked out recently\footnote{we thank B. van Geemen for attracting
our attention to ref. \cite{VGDJ} and for pointing out that this work
may be used to fix indirectly the precise form of the Hitchin map}
in \cite{VGDJ}.
\vskip 0.4cm

The quantization of the genus 2 Hitchin system
is briefly discussed in Conclusions, where we also mention other possible
directions for further research. Four appendices which close the paper
contain some of more technical material.
\vskip 0.4cm

We would like to end the presentation of our paper
by expressing some regrets. We apologize to Ernst Eduard Kummer
and other nineteenth century giants for our insufficient
knowledge of their classic work. The apologies
are also due to few contemporary algebraic geometers who could
be interested in the present work for an analytic character
of our arguments. To the specialist in integrability
we apologize for the yet incomplete analysis of the integrable
system studied here and, finally, we apologize to ourselves
for not having finished this work 2 years ago.
\vskip 0.9cm

\nsection{Hitchin's construction}
\vskip 0.5cm

Let us assume, for simplicity, that the complex Lie group $G$ is simple,
connected and simply connected. We shall denote by ${\bf g}$
its Lie algebra. The complex curve $\Sigma$ will be
assumed smooth, compact and connected. Topologically,
all $G$-bundles on $\Sigma$ are trivial and the complex
structures in the trivial bundle may be described by giving
operators $\bar\partial+A$ where $A$ are
smooth ${\bf g}$-valued 0,1-forms on $\Sigma$ \cite{AtiyBott}.
Let $\cal{A}$ denote the space of such forms
(i.e. of chiral gauge fields). The group $\cal G$ of
local (chiral) gauge transformations composed of smooth maps
$h$ from $\Sigma$ to $G$ acts on operators $\bar\partial+A$
by conjugation and on the gauge fields $A$ by
\begin{eqnarray}
A\ \longmapsto\ {}^h\hspace{-0.14cm}A\hspace{0.05cm}\equiv\hspace{0.05cm}
h\hspace{0.05cm} A\hspace{0.05cm} h^{-1}\hspace{0.05cm}+\hspace{0.05cm} h\hspace{0.05cm} \bar\partial h^{-1}\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Two holomorphic $G$-bundles are equivalent iff the
corresponding gauge fields are in the same orbit of ${\cal G}$.
Hence the space of orbits ${\cal A}/{\cal G}$ coincides with
the (moduli) space of inequivalent holomorphic $G$-bundles.
It may be supplied with a structure of a variety provided one
gets rid of bad orbits. This may be achieved by limiting
the considerations to (semi)stable
bundles, i.e. such that the vector bundle associated
with the adjoint representations of $G$ contains only
holomorphic subbundles with negative (non-positive)
first Chern number. For $\gamma>1$, the moduli space
${\cal N}_s\equiv{\cal A}_s/{\cal G}$
of stable $G$-bundles is a smooth complex variety with
a natural compactification to a variety ${\cal N}_{ss}$,
the (Seshadri-)moduli space of semistable bundles \cite{NarRa}.
\vskip 0.4cm

The complex cotangent bundle $T^*{\cal N}_s$ may be obtained
from the infinite-dimensional bundle $T^*{\cal A}_s$ by
the symplectic reduction. $T^*{\cal A}_s$ may be realized as
the space of pairs $(A,\Phi)$ where $\Phi$
is a (possibly distributional) ${\bf g}$-valued 1,0-form on $\Sigma$,
$A\in{\cal A}_s$ and the duality with the vectors
$\delta A$ tangent to ${\cal A}$ is given by
\begin{eqnarray}
\int_{_\Sigma} tr\ \Phi\wedge\delta A
\nonumber
\end{eqnarray}
with $tr$ standing for the Killing form.
The action of the local gauge group ${\cal G}$ on ${\cal A}_s$
lifts to a symplectic action on $T^*{\cal A}_s$ by
\begin{eqnarray}
\Phi\ \longmapsto\ {}^h\hspace{-0.05cm}\Phi\hspace{0.05cm}\equiv\hspace{0.05cm}
h\hspace{0.05cm}\Phi\hspace{0.05cm} h^{-1}\ .
\nonumber
\end{eqnarray}
The moment map $\mu$ for the action of ${\cal G}$ on $T^*{\cal N}_s$
is
\begin{eqnarray}
\mu(A,\Phi)\ =\ \bar\partial\Phi+A\wedge\Phi
+\Phi\wedge A\ \equiv\ \bar\partial_{_{A}}\Phi\ .
\nonumber
\end{eqnarray}
Note that it takes values in ${\bf g}$-valued 2-forms on $\Sigma$.
These may be naturally viewed as elements of the space
dual to the Lie algebra of ${\cal G}$. The symplectic reduction
of $T^*{\cal A}_s$ realizes $T^*{\cal N}_s$ as the
space of ${\cal G}$-orbits in the zero level of $\mu$:
\begin{eqnarray}
T^*{\cal N}_s\ \cong\ \mu^{-1}(\{0\})\hspace{0.05cm}/\hspace{0.05cm}{\cal G}\ .
\nonumber
\end{eqnarray}
\vskip 0.4cm

For a homogeneous $G$-invariant polynomial $P$ on ${\bf g}$ of
degree $d_P$, the gauge invariant expression \hspace{0.05cm}$P(\Phi)$
defines a section of the bundle $K^{d_P}$ of $d_P$-differentials
on $\Sigma$. If $\Phi$ is in the zero level of $\mu$
then \hspace{0.05cm}$P(\Phi)$ is also holomorphic. Hence the map
\hspace{0.05cm}$\Phi\hspace{0.05cm}\mapsto\hspace{0.05cm} P(\Phi)\hspace{0.05cm}$ induces a map
\begin{eqnarray}
{\cal H}_P\hspace{0.05cm}:\ T^*{\cal N}_s\ \longrightarrow\ H^0(K^{d_P})
\nonumber
\end{eqnarray}
into the finite dimensional vector space of holomorphic
differentials of degree $d_P$ on $\Sigma$. The components
of such vector-valued Hamiltonians clearly Poisson-commute
since upstairs (on $T^*{\cal A}_s$) they depend only
on the momentum variables $\Phi$. By a beautiful
argument, Hitchin showed \cite{Hitch} that taking
all polynomials $P$ one obtains a complete system
of Hamiltonians in involution and that the
collection of maps ${\cal H}_P$ defines in generic
points a foliation of $T^*{\cal N}_s$ into (open subsets of)
abelian varieties.
\vskip 0.4cm

Let us briefly sketch Hitchin's argument for $G=SL_2$.
There is only one (up to normalization) non-trivial
invariant polynomial $P_2$ on $sl_2$ given by, say,
half of the Killing
form. ${\cal H}\equiv{\cal H}_{P_2}$ maps into the space of quadratic
differentials. A non-trivial holomorphic quadratic differential
$\rho$ determines a (spectral) curve $\Sigma'\subset K$ given
by the equation
\begin{eqnarray}
\xi^2=\rho(\pi(\xi))
\label{1}
\end{eqnarray}
where \hspace{0.05cm}$\xi\in K$ and $\pi$ is the projection of $K$
on $\Sigma$. The map $\xi\mapsto-\xi$ gives an involution
$\sigma$ of $\Sigma'$. Restriction of $\pi$ to $\Sigma'$
is a 2-fold covering of $\Sigma$ ramified over $4(\gamma-1)$
points fixed by $\sigma$, the zeros of $\rho$.
$\Sigma'$ has genus $\gamma'=4\gamma-3$. \hspace{0.025cm} If \hspace{0.025cm}$\rho\hspace{0.025cm}=\hspace{0.025cm}
{_1\over^2}\hspace{0.05cm} tr\hspace{0.05cm}\s(\Phi)^2\hspace{0.025cm}$ then
relation (\ref{1}) coincides with the eigen-value equation
\begin{eqnarray}
det\hspace{0.05cm}(\Phi-\hspace{0.05cm}\xi\cdot I)\hspace{0.05cm}=\hspace{0.05cm} 0
\nonumber
\end{eqnarray}
for the Lax matrix $\Phi$.
Let for each $0\not=\xi\in\Sigma'$, $l_\xi$ denote
the corresponding eigen-subspace of $\Phi$. By continuity,
$l_\xi$ extend to vanishing $\xi$ in $\Sigma'$ and
\hspace{0.05cm}$\cup_{_{\xi}}l_{\xi}$ forms a line subbundle $l$
of $\Sigma'\times\NC^2$. In fact, $l$ is a holomorphic subbundle
with respect to the complex structure defined on
$\Sigma'\times\NC^2$ by $\bar\partial+A\circ\pi$. The degree
of $l$ is $-2(\gamma-1)$. Besides,
\begin{eqnarray}
l\hspace{0.05cm}(\sigma^*l)\hspace{0.05cm}=\hspace{0.05cm}\pi^*K^{-1}\hspace{0.05cm}.
\label{K}
\end{eqnarray}
Conversely, given $\Sigma'$ and a holomorphic
line bundle $l$ of degree $-2(\gamma-1)$ on it satisfying
(\ref{K}), we may recover a rank 2 holomorphic bundle $E$
of trivial determinant over $\Sigma$ as a pushdown of $l$
to $\Sigma$. Thus for $0\not=\xi\in\Sigma'$, $E_{\pi(\xi)}
=l_\xi\oplus l_{-\xi}$. $E$ corresponds to a unique
holomorphic $SL_2$-bundle which, if stable (what happens
on an open subset of $l$'s) defines a point in the moduli
space ${\cal N}_s$. A holomorphic 1,0-form
with values in the traceless endomorphisms of $E$
acting as multiplication by $\pm\xi$ on
$l_{\pm\xi}\subset E_{\pi(\xi)}$ defines then
a unique covector of $T^*{\cal N}_s$. Thus $\Sigma'$ encodes
the values of the quadratic Hitchin Hamiltonian ${\cal H}$
(i.e.\hspace{0.05cm}\s of the action variables) whereas the line bundles
$l$ satisfying relation (\ref{K}) form the abelian (Prym)
variety (of the angle variables) describing the level set
of ${\cal H}$.
\vskip 0.9cm

\nsection{$SL_2$ moduli space at genus 2}
\vskip 0.5cm

We shall present briefly the description
of the moduli space ${\cal N}_s$ for $G=SL_2$
and $\gamma=2$ which was worked out in \cite{NarRa}.
\vskip 0.4cm

Let us start by recalling some basic facts about
theta functions. We shall use a coordinate rather
than an abstract language. The space of degree $\gamma-1$
holomorphic line bundles
forms a Jacobian torus $J^{\gamma-1}$ of complex dimension
$\gamma$. Fixing a marking (a symplectic homology basis
$(A_a,B_b),\ {a},{b}=1,\dots,\gamma$),
we may identify $J^{\gamma-1}$ with $\NC^\gamma/(\NZ^\gamma
+\tau\NZ^\gamma)$. $\tau\equiv(\tau^{{a}{b}})$
is the period matrix,
i.e. $\tau^{{a}{b}}=\int_{B_{{b}}}\omega^{a}$
where $\omega^{a}$ are the basic holomorphic
forms on $\Sigma$ normalized so that $\int_{A_{a}}
\omega^{b}=\delta^{{a}{b}}$. The point $0\in\NC^\gamma$
corresponds in $J^{\gamma-1}$ to a (marking dependent) spin
structure $S_0$, i.e. a degree 1 bundle such that
$S_0^2=K$. $u\in\NC^\gamma$ describes the line bundle
$V(u)S_0$ where $V(u)$ is the flat line bundle
with the twists ${\rm e}^{2\pi i u^{b}}$ along the $B_{b}$
cycles. The set of degree 1 bundles $l$ with non-trivial
holomorphic sections forms a divisor $\Theta$ of a holomorphic
line bundle $L_\Theta$ over $J^{\gamma-1}$. Holomorphic
sections of the $k$-th power ($k>0$) of $L_\Theta$
are called theta function of order $k$. With the
use of a marking, they may be represented by holomorphic
functions $u\mapsto\theta(u)$ on $\NC^2$ satisfying
\begin{eqnarray}
\theta(u+p+\tau q)\hspace{0.05cm}=\hspace{0.05cm}{\rm e}^{-\pi i k\hspace{0.05cm} q\cdot\tau q-2\pi i
k\hspace{0.05cm} q\cdot u}\ \theta(u)
\end{eqnarray}
for $p,q\in\NZ^\gamma$. The functions
\begin{eqnarray}
\theta_{k,e}(u)\hspace{0.05cm}=\hspace{0.05cm}\sum\limits_{n\in{\dl Z}^\gamma}
{\rm e}^{\hspace{0.05cm}\pi i k\hspace{0.05cm}
(n+e/k)\cdot\tau(n+e/k)\hspace{0.05cm}+\hspace{0.05cm} 2\pi i k\hspace{0.05cm}(n+e/k)\cdot u}
\end{eqnarray}
where $e\in\NZ^\gamma/k\NZ^\gamma$ form a basis of
the theta functions of order $k$. Hence $dim\hspace{0.05cm} H^0(L_\Theta^k)
=k^\gamma.$ In particular, the Riemann theta function
$\theta_{1,0}(u)\equiv\vartheta(u)$ represents
the unique (up to normalization) non-trivial holomorphic
section of $L_\Theta$. It vanishes on the set
$$\{\sum\limits_{i=1}^{\gamma-1}\smallint_{x_0}^{x_i}\omega-
\Delta\ \vert\ x_1\in\Sigma,\dots,x_{\gamma-1}\in\Sigma\}$$
representing the divisor $\Theta$. Here $\Delta\in\NC^\gamma$
denotes the ($x_0$-dependent) vector of Riemann constants.
All theta functions of order 1 and 2 are even functions
of $u$.
\vskip 0.4cm

For $\gamma=2$, the divisor $\Theta$ is formed by the bundles
${\cal O}(x)$ with divisors $x\in\Sigma$. \hspace{0.025cm}${\cal O}(x)=
V(\int_{x_0}^x\omega-\Delta)S_0$. The pullback
of the theta bundle $L_\Theta$ by means of the map
$x\mapsto{\cal O}(x)$ is equivalent to the canonical bundle $K$.
The equivalence assigns 1,0-forms to functions representing
sections of the pullback of $L_\Theta\hspace{0.025cm}$:
\begin{eqnarray}
\epsilon^{{a}{b}}\hspace{0.025cm}\partial_{b}\vartheta(
\smallint_{x_0}^x\omega-\Delta)\quad\mapsto\quad
\omega^a(x)\hspace{0.05cm}.
\label{ide}
\end{eqnarray}
This is consistent since vanishing
of $\vartheta(\int\limits_{x_0}^x\omega-\Delta)$ implies that
\begin{eqnarray}
\partial_{a}\vartheta(\smallint_{x_0}^x\omega-\Delta)\ \omega^{a}(x)=0\ .
\nonumber
\end{eqnarray}
Hence any multivalued function on $\Sigma$ picking up a factor
${\rm e}^{-\pi i\tau^{{a}{a}}-2\pi i(\int_{x_0}^x\omega^{a}
-\Delta^{a})}$ when $x$ goes around the $B_{a}$
cycle and univalued around the $A_{a}$ cycles may be
identified with a 1,0-form on $\Sigma$.
\vskip 0.4cm

As already suggested by the discussion at the end of
Sect.\hspace{0.05cm}\s 2, for the $SL_2$ group it is more convenient to
use the language of holomorphic vector bundles (of rank 2 and
trivial determinant) than to work with principal
$SL_2$-bundles. Of course the first ones are
just associated to the second ones by the fundamental
representation of $SL_2$. Any stable rank 2
bundle $E$ with trivial determinant is an extension
of a degree 1 line bundle $l$
(\cite{NarRa}, Lemmas 5.5 and 5.8), i.e. it appears
in an exact sequence of holomorphic vector bundles
\begin{eqnarray}
0\hspace{0.025cm}\longrightarrow\hspace{0.025cm} l^{-1}\smash{\mathop{\longrightarrow}^\sigma}
\hspace{0.025cm} E\hspace{0.025cm} \smash{\mathop{\longrightarrow}^\varpi}
\hspace{0.05cm} l\hspace{0.025cm} \longrightarrow\hspace{0.025cm} 0\ .
\label{ES}
\end{eqnarray}
The inequivalent extensions (\ref{ES}) are classified
by the cohomology classes in $H^1(l^{-2})$.
This may be seen as follows.
Taking a section of $\varpi$, \hspace{0.025cm} i.e.\hspace{0.05cm}\s a smooth bundle
homomorphism $s:\hspace{0.05cm} l\rightarrow E$ such that $\varpi\circ s=id_l$,
we infer that $\varpi\hspace{0.05cm}\bar\partial s=0$ and hence that $\bar\partial s=\sigma\hspace{0.05cm} b$
for $b$ a 0,1-form with values in $Hom(l,l^{-1})=l^{-2}$,
\hspace{0.025cm} i.e. $b\in\wedge^{01}(l^{-2})$. $b$ is determined up to
$\bar\partial\varphi$ where $\varphi$ is a smooth section of $l^{-2}$,
i.e. $\varphi\in\Gamma(l^{-2})$. The class $[b]$
in $\wedge^{01}(l^{-2})/\Gamma(l^{-2})\hspace{0.05cm}\cong\hspace{0.05cm}
H^1(l^{-2})$ determines the extension (\ref{ES}) up
to equivalence. Each $b$ corresponds to an extension:
one may simply take $E$ equal to $l^{-1}\oplus l$
with the $\bar\partial$-operator given by $\bar\partial_{_{l^{-1}\oplus\hspace{0.025cm} l}}+
\hspace{0.05cm}(\matrix{_0&_b\cr^0&^0})$. \hspace{0.025cm} Proportional $[b]$
correspond to equivalent bundles $E$. If $E$ is a stable
bundle then the extension (\ref{ES}) is necessarily
nontrivial, i.e. $[b]\not=0$.
\vskip 0.4cm

Let $C_E$ denote the set of degree 1 line
bundles $l$ s.t. \hspace{0.05cm}$H^0(l\otimes E)\not=0\hspace{0.05cm}$
(equivalently, s.t. $E$ is an extension of $l$).
This is a complex 1-dimensional variety. It was shown
in \cite{NarRa} that $C_E$ characterizes the bundle
$E$ up to isomorphism and that there exists a
theta function $\theta$ of the 2$^{\rm nd}$ order which
vanishes exactly on $C_E$. The assignment
\hspace{0.05cm}$E\hspace{0.05cm}\mapsto\hspace{0.05cm}\NC^*\theta\hspace{0.05cm}$ gives an injective map
\begin{eqnarray}
m:\hspace{0.05cm}{\cal N}_s\hspace{0.05cm}\longrightarrow\hspace{0.05cm}\NP H^0(L_\Theta^2)\ .
\label{map}
\end{eqnarray}
Let $V(u_1)S_0\equiv l_{u_1}\in C_E$. $E$ may be
realized as an extension of $l_{u_1}$ which is characterized
by $[b]\in H^1(l_{u_1}^{-2})$. Then one may take
\begin{eqnarray}
\theta(u)\ = \int_{_\Sigma}\hspace{0.05cm} K(x;u_1,u)
\wedge b(x)\ .
\label{thetaE}
\end{eqnarray}
where
\begin{eqnarray}
K(x;u_1,u)\ =\ \vartheta(\smallint_{x_0}^x\omega
-u_1-u-\Delta)\ \vartheta(\smallint_{x_0}^x
\omega-u_1+u-\Delta)\cr
\cdot\left(\epsilon^{{a}{b}}\hspace{0.05cm}\partial_{b}\vartheta
(\smallint_{x_0}^x\omega-\Delta)\right)^{-1}
\omega^{a}(x)
\label{Kk}
\end{eqnarray}
(it does not depend on the choice of $a=1,2$). \hspace{0.025cm}
Let us explain the above formulae. \hspace{0.05cm}$K(x;u_1,u)\hspace{0.025cm}$,
\hspace{0.025cm} in its dependence
on \hspace{0.05cm}$x\hspace{0.025cm}$, \hspace{0.025cm} is a multivalued holomorphic 1,0-form.
More exactly, the function
\begin{eqnarray}
x\ \mapsto\ \vartheta(\smallint_{x_0}^x\omega-u_1-u-\Delta
)
\label{s_2}
\end{eqnarray}
is multivalued around the $B_{a}$-cycles picking up the factor
$${\rm e}^{-\pi i\tau^{{a}{a}}\hspace{0.05cm}-\s2\pi i(\int_{x_0}^x
\omega^{a}-u_1^{a}-u^{a}-\Delta^{a})}$$
when \hspace{0.05cm}$x\hspace{0.05cm}$ goes around \hspace{0.05cm}$B_{a}\hspace{0.05cm}$ so that it describes
an element \hspace{0.05cm}$s_2\in H^0(l_{u_1} l_{u})\hspace{0.05cm}$
(non-vanishing if \hspace{0.05cm}$u_1+u\hspace{0.05cm}\not\in\hspace{0.05cm}\NZ^2+\tau\NZ^2\hspace{0.025cm}$).
\hspace{0.025cm} Similarly,
$$x\ \mapsto\ \vartheta(\smallint_{x_0}^x\omega
-u_1+u-\Delta)\
\left(\epsilon^{{a}{b}}\hspace{0.025cm}\partial_{b}\vartheta
(\smallint_{x_0}^x\omega-\Delta)\right)^{-1}
\omega^{a}(x)$$
picks up the factor $${\rm e}^{\s2\pi i(u_1^{a}-u^{a})}$$
when $x$ goes around $B_{a}$ and describes a
holomorphic 1,0-form $\chi$ with values in $l_{u_1}l_u^{-1}\hspace{0.05cm}$
(non-vanishing if \hspace{0.05cm}$u_1-u\hspace{0.05cm}\not\in\hspace{0.05cm}\NZ^2+\tau\NZ^2\hspace{0.05cm}$).
\hspace{0.05cm} The product \hspace{0.05cm}$s_2\chi=K(\hspace{0.025cm}\cdot\hspace{0.05cm};u_1,u)\hspace{0.05cm}$
is a holomorphic 1,0-form with values in \hspace{0.05cm}$l_{u_1}^{\m2}\hspace{0.05cm}$
and it may be paired with \hspace{0.05cm}$b\in\wedge^{01}(l_{u_1}^{-2})\hspace{0.05cm}$ via
the integral over \hspace{0.05cm}$x\hspace{0.05cm}$ on the r.h.s. of Eq.\hspace{0.05cm}\s(\ref{thetaE}).
The integral is independent of the choice of the representative $b$
of the cohomology class $[b]$. In its dependence on $u$,
\hspace{0.025cm}$K(x;u_1,u)$ is a theta
function of the 2$^{\rm nd}$ order and so is $\theta(u)$.
In Appendix 1 we check explicitly that $\theta$ given
by Eq.\hspace{0.05cm}\s(\ref{thetaE}) possesses the required property.
\vskip 0.4cm

The product of the two shifted Riemann
theta functions \hspace{0.05cm}$\vartheta(u'-u)\hspace{0.05cm}
\vartheta(u'+u)\hspace{0.05cm}$ is a theta function
of the 2$^{\rm nd}$ order both in $u'$ and in $u$
(and it is invariant under the interchange
$u'\leftrightarrow u$). Let $\iota$ denote the (marking
dependent) linear isomorphism between the spaces
$H^0(L_\Theta^2)^*$ and $H^0(L_\Theta^2)$ defined by
\begin{eqnarray}
\iota(\phi)(u)\hspace{0.05cm}=\hspace{0.05cm}{\langle\hspace{0.025cm}}\vartheta(\hspace{0.05cm}\cdot\hspace{0.05cm}-u)
\hspace{0.05cm}\hspace{0.025cm}\vartheta(\hspace{0.05cm}\cdot\hspace{0.05cm}+u)\hspace{0.05cm}\hspace{0.025cm},
\hspace{0.025cm}\hspace{0.05cm}\s\phi{\hspace{0.025cm}\rangle}
\label{lini}
\end{eqnarray}
An easy calculation shows that
\begin{eqnarray}
\vartheta(u'-u)\hspace{0.05cm}\s\vartheta(u'+u)
\ =\ \sum\limits_{e}\theta_{2,e}(u')\hspace{0.05cm}\s\theta_{2,e}(u)\hspace{0.05cm}.
\end{eqnarray}
Hence $\iota$ interchanges the basis $(\theta_{2,e})$
of $H^0(L_\Theta^2)$ with the dual basis $(\theta_{2,e}^{\hspace{0.05cm}*})$
of $H^0(L_\Theta^2)^*$. Denote by $\phi_{u}$ the linear form
on $H^0(L_\Theta^2)$ that computes the value of the theta
function at point $u\in\NC^2$. The Kummer quartic
${\cal K}^*\subset H^0(L_\Theta^2)^*$, \hspace{0.05cm}${\cal K}^*
=\{\hspace{0.025cm}\NC^*\phi_{u'}\hspace{0.05cm}\vert\hspace{0.05cm} u'\in\NC^2\hspace{0.025cm}\}\hspace{0.025cm}$
is mapped by the isomorphism $\iota$ into a quartic
\hspace{0.05cm}${\cal K}\subset H^0(L_\Theta^2)\hspace{0.05cm}$ of theta functions
proportional to
\begin{eqnarray}
u\ \mapsto\ \vartheta(u'-u)\hspace{0.05cm}\s\vartheta(u'+u)
\nonumber
\end{eqnarray}
for some $u'\in\NC^2$.
\vskip 0.4cm

One may define a projective action of $(\NZ/2\NZ)^4$
on $H^0(L_\Theta^2)$ by assigning to an element
$(e,e')\in(\NZ/2\NZ)^4$, with $e,e'=(0,0),\hspace{0.05cm}(1,0),
\hspace{0.05cm}(0,1)$ or $(1,1)$, a linear transformation $U_{e,e'}$ s.t.
\begin{eqnarray}
(U_{e,e'}\theta)(u)\ =\ {\rm e}^{\hspace{0.025cm}{1\over2}\hspace{0.025cm}\pi i\hspace{0.05cm}
e'\cdot\hspace{0.025cm}\tau\hspace{0.025cm} e'\hspace{0.05cm}+\hspace{0.05cm} 2\pi i\hspace{0.05cm} e'\cdot\hspace{0.025cm} u}\hspace{0.05cm}\s
\theta(u+{_1\over^2}(e+\tau e'))\hspace{0.05cm}.
\label{disa}
\end{eqnarray}
The relation \hspace{0.05cm}$U_{e_1,e_1'}\hspace{0.05cm} U_{e_2,e_2'}\ =\
(-1)^{e_1\cdot e_2'}\hspace{0.05cm} U_{e_1+e_2,\hspace{0.025cm} e_1'+e_2'}\hspace{0.025cm}$
holds so that $U$ lifts to the Heisenberg group.
In the action on the basic theta functions,
\begin{eqnarray}
U_{e_1,e_1'}\hspace{0.025cm}\theta_{2,e}\ =\ (-1)^{e_1\cdot\hspace{0.025cm} e}
\hspace{0.05cm}\s\theta_{2,\hspace{0.025cm} e+e_1'}\hspace{0.05cm}.
\label{acone}
\end{eqnarray}
The marking-dependence of the isomorphism
$\iota$ of Eq.\hspace{0.05cm}\s(\ref{lini}) is given by the action
of $(\NZ/2\NZ)^4$. It is easy to check that this
action preserves \hspace{0.05cm}${\cal K}$ and that the transposed action
of $(\NZ/2\NZ)^4$ preserves ${\cal K}^*$. The $(\NZ/2\NZ)^4$
symmetry of the Kummer quartics allows to find easily
their defining equation, see Appendix 3.
\vskip 0.4cm

It was shown in \cite{NarRa} that the image of ${\cal N}_s$
under the map (\ref{map}) contains all non-zero
theta functions of the 2$^{\rm nd}$ order except the
ones in the the Kummer quartic ${\cal K}$.
The latter correspond, however, to the (Seshadri equivalence
classes of) semistable but not stable bundles so that the map
$m$ extends to an isomorphism between ${\cal N}_{ss}$ and
$\NP H^0(L_\Theta^2)$ showing that ${\cal N}_{ss}$ is a smooth
projective variety.
\vskip 0.9cm

\nsection{Cotangent bundle}
\vskip 0.5cm

Let us describe the cotangent space of
${\cal N}_s$ at point $E$. The covectors tangent
to ${\cal N}_s$ at $E$ may be identified with holomorphic
1,0-forms $\Psi$ with values in the bundle of traceless
endomorphisms of $E$. We may assume that $E$ is an
extension of a line bundle $l$ of degree 1 realized as
$l^{-1}\oplus l$ with $\bar\partial_{_E}=\bar\partial_{_{l^{-1}\oplus\hspace{0.025cm} l}}
\hspace{-0.08cm}+B$ where \hspace{0.05cm}$B=(\matrix{_0&_b\cr^0&^0})\hspace{0.025cm}$.
\hspace{0.05cm} Then
\begin{eqnarray}
\Psi\ =\ \left(\matrix{-\mu&\nu\cr\eta&\mu}\right)
\label{Lax}
\end{eqnarray}
where $\mu\in\wedge^{10}$, $\nu\in\wedge^{10}(l^{-2})$,
$\eta\in\wedge^{10}(l^2)$ and
\begin{eqnarray}
\bar\partial_{_{l^2}}\eta=0\hspace{0.05cm},\hspace{1cm}\bar\partial\mu=-\eta\wedge b\hspace{0.05cm},
\hspace{1cm}\bar\partial_{_{l^{-2}}}\nu=2\mu\wedge b\ .
\label{rel}
\end{eqnarray}
It is easy to relate the above description of covectors
tangent to ${\cal N}_s$ to the one of Sect.\hspace{0.05cm}\s2.
Let ${\cal U}:\hspace{0.05cm} l^{-1}\oplus l\rightarrow\Sigma\times\NC^2$ be
a smooth isomorphism of rank 2 bundles with trivial determinant.
Then ${\cal U}\hspace{0.05cm}\bar\partial_{_E}{\cal U}^{-1}=\bar\partial+A$ for a certain $sl_2$-valued 0,1-form
$A$ and $\Phi={\cal U}\Psi{\cal U}^{-1}$
satisfies $\bar\partial_{_{A}}\Phi=0$. The ${\cal G}$ orbit
of $(A,\Phi)$ is independent of the choice of ${\cal U}$
and the quadratic Hitchin Hamiltonian takes value
${1\over 2}\hspace{0.05cm} tr\hspace{0.05cm}(\Phi)^2$ on it. The latter expression
is clearly equal to ${_1\over^2}\hspace{0.05cm} tr\hspace{0.05cm}(\Psi)^2
=\hspace{0.05cm}\mu^2+\eta\hspace{0.05cm}\nu$ which, as easily follows from relations
(\ref{rel}), defines a holomorphic quadratic differential. Hence
\begin{eqnarray}
{\cal H}(E,\Psi)\hspace{0.05cm}=\hspace{0.05cm} \mu^2+\eta\hspace{0.05cm}\nu\ .
\label{hit}
\end{eqnarray}
We would like to express the latter using the theta function
description of $T^*{\cal N}_{ss}=T^*\NP H^0(L_{\Theta}^2)$
where the covectors tangent to ${\cal N}_{ss}$ at $\NC^*\theta$
are represented by linear forms $\phi$ on $H^0(L_{\Theta}^2)$
\hspace{0.025cm} s.t. ${\langle\hspace{0.025cm}}\theta,\hspace{0.025cm}\phi{\hspace{0.025cm}\rangle}=0\hspace{0.025cm}$.

\vskip 0.4cm

Let $l=l_{u_1}\in C_E$, \hspace{0.025cm} i.e. $\theta(u_1)=0$
for the theta function corresponding to $E$.
We shall assume that $l^2\not=K$ i.e.
that \hspace{0.05cm}$2u_1\hspace{0.05cm}\not\in\hspace{0.05cm}\NZ^2+\tau\NZ^2$.
\hspace{0.025cm} An infinitesimal variation $\delta E$ of the bundle $E$
in ${\cal N}_s$ may be achieved by changing
\hspace{0.05cm}$\bar\partial_{_E}=\bar\partial_{_{l^{-1}\oplus\hspace{0.025cm} l}}+B\hspace{0.05cm}$ with \hspace{0.05cm}$B=
(\matrix{_0&_b\cr^0&^0})\hspace{0.05cm}$ to
\begin{eqnarray}
\bar\partial_{_{l^{-1}\oplus\hspace{0.025cm} l}}+\left(\matrix{\pi\hspace{0.025cm}\delta
u_1({\rm Im}\hspace{0.05cm}\tau)^{-1}\bar\omega
&b+\delta b\cr0&-\pi\hspace{0.025cm}\delta u_1({\rm Im}\hspace{0.05cm}\tau)^{-1}\bar\omega}
\right)\ \equiv\ \bar\partial_{_E}+\hspace{0.05cm}\delta B
\label{DB}
\end{eqnarray}
(all other variations of $\bar\partial_{_E}$ may be obtained from (\ref{DB})
by infinitesimal gauge transformations). Clearly
\begin{eqnarray}
{\langle\hspace{0.025cm}}\delta E\hspace{0.05cm},\hspace{0.05cm}\Psi{\hspace{0.025cm}\rangle}\ =\ \int_{_\Sigma}
tr\hspace{0.05cm}\s\Psi\wedge\delta B\ =\ -\hspace{0.05cm} 2\pi
\delta u_1({\rm Im}\hspace{0.05cm}\tau)^{-1}\int_{_\Sigma}\mu\wedge\bar
\omega\hspace{0.05cm}+\hspace{0.05cm}\int_{_\Sigma}\eta\wedge\delta b\ .
\label{25}
\end{eqnarray}
Note that the line bundle $l_{u_1}$ with the $\bar\partial$-operator
changed to $\bar\partial_{_{l_{u_1}}}\hspace{-0.2cm}
-\pi\hspace{0.025cm}\delta u_1({\rm Im}\hspace{0.05cm}\tau)^{-1}
\bar\omega$ is equivalent to $l_{u_1+\delta u_1}\equiv l'$ and the
equivalence is established by multiplication by the
multivalued function
$x\mapsto{\rm e}^{\hspace{0.05cm} 2\pi i\hspace{0.05cm}\delta u_1({\rm Im}\hspace{0.05cm}\tau)^{-1}
\int_{x_0}^x{\rm Im}\hspace{0.05cm}\omega}$. Hence $l^{-1}\oplus l$
with the $\bar\partial$-operator given by Eq.\hspace{0.05cm}\s(\ref{DB}) is equivalent to
${l'}^{-1}\oplus l'$ with the $\bar\partial$-operator $\bar\partial_{_{{l'}^{-1}\oplus
\hspace{0.025cm} l'}}+\hspace{0.05cm}(\matrix{_0&_{b+\delta'b}\cr^0&^0})\hspace{0.05cm}$ where
\hspace{0.05cm}$\delta'b(x)\ =\ \delta b\hspace{0.05cm} -\hspace{0.05cm} 4\pi i\hspace{0.05cm}\delta u_1({\rm Im}
\hspace{0.05cm}\tau)^{-1}(\smallint\limits_{x_0}^x{\rm Im}\hspace{0.05cm}\omega)\hspace{0.05cm} b(x)\hspace{0.025cm}.$
\hspace{0.05cm} The last bundle corresponds by the relation (\ref{thetaE})
to the theta function
\begin{eqnarray}
(\theta+\delta\theta)(u)\ =\ \int_{_\Sigma}
K(x;\hspace{0.025cm} u_1+\delta u_1,\hspace{0.025cm} u)\wedge (b(x)+\delta'b(x))\ .
\nonumber
\end{eqnarray}
Hence $\delta E$ is represented by the variation
\begin{eqnarray}
\delta\theta(u)\ =\
-\ 2\pi\hspace{0.025cm}\delta u_1^{a}\hspace{0.05cm} ({\rm Im}\hspace{0.05cm}\tau)^{-1}_{\hspace{0.05cm}{a}{b}}
\int_{_\Sigma}L^b(x;u_1,u)\wedge b(x)\hspace{0.05cm}
+\hspace{0.05cm}\int_{_\Sigma}K(x;u_1,u)\wedge
\delta b(x)
\label{nd}
\end{eqnarray}
of the theta function, where
\begin{eqnarray}
L^{a}(x;u_1,u)\ =\ K(x;u_1,u)\hspace{0.05cm}
\smallint_{x_0}^x(\omega^{a}-\bar\omega^{a})
\hspace{0.05cm}-\hspace{0.05cm}{_1\over^{2\pi}}{\rm Im}\hspace{0.05cm}\tau^{{a}{b}}
\hspace{0.05cm}\partial_{u_1^{b}}K(x;u_1,u)\ .
\label{gal}
\end{eqnarray}
Note that as functions of $x$, \hspace{0.05cm}$L^a(x;u_1,u)$
are 1,0-forms with values in $l_{u_1}^2$
(as are $K(x;u_1,u)$). They are not holomorphic:
\begin{eqnarray}
\bar\partial_x\hspace{0.025cm} L^{a}(x;u_1,u)\
=\ K(x;u_1,u)\wedge\bar\omega^{a}(x)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
As functions of $u$, \hspace{0.025cm}$L^{a}(x;u_1,u)$ are theta
functions of the 2$^{\rm nd}$ order.
\vskip 0.4cm

We would like to find an explicit form of the Lax matrix $\Psi$
representing the linear form $\phi$ on $H^0(L_{\Theta}^2)$
s.t. ${\langle\hspace{0.025cm}}\theta,\hspace{0.025cm}\phi{\hspace{0.025cm}\rangle}=0$. We shall
achieve this goal partially, finding the entries $\eta$
and $\mu$ of the matrix (\ref{Lax}). The correspondence
between $\Psi$ and $\phi$ is determined by the equality
\begin{eqnarray}
{\langle\hspace{0.025cm}}\delta E\hspace{0.05cm},\hspace{0.05cm}\Psi{\hspace{0.025cm}\rangle}\
=\ {\langle\hspace{0.025cm}}\delta\theta\hspace{0.05cm},\hspace{0.05cm}\phi{\hspace{0.025cm}\rangle}
\nonumber
\end{eqnarray}
Since the left hand side is given by Eq.\hspace{0.05cm}\s(\ref{25})
and $\delta\theta$ by Eq.\hspace{0.05cm}\s(\ref{nd}), we obtain
\begin{eqnarray}
&&\hspace{1cm}-\hspace{0.05cm} 2\pi\hspace{0.025cm}
\delta u_1({\rm Im}\hspace{0.05cm}\tau)^{-1}\int_{_\Sigma}\mu\wedge\bar
\omega\hspace{0.05cm}+\hspace{0.05cm}\int_{_\Sigma}\eta\wedge\delta b\cr
&&=\
-\m2\pi\hspace{0.025cm}\delta u_1^{a}\hspace{0.05cm} ({\rm Im}\hspace{0.05cm}\tau)^{-1}_{\hspace{0.05cm}{a}{b}}
\int_{_\Sigma}\hspace{-0.2cm}{\langle\hspace{0.025cm}}L^b(x;u_1,\hspace{0.05cm}\cdot\hspace{0.05cm})\hspace{0.025cm},
\hspace{0.05cm}\phi{\hspace{0.025cm}\rangle}\wedge\hspace{0.025cm} b(x)\hspace{0.05cm}
+\hspace{0.05cm}\int_{_\Sigma}\hspace{-0.2cm}
{\langle\hspace{0.025cm}}K(x;u_1,\hspace{0.05cm}\cdot\hspace{0.05cm})\hspace{0.025cm},\hspace{0.05cm}\phi{\hspace{0.025cm}\rangle}\wedge\hspace{0.025cm}\delta b(x)\hspace{0.05cm}.
\hspace{0.5cm}
\label{equll}
\end{eqnarray}
Taking $\delta u_1=0$ we infer that
\begin{eqnarray}
\eta(x)\ =\ {\langle\hspace{0.025cm}}K(x;u_1,\hspace{0.05cm}\cdot\hspace{0.05cm})\hspace{0.05cm},\hspace{0.05cm}\phi{\hspace{0.025cm}\rangle}
\label{eta}
\end{eqnarray}
is the lower left entry of the matrix $\Psi$ corresponding
to the linear form $\phi$.
\vskip 0.4cm

It is easy to find the entry $\mu$ of $\Psi$
representing the linear form $\phi_{u_1}$ (recall
that $\phi_{u_1}$ computes the value of a theta
function in $H^0(L_\Theta^2)$ at point $u_1$).
Since $K(x;u_1,u_1)=0\hspace{0.025cm}$, it follows from Eq.\hspace{0.05cm}\s(\ref{eta})
that $\eta=0$ in this case. Eq.\hspace{0.05cm}\s(\ref{equll}) reduces
then to
\begin{eqnarray}
&&-\hspace{0.05cm} 2\pi\hspace{0.025cm}\delta u_1({\rm Im}\hspace{0.05cm}\tau)^{-1}
\int_{_\Sigma}\mu\wedge\bar
\omega\hspace{0.05cm}=\hspace{0.05cm}\delta u_1^{a}\int_{_\Sigma}
\partial_{u_1^{a}} K(x;u_1,u_1)\wedge b(x)\hspace{0.05cm}\cr
&&=\hspace{0.05cm} -\hspace{0.05cm}\delta u_1^{a}\int_{_\Sigma}
\partial_{u^{a}} K(x;u_1,u_1)\wedge b(x)\hspace{0.05cm}
=\hspace{0.05cm}-\hspace{0.05cm}\delta u_1^{a}\hspace{0.05cm}\s\partial_{a}
\theta(u_1)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
This fixes $\mu$ uniquely:
\begin{eqnarray}
\mu\ =\ {_i\over^{4\pi}}\hspace{0.05cm}\partial_{a}\theta(u_1)\hspace{0.05cm}\s
\omega^{a}\ .
\label{mu}
\end{eqnarray}
Let us check that there exists $\nu\in\wedge^{10}(l_{u_1}^{-2})$
such that the last equation of (\ref{rel}) holds. For this
it is necessary and sufficient that
\begin{eqnarray}
\int_{_\Sigma}{\kappa}\hspace{0.025cm}\hspace{0.05cm}\mu\wedge b\ =\ 0
\label{cons}
\end{eqnarray}
for a non-zero holomorphic section ${\kappa}$ of \hspace{0.025cm}$l_{u_1}^2=
V(2u_1)K$
(\hspace{0.025cm}$dim\hspace{0.05cm} H^0(l_{u_1}^2)=1$ if \hspace{0.05cm}$2u_1\hspace{0.05cm}\not\in\hspace{0.05cm}\NZ^2+\tau\NZ^2$).
\hspace{0.025cm} But such a section may be represented by the function
\begin{eqnarray}
x\ \mapsto \ \vartheta(\smallint_{x_0}^x\omega-2u_1-\Delta)
\nonumber
\end{eqnarray}
so that, recalling the definition (\ref{Kk}), we obtain
\begin{eqnarray}
\int_{_\Sigma}{\kappa}\hspace{0.025cm}\hspace{0.05cm}\omega^{a}\wedge b\ =\
\int_{_\Sigma}
\epsilon^{{a}{b}}\hspace{0.05cm}\partial_{u^{b}} K(x;u_1,u_1)\wedge b(x)
\ =\ \epsilon^{{a}{b}}\hspace{0.05cm}\s\partial_{b}\theta(u_1)\ .
\label{kob}
\end{eqnarray}
Hence the relation (\ref{cons}) follows for $\mu$ given by
Eq.\hspace{0.05cm}\s(\ref{mu}). The 1,0-form $\nu$ satisfying the last relation
of (\ref{rel}) is now unique since $H^0(l_{u_1}^{-2}K)=\{0\}$.
\vskip 0.4cm

We would like to find the entry $\mu$
of $\Psi$ corresponding to more general linear forms
$\phi$ s.t. ${\langle\hspace{0.025cm}}\theta,
\hspace{0.025cm}\phi{\hspace{0.025cm}\rangle}=0$. Recall that $\theta$
with $\theta(u_1)=0$ may be given by formula (\ref{thetaE})
with $b\in\wedge^{0,1}(l_{u_1}^{-2})$. Note that any
2$^{\rm nd}$-order theta function \hspace{0.05cm}$\delta\theta$
vanishing at $u_1$ and not in the Kummer quartic ${\cal K}$
may be written as
\begin{eqnarray}
\delta\theta(u)\ =\ \int_{_\Sigma}K(x;u_1,u)\wedge
\delta b(x)
\label{dth}
\end{eqnarray}
with $\delta b\in\wedge^{01}(l_{u_1}^{-2})$ since
it corresponds to an extension of $l_{u_1}$.
The space of $\delta\theta$ vanishing at $u_1$
is 3-dimensional, as well as the space
$H^1(l_{u_1}^{-2})$ of classes $[\delta b]$
and the assumption that $\delta\theta\not\in{\cal K}$
is obviously superfluous. Set for a linear form $\psi$ on
$H^0(L_{\Theta}^2)$,
\begin{eqnarray}
\eta_\psi(x)\hspace{0.05cm}=\ {\langle\hspace{0.025cm}}K(x;u_1,\hspace{0.05cm}\cdot\hspace{0.05cm})\hspace{0.025cm},\hspace{0.05cm}\psi{\hspace{0.025cm}\rangle}\ .
\label{etab}
\end{eqnarray}
$\eta_\psi$ defines a holomorphic 1,0-form with values
in $l_{u_1}^2$. We have
\begin{eqnarray}
{\langle\hspace{0.025cm}}\delta\theta\hspace{0.025cm},\hspace{0.025cm}\psi{\hspace{0.025cm}\rangle}\ =\hspace{0.05cm}\int_{_\Sigma}
\eta_\psi\wedge\delta b
\label{dta}
\end{eqnarray}
for \hspace{0.05cm}$\delta\theta\hspace{0.05cm}$ given by Eq.\hspace{0.05cm}\s(\ref{dth}). By
dimensional count, the map $\psi\mapsto\eta_\psi$ is onto
$H^0(l_{u_1}^2K)$ with the 1-dimensional kernel spanned
by $\phi_{u_1}$. Specifying Eq.\hspace{0.05cm}\s(\ref{dta}) to $\delta\theta
\propto\theta$, we obtain the relation
\begin{eqnarray}
{\langle\hspace{0.025cm}}\theta\hspace{0.025cm},\hspace{0.025cm}\psi{\hspace{0.025cm}\rangle}\ =\hspace{0.05cm}\int_{_\Sigma}\eta_\psi
\wedge b
\label{dtE}
\end{eqnarray}
which determines the class $[b]\in H^1(l_{u_1}^{-2})$ in terms
of $\theta$. On the other hand, taking $\psi=\phi$
in Eq.\hspace{0.05cm}\s(\ref{etab}), we infer that $\eta=0\hspace{0.025cm}$ if and only
if $\phi$ is proportional to $\phi_{u_1}$,
the case studied before.
\vskip 0.4cm

If $\eta_\phi\not=0$ then $\mu$ depends on the choice
of the representative $b$ in the class $[b]\in
H^1(l_{u_1}^{-2})$ characterizing $E$ as the extension
of $l_{u_1}$. Under the transformation $b\mapsto b+\bar\partial\varphi$
where $\varphi$ is a section of $l_{u_1}^{-2}$,
\begin{eqnarray}
\eta\mapsto\eta\hspace{0.025cm},\ \quad\mu\mapsto\mu+\varphi\hspace{0.025cm}\eta\hspace{0.025cm},\ \quad
\nu\mapsto\nu-2\hspace{0.025cm} \varphi\hspace{0.025cm}\mu-\varphi^2\eta\hspace{0.05cm}.
\nonumber
\end{eqnarray}
The pairing of the theta functions $L^a(x;u_1,\hspace{0.05cm}\cdot\hspace{0.05cm})$
of Eq.\hspace{0.05cm}\s(\ref{gal}) with the linear form $\phi$
gives two 1,0-forms with values in $l_{u_1}^2$:
\begin{eqnarray}
\chi^{a}(x)\ =\ {\langle\hspace{0.025cm}}L^{a}(x;\hspace{0.025cm}\cdot\hspace{0.05cm},u_1)\hspace{0.025cm},
\hspace{0.05cm}\phi{\hspace{0.025cm}\rangle}\quad\ \
{\rm s.t.}\quad\ \bar\partial\chi^{a}\hspace{0.05cm}
=\hspace{0.05cm}\eta\wedge\bar\omega^{a}\hspace{0.05cm}.
\label{cih}
\end{eqnarray}
Specifying the equality (\ref{equll}) to the case
with $\delta b=0$, we infer the relation
\begin{eqnarray}
\int_{_\Sigma}\mu\wedge\bar\omega^a\ =\ \int_{_\Sigma}\chi^a
\wedge b
\label{cfm}
\end{eqnarray}
which, together with the equation
\begin{eqnarray}
\bar\partial\mu=-\eta\wedge b
\label{cfm1}
\end{eqnarray}
determines $\mu$ completely. In Appendix 2, we show
that $\mu$ fixed this way satisfies the relation
$\int_{_\Sigma}{\kappa}\hspace{0.05cm}\mu\wedge b=0$ and hence defines
a unique 1,0-form $\nu$ with values in
$l_{u_1}^{-2}$ s.t. $\bar\partial\nu=2\hspace{0.025cm}\mu\wedge b$.
\vskip 0.9cm

\nsection{Hitchin Hamiltonians}
\vskip 0.5cm

From the relation (\ref{hit}) and the explicit form of
$\Psi$ corresponding to $\phi_{u_1}$ ($\eta$ vanishing, $\mu$
given by Eq.\hspace{0.05cm}\s(\ref{mu})\hspace{0.025cm}),
one obtains
\begin{eqnarray}
{\cal H}(\hspace{0.025cm}\theta\hspace{0.025cm},\hspace{0.05cm} a_1\hspace{0.025cm}\phi_{u_1})\ =\ -\hspace{0.025cm}{_1
\over^{16\pi^2}}\hspace{0.05cm} a_1^2\hspace{0.05cm}\s
(\hspace{0.05cm} \partial_{a}\theta(u_1)\hspace{0.05cm}\s\omega^{a}\hspace{0.05cm})^2\hspace{0.05cm}.
\label{first}
\end{eqnarray}
The right hand side is a quadratic differential.
Eq.\hspace{0.05cm}\s(\ref{first}), whose projective
version was first obtained in \cite{VGP},
is consistent with the rescaling $\theta\mapsto
t\hspace{0.05cm}\theta$ and $\phi\mapsto t^{-1}\phi$ for $t\in\NC^*$.
It describes the value of the Hitchin map
${\cal H}$ on the special covectors, namely those
represented by the pairs $(\theta\hspace{0.025cm},\hspace{0.05cm}\phi)$
s.t. $\NC^*\phi$ is in the intersection ${\cal K}^*_E$
of the Kummer quartic ${\cal K}^*$ with the plane
${\langle\hspace{0.025cm}}\theta,\hspace{0.025cm}\phi{\hspace{0.025cm}\rangle}=0$.
The linear span of ${\cal K}^*_E$ gives the whole cotangent
space $T^*_E{\cal N}_{ss}$. Indeed, any theta function of
the 2$^{\rm nd}$ order $\delta\theta$ which vanishes
on $C_E$ has to be proportional to $\theta$ and defines
a zero vector in $T_E{\cal N}_{ss}$.
${\cal K}^*_E$ is itself a quartic. Hence the restriction
of the quadratic polynomial ${\cal H}$ to six lines in ${\cal K}^*_E$
in a general position determines ${\cal H}$ completely.
\vskip 0.4cm

It is possible to find a more explicit description
of the values of ${\cal H}$ away from ${\cal K}^*_E$ and this is
the main aim of the rest of the present section.
Suppose then that the entry $\eta$ in $\Psi$
does not vanish. Let $x_i$, $i=1,\dots,4$, be its four zeros.
We shall assume that $\eta$ cannot be written
as $\kappa\hspace{0.025cm}\omega$ for $\kappa\in H^0(l_{u_1}^2)$ and
$\omega\in H^0(K)$. This is true for
generic $\phi$. In this case, \hspace{0.05cm}$\eta\hspace{0.025cm}=\hspace{0.025cm}
a_2\hspace{0.025cm}\eta_{\phi_{u_2}}\hspace{0.05cm}$ for some $a_2\in\NC^*$
and for $u_2$ satisfying
\begin{eqnarray}
u_1+u_2\hspace{0.05cm}=\hspace{0.05cm}\smallint_{x_0}^{x_1}\omega
+\smallint_{x_0}^{x_2}\omega-2\Delta\ \quad{\rm and}\ \quad
u_1-u_2\hspace{0.05cm}=\hspace{0.05cm} \smallint_{x_0}^{x_3}\omega
+\smallint_{x_0}^{x_4}\omega-2\Delta\hspace{0.05cm},
\label{upm}
\end{eqnarray}
$u_1\pm u_2\hspace{0.05cm}\not\in\hspace{0.05cm}\NZ+\tau\NZ$.
\hspace{0.05cm} Indeed, \hspace{0.025cm}$\eta_{\phi_{u_2}}(x)\hspace{0.025cm}$
is a holomorphic section of $l_{u_1}^2K$
represented by the multivalued function
\hspace{0.05cm}$\vartheta(\smallint_{x_0}^x
\omega-u_1-u_2-\Delta)\hspace{0.05cm}\hspace{0.025cm}\vartheta(\smallint_{x_0}^x\omega
-u_1+u_2-\Delta)\hspace{0.05cm}$ vanishing exactly
at $x_i$ and such a section is unique up to normalization.
We infer that in the action on the theta functions
of Eq.\hspace{0.05cm}\s(\ref{dth}), the linear
forms $\phi$ and $a_2\hspace{0.025cm}\phi_{u_2}$
coincide. Since Eq.\hspace{0.05cm}\s(\ref{dth}) gives all theta functions
vanishing at $u_1$, it follows that
\begin{eqnarray}
\phi\hspace{0.05cm} =\hspace{0.05cm} a_1\hspace{0.05cm}\phi_{u_1}\hspace{0.05cm}+\hspace{0.05cm} a_2\hspace{0.05cm}\phi_{u_2}
\label{alpha}
\end{eqnarray}
for some $a_1\in\NC$. Let us stress that, to fix normalizations,
$u_1$ and $u_2$ should be viewed as elements of $\NC^2$
with $x_i$ in relations (\ref{upm}) belonging to the covering space
$\widetilde\Sigma$ of $\Sigma$. The relation
${\langle\hspace{0.025cm}}\theta\hspace{0.025cm},\hspace{0.025cm}\phi{\hspace{0.025cm}\rangle}\hspace{0.05cm}=0$
implies that $\theta(u_2)=0$.
\vskip 0.4cm

Summarizing, we have shown that a generic pair $(\theta,\phi)$
s.t. ${\langle\hspace{0.025cm}}\theta,\hspace{0.025cm}\phi{\hspace{0.025cm}\rangle}=0$
may be obtained by first choosing $u_1$ and $u_2$
s.t. $2u_1,\hspace{0.025cm} 2u_2,\hspace{0.025cm} u_1\pm u_2\hspace{0.05cm}\not\in\hspace{0.05cm}\NZ+\tau\NZ$
\hspace{0.025cm} and then taking $\theta$ from the 2-dimensional space of theta
functions vanishing at $u_1$ and $u_2$ and $\phi$ from
the orthogonal subspace.
The zeros $x_i$ of $\eta$ are determined from Eqs.\hspace{0.05cm}\s(\ref{upm})
(as the zeros of $\vartheta(\smallint_{x_0}^x
\omega-u_1\pm u_2-\Delta)$). For simplicity, we shall
assume that they are distinct (this is true for generic $\phi$).
Then the differentials $\partial\eta(x_i)\in(l_{u_1}^2K^2)_{x_i}$
do not vanish.
\vskip 0.4cm

A quadratic differential $\rho\in H^0(K^2)$
is determined by its values at four points $x_i$ which form
a divisor of $l_{u_1}^2K\not=K^2$. Since ${dim}\hspace{0.025cm} H^0(K^2)=3$,
there is one linear relation satisfied by all $\rho(x_i)\hspace{0.025cm}$:
\begin{eqnarray}
\sum\limits_{i=1}^4\rho(x_i)\hspace{0.05cm}{\kappa}(x_i)\hspace{0.05cm}\partial\eta(x_i)^{-1}\hspace{0.05cm}=\hspace{0.05cm} 0
\nonumber
\end{eqnarray}
for $0\not={\kappa}\in H^0(l_{u_1}^2)$. \hspace{0.025cm} It expresses the fact that
the sum of residues of the meromorphic 1,0-form $\rho\hspace{0.025cm}{\kappa}
\hspace{0.025cm}\eta^{-1}$ has to vanish. For $\rho={\cal H}(\theta,\phi)=\mu^2+\eta
\nu$,
\begin{eqnarray}
\rho(x_i)\hspace{0.05cm}=\hspace{0.05cm}\mu(x_i)^2
\nonumber
\end{eqnarray}
so that it is enough to know $\mu(x_i)$ in order to determine
\hspace{0.025cm}${\cal H}(\theta,\phi)$. \hspace{0.025cm} Note that although
the 1,0-form $\mu$ depends on the choice of the representative $b$
of the class $[b]\in H^1(l_{u_1}^{-2})$ defined by Eq.\hspace{0.05cm}\s(\ref{dtE}),
the values $\mu(x_i)$ are invariant since under $b\mapsto b+\bar\partial\varphi$
the 1,0-form $\mu$ changes to $\mu+\varphi\hspace{0.025cm}\eta$.
\vskip 0.4cm

It remains to find $\mu(x_i)$. Consider the meromorphic function
$\eta_\psi\eta^{-1}$. Viewed as a distribution,
$\bar\partial(\eta_\psi\eta^{-1})$ is supported at the poles
of $\eta_\psi\eta^{-1}$ and
\begin{eqnarray}
\int_{_\Sigma}\mu\wedge\bar\partial(\eta_\psi\eta^{-1})\hspace{0.05cm}=\hspace{0.05cm}
-2\pi i\sum\limits_{i=1}^4\mu(x_i)\hspace{0.025cm}\eta_\psi(x_i)\hspace{0.05cm}\partial\eta(x_i)^{-1}
\nonumber
\end{eqnarray}
for any (smooth) 1,0-form $\mu$. \hspace{0.025cm} In particular, for $\mu$
satisfying Eq.\hspace{0.05cm}\s(\ref{cfm1}) we obtain
\begin{eqnarray}
\sum\limits_{i=1}^4\mu(x_i)\hspace{0.05cm}\eta_\psi(x_i)\hspace{0.05cm}\partial\eta(x_i)^{-1}
\hspace{0.05cm}=\hspace{0.05cm}{_1\over^{2\pi i}}\int_{_\Sigma}\eta_\psi\wedge b\hspace{0.05cm}
=\hspace{0.05cm}{_1\over^{2\pi i}}\hspace{0.05cm}\hspace{0.025cm}{\langle\hspace{0.025cm}}\theta,\hspace{0.025cm}\psi{\hspace{0.025cm}\rangle}\hspace{0.05cm}.
\label{21}
\end{eqnarray}
Recall that $\eta_\psi$ run through the three-dimensional
space $H^0(l_{u_1}^2K)$. If $\eta_\psi(x_i)=0$ for all $i$
then $\eta_\psi$ has to be proportional to
$\eta=a_2\hspace{0.025cm}\eta_{\phi_{u_2}}$. Hence vectors $(\eta_\psi(x_i))$
form a 2-dimensional subspace in $\mathop{\oplus}
\limits_i(l_{u_1}^2K)_{x_i}$ and equations (\ref{21})
determine vector $(\mu(x_i))\in\mathop{\oplus}\limits_i K_{x_i}$
up to a 2-dimensional ambiguity spanned by $(\omega^a(x_i))$
(indeed, as the residues of the meromorphic 1,0-form
$\eta_\psi\eta^{-1}\omega^a$, the numbers
$\omega^a(x_i)\hspace{0.025cm}\eta_\psi(x_i)\hspace{0.05cm}\partial\eta(x_i)^{-1}$ sum to zero).
It is clearly enough to take for $\psi$ in Eq.\hspace{0.05cm}\s(\ref{21})
any two linear forms independent of $\phi_{u_1}$ and $\phi_{u_2}$.
In the generic situation, we may choose the forms
$\partial_a\phi_{u_1}$ defined by
\begin{eqnarray}
{\langle\hspace{0.025cm}}\theta\hspace{0.025cm},\hspace{0.05cm}\partial_a\phi_{u_1}{\hspace{0.025cm}\rangle}\hspace{0.05cm}=\hspace{0.05cm}\partial_{a}\theta(u_1)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Denoting the corresponding 1,0-forms $\eta_\psi$ by $\eta'_a$,
we obtain 2 relations for $\mu(x_i)\hspace{0.025cm}$:
\begin{eqnarray}
\sum\limits_{i=1}^4\mu(x_i)\hspace{0.05cm}\eta'_a(x_i)\hspace{0.05cm}\partial\eta(x_i)^{-1}
\hspace{0.05cm}=\hspace{0.05cm}{_1\over^{2\pi i}}\hspace{0.05cm}\partial_a\theta(u_1)\hspace{0.05cm}.
\label{22}
\end{eqnarray}
Alternatively, we may choose for $\psi$ the linear forms
$\partial_a\phi_{u_2}$ corresponding to 1,0-forms $\eta''_a$.
This gives the relations
\begin{eqnarray}
\sum\limits_{i=1}^4\mu(x_i)\hspace{0.05cm}\eta''_a(x_i)\hspace{0.05cm}\partial\eta(x_i)^{-1}
\hspace{0.05cm}=\hspace{0.05cm}{_1\over^{2\pi i}}\hspace{0.05cm}\partial_a\theta(u_2)\hspace{0.05cm}.
\label{24}
\end{eqnarray}
$\eta''_a$ must be linearly dependent from $\eta'_a$ and $\eta$
(in the generic situation):
\begin{eqnarray}
\eta''_a\ =\ D_a^b\hspace{0.05cm}\eta'_b\hspace{0.05cm}+\hspace{0.05cm}\eta
\label{lind}
\end{eqnarray}
leading via Eqs.\hspace{0.05cm}\s(\ref{22}) and (\ref{24}) to the relation
\begin{eqnarray}
\partial_a\theta(u_2)\ =\ D_a^b\hspace{0.05cm}\s\partial_b\theta(u_1)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
\vskip 0.4cm

We need 2 more equations to determine $\mu(x_i)$. They may be
obtained from Eqs.\hspace{0.05cm}\s(\ref{cfm}) fixing the holomorphic
contributions to $\mu$. Indeed, using the 2$^{\rm nd}$ equation
in (\ref{cih}), and Eq.\hspace{0.05cm}\s(\ref{cfm1}) we infer that
\begin{eqnarray}
\int_{_\Sigma}\mu\wedge\bar\omega^a\hspace{0.05cm}=\hspace{0.05cm}\int_{_\Sigma}(\mu\hspace{0.025cm}\eta^{-1})
\hspace{0.05cm}\eta\wedge\bar\omega^a\hspace{0.05cm}=\hspace{0.05cm}\int_{_\Sigma}(\mu\hspace{0.025cm}\eta^{-1})\hspace{0.05cm}\bar\partial\chi^a
\hspace{0.05cm}=\hspace{0.05cm}\int_{_\Sigma}\chi^a\wedge\bar\partial(\mu\hspace{0.025cm}\eta^{-1})\cr
=\hspace{0.05cm}\int_{_\Sigma}\chi^a\wedge b\hspace{0.05cm}-\s2\pi i\sum\limits_{i=1}^4
\mu(x_i)\hspace{0.05cm}\chi^a(x_i)\hspace{0.05cm}\partial\eta(x_i)^{-1}
\label{tbr}
\end{eqnarray}
so that Eq.\hspace{0.05cm}\s(\ref{cfm}) implies that
\begin{eqnarray}
\sum\limits_{i=1}^4
\mu(x_i)\hspace{0.05cm}\chi^a(x_i)\hspace{0.05cm}\partial\eta(x_i)^{-1}\hspace{0.05cm}=\s0\hspace{0.05cm}.
\label{mie}
\end{eqnarray}
These are the two missing equations. To see this,
repeat the calculation (\ref{tbr}) for $\mu$
replaced by $\omega^b$. This gives the relation
\begin{eqnarray}
{_1\over^\pi}\hspace{0.05cm}{\rm Im}\hspace{0.025cm}\tau^{ab}\hspace{0.05cm}=\hspace{0.05cm}\sum\limits_{i=1}^4
\omega^b(x_i)\hspace{0.05cm}\chi^a(x_i)\hspace{0.05cm}\partial\eta(x_i)^{-1}\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Suppose now that \hspace{0.025cm}$d_a\hspace{0.025cm}\chi^a(x_i)+e\hspace{0.05cm}\eta_\psi(x_i)=0$
\hspace{0.05cm} for $i=1,\dots,4$. It follows that
\begin{eqnarray}
0\hspace{0.05cm}=\hspace{0.05cm}\sum\limits_{i=1}^4
\omega^b(x_i)\left(d_a\chi^a(x_i)
+e\hspace{0.025cm}\eta_\psi(x_i)\right)\partial\eta(x_i)^{-1}
\hspace{0.05cm}=\hspace{0.05cm} {_1\over^\pi}\hspace{0.05cm}{\rm Im}\hspace{0.025cm}\tau^{ab}\hspace{0.05cm} d_a
\nonumber
\end{eqnarray}
so that $d_a=0$. Hence the vectors $(\chi^a(x_i))$ span
a 2-dimensional subspace of $\mathop{\oplus}\limits_iK_{x_i}$
transversal to the 2-dimensional subspace spanned
by the vectors $(\eta_\psi(x_i))$ and the linear equations
(\ref{21}) and (\ref{mie}) determine $\mu(x_i)$ completely.
\vskip 0.4cm

It is enough to consider the case $\phi=\phi_{u_2}$.
\hspace{0.025cm} Indeed, the shift \hspace{0.05cm}$\phi\hspace{0.05cm}\mapsto\hspace{0.05cm}\phi\hspace{0.025cm}+\hspace{0.025cm} a_1
\hspace{0.025cm}\phi_{u_1}\hspace{0.05cm}$ results in the change
\begin{eqnarray}
\mu\ \ \ \mapsto\ \ \ \mu\hspace{0.05cm}+\hspace{0.05cm}{_{i}\over^{4\pi}}\hspace{0.05cm} a_1\hspace{0.05cm}
\hspace{0.05cm}\partial_a\theta(u_1)\hspace{0.025cm}\hspace{0.05cm}\omega^a\hspace{0.05cm},
\nonumber
\end{eqnarray}
see Eq.\hspace{0.05cm}\s(\ref{mu}). Identifying 1,0-forms with
multivalued functions by the relation (\ref{ide})
and setting \hspace{0.025cm}$\chi_a=2\pi\hspace{0.025cm}({\rm Im}\hspace{0.025cm}\tau)^{-1}_{ab}
\chi^b$, \hspace{0.05cm}${w}_i=\smallint_{x_0}^{x_i}\omega\hspace{0.025cm}-\hspace{0.025cm}\Delta$,
\hspace{0.05cm}$G_1=G_{12}=-G_2$ and $G_3=G_{34}=-G_4$ where
\begin{eqnarray}
G_{ij}\ =\ {det}\left(\matrix{\partial_1\vartheta({w}_i)&
\partial_1\vartheta({w}_j)\cr\partial_2\vartheta({w}_i)&
\partial_2\vartheta({w}_j)}\right)\hspace{0.05cm},
\nonumber
\end{eqnarray}
we obtain
\begin{eqnarray}
&&\partial\eta(x_1)\hspace{0.05cm}=\hspace{0.05cm} G_1\
\vartheta({w}_1-{w}_3-{w}_4)\hspace{0.05cm},
\quad\quad\quad
\ \chi_a(x_1)\hspace{0.05cm} =\hspace{0.05cm} -\partial_a\vartheta({w}_2)\
\vartheta({w}_1-{w}_3-{w}_4)\hspace{0.05cm},\cr
&&\partial\eta(x_2)\hspace{0.05cm} =\hspace{0.05cm} G_2\ \vartheta({w}_2
-{w}_3-{w}_4)\hspace{0.05cm},
\quad\quad\quad
\ \chi_a(x_2)\hspace{0.05cm} =\hspace{0.05cm} -\partial_a\vartheta({w}_1)\
\vartheta({w}_2-{w}_3-{w}_4)\hspace{0.05cm},\cr
&&\partial\eta(x_3)\hspace{0.05cm} =\hspace{0.05cm} G_3\
\vartheta({w}_3-{w}_1-{w}_2)\hspace{0.05cm},
\quad\quad\quad
\ \chi_a(x_3)\hspace{0.05cm} =\hspace{0.05cm} -\partial_a\vartheta({w}_4)\
\vartheta({w}_3-{w}_1-{w}_2)\hspace{0.05cm},\cr
&&\partial\eta(x_4)\hspace{0.05cm} =\hspace{0.05cm} G_4\
\vartheta({w}_4-{w}_1-{w}_2)\hspace{0.05cm},
\quad\quad\quad
\ \chi_a(x_4)\hspace{0.05cm} =\hspace{0.05cm} -\partial_a\vartheta({w}_3)\
\vartheta({w}_4-{w}_1-{w}_2)\hspace{0.05cm},\cr\cr
&&\eta'_a(x_1)\hspace{0.05cm} =\hspace{0.05cm} \partial_a\vartheta({w}_1)\
\vartheta({w}_2+{w}_3+{w}_4)\hspace{0.05cm},\quad
\eta''_a(x_1)\hspace{0.05cm} =\hspace{0.05cm}\ \ \partial_a\vartheta({w}_2)\
\vartheta({w}_1-{w}_3-{w}_4)\hspace{0.05cm},\cr
&&\eta'_a(x_2)\hspace{0.05cm} =\hspace{0.05cm} \partial_a\vartheta({w}_2)\
\vartheta({w}_1+{w}_3+{w}_4)\hspace{0.05cm},\quad
\eta''_a(x_2)\hspace{0.05cm} =\hspace{0.05cm}\ \ \partial_a\vartheta({w}_1)\
\vartheta({w}_2-{w}_3-{w}_4)\hspace{0.05cm},\cr
&&\eta'_a(x_3)\hspace{0.05cm} =\hspace{0.05cm} \partial_a\vartheta({w}_3)\
\vartheta({w}_1+{w}_2+{w}_4)\hspace{0.05cm},\quad
\eta''_a(x_3)\hspace{0.05cm} =\hspace{0.05cm} -\partial_a\vartheta({w}_4)\
\vartheta({w}_3-{w}_1-{w}_2)\hspace{0.05cm},\cr
&&\eta'_a(x_4)\hspace{0.05cm} =\hspace{0.05cm} \partial_a\vartheta({w}_4)\
\vartheta({w}_1+{w}_2+{w}_3)\hspace{0.05cm},
\quad\eta''_a(x_4)\hspace{0.05cm} =\hspace{0.05cm} -\partial_a\vartheta({w}_3)\
\vartheta({w}_4-{w}_1-{w}_2)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Given these values, it is easy to find the explicit
form of the matrix $(D^b_a)$ appearing
in the relation between the derivatives
of $\partial_a\theta$ at $u_1$ and $u_2$
by specifying Eq.\hspace{0.05cm}\s(\ref{lind})
to two of the points $x_i$. One form of these relations is
\begin{eqnarray}
&&\partial_2\vartheta({w}_3)\hspace{0.05cm}\partial_1\theta(u_2)
\hspace{0.05cm}-\hspace{0.05cm}\partial_1\vartheta({w}_3)\hspace{0.05cm}\partial_2\theta(u_2)\hspace{0.05cm}\hspace{0.025cm}\cr\cr
&&\hspace{2cm}=\hspace{0.05cm}-\hspace{0.025cm}{_{\vartheta({w}_3-{w}_1-{w}_2)}
\over^{\vartheta({w}_1+{w}_2+{w}_4)}}
\hspace{0.05cm}\s(\partial_2\vartheta({w}_4)\hspace{0.05cm}\partial_1\theta(u_1)
\hspace{0.05cm}-\hspace{0.05cm}\partial_1\vartheta({w}_4)\hspace{0.05cm}\partial_2\theta(u_1))\hspace{0.05cm},\cr\cr
&&\partial_2\vartheta({w}_4)\hspace{0.05cm}\partial_1\theta(u_2)
\hspace{0.05cm}-\hspace{0.05cm}\partial_1\vartheta({w}_4)\hspace{0.05cm}\partial_2\theta(u_2)\hspace{0.05cm}\cr\cr
&&\hspace{2cm}=\hspace{0.05cm}-\hspace{0.025cm}{_{\vartheta({w}_4-{w}_1-{w}_2)}
\over^{\vartheta({w}_1+{w}_2+{w}_3)}}
\hspace{0.05cm}\s(\partial_2\vartheta({w}_3)\hspace{0.05cm}\partial_1\theta(u_1)
\hspace{0.05cm}-\hspace{0.05cm}\partial_1\vartheta({w}_3)\hspace{0.05cm}\partial_2\theta(u_1))\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Let us denote
\hspace{0.05cm}$\widetilde\mu(x_i)\hspace{0.025cm}=\hspace{0.025cm}\mu(x_i)/G_i\hspace{0.025cm}$.
\hspace{0.025cm} Eqs.\hspace{0.05cm}\s(\ref{mie}) have the general solution
\begin{eqnarray}
(\widetilde\mu(x_1),\dots,\widetilde\mu(x_4))\ =\
g_1\hspace{0.05cm}\hspace{0.025cm}(G_{34},0,G_{23},-G_{24})\hspace{0.05cm}+\hspace{0.05cm}
g_2\hspace{0.05cm}\hspace{0.025cm}(0,G_{34},G_{13},-G_{14})
\nonumber
\end{eqnarray}
and Eqs.\hspace{0.05cm}\s(\ref{24}) fix the values of $g_1$ and $g_2$ to
\begin{eqnarray}
&&g_1\ =\ -\hspace{0.025cm}{\partial_2\vartheta({w}_1)\hspace{0.05cm}\partial_1\theta(u_2)
\hspace{0.025cm}-\hspace{0.025cm}\partial_1\vartheta({w}_1)\hspace{0.05cm}\partial_2\theta(u_2)\over
{4\pi i\hspace{0.05cm} G_{12}\hspace{0.025cm} G_{34}}}\cr\cr
&&g_2\ =\ \hspace{0.05cm}\s\hspace{0.025cm}{\partial_2\vartheta({w}_2)\hspace{0.05cm}\partial_1\theta(u_2)
\hspace{0.025cm}-\hspace{0.025cm}\partial_1\vartheta({w}_2)\hspace{0.05cm}\partial_2\theta(u_2)\over{4\pi i\hspace{0.05cm}
G_{12}\hspace{0.025cm} G_{34}}}\hspace{0.05cm}.
\nonumber
\end{eqnarray}
This leads to the following simple result:
\begin{eqnarray}
\mu(x_i)\hspace{0.05cm}=\hspace{0.05cm}\pm\hspace{0.05cm}{_i\over^{4\pi}}\hspace{0.05cm}\hspace{0.025cm}(\hspace{0.025cm}\partial_2\vartheta({w}_i)
\hspace{0.05cm}\partial_1\theta(u_2)\hspace{0.05cm}-\hspace{0.05cm}\partial_1\vartheta({w}_i)\hspace{0.05cm}\partial_2
\theta(u_2)\hspace{0.025cm})
\label{u1u2}
\end{eqnarray}
or, in a more abstract notation from the introduction,
\begin{eqnarray}
\mu(x_i)\hspace{0.05cm}=\hspace{0.05cm}\pm\hspace{0.05cm}{_i\over^{4\pi}}\hspace{0.05cm}\hspace{0.025cm} d\theta(l_{u_2})
\nonumber
\end{eqnarray}
with the plus sign for $i=1,2$ and the minus one for $i=3,4$.
\vskip 0.4cm

Since the Hitchin Hamiltonian is quadratic in $\phi$
and its values on $\phi_{u_1}$ and $\phi_{u_2}$
are given by Eq.\hspace{0.05cm}\s(\ref{first}), it follows that
\begin{eqnarray}
&&{\cal H}(\hspace{0.025cm}\theta\hspace{0.05cm},\hspace{0.025cm}\hspace{0.05cm} a_1\hspace{0.05cm}\phi_{u_1}\hspace{0.025cm}+\hspace{0.025cm} a_2
\hspace{0.05cm}\phi_{u_2})\cr\cr
&&\hspace{1cm}=\ a_1^2\hspace{0.05cm}{\cal H}(\hspace{0.025cm}\theta\hspace{0.025cm},\hspace{0.05cm}\phi_{u_1})\hspace{0.05cm}
+\hspace{0.05cm} a_2^2\hspace{0.05cm}{\cal H}(\theta\hspace{0.025cm},\hspace{0.05cm}\phi_{u_2})
\hspace{0.05cm}+\s2\hspace{0.025cm} a_1a_2\hspace{0.05cm}(\hspace{0.025cm} c_1\hspace{0.025cm}(\omega^1)^2\hspace{0.05cm}
+\hspace{0.05cm} c_2\hspace{0.025cm}\omega^1\omega^2
\hspace{0.05cm}+\hspace{0.05cm} c_3\hspace{0.025cm}(\omega^2)^2\hspace{0.025cm})\hspace{0.05cm}.\hspace{0.8cm}
\nonumber
\end{eqnarray}
The mixed term may be found from the linear equations
\begin{eqnarray}
&&{_i\over^{4\pi}}\hspace{0.05cm}(\partial_2\vartheta({w}_i)\hspace{0.05cm}\partial_1\theta(u_1)-
\partial_1\vartheta({w}_i)\hspace{0.05cm}\partial_2\theta(u_1))
\hspace{0.05cm}\s\hspace{0.025cm}\widetilde\mu(x_i)\hspace{0.05cm}\s G_i\cr\cr
&&\hspace{1.5cm}=\ c_1\hspace{0.05cm}\partial_2\vartheta
({w}_i)\hspace{0.05cm}\partial_2\vartheta({w}_i)
\hspace{0.05cm}-\hspace{0.05cm} c_2\hspace{0.05cm}\partial_2\vartheta({w}_i)\hspace{0.05cm}\partial_1\vartheta({w}_i)
\hspace{0.05cm}+\hspace{0.05cm} c_3\hspace{0.05cm}\partial_1\vartheta({w}_i)\hspace{0.05cm}\partial_1\vartheta({w}_i)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Their explicit solution leads to the expression
\begin{eqnarray}
&&\hspace{-0.6cm}{\cal H}(\hspace{0.025cm}\theta\hspace{0.05cm},\hspace{0.025cm}\hspace{0.05cm} a_1\hspace{0.05cm}\phi_{u_1}\hspace{0.025cm}
+\hspace{0.025cm} a_2\hspace{0.05cm}\phi_{u_2})
\ =\ -\hspace{0.025cm}{_1\over^{16\pi^2}}\hspace{0.05cm}\s(\hspace{0.025cm} a_1
\hspace{0.05cm}\partial_a\theta(u_1)\hspace{0.05cm}\omega^a
\hspace{0.05cm}+\hspace{0.05cm} a_2\hspace{0.05cm}\partial_a\theta(u_2)\hspace{0.05cm}\omega^a\hspace{0.025cm})^2\hspace{0.05cm}\cr
&&+\hspace{0.05cm}\s{_{a_1\hspace{0.025cm} a_2}\over^{4\pi^2\hspace{0.05cm} G_{13}\hspace{0.025cm} G_{23}}}
\hspace{0.05cm}\s(\partial_2\vartheta({w}_3)\hspace{0.05cm}\partial_1\theta(u_1)
-\partial_1\vartheta({w}_3)\hspace{0.05cm}\partial_2\theta(u_1))\hspace{0.05cm}\clabel{ost}\cr
&&\ \ \ \cdot\ (\partial_2\vartheta({w}_3)\hspace{0.05cm}\partial_1\theta(u_2)
-\partial_1\vartheta({w}_3)\hspace{0.05cm}\partial_2\theta(u_2))
\ \partial_a\vartheta({w}_1)\hspace{0.05cm}\hspace{0.025cm}\partial_b
\vartheta({w}_2)\hspace{0.05cm}\hspace{0.025cm}\omega^a\omega^b\hspace{0.05cm}.
\end{eqnarray}
The second term on the right hand side hand side
is a quadratic differential that vanishes at $x_1$ and $x_2$
and is equal to \hspace{0.05cm}${a_1\hspace{0.025cm} a_2\over 4\pi^2}\hspace{0.05cm}\s
\partial_a\theta(u_1)\hspace{0.05cm}\hspace{0.025cm}\partial_b\theta(u_2)\hspace{0.05cm}\omega^a\omega^b\hspace{0.05cm}$
at $x_3$ and $x_4$ so that
\begin{eqnarray}
{\cal H}(\theta,\hspace{0.025cm}\phi)(x_i)\ =\
-\hspace{0.025cm}{_1\over^{16\pi^2}}\hspace{0.05cm}\s(\hspace{0.025cm} a_1
\hspace{0.05cm}\partial_a\theta(u_1)\hspace{0.05cm}\omega^a(x_i)
\hspace{0.05cm}\pm\hspace{0.05cm} a_2\hspace{0.05cm}\partial_a\theta(u_2)\hspace{0.05cm}\omega^a(x_i)\hspace{0.025cm})^2\hspace{0.05cm}
\label{frr}
\end{eqnarray}
where sign plus should be taken for $x_1$ and $x_2$ and
sign minus for $x_3$ and $x_4$. This is the result
(\ref{fr}) described in Introduction.
\vskip 0.9cm

\nsection{Self-duality}
\vskip 0.5cm

We would like to compare the values of the Hitchin Hamiltonians
on the dual pairs $(\theta,\phi)$ and $(\theta',\hspace{0.025cm}\phi')$ where
$\theta'=\iota(\phi)$ and $\phi'=\iota^{-1}(\theta)$
with $\iota$ defined by Eq.\hspace{0.05cm}\s(\ref{lini}).
Recall that, given $u_1$ s.t. $\theta(u_1)=0$, we associated to
the linear form \hspace{0.025cm}$\phi\hspace{0.025cm}$ a 1,0-form $\eta$
by Eq.\hspace{0.05cm}\s(\ref{eta}).  Viewed as a holomorphic
section of \hspace{0.05cm}$l_{u_1}^2K$,
\begin{eqnarray}
\eta(x)\ =\ {\langle\hspace{0.025cm}}\vartheta(\smallint_{x_0}^x\omega
-u_1-\hspace{0.05cm}\cdot\hspace{0.05cm}-\Delta)\ \vartheta(\smallint_{x_0}^x
\omega-u_1+\hspace{0.05cm}\cdot\hspace{0.05cm}-\Delta)\hspace{0.05cm}\s,\ \phi{\hspace{0.025cm}\rangle}\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Let us denote
\begin{eqnarray}
u_i'\hspace{0.05cm}=\hspace{0.05cm}\smallint_{x_0}^{x_i}\omega\hspace{0.025cm}-\hspace{0.025cm} u_1-\Delta\hspace{0.05cm}.
\label{uip}
\end{eqnarray}
The vanishing of $\eta(x_i)$ implies then that
the linear form $\phi$ annihilates the theta functions
\begin{eqnarray}
u\ \ \mapsto\ \ \vartheta(u'_i-u)\hspace{0.05cm}\hspace{0.025cm}\vartheta
(u'_i+u)\hspace{0.05cm}=\iota(\phi_{u'_i})(u)
\label{4theta}
\end{eqnarray}
and also, if we rewrite $\eta(x_i)$ as $\iota(\phi)(u'_i)$,
that $\theta'(u'_i)=0$. Since $\phi=
a_1\hspace{0.025cm}\phi_{u_1}+a_2\hspace{0.025cm}\phi_{u_2}$ and $\phi_{u_1}$ annihilates
the theta functions (\ref{4theta}) as well, it follows that
they belong to $\Pi$. Hence $\NC^*\iota(\phi_{u'_i})$ are
the 4 points of intersection of the line $\NP\Pi$ with the Kummer
quartic ${\cal K}$. Equivalently, $\NC^*\phi_{u'_i}$ are the points
of intersection of $\NP\hspace{0.025cm}{\Pi'}^\perp$ with ${\cal K}^*$.
In the generic situation, any pair of theta functions
$\phi_{u'_i}$ spans ${\Pi'}^\perp$ and since
$\phi'\in{\Pi'}^\perp$, we may write
\begin{eqnarray}
\phi'\ =\ a'_1\hspace{0.05cm}\phi_{v_1}\hspace{0.05cm}+\hspace{0.05cm} a'_2\hspace{0.05cm}\phi_{v_2}
\label{aap}
\end{eqnarray}
or, equivalently,
\begin{eqnarray}
\theta\ =\ a'_1\hspace{0.05cm}\iota(\phi_{v_1})\hspace{0.05cm}
+\hspace{0.05cm} a'_2\hspace{0.05cm}\iota(\phi_{v_2})\hspace{0.05cm}.
\label{aapp}
\end{eqnarray}
\vskip 0.3cm

The involution $l\mapsto l^{-1}K$ of the Jacobian $J^1$
lifts to $\NC^2$ to the flip of sign of $u$. By restriction
to the bundles ${\cal O}(x)$, it induces the involution $x\mapsto x'$
of $\Sigma$ which leaves 6 Weierstrass points invariant.
The latter involution lifts to an involution
(without fixed points) of the covering space
$\widetilde\Sigma$ determined by the equation
\begin{eqnarray}
\smallint_{x_0}^{x}\omega\hspace{0.05cm}-\hspace{0.05cm}\Delta\
=\ -\smallint_{x_0}^{x'}\omega\hspace{0.05cm}+\hspace{0.05cm}\Delta\hspace{0.05cm}.
\label{invo}
\end{eqnarray}
Definitions (\ref{uip}) together with Eqs.\hspace{0.05cm}\s(\ref{upm})
give the relations
\begin{eqnarray}
u'_1-u'_2\hspace{0.05cm}=\hspace{0.05cm}\smallint_{x_0}^{x_1}\omega
-\smallint_{x_0}^{x_2}\omega\ \quad{\rm and}\ \quad
u'_1+u'_2\hspace{0.05cm}=\hspace{0.05cm} -\smallint_{x_0}^{x_3}\omega
-\smallint_{x_0}^{x_4}\omega+2\Delta
\nonumber
\end{eqnarray}
holding in $\NC^2$, with $x_i\in\widetilde\Sigma$.
They may be rewritten as
\begin{eqnarray}
u'_1-u'_2\hspace{0.05cm}=\hspace{0.05cm}\smallint_{x_0}^{x_1}\omega
+\smallint_{x_0}^{x'_2}\omega\hspace{0.05cm}-\s2\Delta\ \quad{\rm and}\ \quad
u'_1+u'_2\hspace{0.05cm}=\hspace{0.05cm} \smallint_{x_0}^{x'_3}\omega
+\smallint_{x_0}^{x'_4}\omega-2\Delta\hspace{0.05cm},
\label{upmq}
\end{eqnarray}
which, upon the flip of the sign of $u'_2$ leaving
$\phi_{u'_2}$ unchanged, provides the dual version
of relations (\ref{upm}) corresponding to points
$x_1,x'_2,x'_3,x'_4\in\widetilde\Sigma$.
Applying the previous result (\ref{frr}) and
using the possibility to exchange a point with
its image under the involution of $\Sigma$ in the argument
of a quadratic differential, we infer that
\begin{eqnarray}
{\cal H}(\theta',\hspace{0.025cm}\phi')(x_i)\ =\
-\hspace{0.025cm}{_1\over^{16\pi^2}}\hspace{0.05cm}\s(\hspace{0.025cm} a'_1
\hspace{0.05cm}\partial_a\theta'(u'_1)\hspace{0.05cm}\omega^a(x_i)
\hspace{0.05cm}\mp\hspace{0.05cm} a_2\hspace{0.05cm}\partial_a\theta'(u'_2)\hspace{0.05cm}\omega^a(x_i)\hspace{0.025cm})^2\hspace{0.05cm}.
\label{frrr}
\end{eqnarray}
The sign minus should be taken for $x_1$ and $x_2$ and sign plus
for $x_3$ and $x_4$. The exchange of signs in comparison
with Eq.\hspace{0.05cm}\s(\ref{frr}) is due to the flip $u'_2\mapsto-u'_2$.
\vskip 0.4cm

In order to compare expressions (\ref{frr}) and (\ref{frrr})
we shall calculate the coefficients $a_{1,2}$ and $a'_{1,2}$
of the linear combinations (\ref{alpha}) and (\ref{aap}).
Note that the definition $\theta'=\iota(\phi)$ implies that
\begin{eqnarray}
\theta'(\smallint_{x_0}^x\omega-u_1-\Delta)\ =\ a_2\hspace{0.05cm}\s\hspace{0.05cm}
\vartheta(\smallint_{x_0}^x\omega-u_1-u_2-\Delta)
\hspace{0.05cm}\s\vartheta(\smallint_{x_0}^x\omega-u_1+u_2-
\Delta)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Taking the derivative over $x$ at $x_1$, we obtain
\begin{eqnarray}
\partial_a\theta'(u'_1)\hspace{0.025cm}\hspace{0.05cm}\omega^a(x_1)\ =
\ -\hspace{0.025cm} a_2\hspace{0.05cm}\s\hspace{0.05cm}\vartheta(w_1-w_3-w_4)\hspace{0.05cm}\s\partial_a
\vartheta(w_2)\hspace{0.05cm}\hspace{0.025cm}\omega^a(x_1)
\nonumber
\end{eqnarray}
where we employed Eqs.\hspace{0.05cm}\s(\ref{upm}) and the abbreviated
notations $w_i=\smallint_{x_0}^{x_i}-\Delta$. Hence
\begin{eqnarray}
a_2\ =\ -\hspace{0.025cm}{_{\partial_a\theta'(u'_1)\hspace{0.025cm}\hspace{0.05cm}\omega^a(x_1)}\over
^{\vartheta(w_1-w_3-w_4)\hspace{0.05cm}\s\partial_a
\vartheta(w_2)\hspace{0.05cm}\hspace{0.025cm}\omega^a(x_1)}}\hspace{0.05cm}.
\label{a22}
\end{eqnarray}
Similarly,
\begin{eqnarray}
\theta'(\smallint_{x_0}^x\omega-u_2-\Delta)\ =\ a_1\hspace{0.05cm}\s\hspace{0.05cm}
\vartheta(\smallint_{x_0}^x\omega-u_1-u_2-\Delta)
\hspace{0.05cm}\s\vartheta(\smallint_{x_0}^x\omega+u_1-u_2-
\Delta)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Taking the derivative at $x=x_1$ and noting that
\hspace{0.025cm}$w_1-u_2=-u'_2\hspace{0.025cm}$, \hspace{0.025cm} we infer that
\begin{eqnarray}
a_1\ =\ {_{\partial_a\theta'(u'_2)\hspace{0.025cm}\hspace{0.05cm}\omega^a(x_1)}\over
^{\vartheta(w_1+w_3+w_4)\hspace{0.05cm}\s\partial_a
\vartheta(w_2)\hspace{0.05cm}\hspace{0.025cm}\omega^a(x_1)}}\hspace{0.05cm}.
\label{a11}
\end{eqnarray}
To calculate $a'_{1,2}$, we note that Eq.\hspace{0.05cm}\s(\ref{aapp})
implies that
\begin{eqnarray}
\theta(\smallint_{x_0}^x\omega-v_1-\Delta)\ =\ a'_2\hspace{0.05cm}\s\hspace{0.05cm}\vartheta(
\smallint_{x_0}^x\omega-u'_1-u'_2-\Delta)\hspace{0.05cm}\s
\vartheta(\smallint_{x_0}^x\omega-u'_1+u'_2-\Delta)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Upon derivation at $x=x_1$ and with the use of relations
(\ref{upmq}) and (\ref{invo}), this gives
\begin{eqnarray}
a'_2\ =\ -\hspace{0.025cm}{_{\partial_a\theta(u_1)\hspace{0.025cm}\hspace{0.05cm}\omega^a(x_1)}\over
^{\vartheta(w_1+w_3+w_4)\hspace{0.05cm}\s\partial_a
\vartheta(w_2)\hspace{0.05cm}\hspace{0.025cm}\omega^a(x_1)}}\hspace{0.05cm}.
\label{a22p}
\end{eqnarray}
Finally, since
\begin{eqnarray}
\theta(\smallint_{x_0}^x\omega+v_2-\Delta)\ =\ a'_1\hspace{0.05cm}\s\vartheta(
\smallint_{x_0}^x\omega-u'_1+u'_2-\Delta)\hspace{0.05cm}\s
\vartheta(\smallint_{x_0}^x\omega+u'_1+u'_2-\Delta)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
and \hspace{0.05cm}$w_1+u'_2=u_2$ we infer that
\begin{eqnarray}
a'_1\ =\ -\hspace{0.025cm}{_{\partial_a\theta(u_2)\hspace{0.025cm}\hspace{0.05cm}\omega^a(x_1)}\over
^{\vartheta(w_1-w_3-w_4)\hspace{0.05cm}\s\partial_a
\vartheta(w_2)\hspace{0.05cm}\hspace{0.025cm}\omega^a(x_1)}}\hspace{0.05cm}.
\label{a11p}
\end{eqnarray}
Substitution of expressions (\ref{a11}),(\ref{a22}),(\ref{a11p})
and (\ref{a22p}) shows equality of the right hand sides of
Eqs.\hspace{0.05cm}\s(\ref{frr}) and (\ref{frrr}) for $x_i=x_1$. Since
there is a full symmetry between points $x_i$ (hidden in
our arbitrary choices of the order and the signs
of $u_j\hspace{0.025cm}$'s and $u'_j\hspace{0.025cm}$'s), the self-duality
\begin{eqnarray}
{\cal H}(\theta,\hspace{0.025cm}\phi)\ =\ {\cal H}(\theta',\hspace{0.025cm}\phi')
\label{seld}
\end{eqnarray}
follows.
\vskip 0.9cm

\nsection{van Geemen-Previato's result and beyond}
\vskip 0.5cm

The genus 2 curves are hyperelliptic.
The map \hspace{0.05cm}$H^0(K)\ni\omega\hspace{0.05cm}\mapsto\hspace{0.05cm}\omega(x)\hspace{0.05cm}$
defines an element of $\NP H^0(K)^*$ and varying $x\in\Sigma$
one obtains a realization of $\Sigma$ as a ramified double cover
\hspace{0.025cm}$\NP H^0(K)^*\cong\NP^1\hspace{0.025cm}$. One may use the 1,0-forms
$\omega^a\in H^0(K)$ to define the homogeneous coordinates
on $\NP H^0(K)^*$. Then
\begin{eqnarray}
\lambda(x)\ =\ {_{\omega^2(x)}\over^{\omega^1(x)}}\ =\
-\hspace{0.025cm}{_{\partial_1\vartheta(\smallint_{x_0}^x\omega-\Delta)}
\over^{\partial_2\vartheta(\smallint_{x_0}^x\omega-\Delta)}}
\label{inho}
\end{eqnarray}
becomes the inhomogeneous coordinate of the image in $\NP^1$
of the point $x\in\Sigma$. \hspace{0.025cm} If $x'$ is the image
of $x$ under the involution \hspace{0.05cm}${\cal O}(x)\mapsto {\cal O}(-x)K={\cal O}(x')\hspace{0.025cm}$,
\hspace{0.05cm} i.e.\hspace{0.05cm}\s\hspace{0.025cm} if
\begin{eqnarray}
\smallint_{x_0}^x\omega+\smallint_{x_0}^{x'}\omega-2\Delta\ \in\
\NZ+\tau\NZ\quad\quad{\rm then}\quad\quad
\lambda(x)\hspace{0.05cm}=\hspace{0.05cm}\lambda(x')\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Hence the involution $x\mapsto x'$ permutes the sheets of the
covering $\Sigma\mapsto\NP^1$ ramified over the 6 Weierstrass
points $x_s$, $s=1,\dots,6$, fixed by the involution. ${\cal O}(x_s)$
is an odd spin structure. i.e.
\begin{eqnarray}
\smallint_{x_0}^{x_s}\omega-\Delta\ =\ E_s\ \
{\rm mod}\hspace{0.05cm}(\NZ^2+\tau\NZ^2)
\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\lambda_s\equiv\lambda(x_s)\hspace{0.05cm}=\hspace{0.05cm}
-\hspace{0.025cm}{_{\partial_1\vartheta(E_s)}
\over^{\partial_2\vartheta(E_s)}}
\label{wp}
\end{eqnarray}
where $E_s={_1\over^2}(e_s+\tau\hspace{0.025cm} e_s')$ with $e_s,\hspace{0.025cm} e_s'=(1,0),\hspace{0.05cm}(0,1)$
or $(1,1)$ such that $e_s\cdot\hspace{0.025cm} e_s'$ is odd. The possibilities
are:
\begin{eqnarray}
&&e_1=(1,0),\ e_1'=(1,0);\quad e_2=(1,1),\ e_2'=(1,0);
\quad e_3=(0,1),\ e_3'=(0,1);\cr\clabel{ordr}\cr
&&e_4=(1,1),\ e_4'=(0,1);\quad e_5=(0,1),\ e_5'=(1,1);
\quad e_6=(1,0),\ e_6'=(1,1).
\end{eqnarray}
and we shall number the Weierstrass points (in a marking-dependent
way) in agreement with this list. $\Sigma$ may be identified
with the hyperelliptic curve given by the equation
\begin{eqnarray}
\zeta^2\ =\ \prod\limits_{s=1}^6(\lambda-\lambda_s)
\label{hell}
\end{eqnarray}
with the involution mapping $(\lambda,\zeta)$ to $(\lambda,-\zeta)$.
The expressions
\begin{eqnarray}
\omega^1=C\hspace{0.05cm}{{d\lambda}\over\zeta}\quad\ \ {\rm and}\quad\ \
\omega^2=C\hspace{0.05cm}{{\lambda\hspace{0.025cm} d\lambda}\over\zeta}\hspace{0.05cm},
\label{newf}
\end{eqnarray}
where $C$ is a constant, give the basis of holomorphic 1,0-forms
of $\Sigma$ (the right hand sides vanish exactly where
the left hand sides do).
\vskip 0.4cm

Let us recall the main result of \cite{VGP} based on
the analysis of the formula (\ref{first}) for the Hitchin
Hamiltonians on the Kummer quartic ${\cal K}^*$. It will be
convenient to identify the pairs $(\theta,\phi)$
s.t. ${\langle\hspace{0.025cm}}\theta,\phi{\hspace{0.025cm}\rangle}=0$
with pairs $(q,p)\in\NC^4\times\NC^4$ s.t. $q\cdot p=0$
by the relations
\begin{eqnarray}
&&\theta\ =\ q_1\hspace{0.05cm}\theta_{2,(0,0)}\hspace{0.05cm}+\hspace{0.05cm} q_2\hspace{0.05cm}\theta_{2,(1,0)}
\hspace{0.05cm}+\hspace{0.05cm} q_3\hspace{0.05cm}\theta_{2,(0,1)}\hspace{0.05cm}+\hspace{0.05cm} q_4\hspace{0.05cm}\theta_{2,(1,1)}\hspace{0.05cm},\cr
&&\phi\ =\ p_1\hspace{0.05cm}\theta_{2,(0,0)}^{\hspace{0.05cm}*}\hspace{0.05cm}+\hspace{0.05cm} p_2\hspace{0.05cm}\theta_{2,(1,0)}
^{\hspace{0.05cm}*}\hspace{0.05cm}+\hspace{0.05cm} p_3\hspace{0.05cm}\theta_{2,(0,1)}^{\hspace{0.05cm}*}\hspace{0.05cm}+\hspace{0.05cm}
p_4\hspace{0.05cm}\theta_{2,(1,1)}^{\hspace{0.05cm}*}\hspace{0.05cm}.
\nonumber
\end{eqnarray}
The symplectic form of $T^*\NP^3$ is the standard
$dp\wedge dq$ and the isomorphism $\iota$ interchanges $p$
and $q$. By examining the values of the quadratic differentials
given by ${\cal H}$ at the Weierstrass points $x_s$, van Geemen and
Previato showed that
\begin{eqnarray}
{\cal Z}_s(q)\ =\ \{\hspace{0.05cm} p\hspace{0.05cm}\s|\hspace{0.05cm}\s q\cdot p=0,\
{\cal H}(q,p)(x_s)=0\hspace{0.05cm}\}
\nonumber
\end{eqnarray}
is a union of a pair of bitangents to ${\cal K}^*$.
Then classical results giving the equations
for bitangents to the Kummer surface permitted the authors
of \cite{VGP} to write an almost explicit formula for ${\cal H}(x_s)$
in the form
\begin{eqnarray}
{\cal H}(q,p)(x_s)\ =\ h_s\hspace{0.05cm}\sum\limits_{t\not= s}{{r_{st}(q,p)}
\over{\lambda_s-\lambda_t}}
\label{VGP}
\end{eqnarray}
where $r_{st}=r_{ts}$ are homogeneous polynomials,
\begin{eqnarray}
&&r_{12}(q,p)\ =\ \ \ \hspace{0.025cm}(q_1p_1+q_2p_2-q_3p_3-q_4p_4)^2\hspace{0.05cm},\cr
&&r_{13}(q,p)\ =\ \ \ \hspace{0.025cm}(q_1p_4-q_2p_3-q_3p_2+q_4p_1)^2\hspace{0.05cm},\cr
&&r_{14}(q,p)\ =\ -(q_1p_4+q_2p_3-q_3p_2-q_4p_1)^2\hspace{0.05cm},\cr
&&r_{15}(q,p)\ =\ -(q_1p_3-q_2p_4-q_3p_1+q_4p_2)^2\hspace{0.05cm},\cr
&&r_{16}(q,p)\ =\ \ \ \hspace{0.025cm}(q_1p_3+q_2p_4+q_3p_1+q_4p_2)^2\hspace{0.05cm},\cr
&&r_{23}(q,p)\ =\ -(q_1p_4-q_2p_3+q_3p_2-q_4p_1)^2\hspace{0.05cm},\cr
&&r_{24}(q,p)\ =\ \ \ \hspace{0.025cm}(q_1p_4+q_2p_3+q_3p_2+q_4p_1)^2\hspace{0.05cm},\cr
&&r_{25}(q,p)\ =\ \ \ \hspace{0.025cm}(q_1p_3-q_2p_4+q_3p_1-q_4p_2)^2\hspace{0.05cm},\clabel{rs}\cr
&&r_{26}(q,p)\ =\ -(q_1p_3+q_2p_4-q_3p_1-q_4p_2)^2\hspace{0.05cm},\cr
&&r_{34}(q,p)\ =\ \ \ \hspace{0.025cm}(q_1p_1-q_2p_2+q_3p_3-q_4p_4)^2\hspace{0.05cm},\cr
&&r_{35}(q,p)\ =\ \ \ \hspace{0.025cm}(q_1p_2+q_2p_1+q_3p_4+q_4p_3)^2\hspace{0.05cm},\cr
&&r_{36}(q,p)\ =\ -(q_1p_2-q_2p_1-q_3p_4+q_4p_3)^2\hspace{0.05cm},\cr
&&r_{45}(q,p)\ =\ -(q_1p_2-q_2p_1+q_3p_4-q_4p_3)^2\hspace{0.05cm},\cr
&&r_{46}(q,p)\ =\ \ \ \hspace{0.025cm}(q_1p_2+q_2p_1-q_3p_4-q_4p_3)^2\hspace{0.05cm},\cr
&&r_{56}(q,p)\ =\ \ \ \hspace{0.025cm}(q_1p_1-q_2p_2-q_3p_3+q_4p_4)^2\hspace{0.05cm}
\end{eqnarray}
and $h_s\in K^2_{x_s}$ could still depend on $q$.
In the original language of pairs $(\theta,\phi)$,
and of the $(\NZ/2\NZ)^4$-action (\ref{acone})
on $H^0(L_\Theta^2)$ one has
\begin{eqnarray}
r_{st}(\theta,\phi)\ =\ \ {\langle\hspace{0.025cm}}U_{e_s,e'_s}U_{e_t,e'_t}\hspace{0.025cm}\theta\hspace{0.025cm},
\hspace{0.05cm}\phi{\hspace{0.025cm}\rangle}\hspace{0.05cm}{\langle\hspace{0.025cm}}U_{e_t,e'_t}U_{e_s,e'_s}
\hspace{0.025cm}\theta\hspace{0.025cm},\hspace{0.05cm}\phi{\hspace{0.025cm}\rangle}
\nonumber
\end{eqnarray}
with $e_s,\hspace{0.025cm} e'_s$ from the list (\ref{ordr}).
The polynomials $r_{st}$ are self-dual:
\begin{eqnarray}
r_{st}(q,p)\ =\ r_{st}(p,q)
\end{eqnarray}
and the self-duality of ${\cal H}$ proven in the present paper forces
coefficients $h_s$ in Eq.\hspace{0.05cm}\s(\ref{VGP}) to be $q$-independent
filling partially the gap left in \cite{VGP}.
An easy but important identity is
\begin{eqnarray}
\sum\limits_{t\not=s}r_{st}(q,p)\ =\ (q\cdot p)^2\ =\ 0
\label{sum}
\end{eqnarray}
for any fixed $s$. It implies that the Hamiltonians
(\ref{VGP}) are preserved up to normalization
by the isomorphisms of the hyperelliptic surfaces induced
by the fractional action \hspace{0.05cm}$\lambda\hspace{0.05cm}\mapsto\hspace{0.05cm}\lambda'=
{a\lambda+b\over c\lambda+d}\hspace{0.05cm}$ of $SL(2,\NC)$ on $\NP^1$.
\vskip 0.4cm

We would still like to fix the values of the constants
$h_s$ in Eqs.\hspace{0.05cm}\s(\ref{VGP}). We claim that they are such
that the Hitchin map is given by Eq.\hspace{0.05cm}\s(\ref{GR}), i.e. that
\begin{eqnarray}
{\cal H}(q,p)\ \ =\ \ -\hspace{0.025cm}{_1\over^{128\hspace{0.025cm}\pi^2}}\hspace{0.05cm}\hspace{0.025cm}\sum\limits_{s,
t=1,\dots,6,\atop s\hspace{0.05cm}\not=\hspace{0.05cm} t}{r_{st}(q,p)\over
(\lambda-\lambda_s)(\lambda-\lambda_t)}\hspace{0.05cm}\hspace{0.025cm}(d\lambda)^2\hspace{0.05cm}.
\label{glr}
\end{eqnarray}
First note that the above formula is consistent with the
$SL(2,\NC)$ transformations. Indeed, relations
(\ref{sum}) imply that
\begin{eqnarray}
\sum\limits_{s\hspace{0.05cm}\not=\hspace{0.05cm} t}{r_{st}\over
(\lambda'-\lambda'_s)(\lambda'-\lambda'_t)}\hspace{0.05cm}\hspace{0.025cm}(d\lambda')^2
\ =\ \sum\limits_{s\hspace{0.05cm}\not=\hspace{0.05cm} t}{r_{st}\over
(\lambda-\lambda_s)(\lambda-\lambda_t)}\hspace{0.05cm}\hspace{0.025cm}(d\lambda)^2
\nonumber
\end{eqnarray}
for $\lambda'={a\lambda+b\over c\lambda+d}$.
Taking, in particular, $\lambda'=\lambda^{-1}$ one verifies
that the quadratic differentials (\ref{glr}) are
regular at infinity. They are also regular at the branching
points since ${d\lambda\over\sqrt{\lambda-\lambda_s}}$
is a local holomorphic differential around $x_s$.
Hence the r.h.s. of Eq.\hspace{0.05cm}\s(\ref{glr}) is indeed
a (holomorphic) quadratic differential. Thus
Eq.\hspace{0.05cm}\s(\ref{glr}) is equivalent to relations (\ref{VGP})
with \hspace{0.05cm}$h_s= {(d\lambda)^2\over(\lambda-\lambda_s)}
\vert_{_{x_s}}\hspace{0.025cm}$, \hspace{0.05cm} modulo an overall normalization.
To prove Eq.\hspace{0.05cm}\s(\ref{glr}) we shall verify it
at a point of the phase space for which ${\cal H}(q,p)(x_s)\not=0$
for $s\not=1$. This will fix $h_s$ for $s\not=1$ and hence
all of them (two quadratic differentials equal at points
$x_s$ with $s\not=1$ have to coincide).
\vskip 0.4cm

Consider a pair $(\theta,\phi_{u_1})$ lying in the product
${\cal K}\times{\cal K}^*$ of the Kummer quartics with
\begin{eqnarray}
\theta(u)\ =\ {\rm e}^{{1\over 2}\hspace{0.025cm}\pi i\hspace{0.05cm} e'_1\cdot\tau\hspace{0.025cm} e'_1\hspace{0.025cm}
+\hspace{0.025cm} 2\pi i\hspace{0.05cm} e'_1\cdot u_1}\ \vartheta(u_1+E_1-u)
\ \vartheta(u_1+E_1+u)\hspace{1.5cm}\cr\cr
=\hspace{0.05cm}\sum\limits_e\hspace{0.05cm}(U_{e_1,e'_1}\theta_{2,e})(u_1)\
\theta_{2,e}(u)
\label{121}
\end{eqnarray}
for $e_1=e'_1=(1,0)$. Note that ${\langle\hspace{0.025cm}}\theta,\phi_{u_1}
\hspace{0.025cm}\rangle=0$.
Eq.\hspace{0.05cm}\s(\ref{first}) together with the relations (\ref{newf})
and the equation
\begin{eqnarray}
\partial_a\theta(u_1)\ =\ -\hspace{0.025cm}{\rm e}^{{1\over 2}\hspace{0.025cm}\pi i\hspace{0.05cm} e_1'\cdot\hspace{0.025cm}\tau
\hspace{0.025cm} e'_1\hspace{0.05cm}+\hspace{0.05cm} 2\pi i\hspace{0.05cm} e'_1\cdot\hspace{0.025cm} u_1}\hspace{0.05cm}\s\hspace{0.025cm}\partial_a\vartheta(E_1)
\ \vartheta(2u_1+E_1)
\nonumber
\end{eqnarray}
results in the identity
\begin{eqnarray}
{\cal H}(\theta,\phi_{u_1})\ =\
-\hspace{0.025cm} {_{C^2}\over^{16\pi^2}}\hspace{0.05cm}\s{\rm e}^{\hspace{0.025cm}\pi i\hspace{0.05cm} e'_1\cdot\hspace{0.025cm}\tau\hspace{0.025cm} e'_1
\hspace{0.05cm}+\hspace{0.05cm} 4\pi i\hspace{0.05cm} e'_1\cdot\hspace{0.025cm} u_1}\hspace{0.05cm}\s(\partial_2\vartheta(E_1))^2\hspace{0.05cm}\s
\vartheta(2u_1+E_1)^2\hspace{0.05cm}\s(\lambda-\lambda_1)^2
\hspace{0.05cm}\s{_{(d\lambda)^2}\over^{\zeta^2}}
\hspace{0.4cm}
\label{123}
\end{eqnarray}
where $C$ is the constant appearing in Eq.\hspace{0.05cm}\s(\ref{newf}).
Note that ${\cal H}(\theta,\phi_{u_1})\not=0$ as long as
\hspace{0.05cm}$\vartheta(2u_1+E_1)\not=0$. It follows
that ${\cal H}(\theta,\phi_{u_1})$ is a quadratic
differential proportional to \hspace{0.05cm}$(\lambda-\lambda_1)^2
\hspace{0.05cm}{(d\lambda)^2\over\zeta^2}$ which has the $4^{\hspace{0.025cm}\rm th}$
order zero at $x_1$. The latter property characterizes it
uniquely up to normalization.
\vskip 0.4cm

It is not difficult to check that Eq.\hspace{0.05cm}\s(\ref{glr}) gives
a quadratic differential with the same property. Indeed,
in the language of $q\hspace{0.025cm}$'s and $p\hspace{0.025cm}$'s, the linear form
\hspace{0.025cm}$\phi_{u_1}$ corresponds to a vector $p\in\NC^4$
and $\theta$ to $q=(p_2,-p_1,p_4,-p_3)$. A straightforward
verification shows that $r_{1\hspace{0.025cm} t}(q,p)=0$ for all $t\not=1$.
This implies that the quadratic differential given
by Eq.\hspace{0.05cm}\s(\ref{glr}) vanishes to the second order
at $x_1$. The condition that it vanishes to the fourth
order is
\begin{eqnarray}
\sum\limits_{s\hspace{0.05cm}\not=\hspace{0.05cm} t,\atop s,t\hspace{0.05cm}\not=\hspace{0.05cm} 1}
r_{st}((p_2,-p_1,p_4,-p_3),\hspace{0.05cm} p\hspace{0.025cm})\
\prod\limits_{v\hspace{0.05cm}\not=\s1,s,t}
(\lambda_1-\lambda_v)\ =\ 0\hspace{0.05cm}.
\nonumber
\end{eqnarray}
A direct calculation shows that this is exactly the equation
(\ref{kuma}) of the Kummer quartic with the coefficients
(\ref{kumb}) so that it holds for
$p$ corresponding to $\phi_{u_1}$. This establishes
proportionality between the Hitchin map and
the right hand side of Eq.\hspace{0.05cm}\s(\ref{glr}) with
a coefficient that may be still curve-dependent.
\vskip 0.4cm

Fixing the overall normalization of the Hitchin map
is more involved. We shall calculate the value
of the quadratic differential on the right hand side
of Eq.\hspace{0.05cm}\s(\ref{123}) at $\lambda=\lambda_2$ and
compare it to the value given by Eq.\hspace{0.05cm}\s(\ref{glr}).
Since this is somewhat technical, we defer the argument
to Appendix 4.
\vskip 0.4cm

The system with Hamiltonians (\ref{VGP}) bears some similarity
to the classic Neumann systems\footnote{we thank M. Olshanetsky
for attracting our attention to this fact}, also anchored
in modular geometry \cite{Mum}\cite{AT}. The Hamiltonians
of a Neumann system have the form
\begin{eqnarray}
{\cal H}_s\ =\ \sum\limits_{1\leq t\not= s\leq n}{J_{st}^2
\over{\lambda_s-\lambda_t}}
\label{Neum}
\end{eqnarray}
where $J_{st}\hspace{0.05cm}=\hspace{0.05cm} q_sp_t-q_tp_s$ are the functions on $T^*\NC^n$
generating the infinitesimal action of the complex group $SO_n$:
\begin{eqnarray}
&&\hbox to 5cm{$\{J_{st},J_{tv}\}\hspace{0.05cm}=\hspace{0.05cm} -J_{sv}$\hfill}{\rm for\ \ }
s,t,v\ \ {\rm different},\cr
&&\clabel{son}\cr
&&\hbox to 5cm{$\{J_{st},J_{vw}\}\hspace{0.05cm}=\s0$\hfill}{\rm for\ \ }
s,t,v,w\ \ {\rm different}.
\end{eqnarray}
The fact that the Hamiltonians (\ref{VGP}) (with constant $h_s$)
Poisson commute reduces, as is well known, to the identities
\begin{eqnarray}
&&\hbox to 5cm{$\{r_{st}+r_{sv}\hspace{0.025cm},\hspace{0.05cm} r_{tv}\}\hspace{0.05cm}=\hspace{0.05cm} 0$\hfill}
{\rm and\  cyclic\ permutations\ thereof}\hspace{0.05cm},\cr
\clabel{cyb1}\cr
&&\hbox to 5cm{$\{r_{st}\hspace{0.025cm},\hspace{0.05cm} r_{vw}\}\hspace{0.05cm}=\s0$\hfill}
{\rm for}\quad\quad\{s,t\}\cap\{v,w\}=\emptyset\hspace{0.05cm}.
\end{eqnarray}
If we set $r_{st}=J_{st}^{\m2}$ for the Neumann system, then
Eqs.\hspace{0.05cm}\s(\ref{cyb1}) follow from the relations (\ref{son}).
It appears that the same algebra stands behind
the fact\footnote{this is the classical version
of the observation of \cite{VGDJ}}
that $r_{st}$ given by Eq.\hspace{0.05cm}\s(\ref{rs}) verify
(\ref{cyb1}). The phase space \hspace{0.05cm}$T^*{\cal N}_{ss}\hspace{0.05cm}\cong\hspace{0.05cm}
\{\hspace{0.05cm}(q,p)\hspace{0.05cm}\hspace{0.025cm}\vert\hspace{0.05cm}\hspace{0.025cm} q\cdot p=0\hspace{0.05cm}\}/\NC^*\hspace{0.025cm}$, \hspace{0.025cm} where
$\NC^*$ acts by $(q,p)\mapsto(tq,\hspace{0.025cm} t^{-1}p)$, may be
identified with the coadjoint orbit of the group $SL_4$
composed of the traceless complex 4$\times$4 matrices
$\vert p\rangle\langle q\vert$ of rank 1. Using the isomorphism
of the complex Lie algebras $sl_4\cong so_6$, we obtain
the functions $J_{st}=-J_{ts}$ on this $SL_4$-orbit
which generate the action of $so_6$ and have the Poisson
brackets given by (\ref{son}). A straightforward check shows
that, for $r_{st}$ of Eq.\hspace{0.05cm}\s(\ref{rs}),
\begin{eqnarray}
r_{st}\ =\ -\m4\hspace{0.025cm} J_{st}^{\hspace{0.025cm} 2}
\end{eqnarray}
so that Eq.\hspace{0.05cm}\s(\ref{cyb1}) follows from the $so_6$-algebra
(\ref{son}).
\vskip 0.4cm

Upon the introduction of the rational
functions \hspace{0.05cm}${r_{st}\over\lambda}\hspace{0.025cm}$,
Eqs.\hspace{0.05cm}\s(\ref{cyb1}) take the form
\begin{eqnarray}
&&\{{_{r_{st}}\over^{\lambda_s-\lambda_t}}\hspace{0.025cm},\hspace{0.05cm}
{_{r_{sv}}\over^{\lambda_s-\lambda_v}}\}\ +\ \{
{_{r_{st}}\over^{\lambda_s-\lambda_t}}\hspace{0.025cm},\hspace{0.05cm}
{_{r_{tv}}\over^{\lambda_t-\lambda_v}}\}
\ +\ \{{_{r_{sv}}\over^{\lambda_s-\lambda_v}}\hspace{0.025cm},
\hspace{0.05cm}{_{r_{tv}}\over^{\lambda_t-\lambda_v}}\}
\ =\ 0\hspace{0.05cm},\cr
\clabel{cyb2}\cr
&&\{{_{r_{st}}\over^{\lambda_s-\lambda_t}}\hspace{0.025cm},\hspace{0.05cm}
{_{r_{vw}}\over^{\lambda_v-\lambda_w}}\}\ =\ 0\
\quad\quad\quad {\rm for}\quad\ \{s,t\}
\cap\{v,w\}=\emptyset\hspace{0.05cm}.
\end{eqnarray}
The first of these identities is, essentially, the classical
Yang-Baxter equation. Note, however, that $r_{st}$,
unlike in the Gaudin and Neumann systems, is not an element
of a product of two copies of a Poisson algebra
of functions: there is no sign of an explicit product
structure, or of a reduction thereof, in our phase space.
The important question is whether $r_{st}$ come from
a rational solution of the CYBE. The conformal field theory
work \cite{Knizh}\cite{Zamo} suggests that the answer may
be positive, at least in some sense.
\vskip 0.4cm

The knowledge of the explicit form of the quadratic differentials
${\cal H}(q,p)$ allows to write the explicit equations for
the genus 5 spectral curve of the $SL_2$ Hitchin system at genus 2,
see Eq.\hspace{0.05cm}\s(\ref{1}). They take the form
\begin{eqnarray}
\zeta^2\ =\ \prod\limits_{s=1}^6(\lambda-\lambda_s)\hspace{0.05cm},\quad\ \
\xi^2\ =\ \sum\limits_{s\hspace{0.05cm}\not=\hspace{0.05cm} t}r_{st}(q,p)
\prod\limits_{v\hspace{0.05cm}\not=\hspace{0.05cm} s,t}(\lambda-\lambda_v)\hspace{0.05cm}.
\label{specc}
\end{eqnarray}
The involution of the spectral curve flips the sign of $\xi$.
To extract explicit formulae for the angle variables describing
the point on the Prym variety of the spectral curve, we would
need, however, a more explicit knowledge of the entire
Lax matrix $\Psi$.
\vskip 0.9cm

\nsection{Conclusions}
\vskip 0.5cm

The main result of the present paper is the proof
of self-duality of the Hitchin Hamiltonians
on the cotangent bundle to the moduli space
of the holomorphic $SL_2$ bundles on a genus 2 complex
curve. The result was based on an expression for
the Hitchin Hamiltonians off the Kummer quartic on
which the values of the Hamiltonians were determined
in \cite{VGP}. Using the self-duality,
we were able to complete the analysis of \cite{VGP}
and to obtain the explicit formula (\ref{GR})
for the Hitchin map (\ref{Hm}) giving the action
variables of the integrable system. The explicit
formula for the angle variables remains still to be found.
An interesting open problem is an extension of the present
work to the case with insertion points.
\vskip 0.4cm

Another important problem related to Hitchin's construction
is the quantization of the corresponding integrable systems.
For the $SL_2$ case such a quantization is essentially
provided by the Knizhnik-Zamolodchikov-Bernard-Hitchin
connection \cite{KZ}\cite{Ber}\cite{Ber2} which describes
the variation of conformal blocks of the $SU_2$ WZW conformal
field theory under the change of the complex structure
of the curve. The (partition function) conformal blocks
are holomorphic sections of the $k^{\hspace{0.025cm}\rm th}$-power
of the determinant line bundle over the moduli space ${\cal N}_{ss}$
($k$ is the level of the WZW theory). In our case, they are simply
$k^{\hspace{0.025cm}\rm th}$-order homogeneous polynomials on $H^0(L_\theta^2)$.
It is easy to quantize the Hitchin Hamiltonians
\begin{eqnarray}
H_s\ =\ \sum\limits_{t\hspace{0.05cm}\not=\hspace{0.05cm} s}{r_{st}\over\lambda_s-\lambda_t}\hspace{0.05cm}.
\nonumber
\end{eqnarray}
If one keeps the original formulae (\ref{rs}) for $r_{st}$
in which $p_i$ stands now for ${1\over i}\hspace{0.05cm}\partial_{q_i}$, the
relations (\ref{cyb1}) or (\ref{cyb2}) still hold after
the replacement of the Poisson brackets by the commutators.
One obtains this way the commuting operators $H_s$ mapping
the space of homogeneous, degree $k$ polynomials
in variables $q$ into itself. Note, however, that now
\begin{eqnarray}
\sum\limits_{t\hspace{0.05cm}\not=\hspace{0.05cm} s}r_{st}\ =\ -\hspace{0.025cm} k(k+4)
\nonumber
\end{eqnarray}
for each fixed $s$ so that the quantization changes the conformal
properties of the Hamiltonians. A direct construction of
the projective version of the KZBH connection for group $SU_2$
and genus 2 has been recently given in ref.\hspace{0.05cm}\s\cite{VGDJ}
by following Hitchin's approach \cite{Hitch2}. It is consistent
with the above {\it ad hoc} quantization of the classical
Hitchin Hamiltonians.
\vskip 0.4cm

The integral formulae for the conformal blocks
\cite{BabF,ReshV,EtinK} or, equivalently,
the integral formulae for the scalar product of
the conformal blocks \cite{FalKG} have been used
at genus 0 and 1 to extract the Bethe Ansatz
eigen-vectors and eigen-values of the quantized
version of the quadratic Hitchin Hamiltonians.
The Bethe-Ansatz type diagonalization of the
quantization of the genus 2 Hitchin Hamiltonians
is among the issues that will have to be examined.
\vskip 0.4cm

Finally, as we stressed in the text, the relations
between the conformal WZW field theory on a genus 2
surface and an orbifold theory in genus 0 requires
further study.
\vskip 0.9cm

\renewcommand{{A\thesection.\arabic{equation}}}{A#1.\arabic{equation}}
\def{A\thesection.\arabic{equation}}{{A#1.\arabic{equation}}}
\nappendix{1}

\vskip 0.5cm

Let us check that $\theta$ given by Eq.\hspace{0.05cm}\s(\ref{thetaE})
vanishes if and only if $$H^0(l_u\otimes E)\hspace{0.05cm}
=\hspace{0.05cm}\{\hspace{0.05cm}(s_1,s_2)\ |\ s_2\in H^0(l_u
l_{u_1}),\ \bar\partial_{_{l_u^{-1}l_{u_1}}}
\hspace{-0.1cm}s_1+s_2\hspace{0.05cm} b=0\hspace{0.05cm}\}\hspace{0.05cm}\ \not=\ \s0\hspace{0.05cm}.$$
For \hspace{0.05cm}$u-u_1\in\NZ^2+\tau\NZ^2\hspace{0.05cm}$ the 1$^{\rm st}$
theta function on the r.h.s. of Eq.\hspace{0.05cm}\s(\ref{Kk}) vanishes
but  \hspace{0.05cm}$l_u=l_{u_1}\hspace{0.05cm}$ and $l_{u_1}\in C_E\hspace{0.025cm}$. Assume
now that \hspace{0.05cm}$u-u_1\hspace{0.05cm}\not\in\hspace{0.05cm}\NZ^2+\tau\NZ^2\hspace{0.025cm}$.
Then \hspace{0.05cm}${dim}\ H^0(l_u^{-1}l_{u_1}K)=1\hspace{0.05cm}$
with a non-zero \hspace{0.05cm}$\chi\in H^0(l_u^{-1}l_{u_1}K)$.
The necessary and sufficient condition for the solvability
of the equation \hspace{0.05cm}$\bar\partial_{_{l_ul_{u_1}^{-1}}}s_1+s_2\hspace{0.025cm} b=0\hspace{0.05cm}$
for a given \hspace{0.05cm}$s_2\in H^0(l_ul_{u_1})\hspace{0.05cm}$ is
\begin{eqnarray}
\int_{_\Sigma}\chi\hspace{0.025cm} s_2\hspace{0.05cm} b=0\hspace{0.05cm}.
\label{soco}
\end{eqnarray}
If \hspace{0.05cm}$u+u_1\in \NZ^2+\tau\NZ^2\hspace{0.05cm}$ then
\hspace{0.05cm}$l_ul_{u_1}=K\hspace{0.05cm}$ and \hspace{0.05cm}${dim}\hspace{0.05cm} H^0(l_ul_{u_1})=2\hspace{0.05cm}$
so that there always is a non-zero solution but
also \hspace{0.05cm}$\theta(u)=0\hspace{0.05cm}$ in this case due to the vanishing of the
$2^{\rm nd}$ theta function on the r.h.s. of Eq.\hspace{0.05cm}\s(\ref{Kk}).
Finally, if \hspace{0.05cm}$u\pm u_1\hspace{0.05cm}\not\in\hspace{0.05cm}\NZ^2+\tau\NZ^2\hspace{0.05cm}$ then
\hspace{0.05cm}$s_2\in H^0(l_ul_{u_1})\hspace{0.05cm}$ has to be proportional
to the element defined by (\ref{s_2}) and the condition
(\ref{soco}) coincides with the equation \hspace{0.05cm}$\theta(u)=0\hspace{0.025cm}$.
\vskip 0.9cm

\nappendix{2}
\vskip 0.5cm

Let us show that the 1,0-form $\mu$ satisfying relations
(\ref{cfm}) and (\ref{cfm1}) automatically fulfills the condition
\begin{eqnarray}
\int_{_\Sigma}{\kappa}\hspace{0.05cm}\mu\wedge b\hspace{0.05cm}=\s0\hspace{0.05cm}.
\label{orth}
\end{eqnarray}
Among the infinitesimal gauge field variations $\delta B$
given by Eq.\hspace{0.05cm}\s(\ref{DB}) there are ones which are equivalent
to infinitesimal gauge transformations:
\begin{eqnarray}
\delta B\hspace{0.05cm}=\hspace{0.05cm}\bar\partial\Lambda\hspace{0.05cm}+\hspace{0.05cm}[B,\Lambda]\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Explicitly, for \hspace{0.05cm}$\Lambda=(\matrix{_{-\sigma}&_\varphi\cr^\kappa
&^\sigma})\hspace{0.05cm}$ with $\sigma$ a function, $\varphi$ a section
of $l_{u_1}^{-2}$ and $\kappa$ a section of $l_{u_1}^2$,
this requires that
\begin{eqnarray}
\bar\partial\kappa\hspace{0.05cm}=\s0\hspace{0.05cm},\quad\quad
\pi\hspace{0.05cm}\delta u_1\hspace{0.025cm}({\rm Im}\hspace{0.025cm}\tau)^{-1}\bar\omega\hspace{0.05cm}=\hspace{0.05cm}
-\bar\partial\sigma+\kappa\hspace{0.025cm} b\hspace{0.05cm},\quad\quad\delta b\hspace{0.05cm}=\hspace{0.05cm}\bar\partial\varphi
+2\hspace{0.025cm}\sigma\hspace{0.025cm} b\hspace{0.05cm}.
\label{three}
\end{eqnarray}
Such variations may only change the normalization
of the theta function $\theta$. Integrating the second of
the above relations against forms $\omega^a$ and using
Eq.\hspace{0.05cm}\s(\ref{kob}) we find that
\begin{eqnarray}
\delta u_1^a\hspace{0.05cm}=\hspace{0.05cm}-{_1\over^{2\pi i}}\hspace{0.05cm}\epsilon^{ab}
\partial_{b}\theta(u_1)
\label{du0}
\end{eqnarray}
for the proper normalization of ${\kappa}$. For such $\delta u_1$
the first term on the right hand side
of Eq.\hspace{0.05cm}\s(\ref{nd}) gives a theta
function vanishing at $u=u_1$ and may be compensated by the
second term. The 3$^{\rm rd}$ equation of (\ref{three})
gives the compensating $\delta b\in\wedge^{01}(l_{u_1}^{-2})$.
Pairing Eq.\hspace{0.05cm}\s(\ref{nd}) with the above $\delta u_1$
and $\delta b$ with the linear form $\phi$, we obtain
the identity
\begin{eqnarray}
{_1\over^i}\hspace{0.05cm}\epsilon^{ab}
\partial_{b}\theta(u_1)\hspace{0.05cm}({\rm Im}\hspace{0.025cm}\tau)_{ac}^{-1}
\int_{_\Sigma}\chi^c\wedge b\hspace{0.05cm}+\s2\int_{_\Sigma}
\sigma\hspace{0.05cm}\eta\wedge b\ =\ 0\ .
\label{cri}
\end{eqnarray}
On the other hand,
\begin{eqnarray}
\int_{_\Sigma}{\kappa}\hspace{0.05cm}\mu\wedge b\hspace{0.05cm}=\hspace{0.05cm}
\int_{_\Sigma}\mu\wedge\bar\partial\sigma\hspace{0.05cm}-\hspace{0.05cm}{_1\over^{2i}}
\hspace{0.05cm}\epsilon^{ab}\partial_b\theta(u_1)\hspace{0.05cm}({\rm Im}\hspace{0.025cm})^{-1}_{ac}
\int_{_\Sigma}\mu\wedge\bar\omega^c\hspace{0.05cm}\cr
=\hspace{0.05cm}-\int_{_\Sigma}\sigma\hspace{0.05cm}\eta\wedge b\hspace{0.05cm}-\hspace{0.05cm}
{_1\over^{2i}}\hspace{0.05cm}\epsilon^{ab}\partial_b\theta(u_1)\hspace{0.05cm}({\rm Im}
\hspace{0.025cm})^{-1}_{ac}\int_{_\Sigma}\chi^c\wedge b\ =\ 0
\nonumber
\end{eqnarray}
where we have subsequently used the 2$^{\rm nd}$ equation
in (\ref{three}) with $\delta u_1$ given by Eq.\hspace{0.05cm}\s(\ref{du0}),
the relation $\bar\partial\mu=-\eta\wedge b$ and Eq.\hspace{0.05cm}\s(\ref{cfm})
fixing $\mu$ and, finally, the identity (\ref{cri}).
\vskip 0.9cm

\nappendix{3}
\vskip 0.5cm

It is not difficult to see that there exist a non-zero
element $P\in S^4 H^0(L_\Theta^2)$, a homogeneous polynomial
of degree 4 on $H^0(L_\Theta^2)^*$, s.t.
\begin{eqnarray}
P(\phi_{u'})\hspace{0.05cm}=\s0
\nonumber
\end{eqnarray}
for all $u'\in\NC^2$. Indeed, $dim\hspace{0.05cm} S^4H^0(L_\Theta^2)
=({7\atop3})=35$ but the map $u'\mapsto P(\phi_{u'})$
defines an even theta function of order 8 and
$dim\hspace{0.05cm} H^0_{\rm even}(L_\Theta^8)=34$.
$P$ is a quartic expression in $\theta_{2,e}(u')$
which vanishes for all $u'$. It has to be preserved
by the $(\NZ/2\NZ)^4$-action (\ref{acone}) and
hence it must be of the form
\begin{eqnarray}
P&=&c_1\hspace{0.05cm}(\theta_{2,(0,0)}^{\s4}+\hspace{0.05cm}\theta_{2,(1,0)}^{\s4}
+\hspace{0.05cm}\theta_{2,(0,1)}^{\s4}+\hspace{0.05cm}\theta_{2,(1,1)}^{\s4})\cr\cr
&+&c_2\hspace{0.05cm}(\theta_{2,(0,0)}^{\s2}\hspace{0.05cm}\hspace{0.025cm}\theta_{2,(1,0)}^{\s2}+
\hspace{0.05cm}\theta_{2,(0,1)}^{\s2}\hspace{0.05cm}\hspace{0.025cm}\theta_{2,(1,1)}^{\s2})\cr\cr
&+&c_3\hspace{0.05cm}(\theta_{2,(0,0)}^{\s2}\hspace{0.05cm}\hspace{0.025cm}\theta_{2,(0,1)}^{\s2}+
\hspace{0.05cm}\theta_{2,(1,0)}^{\s2}\hspace{0.05cm}\hspace{0.025cm}\theta_{2,(1,1)}^{\s2})\cr\cr
&+&c_4\hspace{0.05cm}(\theta_{2,(0,0)}^{\s2}\hspace{0.05cm}\hspace{0.025cm}\theta_{2,(1,1)}^{\s2}+
\hspace{0.05cm}\theta_{2,(1,0)}^{\s2}\hspace{0.05cm}\hspace{0.025cm}\theta_{2,(0,1)}^{\s2})\cr\cr
&+&c_5\hspace{0.05cm}\theta_{2,(0,0)}\hspace{0.05cm}\hspace{0.025cm}\theta_{2,(1,0)}\hspace{0.05cm}\hspace{0.025cm}\theta_{2,(0,1)}
\hspace{0.05cm}\hspace{0.025cm}\theta_{2,(1,1)}\hspace{0.05cm}.
\nonumber
\end{eqnarray}
It is not difficult to calculate the values of coefficients
$c_i$. Denoting $\alpha\equiv\theta_{2,(0,0)}(0)$,
\hspace{0.05cm}$\beta\equiv\theta_{2,(1,0)}(0)$,
\hspace{0.05cm}$\gamma\equiv\theta_{2,(0,1)}(0)\hspace{0.05cm}$
and \hspace{0.05cm}$\delta\equiv\theta_{2,(1,1)}(0)$,
one has
\begin{eqnarray}
c_1&=&\ \ \hspace{0.025cm}(\alpha^2\beta^2-\gamma^2\delta^2)
(\alpha^2\gamma^2-\beta^2\delta^2)(\alpha^2\delta^2
-\beta^2\gamma^2)\hspace{0.05cm},\cr\cr
c_2&=&-(\alpha^4+\beta^4-\gamma^4-\delta^4)
(\alpha^2\gamma^2-\beta^2\delta^2)(\alpha^2\delta^2
-\beta^2\gamma^2)\hspace{0.05cm},\cr\clabel{em}\cr
c_3&=&-(\alpha^4-\beta^4+\gamma^4-\delta^4)
(\alpha^2\beta^2-\gamma^2\delta^2)(\alpha^2\delta^2
-\beta^2\gamma^2)\hspace{0.05cm},\cr\cr
c_4&=&-(\alpha^4-\beta^4-\gamma^4+\delta^4)
(\alpha^2\beta^2-\gamma^2\delta^2)(\alpha^2\gamma^2
-\beta^2\delta^2)\hspace{0.05cm},\cr\cr
c_5&=&\ \ \m2\hspace{0.025cm} \alpha\beta\gamma\delta\hspace{0.025cm}
[(\alpha^4-\beta^4+\gamma^4-\delta^4)^2-\hspace{0.025cm}
4(\alpha^2\gamma^2-\beta^2\delta^2)^2]\hspace{0.05cm}.
\end{eqnarray}
If we use the basis dual to $(\theta_{2,e})$ to
identify $\phi\in H^0(L_\Theta^2)^*$ with a vector
$p=(p_1,p_2,p_3,p_4)\in\NC^4$, the equation
of the Kummer quartic ${\cal K}^*$ becomes
\begin{eqnarray}
c_1\hspace{0.05cm}(p_1^4+p_2^4+p_3^4+p_4^4)\hspace{0.05cm}+\hspace{0.05cm} c_2\hspace{0.05cm}
(p_1^2\hspace{0.025cm} p_2^2\hspace{0.05cm}+\hspace{0.05cm} p_3^2\hspace{0.025cm} p_4^2)
\hspace{0.05cm}+\hspace{0.05cm} c_3\hspace{0.05cm}(p_1^2\hspace{0.025cm} p_3^2\hspace{0.05cm}+\hspace{0.05cm} p_2^2\hspace{0.025cm} p_4^2)\hspace{0.05cm}\hspace{0.025cm}\cr
\clabel{kuma}\cr
+\hspace{0.05cm} c_4\hspace{0.05cm}(p_1^2\hspace{0.025cm} p_4^2\hspace{0.05cm}+\hspace{0.05cm} p_2^2\hspace{0.025cm} p_3^2)
\hspace{0.05cm}+\hspace{0.05cm} c_5\hspace{0.05cm} p_1\hspace{0.025cm} p_2\hspace{0.025cm} p_3\hspace{0.025cm} p_4\ =\ 0\hspace{0.05cm}.
\end{eqnarray}
Similarly, identifying $\theta\in H^0(L_\Theta^2)$ with
$q=(q_1,q_2,q_3,q_4)\in\NC^4$ with the help of the basis
$(\theta_{2,e})$, the same equation with $p$ replaced by
$q$ defines the Kummer quartic ${\cal K}$, compare \cite{Kummer},
page 81.
\vskip 0.4cm

We shall also need another well known presentation of the above
equation using the inhomogeneous coordinates of the Weierstrass
points $\lambda_s$ given by Eq.\hspace{0.05cm}\s(\ref{wp}). It is usually
obtained by beautiful geometric considerations about quadratic
line complexes, see \cite{GH}. It may be also obtained
analytically by observing that the multivalued functions
\begin{eqnarray}
x\ \mapsto\ \theta_{2,e}(\smallint_{x_0}^x\omega-\Delta)
\nonumber
\end{eqnarray}
transform like bilinears in \hspace{0.05cm}$\partial_a\vartheta(
\smallint_{x_0}^x\omega-\Delta)\hspace{0.025cm}$, \hspace{0.05cm} i.e.\hspace{0.05cm}\s that they
represent quadratic differentials. It follows that
\begin{eqnarray}
\sum\limits_e\theta_{2,e}(E_s)\hspace{0.05cm}\s\theta_{2,e}(
\smallint_{x_0}^x\omega-\Delta)\ =\ \vartheta(E_s+
\smallint_{x_0}^x\omega-\Delta)\hspace{0.05cm}\s\vartheta(E_s-
\smallint_{x_0}^x\omega+\Delta)\hspace{0.05cm}\cr
=\ D_s\hspace{0.05cm}\s\left(\partial_1\vartheta(E_s')\hspace{0.05cm}\hspace{0.025cm}\partial_2\vartheta(
\smallint_{x_0}^x\omega-\Delta)\hspace{0.05cm}-\hspace{0.05cm}
\partial_2\vartheta(E_s')\hspace{0.05cm}\hspace{0.025cm}\partial_1\vartheta(
\smallint_{x_0}^x\omega-\Delta)\right)\hspace{0.05cm}\clabel{rlti}\cr
\ \cdot\hspace{0.05cm}\left(\partial_1\vartheta(E_s'')\hspace{0.05cm}\hspace{0.025cm}\partial_2\vartheta(
\smallint_{x_0}^x\omega-\Delta)\hspace{0.05cm}-\hspace{0.05cm}
\partial_2\vartheta(E_s'')\hspace{0.05cm}\hspace{0.025cm}\partial_1\vartheta(
\smallint_{x_0}^x\omega-\Delta)\right)\hspace{0.05cm}
\end{eqnarray}
where $E_s={_1\over^2}(e_s+\tau e'_s)\hspace{0.05cm}$ is an odd characteristics
from the list (\ref{ordr}) and $E_s'\hspace{0.025cm},\hspace{0.05cm} E_s''$ are the
two other ones s.t. $E_s+E_s'=E_s''\hspace{0.05cm}\hspace{0.025cm}{\rm mod}\hspace{0.025cm}(\NZ^2+\tau\NZ^2)$.
The odd characteristics $E_s,\hspace{0.025cm} E_s',\hspace{0.025cm} E_s''$ are
either a permutation of $E_1,\hspace{0.025cm} E_4,\hspace{0.025cm} E_5$ or
a permutation of $E_2,\hspace{0.025cm} E_3,\hspace{0.025cm} E_6$.
The relations (\ref{rlti}) hold since both sides represent
a quadratic differential with double zeros at the Weierstrass
points corresponding to $E_s'$ and $E_s''$. One may obtain
expressions for the coefficients $D_s$ by the de l'Hospital
rule applied twice at those points. Specifying then
$\smallint_{x_0}^x\omega-\Delta$ to $E_s$ or to 3 remaining
odd characteristics one obtains relations for
quadratic combinations of $\theta_{2,e}(0)$ of the form
\hspace{0.05cm}$\pm\alpha^2\pm\beta^2\pm\gamma^2\pm\delta^2\hspace{0.05cm}$
with 2 plus and 2 minus signs as well as for
\hspace{0.05cm}$\alpha\beta\pm\gamma\delta\hspace{0.025cm}$,
\hspace{0.05cm}$\alpha\gamma\pm\beta\delta\hspace{0.025cm}$
and \hspace{0.05cm}$\alpha\delta\pm\beta\gamma\hspace{0.025cm}$.
These relations may be used to compute the ratios
of the coefficients $c_i$ (\ref{em})
which become functions of $\lambda_s$ only.
One obtains this way an alternative expression
for the coefficients $c_i$
\begin{eqnarray}
c_1&=&(\lambda_1-\lambda_2)(\lambda_3-\lambda_4)
(\lambda_5-\lambda_6)\hspace{0.05cm},\cr\cr
c_2&=&2(\lambda_1-\lambda_2)
((\lambda_3-\lambda_5)(\lambda_4-\lambda_6)
+(\lambda_3-\lambda_6)(\lambda_4-\lambda_5))\hspace{0.05cm},\cr\cr
c_3&=&-2(\lambda_3-\lambda_4)((\lambda_1-\lambda_5)
(\lambda_2-\lambda_6)
+(\lambda_1-\lambda_6)(\lambda_2-\lambda_5))\hspace{0.05cm},
\clabel{kumb}\cr\cr
c_4&=&2(\lambda_5-\lambda_6)((\lambda_1-\lambda_3)
(\lambda_2-\lambda_4)
+(\lambda_1-\lambda_4)(\lambda_2-\lambda_3))\hspace{0.05cm},\cr\cr
c_5&=&-2(\lambda_1-\lambda_3)((\lambda_4-\lambda_5)
(\lambda_2-\lambda_6)
+(\lambda_4-\lambda_6)(\lambda_2-\lambda_5))\cr
&&-2(\lambda_1-\lambda_4)((\lambda_3-\lambda_5)
(\lambda_2-\lambda_6)
+(\lambda_3-\lambda_6)(\lambda_2-\lambda_5))\cr
&&-2(\lambda_1-\lambda_5)((\lambda_2-\lambda_4)
(\lambda_3-\lambda_6)
+(\lambda_2-\lambda_3)(\lambda_4-\lambda_6))\cr
&&-2(\lambda_1-\lambda_6)((\lambda_2-\lambda_4)
(\lambda_3-\lambda_5)
+(\lambda_2-\lambda_3)(\lambda_4-\lambda_5))\hspace{0.05cm}.
\end{eqnarray}
equivalent to the previous one up to normalization.
Note that the $SL(2,\NC)$ transformations
\hspace{0.05cm}$\lambda_s\mapsto{a\lambda_s+b\over c\lambda_s+d}\hspace{0.05cm}$
preserve the form the quartic equation.
The virtue of the analytic approach is that it
also provides useful expressions for the non-homogeneous
ratios like e.g.
\begin{eqnarray}
{\alpha\beta+\gamma\delta\over\alpha^2\gamma^2-\beta^2\delta^2}
\ =\ -\hspace{0.05cm}{{\rm e}^{-\hspace{0.025cm}{1\over 2}\hspace{0.025cm} \pi i\hspace{0.05cm}(1,0)\cdot\tau\hspace{0.025cm}(1,0)}\over
2\hspace{0.05cm}\hspace{0.025cm} C^2\hspace{0.05cm}\s(\partial_2\vartheta(E_1))^2}\ {(\lambda_2-\lambda_5)
(\lambda_2-\lambda_6)(\lambda_3-\lambda_4)\over\lambda_1-\lambda_2}\hspace{0.05cm}.
\label{tbus}
\end{eqnarray}
$C^2$ is given by the equations
\begin{eqnarray}
C^2\hspace{0.05cm}=\hspace{0.05cm}{_1\over^2}\hspace{0.05cm}{_{(\partial_1\vartheta)^3\hspace{0.05cm}\partial_2^3\vartheta
\hspace{0.05cm}-\hspace{0.05cm} 3\hspace{0.05cm}(\partial_1\vartheta)^2\hspace{0.05cm}\partial_2\vartheta
\hspace{0.05cm}\partial_1\partial_2^2\vartheta
\hspace{0.05cm}+\hspace{0.05cm} 3\hspace{0.05cm}\partial_1\vartheta\hspace{0.05cm}(\partial_2\vartheta)^2
\hspace{0.05cm}\partial_1^2\partial_2\vartheta
\hspace{0.05cm}-\hspace{0.05cm}(\partial_2\vartheta)^3
\hspace{0.05cm}\partial_1^3\vartheta}\over^{(\partial_2\vartheta)^4}}\bigg\vert_{E_s}
\ \prod\limits_{t\hspace{0.05cm}\not=\hspace{0.05cm} s}(\lambda_s-\lambda_t)\hspace{0.4cm}
\nonumber
\end{eqnarray}
holding for any fixed $s$. It is not difficult to see by
differentiating twice Eq.\hspace{0.05cm}\s(\ref{inho}) at $x=x_s$
that $C$ is the same constant that appears in Eq.\hspace{0.05cm}\s(\ref{newf}).
The expression (\ref{tbus}) is used below to fix the
normalization of the Hitchin map.
\vskip 0.9cm

\nappendix{4}
\vskip 0.5cm

We shall show here that the overall normalization of the Hitchin map
is as in Eq.\hspace{0.05cm}\s(\ref{glr}). Since
\begin{eqnarray}
&&{\rm e}^{\hspace{0.025cm}\pi i\hspace{0.05cm} e_1'\cdot\tau\hspace{0.025cm} e_1'\hspace{0.05cm}+\hspace{0.05cm} 4\pi i\hspace{0.05cm} e_1'
\cdot u_1}\hspace{0.05cm}\s\vartheta(2u_1+E_1)^2\hspace{0.05cm}\cr\cr
&&=\hspace{0.05cm}-\hspace{0.025cm}{\rm e}^{\hspace{0.025cm}\pi i\hspace{0.05cm} e_1'\cdot\tau\hspace{0.025cm} e_1'}
\hspace{0.05cm}\s\vartheta(2u_1+E_1)\hspace{0.05cm}\s\vartheta(2u_1-E_1)
=\hspace{0.05cm}-\hspace{0.025cm}{\rm e}^{\hspace{0.025cm}\pi i\hspace{0.05cm} e_1'\cdot\hspace{0.025cm}\tau\hspace{0.025cm} e_1'}\hspace{0.05cm}\sum\limits_e
\theta_{2,e}(E_1)\hspace{0.05cm}\s\theta_{2,e}(2u_1)\hspace{0.05cm}\cr
&&=\hspace{0.05cm}-\hspace{0.025cm}{\rm e}^{\hspace{0.025cm}{1\over 2}\hspace{0.025cm}\pi i\hspace{0.05cm}(1,0)\cdot\tau(1,0)}
\hspace{0.05cm}\sum\limits_e(-1)^{(1,0)\cdot e}\hspace{0.05cm}\s\theta_{2,e+(1,0)}(0)
\hspace{0.05cm}\s\theta_{2,e}(2u_1)\hspace{0.05cm},
\nonumber
\end{eqnarray}
the coefficient of ${(d\lambda)^2\over\zeta^2}$
on the right hand side of Eq.\hspace{0.05cm}\s(\ref{123}) takes
at $\lambda=\lambda_2$ the value
\begin{eqnarray}
\hspace{0.025cm}{_{C^2}\over^{16\pi^2}}\hspace{0.05cm}\s{\rm e}^{\hspace{0.025cm}{1\over 2}\hspace{0.025cm}\pi i\hspace{0.05cm}(1,0)
\cdot\tau(1,0)}\ (\partial_2\vartheta(E_1))^2\
(\lambda_1-\lambda_2)^2\hspace{0.05cm}\s(\beta\hspace{0.025cm}\theta_{2,(0,0)}(2u_1)
\hspace{0.05cm}-\hspace{0.05cm}\alpha\hspace{0.025cm}\theta_{2,(1,0)}(2u_1)\cr
+\hspace{0.05cm}\delta\hspace{0.025cm}\theta_{2,(0,1)}(2u_1)\hspace{0.05cm}
-\hspace{0.05cm}\gamma\hspace{0.025cm}\theta_{2,(1,1)}(2u_1))
\label{ha}
\end{eqnarray}
in the notations of Appendix 3.
This coefficient should coincide with the one obtained from the
right hand side of Eq.\hspace{0.05cm}\s(\ref{glr}) which is equal to
\begin{eqnarray}
-\hspace{0.025cm}{_1\over^{64\pi^2}}\sum\limits_{t\hspace{0.05cm}\not=\s2}r_{2\hspace{0.025cm} t}(q,p)\hspace{0.05cm}
\prod\limits_{v\hspace{0.05cm}\not=\s2,t}(\lambda_2-\lambda_v)
\label{ho}
\end{eqnarray}
calculated at $(q,p)$ corresponding to $(\theta,\phi_{u_1})$
with $\theta$ given by Eq.\hspace{0.05cm}\s(\ref{121}). The respective
values of $r_{st}$ are:
\begin{eqnarray}
r_{1\hspace{0.025cm} t}&=& 0\hspace{0.05cm},\cr\cr
r_{23}&=&2\hspace{0.05cm}(-\hspace{0.025cm}\alpha\gamma^2
\hspace{0.025cm}\theta_{2,(0,0)}(2u_1)\hspace{0.05cm}-\hspace{0.05cm}\beta\delta^2\hspace{0.05cm}\theta_{2,(1,0)}(2u_1)
\hspace{0.05cm}-\hspace{0.05cm}\gamma\alpha^2\hspace{0.05cm}\theta_{2,(0,1)}(2u_1)\cr
&&-\hspace{0.05cm}\delta\beta^2\hspace{0.05cm}\theta_{2,(1,1)}(2u_1)
\hspace{0.05cm}-\hspace{0.05cm}\beta\gamma\delta\hspace{0.05cm}\theta_{2,(0,0)}(2u_1)
\hspace{0.05cm}-\hspace{0.05cm}\alpha\gamma\delta\hspace{0.05cm}\theta_{2,(1,0)}(2u_1)\cr
&&-\hspace{0.05cm}\alpha\beta\delta\hspace{0.05cm}\theta_{2,(0,1)}(2u_1)
\hspace{0.05cm}-\hspace{0.05cm}\alpha\beta\gamma\hspace{0.05cm}\theta_{2,(1,1)}(2u_1)\hspace{0.025cm})\hspace{0.05cm},\cr\cr
r_{24}&=&2\hspace{0.05cm}(\hspace{0.025cm}\alpha\gamma^2
\hspace{0.05cm}\theta_{2,(0,0)}(2u_1)\hspace{0.05cm}+\hspace{0.05cm}\beta\delta^2\hspace{0.05cm}\theta_{2,(1,0)}(2u_1)
\hspace{0.05cm}+\hspace{0.05cm}\gamma\alpha^2\hspace{0.05cm}\theta_{2,(0,1)}(2u_1)\cr
&&+\hspace{0.05cm}\delta\beta^2\hspace{0.05cm}\theta_{2,(1,1)}(2u_1)
\hspace{0.05cm}-\hspace{0.05cm}\beta\gamma\delta\hspace{0.05cm}\theta_{2,(0,0)}(2u_1)
\hspace{0.05cm}-\hspace{0.05cm}\alpha\gamma\delta\hspace{0.05cm}\theta_{2,(1,0)}(2u_1)\cr
&&-\hspace{0.05cm}\alpha\beta\delta\hspace{0.05cm}\theta_{2,(0,1)}(2u_1)
\hspace{0.05cm}-\hspace{0.05cm}\alpha\beta\gamma\hspace{0.05cm}\theta_{2,(1,1)}(2u_1)\hspace{0.025cm})\hspace{0.05cm},\clabel{00}\cr\cr
r_{25}&=&2\hspace{0.05cm}(\hspace{0.025cm}\alpha\delta^2
\hspace{0.05cm}\theta_{2,(0,0)}(2u_1)\hspace{0.05cm}+\hspace{0.05cm}\beta\gamma^2\hspace{0.05cm}\theta_{2,(1,0)}(2u_1)
\hspace{0.05cm}+\hspace{0.05cm}\gamma\beta^2\hspace{0.05cm}\theta_{2,(0,1)}(2u_1)\cr
&&+\hspace{0.05cm}\delta\alpha^2\hspace{0.05cm}\theta_{2,(1,1)}(2u_1)
\hspace{0.05cm}+\hspace{0.05cm}\beta\gamma\delta\hspace{0.05cm}\theta_{2,(0,0)}(2u_1)
\hspace{0.05cm}+\hspace{0.05cm}\alpha\gamma\delta\hspace{0.05cm}\theta_{2,(1,0)}(2u_1)\cr
&&+\hspace{0.05cm}\alpha\beta\delta\hspace{0.05cm}\theta_{2,(0,1)}(2u_1)
\hspace{0.05cm}+\hspace{0.05cm}\alpha\beta\gamma\hspace{0.05cm}\theta_{2,(1,1)}(2u_1)\hspace{0.025cm})\hspace{0.05cm},\cr\cr
r_{26}&=&2\hspace{0.05cm}(-\hspace{0.025cm}\alpha\delta^2
\hspace{0.05cm}\theta_{2,(0,0)}(2u_1)\hspace{0.05cm}-\hspace{0.05cm}\beta\gamma^2\hspace{0.05cm}\theta_{2,(1,0)}(2u_1)
\hspace{0.05cm}-\hspace{0.05cm}\gamma\beta^2\hspace{0.05cm}\theta_{2,(0,1)}(2u_1)\cr
&&-\hspace{0.05cm}\delta\alpha^2\hspace{0.05cm}\theta_{2,(1,1)}(2u_1)
\hspace{0.05cm}+\hspace{0.05cm}\beta\gamma\delta\hspace{0.05cm}\theta_{2,(0,0)}(2u_1)
\hspace{0.05cm}+\hspace{0.05cm}\alpha\gamma\delta\hspace{0.05cm}\theta_{2,(1,0)}(2u_1)\cr
&&+\hspace{0.05cm}\alpha\beta\delta\hspace{0.05cm}\theta_{2,(0,1)}(2u_1)
\hspace{0.05cm}+\hspace{0.05cm}\alpha\beta\gamma\hspace{0.05cm}\theta_{2,(1,1)}(2u_1)\hspace{0.025cm})\hspace{0.05cm}.
\nonumber
\end{eqnarray}
Multiplying the coefficients at subsequent
$\theta_{2,e}(2u_1)$ in expression (\ref{ha})
by $\alpha,\hspace{0.05cm}-\beta,\hspace{0.05cm}\gamma$ and $-\delta$, respectively,
and summing them up we obtain
\begin{eqnarray}
\hspace{0.025cm}{_{C^2}\over^{8\pi^2}}\hspace{0.05cm}\s{\rm e}^{\hspace{0.025cm}{1\over 2}\hspace{0.025cm}\pi i\hspace{0.05cm}
(1,0)\cdot\tau(1,0)}
\ (\partial_2\vartheta(E_1))^2\
(\lambda_1-\lambda_2)^2\hspace{0.05cm}\s(\alpha\beta+\gamma\delta)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
A similar operation on expression (\ref{ho}) gives
\begin{eqnarray}
-\hspace{0.025cm}{_1\over^{16\pi^2}}\hspace{0.05cm}\s(\lambda_1-\lambda_2)
(\lambda_2-\lambda_5)
(\lambda_2-\lambda_6)(\lambda_3-\lambda_4)
\hspace{0.05cm}(\alpha^2\gamma^2-\beta^2\delta^2)\hspace{0.05cm}.
\nonumber
\end{eqnarray}
The equality of the two expressions follows from
Eq.\hspace{0.05cm}\s(\ref{tbus}). This verifies the correctness
of the overall normalization of the Hitchin map in
Eq.\hspace{0.05cm}\s(\ref{glr}).
\vskip 0.9cm