Singularity, complexity, and quasi--integrability of rational mappings

Abstract:
  We investigate global properties of the mappings entering the description of
symmetries of integrable spin and vertex models, by exploiting their nature of
birational transformations of projective spaces. We give an algorithmic
analysis of the structure of invariants of such mappings. We discuss some
characteristic conditions for their (quasi)--integrability, and in particular
its links with their singularities (in the 2--plane). Finally, we describe some
of their properties {\it qua\/} dynamical systems, making contact with
Arnol'd's notion of complexity, and exemplify remarkable behaviours.

\section{Introduction}
We want to analyze in detail some realizations of
Coxeter groups~\cite{CoMo65,Hu90}  by birational transformations of projective
spaces which  have been shown to appear in the description of the symmetries of
quantum integrable
systems~\cite{Ba81,Ba82,Fa82}~\cite{bmv1,bmv1b,bmv2,bmv2b,prl}.

The first motivation to look at these realizations resides of course in their
relations with the star--triangle and the Yang--Baxter equations or their
higher dimensional generalizations such as the tetrahedron equations.
A characteristic feature of the orbits of  the known solutions of the
Yang-Baxter equations under these groups is  that they are  confined to
subvarieties of high codimension of the parameter space (actually curves), {\em
signaling the existence of an  unexpectedly large number of algebraically
independent invariants\/}.
The discovery and the analysis of the possible invariants is a decisive step in
the study of the Yang--Baxter (tetrahedron,...)  equations, in particular for
what concerns the so--called baxterization problem~\cite{Jo90}.

Another motivation  is to use these realizations to construct  discrete time
evolution maps, as it is usual in the study {\it a la\/} Poincar\'e of
dynamical systems, by iterating some element of the group.
One of the main   questions in this setting is again to  bring to the light the
possible  presence of invariants and invariant
tori~\cite{henri92,henri52,Wi81}~\cite{PaNiCa90,QuRoTh88,QuRoTh89,Ra91,Ra91b}~\cite{bmv1,bmv1b,bmv2,bmv2b,prl}.

If a realization admits algebraic invariants, we will say it has a property
of quasi--integrability.

In~\cite{bmv1,bmv1b}  it was shown, among other things how the direct graphical
investigation of the group action by means of numerical calculations (``drawing
the picture'')  leads to a nice representation of the orbits.
We want to elaborate on the problem and discuss   the deeper structure of the
non linear realizations by  birational transformations of projective spaces.

We first recall which kind of infinite Coxeter groups arise in the theory of
solvable models of statistical mechanics and  collect  some facts about
birational maps which we shall  use in the sequel.

We then analyze  the cohomological structure of the possible invariants.
We  give the general (albeit formal) solution to the problem in terms of an
algorithmic research of  linear systems on ${\Bbb P}_n$ satisfying a covariance
property with respect to the realizations we handle.
 We then specialize to the case $n=2$,  analyze the singular locus of the
group, and discuss some illustrative examples.

In the final section we make contact with Arnol'd's notion of
complexity~\cite{Ar90,Ve92}, which measures  the growth of the topological
non--triviality of the intersection  of a fixed subvariety   and the image of
another one  under an iteration map.
We will argue how our notion of quasi--integrability is related to a {\em
polynomial \/} growth of the complexity while the generic one is exponential.

\section{Coxeter groups and birational realizations}

One of the outcomes of~\cite{bmv1}-\cite{prl} is the construction of a number
of groups generated by involutions and of various  rational realizations on
projective spaces.

Consider the  Coxeter group $G$  engendered by $\nu$  involutions $I_1, I_2,
\dots, I_k, \; (k=1\dots \nu)$, {\em verifying no relations other than the
involution property}.
The group $G$ is infinite and there are two essentially different situations.

If $\nu=2$, the group is the infinite dihedral group ${\Bbb Z}_2 \ltimes {\Bbb
Z}$, and all elements may uniquely be written $I_1^\alpha(I_1I_2)^q$, with
$\alpha=0,1$ and $q\in {\Bbb Z}$. The number of elements of given lenght $l$ is
2.

If $\nu \geq 3$, the number of elements of lenght $l$ grows exponentially with
$l$, and the group is in a sense bigger (still countable).

As an example for the groups described  in~\cite{bmv1}-\cite{prl},  the number
$\nu$ of generators depends on the dimension $d$ of the lattice: it is just
$2^{d-1}$ so that if $d=2$,  $G$ is generated by two involutions  and   if
$d\ge 3$, $G$ is generated by more than three involutions.

One may then construct  various  realizations $\Gamma$ of $G$ by explicit
transformations of some projective space.
They are  obtained by specifying the realization of the generators.
Since it is precisely the realizations that we want to study here, and
especially the problem of the existence of invariants, we will mainly talk
about $\Gamma$ and not $G$, and   use the same  notation $I_k$ for the
generators of $G$ and their representatives in $\Gamma$.

The realizations $\Gamma$ we consider are essentially obtained from operations on
matrices, especially matrix inversions, and transpositions of their entries,
the  matrices being  originally matrices of Boltzmann weigths of statistical
mechanical spin and vertex models on the lattice or $R$--matrices of
2--dimensional field theories. The projective space we consider is just the
space of entries of the matrices up to a common multiplicative factor.

Let us describe here typical examples of such  realizations.

Suppose $m$ is a $q\times q$ matrix.
The ordinary matrix inverse $I$ defines an involutive rational transformation
of ${\Bbb P}_{q^2-1}$, which reads in homogeneous coordinates:
\begin{equation}	\label{inver}
I: \quad m_{ij} \longrightarrow \mbox{ cofactor of }m_{ij}
\end{equation}
We may  also consider  the element by element inverse (so called Hadamard
inverse):
\begin{equation}	\label{hada}
J: \quad m_{ij}  \longrightarrow 1/m_{ij}
\end{equation}
These two inverses appear in the study of spin models.
Notice that $I^2$=$J^2$=1 and there is no other relation between $I$ and $J$.
In particular $I$ and $J$ do not commute and $\varphi=IJ$ is of infinite order.

It is of course possible to define all kinds of block inverses, the size of the
blocks ranging from  the full matrix size (for $I$) and 1 (for $J$).

One may also mix $I$ with linear transformations.
This  happens already  when $I$ and $J$  defined by (\ref{inver}) and
(\ref{hada}) are collineated, i.e when there exists a linear transformation $C$
such that $I=C^{-1}JC$ (see~\cite{bmv1}).
We may also mix $I$ with permutations of the entries.

Let us describe here the  transpositions of entries which appear in the study
of  vertex models on a $d$--dimensional lattice~\cite{bmv2b,prl}.

Suppose $M$ is a multiindex matrix of size  $q^d \times q^d$ written in the
form $M^{i_1 i_2 \dots i_d}_{j_1 j_2 \dots j_d}$.
There exist  $d$ different partial transpositions $t_1,\, t_2, \dots, t_d$ with
the evident definition:
\begin{eqnarray}
        (t_k \, M)^{i_1 \dots i_k \dots i_d}_{j_1 \dots j_k\dots j_d}=
                M^{i_1  \dots j_k\dots  i_d}_{j_1 \dots i_k \dots j_d} .
\end{eqnarray}
We clearly have a product and an inverse $I$ for these multiindex matrices.
We may define $2^{(d-1)}$ new inversions by:
$$
        I_\kappa = t_{\alpha_1}\dots t_{\alpha_s}\,  I \,
                t_{\alpha_{s+1}}  \dots t_{\alpha_d} =
        t_{\alpha_{s+1}}  \dots t_{\alpha_d} \, I \,
                t_{\alpha_1}\dots t_{\alpha_s}
$$
where $\kappa=(\{ \alpha_1,\dots \alpha_s\},\{ \alpha_{s+1},\dots \alpha_d\})$
is a partition of $\{ 1,\dots, d\}$.

These various  inverses yield involutive (bi)rational mappings of
${\Bbb P}_{q^{2d}-1}$.
They are related by collineations, easy to write from the representation of the
partial transpositions.
Note that the  product  of all $t_k$'s is the full transposition $t$ and
commutes with all the inverses.
Such realizations appear in the study of vertex models.

We may further enrich the representations by imposing constraints on the
entries of the matrices, provided the transformations are compatible with these
constraints (see~\cite{bmv1} for the notion of admissible patterns).
This yields realizations on projective spaces of lower dimensions.

Needless to say that along these ideas, one may construct a variety of
involutions acting on various projective spaces.

We stress that, at the level of the  realization, there may exist additional
relations between the generators, possibly making it  finite (as in example
(\ref{fm})).

\section{Some facts about rational mappings} In this section we
collect
some results about rational and birational mappings between algebraic
varieties  (see for example~ \cite{Wa50,Mu78,GrHa78,Sh77}).
\begin{defi}
 A correspondence  $\varphi_Z:X\to Y$ between algebraic varieties $X$ and $Y$
is an algebraic subset  $Z\subset X\times Y$.
$\varphi_Z$ is  a rational map if there is a Zariski open set $U\subset X$ on
which the correspondence is one to one.
\end{defi}
If $Z\subset X\times Y$ is a rational map,  the inverse correspondence is
defined  by the  graph $Z^{-1}:=\{(y,x)\in Y\times X\ |(x,y)\in Z\}$.
 If the correspondences $Z$ and $Z^{-1}$ are both
rational mappings, then $Z$ (or  $\varphi_Z$) is called a
birational transformation.
A  birational map is a  biholomorphism except on subvarieties of codimension at
least two.

A linear system $D$ on $X$ is a non empty linear subspace of the space
of global sections of some line bundle ${\cal L}$ over $X$.
Its base locus is the set  of common zeroes of all sections in $D$.
A remarkable result is that \cite{Mu78} there exists a one to one
correspondence between linear systems on $X$ of  dimension $d$  with base locus
of codimension not less that 2  and rational maps $X\mapright{\phi}{\Bbb P}_{d-1}$
up to  projective automorphisms of ${\Bbb P}_{d-1}$.

For what we are concerned with, the paradigm  of  rational map is the
so--called $\sigma$--process or blow up of a point.

We refer to~\cite{GrHa78,Sh77} for the general definitions.
We shall call Hadamard inversion and generically denote by $J$ the prototypical
birational mapping in  ${\Bbb P}_n$.
Let $x_0,x_1,\dots, x_n$ and $y_0,y_1,\dots, y_n$ be coordinates in two
different copies of ${\Bbb P}_n$ and let us consider the algebraic set
$Z\subset{\Bbb P}_n\times{\Bbb P}_n$ given by the $n$ equations $x_0 y_0 = x_1 y_1
=\dots= x_n y_n$.
By definition, the graph of $J$ is $Z$.

It is valuable to specialize to $n=2$: outside the triangle $x_0x_1x_2=0$, $Z$
is the graph of the map $[x_0,x_1,x_2]{\sim\hskip -2.5pt\sim\hskip -2.5pt\sim\hskip -4.2pt\triangleright }[1/x_0,1/x_1,1/x_2]$.
The generic point on the line $x_i=0$ is sent into the point $p_i$ whose only
non--vanishing coordinate is $y_i$, while to the points $p_i$  corresponds the
entire line $y_i=0$.
One can say that $J$ blows up $p_i$ to the line $x_i=0$ and blows down the line
$x_i=0$ to the point $p_i$.

Finally, we recall  the following  properties, which will be used in what
follows.
{}From the description of a  birational mapping as an algebraic set in the
product $X\times Y$ it is apparent that $\varphi$ blows up $p$ to a divisor $D$
if and only if $\varphi^{-1}$ blows down $D$ to $p$, and it is also evident
that different points $p_1$ and $p_2$ cannot be blown up to the same divisor
$D$ (otherwise  the inverse map would not be rational).

Also, it is a standard result that blowing up a point adds a free factor ${\Bbb Z}$
in the Picard group of $X$.
It follows that  if $\varphi$ is a birational map in ${\Bbb P}_2$ which blows up $n$
points, then it must blow down exactly $n$ exceptional divisors,  as explained
above.

\section{Invariants and quasi--integrability}
Let $\Gamma$ be a  group of birational transformations in ${\Bbb P}_n$.
A meromorphic function $\Delta:{\Bbb P}_n\to {\Bbb P}_1$ deserves to be called a
$\Gamma$--invariant if it satisfies
\begin{equation}
\Delta(g(x))=\Delta(x)\quad \forall g \in \Gamma.
\end{equation}
Since a meromorphic function on an algebraic variety can be   thought of as the
ratio of two  sections of a suitable line bundle, we are naturally lead to the
following scheme.

A $\Gamma$ one--cocycle is a collection of sections  $a(g,x)$ of some line bundles
over ${\Bbb P}_n$ satisfying the cocycle condition:
\begin{equation}
\label{coc}
a(g_1  \;  g_2,x)= a(g_1,g_2 x)\cdot a(g_2,x)
\end{equation}
A section  $\sigma$ of a  line bundle  will be called {\em $a$--covariant} (for
some cocycle a) if   the equation
\begin{equation}
\label{cob}
\sigma(g x)=a(g,x)\cdot\sigma(x)
\end{equation}
holds.
This equation may be reformulated in group cohomology terms~\cite{HoSe53b}  as
 $ a = \delta \; \sigma $  meaning  that $a$ is actually a coboundary.

Finding a $\Gamma$--invariant is equivalent to finding two sections $\sigma_1 $ and
$\sigma_2$ verifying equation (\ref{cob}) for the {\em  same} $a$.
This means  that we are interested in  the coboundaries of the $0$--cochains
 rather than the cohomology groups. The strategy is to find  1-cocycles
admitting a sufficient number of primitives.

 As a side remark, we notice  that these equations, which will play a prominent
role in the sequel, are well defined at all points of ${\Bbb P}_n$.
In fact for any  birational transformation $g$, the singular locus is a
subvariety of codimension greater than or equal to two in ${\Bbb P}_n$, and the
equations above admit a unique holomorphic extension to the whole of ${\Bbb P}_n$,
even if they have meaning in the point set sense only on the nonsingular locus
of $g$.

We are interested in the case where  $\Gamma$ is a Coxeter group, i.e.
is  generated by $\nu$  involutions $I_k$ of degree $d_k$ (the degree is a
natural notion in terms of the homogeneous coordinates   $[x_0,\ldots,x_n]$).
A  $\Gamma$ one--cocycle will be
completely specified by the assignment of $\nu$ sections $a(I_k,x)$.
Remarkably,  the possible values of $a(I_k,x)$ may be found explicitely.

Each involution $I_k$ defines a  characteristic polynomial $\phi_k$ of degree
$d_k^2-1$  in the following manner. The $I_k$ being involutions,  $I_k^2$
appears as the multiplication by a degree $d_k^2-1$ polynomial
$\phi_k(x_0,\ldots,x_n)$. We then have the following
\begin{lem}
If $a(g,x)$ is a trivial
$\Gamma$ cocycle, the sections $a(I_k,x)$ divide a suitable power of
$\phi_i$ for $i=1,\ldots ,\nu$.
\end{lem}
{\par\noindent } {\bf Proof.} Suppose $a =\delta \; \sigma$. Then, from  the coboundary
equation (\ref{cob}) for $g=I_i^2$ we get that $a(I^2_i,x)=\phi(x)^m$, with
$m=deg(\sigma)$. The assertion follows from the cocycle condition (\ref{coc}).
\begin{defi}
We shall say that  $\Gamma$ is   {\em collineated\/}  when
its generators are all conjugated by means of elements of $PGL(n+1,{\Bbb C})$
to a standard one, $K$.
\end{defi}

This is the case for a number of models among which the Baxter model ($n=3$)
and the examples of section \ref{examples}.

If  the characteristic polynomial $\phi_K$  factorizes into
$\phi_K=\prod_{l=0}^j p_l^{d_l}$,
then it  is clear that for every $i$ there exist a set of global homogeneous
coordinates $[X_0^{(i)},\ldots,X_n^{(i)}]$ in which $a(I_i,x)$ will be the
product of the same polynomials $p_l$, possibly weighted with different
exponents $d^\prime_l$.

{\par\noindent } {\bf Remark}. When $K$ is the Hadamard inversion $J$  in ${\Bbb P}_n$  a
straightforward computation shows that
$\phi_J= \prod_{l=0}^n x_l^{(n-1)}$ so that we get

\begin{theorem}
A coboundary for a collineated Coxeter group of birational transformations with
generators conjugated to the Hadamard inversion $J$ in ${\Bbb P}_n$ is a monomial
in some suitable homogeneous coordinates, with coefficient $\pm 1$.
\end{theorem}

It  is apparent that our  cohomological setting gives an algorithmic
prescription for the search for invariants: find first the characteristic
polynomials $\phi_k$, which is straightforward, then the possible coboundaries,
which is a factorization problem, and check how many primitives they have,
which amounts to solving a linear problem.

\bigskip

The realization  $\Gamma$ in  ${\Bbb P}_n$ admits  $p$ invariants $\Delta_1,\dots
\Delta_p$ if  there exists  an $a$--covariant linear system $L^d_p$ of
projective dimension $p$ and  degree $d$ for some cocycle $a$.
It is not guaranteed that the orbits of a realization admitting $p$ invariants
lie on subvarieties of dimension $n-p$. Indeed the question of the {\em
algebraic \/} independence of the invariants has to be examined
further~\cite{prl2}.

\bigskip

One has to realize how exceptional is  the existence  of any invariant
 of the birational realization, as the following argument shows.

Let us  consider  covariance with respect to  one generator $K$ of $\Gamma$.
Suppose $K$ is of degree  $d_K$,  and  suppose we are looking for an invariant
of degree $m$.
Clearly from equation (\ref{cob})  the degree $q$ of the cocycle is related to
$d_K$ and $m$ by  $m(d_K-1)=q$.
The dimension of the space  ${\cal P}(m,n)$   of homogeneous polynomials of
degree $m$ in $n$ variables  is  $n+m-1\choose m-1$.
The requirement of  $K$--covariance selects a  linear system $L^K$ in   ${\cal
P}(m,n)$, of generic  dimension at most ${1\over 2} {n+m-1\choose m-1}$.
Imposing  the same condition for another generator $I$ leads to look for the
intersection of hyperplanes of at most complementary dimensions in ${\cal
P}(m,n)$. This intersection is generically empty. As a consequence we have
\begin{theorem}
The action of a generic  Coxeter group of birational transformations in ${\Bbb P}_n$
does  not admit any non--trivial invariant.
\end{theorem}
{\par\noindent }
In the particular case of collineated realizations, this leads to the further
property, which we prove for $\nu=2$ for simplicity.
\begin{theorem}
 The set of {\em collineated}   groups admitting a non--trivial invariant has
the structure of  a quasi--projective variety.
\end{theorem}
{\par\noindent }{\bf Proof}.
The group $PGL(n+1,{\Bbb C})$ admits a natural structure
of quasi--projective variety, since it is  identified with
the complement of the degree $(n+1)$ hypersurface $det A =0$ in
${\Bbb P}^{(n+1)^2-1}$.
Choosing  $K$ as prototypical generator of $\Gamma$,
$I$ is specified (in the collineated case) by the choice of an element $C\in
PGL(n+1,{\Bbb C})$ by  $I=C^{-1}KC$.
If for any polynomial $P$ we define $P_C:=P(C^{-1}x)$ and set  $y=Cx$, then
the covariance equation with respect to  $I$ reads
$$P_C(Ky)=a_I(C^{-1}y)P_C(y)$$
\noindent
For each possible cocycle $a$, this  is an algebraic equation
in the parameters $t_a$ of the matrix $C$.
The coordinates of any basis $\{e_\alpha\}$ in $L^I$ with respect to a basis in
${\cal P}(m,n)$ completed from a basis of $L^K$ may be expressed as algebraic
functions of the $t_a$'s.
The condition of nontriviality of the intersection is a condition on the rank
of the matrix
$$B:=\left[\matrix{
	{\bf 1}				&{\bf 0}	\cr
	e_\alpha^i(t_1,..t_{(n+1)^2-1})	&\dots		\cr}
\right]$$
which in turn is an algebraic condition on the
$e_\alpha^i(t_1,..t_{(n+1)^2-1})$, concluding  the proof.

\section{Realizations in the 2-plane}

\label{cp2}
Let us
discuss mappings in ${\Bbb P}_2$. Here our analysis is made quite complete
by the fact that all rational maps from an algebraic surface can be
described in terms of $\sigma$ processes  and also that singularities of
birational maps can occur only at points.
Moreover  a theorem of M. Noether~\cite{Sh77} assures that the group of
birational maps is generated by the inversion $J$ together with the projective
group $PGL(3,C)$.
We will restrict ourselves here to $\nu=2$, i.e. $\Gamma$ is generated by two
involutions $I$ and $K$.

\begin{defi}The singular locus of $\Gamma$, $S(\Gamma)$ is
defined as $$ S(\Gamma)=\{x\in{\Bbb P}_2 | \exists g\in \Gamma\  {\rm s.t. \ } g\
{\rm blows\ } x\ {\rm up\ } \}$$
\end{defi}
\noindent Since $\Gamma$ is generated  by  $I$ and $K$, $S(\Gamma)$ is obtained by the
action of $\Gamma$ on the singular points of $I$ and $K$.

\begin{defi}
We will  say that $\Gamma$  is properly singular if $S(\Gamma)$ contains at least four
points in general position, i.e. such that no three of them are aligned.
\end{defi}

\noindent Let  $\Pi(\Gamma)$ be the set of singular divisors:
$$\Pi(\Gamma)=\{\pi\in Div({\Bbb P}_2) |\exists g\in \Gamma\  {\rm s.t. \ } g\
{\rm blows\ }\ \pi \mbox{  down to a  point  }\}$$ Let us suppose
that  $\Gamma$ admits a rational invariant $\Delta$ and let us look at what happens
at points in $S(\Gamma)$.
The equation $\Delta(g(x))= \Delta(x)$ is clearly meaningless at the
indetermination points of $\Delta$.
Noticing that $\forall x\in S(\Gamma)$, $\exists \pi_x\in\Pi(\Gamma)$ such that $\Pi_x$
is blown down to $x$ by some element $g \in \Gamma$,  we can conclude that
$x\in S(\Gamma)$ is either an indetermination point of $\Delta$ or $\Delta$ is
constant along the corresponding divisor $\pi_x$,  i.e. $\pi_x$ belongs to the
pencil of curves $\Delta= const$.
Then we can state the

\begin{prop}
If $S(\Gamma)$ is infinite, and $\Gamma$ is properly singular,  then  $\Gamma$ does not
admit any invariant.
\end{prop}
{\par\noindent }{\bf Proof.} With the above
observation  in mind, the only case  we must rule out is that
$\Gamma$ admits an infinite number of singular points and only a finite number of
singular divisors, since by Bezout's theorem, any rational degree $d$ invariant
admits at most $d^2$ indetermination points. Let us suppose that $\#S(\Gamma)=
\infty$ and $\# \Pi(\Gamma)=N$ and let us consider the set
${\cal P}= \{ \Pi_\alpha \}$ $(\alpha = 1 \dots 2^N)$  of  parts  of $\Pi(\Gamma)$.
We can make a partition of $\Gamma$ into $2^N$ disjoint subset
$\Gamma=\bigcup_{\alpha=1}^N \Gamma_\alpha$,  where
$$\Gamma_\alpha=\{g\in \Gamma \vert  g \mbox{ blows down exactly all the divisors in }
\Pi_\alpha\}$$
\noindent
At least one of the  $\Gamma_\alpha$, say $\Gamma_0$ is infinite.
If we then consider any pair of elements  $(h_0, l_0)$ in
$\Gamma_0$,   the product $h_0 \cdot l_0^{-1}$ is a birational map without
singularities, and hence an element $D\in PGL(3,{\Bbb C})$.
Since  $\Gamma_0$ is infinite, we can arrange things so that $D$ is a non trivial
word  of even lenght.
Since  words of odd length in $\Gamma$ are involutions one has:
\begin{equation}
\nonumber
D K D = K \qquad D  I D =  I
\end{equation}
 It follows that $D$ permutes the points of $S(\Gamma)$, and some  power $D^k$ must
be  the identity.
As a consequence,  there is some non--trivial product $(IK)^l={\bf 1}$, meaning
that actually $\Gamma$ is  finite, and contradicting the infiniteness of $S(\Gamma)$.
{\par\noindent }{\bf Remark.} The above proposition  proves that a necessary condition
for existence of an invariant is the  finiteness of the singular locus.
It is tempting to conjecture that this is also a sufficient condition for any
properly singular realization.
This is unfortunately not the case, as the  example of section (\ref{fdm}) will
show.

\bigskip

The next step is to try to relate the singular
locus of the group with the singular locus of the would--be invariant. Let
the $\Delta$ be a rational invariant for $\Gamma$ and let $S(\Delta)$ be the
set of its indetermination points.
If $y\in S(\Delta)$ and $y\notin S(\Gamma)$, the (finite) orbit $\Gamma_y$ of $y$ is
made out of indetermination points of $\Delta$, on which  every element $g\in
\Gamma$ is regular.
Since $\Gamma_y$  is of even order, say  $2l$, $y$ is a fixed point for
$(IK)^{n\cdot l}, \quad n\in{\Bbb Z}$.
Let $\tilde{\Bbb P}_2^y\mapright{pr}{\Bbb P}_2$ be the 2--plane ${\Bbb P}_2$ blown up at $y$.
Since we are working with ${\Bbb P}_2$,  $\Delta$ extends to a function
$\tilde\Delta$ on  $\tilde{\Bbb P}_2^y$ with no indetermination points in a
neighbourhood of the  exceptional divisor $E=pr^{-1}(y)$.
The restriction $\varphi =  \tilde\Delta\vert_E$ is a meromorphic map on
${\Bbb P}_1$ whose value is  determined by the limits of $\Delta(x)$ along lines in
${\Bbb P}_2$ passing through $y$.

We will say that  $\Gamma$ is  non--degenerate if  for every $g\in \Gamma$  the tangent
map to $g$ is non--nilpotent at every isolated fixed point of $g$.
Then we can  state the
\begin{prop}
Let $\Gamma$ be a non--degenerate Coxeter group of birational transformations of
${\Bbb P}_2$.
If $\Gamma$ admits an invariant $\Delta$, then $S(\Delta)\subset S(\Gamma)$.
\end{prop}	{\bf Proof.} Suppose  $y\in S(\Delta)$ and $y\notin S(\Gamma)$; the
non--degeneracy of $\Gamma$ ensures that there is at least one line in $P
T_{{\Bbb P}_2}(y)$ whose orbit under the tangents to $(IK)^{n\cdot l},\quad
l\in{\Bbb Z}$ is infinite (recall that $(IK)^l$ is a minimal length  element in
$\Gamma$ which fixes $y$).
Hence $\varphi$ as defined above assumes the same value on an infinite
number of points, i.e. it is constant over the whole of $E_y$.
But this in turn implies that $\Delta$ is well defined at $y$ which contradicts
$y \in S(\Delta)$ and ends the proof.
\label{cex}

Notice that the converse inclusion  $S(\Gamma)\subset S(\Delta)$ does not hold in
general. If  the inclusion is strict, we have seen that for any $x\in S(\Gamma)$
and not in $S(\Delta)$, the invariant $\Delta$ is constant along some divisor
$\pi_x$. This gives a useful information about the invariant (see example
\ref{at}).

\bigskip

As for what the  singularities of the generic curve $\Sigma_\lambda$ of the
pencil are related  to the singular orbit $S(\Gamma)$ the following considerations
hold.
By Bertini's theorem, generic curves are smooth outside the base locus
$S(\Delta)$ whence the chain of  inclusions
\begin{equation}\label{incl}
S(\Sigma_\lambda)\subset S(\Delta)\subset S(\Gamma)
\end{equation}
Moreover, exploiting the genus formula
for singular curves  one can give relations{~\footnote{We
thank M. Talon for useful remarks on this point.} between the degree $d$ of
the invariant and $S(\Gamma)$.
If $\Sigma$ is a degree $d$ curve in ${\Bbb P}_2$   with singular locus
$Sing(\Sigma)$,  then \cite{Wa50}
$$
g(\Sigma)={{(d-1)\cdot (d-2)}\over 2}-\sum_{p\in {\rm Sing}(\Sigma)}\delta_p
$$
with $\delta_p$ depending on the type of the singularity at p.
Since  $\Sigma_\lambda$ is irreducible and admits $\Gamma$ as an infinite group of
automorphisms, thanks to the inclusion (\ref{incl}) and to the fact that
$\delta_p = 0$ if $p$ is not in $Sing(\Sigma_\lambda)$, one has
$${{(d-1)\cdot (d-2)}\over 2}-\sum_{p\in S(\Gamma)}\delta_p=\cases{0\cr 1\cr}$$
\noindent
This relation shows that there is a balance between the degree of
the invariants and the number and nature of the singular points of the generic
curve, and consequently of $\Gamma$.

\section{ Examples}
\label{examples}
We  describe  here some specific examples  in ${\Bbb P}_2$ with two generators $I$
and $J$. The physical origin of the models we will be dealing with
is to be found notably among two--dimensional spin  models with interaction
along the edges~\cite{Ba81}, and this explains the terminology we use.

We fix  $J$ to be the Hadamard inversion $[x_i]{\sim\hskip -2.5pt\sim\hskip -2.5pt\sim\hskip -4.2pt\triangleright } [1/x_i]$,
and $I$ to be  $I=C^{-1} J C$ with the collineation matrix
$C\in PGL(3,{\Bbb C})$. We will concentrate on the parametrization by $C$ and
examine some algebraic families in $PGL(3,{\Bbb C})$. For some of them $\Gamma $
admits a non trivial algebraic invariant.

We can associate to $\Gamma$ a diagram  $D_\Gamma$ whose vertices are the points in
$S(\Gamma)$ and where two vertices $p_1$ and $p_2 \in S_\Gamma$ are joined by an edge
if either $p_1=J p_2$ or $p_1=I p_2$. The edges are oriented if $I$ (resp. $J$)
can be applied in one direction only.
The diagram does not characterize the examples, but is a useful tool. In
particular the families we give here have been obtained by deformations of
given collineation,  demanding the stability of the topology of the diagram.

The singular points of $J$ are $P_1=[1,0,0]$, $P_2=[0,1,0]$, $P_3=[0,0,1]$, and
the one of $I$ are $Q_i=C^{-1}(P_i), (i=1,2,3)$.

One should notice that a number of the  families we produce in this way  fall
into the general form found in~\cite{Ja91} (i.e. verifying $C([1,0,0])=[1,1,1]$
and $C^2([1,0,0]=[1,0,0]$), but with {\em  non integer entries}.
This general form depends on four parameters:
\begin{equation}
\label{jaeger}
\left[\matrix{	2&	\alpha&		\beta	\cr
		2&	-1+\gamma&	-1-\gamma	\cr
		2&	-1-\delta&	-1+\delta	\cr}\right]
\end{equation}

\subsection{The $Z_5$ family}
The general $Z_5$ (five-state chiral Potts) model is described by a $5\times 5$
cyclic  matrix.
We  may consider its 2--parameter reduction obtained by imposing that the
matrix is symmetric.
It falls into  a family of quasi--integrable  models parameterized by
a complex number $q$, which (whenever this makes sense)  is identifiable with
the square root  of the number of states~\cite{defo92}.
The collineation matrix is
\begin{equation}
C_{Z_5}(q)=\left[\matrix{
		2 &	{{q^2-1}}	&{{q^2-1}}\cr
		2 & 	-1+q		& -1-q \cr
		2 &	-1-q 		& -1+q \cr	}\right]
\end{equation}
Its  singular set is made of {\em ten} points:$P_1,P_2,P_3$ which are the usual
singular points of $J$,  $Q_1=[1,1,1]$ and $Q_\pm=[-4, {{1\pm {q}}},{{1\mp
{q}}}]$, the singular points of $I$, and $R_2=I(P_2),\>
R_3=I(P_3),\>R_\pm=J(Q_\pm)$.
All these points are  indetermination points for the invariant
$$\Delta_{Z_q}= {{(x-z)(y-z)((q-1)(x^2+y^2)+2(q+1)xy)}\over{(2+(q-2)
(zx+zy)+2xy)(x-y)^2}}$$
Notice that here $S(\Delta)=S(\Gamma)$. The diagram $D_\Gamma$ is the following:
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        \put(25,37){\small I}\put(13,27){P$_1$}

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\put(125,37){\small J}\put(123,27){R$_3$}

\put(70,10){\circle*{2}}
\put(70,15){\circle{10}}
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\put(120,15){\circle{10}}
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\put(50,30){\line(1,0){20}}\put(60,31){\small I}
\put(50,30){\vector(1,0){15}}

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\end{picture}
\hfill\null\par

\subsection{The BMV family}
The matrix of (Boltzmann) weights of the BMV model \cite{bmv1} is
$$
\left[\matrix{
	x&	y&	z&	y&	z&	z\cr
	z&	x&	y&	z&	y&	z\cr
	y&	z&	x&	z&	z&	y\cr
	y&	z&	z&	x&	z&	y\cr
	z&	y&	z&	y&	x&	z\cr
	z&	z&	y&	z&	y&	x\cr
}\right]
$$
The inversion $I$ is just the matrix inversion. This model
pertains to the one--parameter family of quasi--integrable models
whose collineation  matrix is
\begin{equation}
C_{BMV}(w)=\left[\matrix{1&w-1&w\cr 1&-1&0\cr 1&0&-1\cr}\right]
\end{equation}
and is reached for the value $w=3$.
The singular locus is made out  of $P_1,P_2,P_3$ and
$$Q_1=[1,1,1], Q_2=[ w-1, -(w+1),w-1], Q_3=[1,1,-1],$$
\noindent
together with an extra point $R= J(Q_2)=[w^2-1,(w-1)^2,w^2-1 ]$. The case
$w=1$ is singular.
The family admits the invariant
$$\Delta_{BMV}(w)={{P_w^2(x,y,z) Q_w(x,y,z)}
\over{(y+z)^4(x-z)^2(x-y)}}$$ where
$$P_w(x,y,z)=(1-w)(z^2-xy)+(w-3)z(x-y),$$ $$
Q_w(x,y,z)=(1-w^2)(y^3-xz^2)+(w^2-4w-1)y^2(x-z)+2(w-1)^2yz(x-y)$$
 Here again  $S(\Delta)=S(\Gamma)$ and the  singular graph is as follows:

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\end{picture}
\hfill\null\par

\subsection{The symmetric Ashkin--Teller model}
\label{at}
By symmetric Ashkin--Teller model we understand
the $4$--state spin model with the cyclic and symmetric matrix of Boltzmann
weights
$$\left[\matrix{
	x_0&	x_1&	x_2&	x_1	\cr
	x_1&	x_0&	x_1&	x_2	\cr
	x_2&	x_1&	x_0&	x_1	\cr
	x_1&	x_2&	x_1&	x_0	\cr
}\right]$$
for which the matrix inversion is collineated
to the Hadamard inversion by means of the matrix
\begin{equation}
C_{AT}=\left[\matrix{
	1&	2&	1	\cr
	1&	0&	-1	\cr
	1&	-2&	1	\cr
}\right]
\end{equation}
The singular points of the matrix inversion are
 $Q_1=[1,1,1]$, $Q_2=[1,0,-1]$,  $ Q_3=[1,-1,+1]$.

The singular diagram is drawn below
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\put(10,30){\circle*{2}}\put(10,35){\circle{10}}
\put(15,37){\small I}\put(4,27){P$_1$}

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\put(85,37){\small J}\put(83,27){Q$_1$}

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\put(85,17){\small J}\put(83,7){Q$_3$}

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\put(40,30){\vector(1,0){6}}\put(50,20){\vector(-1,0){6}}
\end{picture}
\hfill\null\par
\vskip 1truecm\noindent
It contains  a loop connecting the two points  $P_2$ and $Q_2$. This means that
the points  $P_2=[0,1,0]$ and $Q_2=C^{-1}(P_2)$ are not singular for $IJ$ and
hence it is no surprise that they are non singular for the invariant
$$ \Delta_{AT}= \displaystyle \frac{y^2-xz}{y(x-z)}.$$
There is a strict inclusion of $S(\Delta) $ in $S(\Gamma)$, since  the points $P_2$
and $Q_2$ are not in $S(\Delta)$.
The corresponding divisors $\Pi_{P_2} =\{ \mbox{ the line } y=0 \}$ (resp
$\Pi_{Q_2} = \{\mbox{  the line } x=z \}$), are indeed curves in the pencil
$\Delta=const$.

\subsection{A finite realization} \label{fm}
It is instructive to consider the group  described by the collineation matrix
\begin{equation}
C_F(q) = \left[
	\matrix{	1&	0&	1	\cr
			1&	q&	-(1+q)	\cr
			1&	0&	-1	\cr
	}\right]
\end{equation}
which is invertible  for $q\neq 0$.
It is  apparent  that $I$ and $J$ share $P_2=[0,1,0]$ as a {\em common\/}
singular point.
Apart from $P_2$,  $I$ admits as singular points $Q_1=[1,1,1]$ and $
Q_3=[q,-(2+q),-q]$.
The singular graph $D_\Gamma$ depicted below contains two  more points
$R=J(Q_3)$ and $S=I(Q_2)$.
This model admits at least {\em two} algebraically independent  invariants,
which can be taken to be
$$\Delta_F^{(1)}={{z^4+x^4}\over{z^2x^2}}$$
$$\Delta_F^{(2)}={{q^2zx(y-z)^2+q[(z^2-xy)^2-z(z-2x)(x^2+y^2)]+2(z^2-xy)^2}
\over{(z-y)^2(z^2+x^2)}}$$
This, together with Bezout's  theorem tells us that the orbit under $\Gamma$ of
any point $p\in {\Bbb P}_2$ is finite and of order not greater that $16$, i.e $\Gamma$
is {\em finite\/}.
A  direct inspection  shows that $(IJ)^4={\bf 1}$ and  there are additional
invariants, of course not algebraically independent from the two previous ones.

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\put(55,31){\small J}
\end{picture}
\hfill\null\par

\subsection{The symmetric $Z_7$ model}
The symmetric $Z_7$ model~\cite{bmv1} may be   defined by the collineation
matrix
\begin{equation}
C_{Z_7}
=\left[\matrix{
	2	&6	&6	\cr
	2	&{{-1-i\sqrt{7}}}	&{{-1+i\sqrt{7}}} \cr
	2	&{{-1+i\sqrt{7}}}	&{{-1-i\sqrt{7}}}\cr
}\right]
\end{equation}

{}From the point of view of dynamical systems the   model shows chaotic
properties in ${\Bbb P}_2$.
It has  an {\em infinite\/} singular orbit $S(\Gamma)$.
According to proposition  5.1, it does not admit any invariant, as it is
confirmed by a direct inspection of the orbits.
This model behaves actually like a generic element of the family
(\ref{jaeger}).

\subsection{A finite diagram model (FDM)}
\label{fdm}
Here  $\Gamma$ is  generated by the Hadamard inverse $J$ and
$I= D^{-1} J D$ with $$D=\left[\matrix{
2	&0	&2\cr
1	&1	&-1\cr
-1	&1	&1\cr }\right]$$
\noindent Notice that $D$ is not of the form (\ref{jaeger}).
The singular diagram is finite   but $\Gamma$ does  not have any invariant.
This provides a remarkable example for what  we said in section(\ref{cex}),
i.e. that the finiteness of the singular  orbit is  only a necessary condition
for the existence of a  non--trivial invariant.
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\hfill\null\par

\section{About complexity}
In this section we will discus the notion of complexity of the groups we have
been considering.

As we said earlier, we may  consider  the group $G$  as generated by
involutions and relations.
If the number $\nu$ of generators is bigger than 2,  $G$ is ``exponentially
big''~\cite{Gr81}.
We will not comment on this here, and limit ourselves to $\nu=2$,  as one would
concentrate on 1--dimensional subgroups of differentiable  groups, and examine
the realizations $\Gamma$.

What we want to point out  is that there is a very diverse behaviour of these
realizations , {\em even in the case $\nu=2$}, manifesting itself in  the
complexity of the  representing transformations. The notion of complexity
we appeal to is a simple form of the  one introduced in~\cite{Ar90}.

Suppose $F$ is a  diffeomorphism of a compact smooth n--manifold $M$.
Let $S_k$ and $R_{l}$ submanifolds of $M$ of dimension respectively $k$ and
$l$.
Let  $S^{m}_k$ be the m$\rm ^{th}$  iterate of $S_k$ by $F$ and $T_{S,R}(m)$
the intersection
$$T_{S,R}(m)=R_l\cap S^{m}_k.$$
Arnol'd \cite{Ar90} defines the {\em complexity\/} $C_{S,R}(m)$
as $$ C_{S,R}(m)=\sum b_p(T_{S,R}(m))$$\noindent the $b_p$'s being the
Betti numbers. He also proves that for a sufficiently generic choice of $S_k$
and $R_l$ the complexity grows at most exponentially with $m$.

The analysis of complexity of plane mappings has already been achieved for the
case of  {\em polynomial and polynomially invertible\/}
transformations~\cite{FrMi89,Ve92}. We are interested in  a wider class of
transformations, as the one described in section(\ref{cp2}) where $\Gamma$ is
generated by two {\em rational \/}  transformations.

Although these transformations  are not diffeomorphisms there is  a natural
measure of the complexity  by  the degree of iterates.

Generically, if the two generating involutions are of degree $u$ and $v$
respectively, the degree of $\varphi=IK$ is $w=uv$,   so that deg
$\varphi^{(k)}=w^k$.
If $\Sigma_1$ and $\Sigma_2$ are two  linear subspaces of ${\Bbb P}_2$ then
$\varphi^{(k)}(\Sigma_1)$ should   be for generic $\Gamma$ a curve of degree $w^k$,
and
the complexity $C_{\Sigma_1,\Sigma_2}(k)$ would then be  exactly $w^k$.

However,  due to the fact that we are working with projective space, there is a
simple mechanism for the lowering of the degree of the iterates
$\varphi^{(k)}$, for one has  to factorize out common factors from the
expressions of the homogeneous coordinates of  $\varphi^{(k)}$.
This provides a variety of behaviours for the degree $d$ as a function of the
order of iteration $k$, lying between  exponential  growth and periodicity,
with the particular case of polynomial (or polynomially bounded) growth.

The outcome of our analysis is that there is a connection between the existence
of an invariant  and a polynomial (as opposed to exponential)  rate of growth,
as shows Table 1, inferred from the results of the direct calculation of the
first few iterations on the examples of section (\ref{examples}) where the
degree of $\varphi$ is 4.

\begin{table}[bt]
\caption{Behaviour of degrees}\label{table1}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline\bf Model &\bf Growth 		& \bf rank of invariants\\
\hline
Z$_5$ 	& $d(k+1)+d(k-1)-2d(k)$ periodic of period  3		&  one\\
BMV	& $d(k+1)+d(k-1)-2d(k)$ bounded				& one\\
Finite	& $d(k)$ periodic of period 4				& two \\
A--T	& $d(k)= 4 k $						& one\\
FDM	& $d(k)= 4^k$ 	(generic)				& none\\
Z$_7$ 	& $d(k+1)-d(k)=f_{2k}$  ($\{f_k\}$ a Fibonacci sequence)& none \\
\hline
\end{tabular}
\end{center}
\end{table}

A more detailed analysis of these properties, together with the study of other
properties of realizations of Coxeter groups {\it qua\/} dynamical systems such
as finite (periodic) orbits is the matter  of further investigations
(see~\cite{bbb6,bmr1}).

{\bf Acknowledgments} We thank M. Talon, O. Babelon, M. Bellon, J-M. Maillard,
and G. Rollet for a number of stimulating discussions.


A.S.Cattaneo, P.Cotta-Ramusino, A.Gamba and M.Martellini

The Donaldson-Witten Invariants in Pure QCD with Order and Disorder 't
  Hooft-like Operators

12 pages, LaTeX file, no macros used. Revised Version

Abstract:
  We study the first-order formalism of pure four-dimensional ${\rm SU}(2)$
Yang--Mills theory with theta-term. We describe the Green functions associated
to electric and magnetic flux operators \`a la 't~Hooft by means of
gauge-invariant non-local operators. These Green functions are related to
Witten's invariants of four-manifolds.

\section{Introduction}
Seiberg and Witten \cite{SW} studied recently the strongly coupled infrared
limit of minimal
$N=2$ super Yang--Mills theory. The key point in their work is the
existence of an electric-magnetic $S$-duality \cite{VW,Gira,Ferr},
which extends the Olive--Montonen duality \cite{OM}.
By taking into account the low energy effective theory, Witten \cite{w1}
introduced an interesting
``monopole equation"\rlap,\footnote{
A monopole equation can be obtained also in the framework of the twisted $N=2$
sigma-model \cite{AF}.}
describing abelian gauge fields weakly coupled to monopoles.
By counting the
solutions of the ``dual'' equation, Witten obtains
new invariants for a compact, simply connected, oriented
4-manifold with a distinguished integral cohomology class.
These invariants are closely related to
Donaldson polynomial invariants.

On the other hand, the infrared phase of the $N=2$ super Yang--Mills
theory is like 't Hooft's ``superinsulator'' (dual superconductor),
where one gets absolute confinement of electric charges
by the monopole-condensation mechanism. This phase is
controlled by a set of topological operators, implicitly
defined by 't Hooft, that create or destroy  topological  quantum
numbers.
We call such operators  ``disorder  operators''.
't~Hooft  argued
that the correlators of the disorder operators  are  a  sort  of
``topological'' Green's function.

In~\cite{cm,ccm,c}
the authors  consider three- and four-dimensional
$BF$-theory and define new
gauge-invariant, non-local operators $M(\Sigma)$ associated to
surfaces $\Sigma$ immersed or embedded in a  four-dimensional  manifold.

It turns  out  that
these operators $M(\Sigma)$ are  an  explicit  realization  of
't~Hooft disorder operators.

In this work we generalize the ideas of
\cite{cm,c,ccfm} and
prove that the first-order formalism of the  standard  pure
bosonic Yang--Mills theory with theta-term (which is a special type
of $BF$-theory) allows us to get
an explicit  realization  of  the  't~Hooft
scenario.

Moreover we show that the  expectation
values  of  the  non-local  operators $M(\Sigma)$ in the weak coupling
regime provide solutions of a ``monopole equation".

When we consider a K\"ahler manifold, then our monopole
equation is related to the one considered by Witten,
provided that some extra relations are fulfilled \cite{w1} (see
\rf{spinor}).

The inclusion in the expectation values of
$M(\Sigma)$ of the ``order parameters'' provided
by Wilson operators ``perturbs'' the above topological
Green functions. The new  correlation functions appear to
be related to Donaldson invariants.

We would like to stress that the main ingredient
here
is the fact that monopole equations select singular abelian
connections. In our picture the
``source" of the monopole equation is provided by a bosonic 2-form,
which appears naturally in the first order formalism of
pure Yang--Mills theory, while in the $N=2$ super Yang--Mills theory \cite{w1}
such a source is given by a bilinear form of a complex Weyl spinor.

Our approach may give a hint on how to perform a
completely non-perturbative calculation  directly  in
standard 4D-QCD, without the need to get it by some spontaneous
symmetry breaking of an $N=2$ super Yang--Mills theory \cite{SW}.
Our result gives more support to the general intuition that in
standard 4D Yang--Mills theory, there should be a  topological  sector
controlled by the topological Green's functions of  suitable
gauge-invariant, non-local, composite operators.

\section{4D-QCD in first order formalism}

Let us start with some mathematical definitions.
Let $X$ be a compact, oriented 4-dimensional  manifold,  and
more specifically a K\"ahler  manifold  with  K\"ahler  form
$\omega$ (i.e. $\omega$ is a closed $(1,1)$-form).
On $X$ we consider a non-abelian gauge theory.
{}From a mathematical point of  view  this  amounts  to  defining  a
complex vector bundle $E$ associated to a principal bundle
with structure group $G$.
We will always consider the case of $G={\rm SU}(2)$.
We will write ${\frak g}_E$  for  the  bundle  associated  to  the
adjoint representation of ${\rm SU}(2)$.
If $A$ denotes a unitary connection on $E$, then
we denote with
$d_A:\Omega_X^p(E)\rightarrow\Omega_X^{p+1}(E)$
the
associated covariant exterior derivative.
Notice that $d_A=d+A$ on $\Omega_X^0(E)$.
Here $A$ stands for an ${\frak su}(2)$-valued one-form
which transforms locally as
$A\rightarrow A_u\equiv u^{-1}Au+u^{-1}du$,
where $u$ is (a local representative of)
an element of the gauge group, {\rm i.e.}\ a smooth map
$u:X\rightarrow {\rm SU}(2)$,
and $\Omega_X^p(E)$
denotes as usual
the space of $p$-forms on $X$ with values in $E$.
The Yang--Mills curvature, i.e. the  curvature  of  the  connection
$A$, is the ${\frak su}(2)$-valued two-form
$F_A=dA+A\wedge A\in \Omega_X^2({\frak g}_E)$
which transforms as
$F_A\rightarrow F_{A_u}\equiv u^{-1}F_A u$.
Notice that for any connection $A$ we have the Bianchi identities
$d_A F_A=0$.
If we fix a  metric  structure  on  $X$  and  denote  by  $*$  the
associated Hodge operator we have the splitting into self-dual and
anti-self-dual parts of the bundle-valued 2-form
\begin{equation}\label{tre}
F_A=F_A^+\oplus F_A^-\in\Omega_X^{2,+}({\frak g}_ E)\oplus
                         \Omega_X^{2,-}({\frak g}_ E)
\end{equation}
We call the connection $A$   anti-self-dual (ASD) if
$F_A^+\equiv \displaystyle{\left({1+*\over 2}\right)}F_A=0$.
Since $X$ is by  hypothesis  a  Hermitian  manifold  we  can  also
decompose the curvature $F_A$ as
\begin{equation}\label{quattro}
F_A=F_A^{(2,0)}\oplus F_A^{(1,1)}\oplus F_A^{(0,2)}
\in\Omega_X^{(2,0)}({\frak g}_E)\oplus\Omega_X^{(1,1)}({\frak g}_E)
\otimes\Omega_X^{(0,2)}({\frak g}_E).
\end{equation}
Now one can show that  the  self-dual  bundle-valued  complexified
2-forms over $X$ are
\cite{dk}:
\begin{equation}\label{cinque}
\Omega_X^{2,+}({\frak g}_E)
=\Omega_X^{(2,0)}({\frak g}_E)\oplus\omega\Omega_X^0({\frak g}_E)
\oplus\Omega_X^{(0,2)}({\frak g}_E)
\end{equation}
with $\omega$ the K\"ahler $(1,1)$-form
($\omega\Omega_X^0({\frak g}_E)=\Omega_X^{(1,1)}({\frak g}_E)$).
In~(\ref{tre}-\ref{cinque}) the
$\Omega_X^{(p.q)}({\frak g}_E)$
denote the sections
$\Gamma(\Lambda_X^{(p,q)}\otimes{\frak g}_E)$
and the $\Omega_X^\pm$
stand for the algebraic projections
$\displaystyle{{1\over 2}(1\pm *)}\Omega_X\equiv \pi_\pm\Omega_X$.

Our physical theory
is the first-order version of the pure  ${\rm SU}(2)$ Yang--Mills
theory with the so-called ``theta-term'' given by the action
functional (in Euclidean signature)
\begin{equation}\label{sei}
S_{BF^+}=\int_X{\rm Tr}(B\wedge F_A^+)
-{g^2\over 4}\int_X{\rm Tr}(B\wedge B),
\end{equation}
$B\in\Omega_X^2({\frak g}_E)$,
and hence $B\rightarrow B_u\equiv u^{-1}B u$
under gauge transformations.
In~\rf{sei}, $g^2$ denotes the bare gauge coupling constant.
The field equations obtained from~\rf{sei} are
\begin{eqnarray}
\label{settea}
F_A^+&=&{g^2\over 2}B, \\
\label{setteb}
(d_A^+)^*B&=&0
\end{eqnarray}
where  the $+$ superscript  denotes  the  self-dual
projection and
$d_A^*:\Omega_X^{p+1}(E)\rightarrow \Omega_X^p(E)$
is the
adjoint operator of the covariant derivative.
Eq.~\rf{settea} clearly implies that $B\in\Omega^{2,+}$
and hence one may set $B=B^+$.
In the following we shall denote by ${\cal M}_c$
the ``classical'' moduli space, that is the space of
solutions  of~(\ref{settea}-\ref{setteb})  modulo  the
gauge group.
Now  using~\rf{settea}  in~\rf{sei}  one   gets   the   action
functional  in  the  second-order  formalism  (in  the   Euclidean
signature):
\begin{equation}\label{nove}
S_{\rm YM}={1\over g^2}\int_X{\rm Tr}(F_A^+\wedge F_A^+)
={1\over 2g^2}\int_X{\rm Tr}(F_A\wedge *F_A)+
{1\over 2g^2}\int_X{\rm Tr}(F_A\wedge F_A).
\end{equation}
At this point one may reabsorb the gauge coupling $g$  into  $F_A$
through the rescaling $A\rightarrow A'=A/g$, so that \rf{settea} now reads
\begin{equation}\label{Fprimo}
F_{A'}^{\prime +} \equiv (dA'+gA'\wedge A')^+ = \frac g2 B.
\end{equation}

Notice that the first term in the right-hand-side of~\rf{nove}
is the standard Yang--Mills functional, while the second one
is the topological charge, the so-called ``theta-term''
associated with the instanton number $k$
\cite{dk}
\begin{equation}\label{dieci}
k(A)=-c_2(E)[X]=-{1\over 8\pi^2}\int_X{\rm Tr}(F_A\wedge F_A).
\end{equation}
Here $c_2(E)$ is the second Chern class of the complex
${\rm SU}(2)$-vector bundle $E$ and we have adopted the standard sign
convention for $k$.
If we denote by $t_a$ the generators of ${\frak su}(2)$ (in the fundamental
representation) we can write
$A=\sum_a A_\mu^a t_a dx^\mu$,
$F_A=\sum_a(F_A)_{\mu\nu}^at_adx^\mu\bigwedge dx^\nu$
in a local coordinate system $\{x^\mu\}$ on $X$.
Furthermore, we fix the normalization for the $t_a$'s according
to
${\rm Tr}(t_at_b)=-{1\over 2}\delta^{ab}$.
For ${\rm SU}(2)$ we get that $t_a=i\sigma_a/2$ $(a=1,2,3)$
with $\sigma^a$ the Pauli matrices.

In the following we shall consider the quantum  theory  associated
to  the  action~\rf{sei}  in  the  frame  of  path-integral
quantization.
For  this  purpose   we   need   to   fix   the   gauge-invariance
of~\rf{sei}.
We use the background covariant ``gauge fixing" condition
\begin{equation}\label{dodici}
d_{A_0}^*a=0=d_{A_0}^*B,
\end{equation}
where we have defined $A=A_0+ a$ with $A_0$ a fixed background
${\rm SU}(2)$-connection.
Notice that we have to impose a constraint on the field $B$, since the theory
acquires on shell a larger symmetry. We will stick to the common
terminology and call this constraint on the field $B$ a ``gauge fixing".
Since~\rf{sei} is just the 4-dimensional Yang--Mills theory
(with theta-term)
\footnote{
In Euclidean signature the
Gibbs measure associated to the
``theta  action-functional''is  usually
written as $\exp(i\theta k(A))$, where $k(A)$ is defined by~\rf{dieci}
and the Lagrange multiplier $\theta$ is the so-called $\theta$-parameter.
Then~\rf{nove} implies
$\theta=4\pi^2$, which is consistent with the fact that
$\theta$ does not  renormalize.
}
written in the first-order formalism, and we know that
a non-abelian gauge theory is asymptotically free, it follows that
the $B$ field is weakly (strongly) coupled to the Yang--Mills
potential $A$ in the ultraviolet (infrared) regime.
Therefore~\rf{Fprimo}, after the replacing of $g$
with the running coupling constant $g(\mu)$
($g(\mu)\sim 0$ at energies $\mu\gg M$ with respect to
some physical energy scale $M$)
becomes the ASD-condition $F_{A'}^{\prime +}\sim 0$
in the ultraviolet regime.

An alternative theory which, in the semiclassical approximation,
is equivalent to the one described by \rf{sei}
is given by the following topological action
\begin{equation}
S_{BF}=\int_X{\rm Tr}(B\wedge F_A)
-{g^2\over 4}\int_X{\rm Tr}(B\wedge B),
\label{BF}
\end{equation}
provided that we assume:
\begin{equation}\label{singular}
B^-=0.
\end{equation}
By assuming the above constraint, we are breaking the ``topological"
symmetries of the $BF$ theory \rf{BF} given by:
\begin{eqnarray}\label{topo}
\delta_t A &=& \frac{g^2}2 \eta\\
\delta_t B &=& -d_A \eta,
\end{eqnarray}
where $\eta$ is a 1-form ghost \cite{cm}\cite{c}.

\section{Quantum theory}

The quantum theory associated to~\rf{sei} is determined  by  the
functional integral
\begin{equation}\label{tredici}
\left\langle 1 \right\rangle_{\cal M}
=\int_{\cal M} d\mu(A,B)
e^{-S_{BF^+}}.
\end{equation}
In~\rf{tredici} $\cal M$ is the orbit space, i.e. the  quotient
of the set of fields $(A,B)$ by the gauge group.
Following the usual procedure, the  functional measure  $d\mu(A,B)$  over
$\cal M$ is  obtained  by  the  standard  Faddeev-Popov  procedure
\begin{equation}\label{quattordici}
\left. d\mu(A,B)\right|_{A=A_0+a}
=[da][dB]\delta(d_{A_0}^*a)\delta(d_{A_0}^*B)J(A,B),
\end{equation}
with $a\in\Omega^1({\frak g}_E)$ and
the functional Jacobian
$J(A,B)=\det'[\delta_\mu(d_{A_0}^*a_u)]
\det'[\delta(d_{A_0}^*B_u)]$
is the standard Faddeev-Popov determinant.
Of course, in  order
to derive~\rf{quattordici} in the background gauge  one  assumes
that the topology of $X$ allows only  irreducible
connections  $A$  on  the  ${\rm  SU}(2)$-bundle  $E$ over $X$.
This means
that there are no covariantly constant scalar fields.
A sufficient condition for that~\cite{dk} is  that
$b_2^+>1$; here $b_2^+$ is the dimension of  the  space
of self-dual harmonic forms over $X$.

Our  key  idea is  to  include   into~\rf{tredici}   some
gauge-invariant   non-local   operator $M(\Sigma)$
(to be defined in the next section),
which is
analogous to the disorder loop variable of 't~Hooft
\cite{th1},
in such a way that one has
\begin{equation}\label{quindici}
\left\langle M(\Sigma) \right\rangle_{\cal M}
=\int_{\tilde{\cal M}} d\tilde\mu_{L_\Sigma}(\tilde A,\tilde B)
e^{-\tilde S_{BF^+}}
\equiv\left\langle 1 \right\rangle_{(\tilde {\cal M},L_\Sigma)}
\end{equation}
where $\tilde{\cal M}$ is a particular case (namely the 0-dimensional
one) of the  Witten
moduli  space  which  gives   the   new   differential-topological
invariants   for   compact,      4-manifolds    recently
discovered by
Witten~\cite{w1}.
Eq.~\rf{quindici} is one of our basic results.
In the following we shall   define  explicitly  in  terms  of
$(A,B)$  a  non-local   gauge   invariant   operator   $M(\Sigma)$
associated   to   a   surface   $\Sigma\subset   X$,   such
that\rf{quindici} holds identically, in the semiclassical
approximation and in the weak coupling limit (up to some  irrelevant
normalization).
In~\rf{quindici} the symbols have the following meaning: the introduction
of the observable $M(\Sigma)$ brings into the picture (singular)
reducible connections that are not allowed in the theory without such
observables. In other words we obtain (abelian) connections
(denote by the symbol $\tilde A$)
on the holomorphic line bundle $L_\Sigma$
over $X$ having first Chern class $c_1(L_\Sigma)$ equal  to  the
K\"ahler class $[\omega]\in H^2(X)$ which is the  Poincar\'e  dual
(PD) of [the homology class of] the surface  $\Sigma\subset  X$, i.e.
$[\Sigma] ={\rm
PD}([\omega])={\rm PD}(c_1(L_\Sigma))$\footnote{From now on we will
drop the square parentheses denoting  (co)homology classes.}.
$\tilde B$ is the 2-form $B$ restricted to a  generic  $
(1,1)$-section of the bundle
$\Lambda^{(1,1),+}_X\otimes L_\Sigma$ over $X$.
Notice that the condition that $\Sigma$ be Poincar\'e dual to
the K\"ahler form $\omega$ means that there is a  singular  1-form
$\eta$ on $X$ such that for any 2-form $\hat\theta$ over  $X$  one
has
\begin{equation}\label{sedici}
\Sigma={\rm PD}(\omega)\iff
\int_{\Sigma\subset X}\hat\theta-\int_X\hat\theta\wedge\omega
=\int_Xd\hat\theta\wedge\eta.
\end{equation}
In~\rf{quindici} the action functional
$\tilde S\equiv S_{BF^+}(\tilde A,\tilde B;L_\Sigma;X)$
is the restriction of~\rf{sei} to the fields
$(\tilde A,\tilde B)$ defined above.
Equivalently one can say that the introduction of the operator
$M(\Sigma)$
into the expectation value~\rf{tredici}
gives an {\em effective} gauge theory which is controlled by a
 split $\>{\rm SU}(2)$-bundle
$E_\Sigma=L_\Sigma\oplus L_\Sigma^{-1}$,
where $\Sigma\rightarrow L_\Sigma$ by the correspondence
$\Sigma={\rm PD}(\omega)={\rm PD}(c_1(L_\Sigma))$.

\section{'t Hooft-like disorder operators}

In a four-dimensional  $BF$-theory, we can define
as in ref.~\cite{cm}, a   non-local   gauge   invariant
operator associated to any closed oriented immersed surface $\Sigma$:
\begin{eqnarray}
\label{diciassette}
{O}(A,B;\Sigma,\gamma,\gamma',\bar x)
&\equiv& 2\pi q_m g\int_{\Sigma}
{\rm Tr}_R\left[
{\rm Hol}_{\bar x}^{y}(\gamma) B(y) {\rm  Hol}_{y}^{\bar  x}(\gamma')
\right]    \\
\nonumber
B(y)&\equiv&\sum_a B^a(y)R_a\equiv\sum_a B_{\mu\nu}^a(y)
R^ady^\mu\wedge dy^\nu
\end{eqnarray}
where $q_m$ will play the r\^ole of a ``magnetic charge"
(as it will be clear after~\rf{phim}),
$\bar   x$   is   a   base   point   on
$\Sigma$
(equivalently, one may remove the point $\bar x$ from $\Sigma$),
$R^a$  is  an  irreducible  representation\footnote{We will consider
later on, only the fundamental representation.}  of
${\rm SU}(2)$,  ${\rm  Tr}_R$  denotes  the  trace  over  $R$ and
$\gamma'\cup\gamma$ is any closed path $C$ with base point
$\bar x$ and passing through $y$.

Furthermore, $\hol{\bar x}{y}{\gamma}$ in~\rf{diciassette}
stands for the holonomy of the ${{\rm SU}(2)}$-connection $A$ along the
path $\gamma$ from ${\bar x}$ to $y$:
\begin{equation}\label{venti}
\hol{{\bar x}}{y}{\gamma}
\equiv P\exp(\int_{\bar x}^y A(x))
\end{equation}
It has been shown in~\cite{cm,c} that the choice of the base point ${\bar x}$
is irrelevant\footnote{
Of course the expectation values of the operators $O$ depend
on the choice of the paths $\gamma,\gamma'$, but this choice is irrelevant
for the computation in this framework of the Donaldson-Witten
invariants.},
and underlying a choice for the
closed path $C$, one can finally set
in this context:
\begin{equation}\label{ventuno}
M(\Sigma)\equiv\exp[O_+(\Sigma)], \qquad
O_+(\Sigma)\equiv O[A,B=B^+;\Sigma,C,\bar x],
\end{equation}
where $O$ is defined by~\rf{diciassette}. Here of course
$O_+$ is the projection by ${1\over 2}(1+*)$ of
the integrand of $O$
to the self-dual one. We also have to consider the standard Wilson loop
observable
\begin{equation}
W(C) \equiv {\,\hbox{\rm Tr}\,}_R\hol{}{}{C}
\end{equation}
where $C$ is a closed loop.

Our next step is to compute
the semiclassical approximation of
$\left\langle M(\Sigma)\right\rangle_{\cal M}$.
We now consider a background {\em quasi-ASD connection $A_0$},
namely a connection satisfying eq.\rf{settea} and \rf{setteb} with $B_0\neq 0$
and $g$ small enough but different from $0$.
The linear version of
the field equations~(\ref{settea},\ref{setteb})
yields the
the following equations for the field fluctuations $A_0\longrightarrow
A_0+a$ and $B_0\longrightarrow B_0 +b$ with $(a,b)\in
\Omega^1\oplus \Omega^{2,+}.$
\begin{eqnarray}
\label{ventiduea}
d_{A_0}^+a &=& {g^2\over 2}b   \\
\label{ventidueb}
(d_{A_0}^+)^*b &=& 0.
\end{eqnarray}

The operator $t_g$ corresponding to~\rf{ventiduea},
\rf{ventidueb} defined by
\begin{eqnarray}\nonumber
t_g:\Omega^1\oplus\Omega^{2,+} &\mapsto & \Omega^1\oplus\Omega^{2,+}  \\
\label{deft}
(a,b) &\rightarrow& (({d_{A_0}^+})^*b,d_{A_0}^+a-{g^2\over 2}b).
\end{eqnarray}
Pure gauge fluctuations $\phi\in \Omega^0$
due to the gauge transformations of the backgrounds $(A_0,B_0)$
are not relevant here
since by choosing:
$a=d_{A_0}\phi$ and $b=[B_0,\phi]$ we have that the condition
$(d_{A_0}^+)^*[B_0,\phi]=0$ is compatible with the equation
\[
\left(d^+_{A_0}d_{A_0}\phi, [B_0,\phi]\right)=
{g^2\over 2}\parallel [B_0,\phi]\parallel ^2
\]
only if we set $\phi=0$.

Hence the dimension of our moduli space will be simply given by the dimension
of $\ker t_g$.
By using the Weitzenb\"ock formulae it is possible to show
that $(a,b)\in\ker t_g$ implies $b=0$.
The same argument should also imply that
$(d_{A_0}^+)^*d_{A_0}^+ a=0$ holds only when $a=0$, provided that $A_0$ is
a quasi-ASD (not ASD) connection.

The vanishing of $\ker t_g$
can be restated by saying that the classical moduli space ${\cal M}_c$ is
zero-dimensional
or discrete.
In other words the inclusion
of the field $B$
(with $g\ne 0$) seems to remove the degeneracy of the instanton
vacua of the classical Yang--Mills theory.
We will essentially find points of the classical moduli
space, given by {\em singular} solutions.
In the semiclassical computation of
$\left\langle M(\Sigma)\right\rangle_{\cal M}$
one may set
${\cal M}\simeq{\cal M}_c$.

{}From the point of view of the field equations, the insertion into
the path-integral~\rf{tredici}
of the operator $M(\Sigma)$ is equivalent to considering
the improved gauge-fixed $BF^+$-action
\begin{equation}\label{ventiquattro}
S_{BF^+}(A,B,d_{A_0}^*A=0=d_{A_0}^*B;X)+O_+(\Sigma)
\equiv S_{BF^+}^{(\Sigma)}(A,B,d_{A_0}^*A=0=d_{A_0}^*B;X,\Sigma\subset X)
\end{equation}
Here $O_+(\Sigma)$ plays the r\^ole of a ``source term'' for~\rf{sei}. Indeed,
when we make the following choice of the
2-form $\hat\theta$ in~\rf{sedici}
\begin{equation}\label{venticinque}
\hat\theta(\Sigma)\equiv 2\pi q_mg{\,\hbox{\rm Tr}\,}_R[\hol{{\bar x}}{y}{\gamma}
B^+(y)\hol{y}{{\bar x}}{\gamma'}],
\qquad
O_+(\Sigma)=\int_\Sigma\hat\theta(\Sigma)
\end{equation}
we have:
\begin{equation}\label{ventisei}
\int_{\Sigma}\hat\theta=\int_X\omega\wedge\hat\theta,
\qquad\omega\equiv {\rm PD}(\Sigma)
\end{equation}
Eq.~\rf{ventisei} follows from the fact that $\theta$ is a closed form. In fact
we have:
\begin{eqnarray}
\label{ventisette}
d\hat\theta(\Sigma)&=& 2\pi q_mg
d{\,\hbox{\rm Tr}\,}_R[\hol{{\bar x}}{y}{\gamma}B^+(y)\hol{y}{{\bar x}}{\gamma'}]  \\
&=&2\pi q_mg{\,\hbox{\rm Tr}\,}_R
[\hol{{\bar x}}{y}{\gamma}d_{A_0} B^+(y)\hol{y}{{\bar x}}{\gamma'}]=0
\nonumber
\end{eqnarray}
Notice that in~\rf{ventisette} we have used the gauge condition
$d_{A_0}^* B=0$ which is the same as $*d_{A_0}*B=0$, and implies $\d_{{A_0}}
B=0$
since $B$ is self-dual.

\section{Moduli space and topological invariants}
\label{sezquattro}

In this section we prove~\rf{quindici} in the semiclassical
approximation. The starting point is given by the
field  equations obtained
after the inclusion
of  a disorder
operator  $M(\Sigma)$ \`a la 't~Hooft.

These field equations are obtained
by applying the functional derivative
$\displaystyle{{ \delta\over{\delta B^a}}}$ to
the action $S^{(\Sigma)}_{BF^+}$
(see (\ref{ventiquattro}-\ref{ventisei})) and read:
\begin{eqnarray}
\nonumber
F_+^{(2,0),a}&=&0+O(g^2)=F_+^{(0,2),a}  \\
\label{ventotto}
F_+^{(1,1),a}(x)&=&-2\pi q_mg\omega(x){\rm Tr}_R
\{\hol{{\bar x}}{x}{\gamma}R^a\hol{x}{{\bar x}}{\gamma'}\}
+O(g^2)=  \\
\nonumber
&=&- 2\pi q_mg\omega(x){\rm Tr}_R\{\hol{}{}{C_x}
R^a\}
+O(g^2).
\end{eqnarray}
Here $C_x$ is
a loop based at $x$,
$\{R^a\}$    is  the basis   of a
representation of
${\rm SU}(2)$,
 $F_+^{(p,q)}$ is the $(p,q)$-part of $F^+$
and the other notation is the same as in the previous section.
The holonomy is non trivial if we think that
the loop $C_x$ includes a ``Dirac singularity"
(see below).
{}From now on we choose the fundamental
representation of ${{\rm SU}(2)}$ and hence we set $R_a\equiv t_a; a=1,2,3$.
As a consequence of equation \rf{ventotto}
the curvature $F_A$ is given, in the limit $g\to 0$, by
the product of a numerical constant times
$\omega$ times the projection of ${\rm Hol}(C)$ into the
Lie algebra of
$SU(2)$. This projection defines a constant direction
in the Lie Algebra, so  we can assume that this
direction is parallel to $t_3.$
This is equivalent to considering a (singular) reducible connection.
We can furthermore choose $\omega$
as a {\em self-dual 2-form}. In conclusion
Eq.~\rf{ventotto} admits a non-trivial  singular reducible\footnote
{
In order to implement the Faddeev-Popov
procedure, we required that
there are no {\em smooth} reducible connections on $X$.
There is no contradiction here, since $\tilde A$ is singular on $\Sigma$.
}
quasi ASD
connection of the form~\cite{fu}:
\begin{equation}
\label{ventinovea}
\tilde A(x)={i\over 2}
\left(
\begin{array}{cc}
\alpha(x) & 0 \\ 0 & -\alpha(x)
\end{array}
\right)
=\alpha(x)t_3,
\end{equation}
where $\alpha$ is a 1-form,
$t_3=\frac{i}2\sigma_3=\frac{i}2{\rm diag}(1,-1)$.
Hence $\alpha$ is an {\em abelian} connection
and the curvature of \rf{ventinovea} is
$ft_3$,   where we have set:
$f\equiv d\alpha$.
Thus
the  introduction   of   the   operators
$M(\Sigma)$ replaces the original non-abelian Yang--Mills
theory [with theta-term]
to an effective theory based on a split $SU(2)$-bundle
$E_\Sigma=L_\Sigma\oplus  L_\Sigma^{-1}$,  with  $L_\Sigma$  a
holomorphic line bundle on $X$  ``parameterized'' by  $\Sigma\subset
X$, in the sense that $c_1(L_\Sigma)=\omega={\rm PD}(\Sigma).$
So the instanton number
$k=-c_2(E_\Sigma)$  is expressed in terms of
the monopole number $\lambda=c_1(L_\Sigma)$, since one has
$k=-c_2(E_\Sigma= L_\Sigma\oplus L_\Sigma^{-1})=c_1(L_\Sigma)^2=
\lambda^2$.
As a consequence of~\rf{ventinovea}, we have:
\begin{equation}
\label{trentaa}
{\rm Tr}(\hol{}{}{C}t_a)=-\sin (g\phi_m/2) \delta_{a,3}
\end{equation}
where
$\phi_m$ is the monopole magnetic flux of  $\tilde F/g$
across  the  ``Dirac  surface''  $\Sigma$.
One may regard the r.h.s.\ of~\rf{ventotto} as an effective field
$\tilde B$, given by
\begin{equation}\label{Btilde}
\tilde B^a=-\pi gq_m \omega {\rm Tr}({\rm Hol}(C)t_3)\delta_{a,3}
\end{equation}
obtained by the insertion of the operator $M(\Sigma)$
into~\rf{tredici} in the limits $g\rightarrow 0$ and
$\hbar\rightarrow 0$. Notice that by \rf{Dirac} the field $\tilde B$ is
actually independent of $g$ and hence contributes also in the weak-coupling
limit.
By  \rf{ventotto} and \rf{Btilde},
the self dual part of the curvature of
the vector potential $\tilde A'=\tilde A/g$
is given by:
\begin{equation}
\label{nuova}
d^+ \tilde A'=(2/g)\tilde B+O(g^2)
\end{equation}
Eqn. \rf{nuova} is the same as \rf{Fprimo} when applied
to the reducible connection $\tilde A$, provided that we perform
the so called weak-strong coupling duality $g\to 1/g.$
Furthermore~\rf{ventidueb} becomes
\begin{equation}\label{trentab}
(d^+_{\tilde A})^*\tilde B=0+O(g)
\end{equation}
where
$\tilde B=\tilde B^+$
 is a
$(1,1)$-section of the twisted self-dual
vector bundle $\Lambda^{(1,1),+}_X\otimes L_\Sigma$ over $X$.
Equations~(\ref{nuova}-\ref{trentab})  are the
monopole equations that, in the case of  a
general K\"ahler manifold $X$, correspond to Witten's monopole
equations \cite{w1}.

The relation between the complex Weyl spinors $M$ and $\bar M$
in Witten's equation and our
field $\tilde B$ is of the form:
\begin{equation}\label{spinor}
\tilde B_{\mu\nu} = \frac i2 \bar M\Gamma_{\mu\nu} M
\end{equation}
where $\Gamma_{\mu\nu}=\frac12[\Gamma_\mu,\Gamma_\nu]$ and the $\Gamma$'s are
the Clifford matrices. In the general case the solutions of our monopole
equations \rf{nuova} and \rf{trentab} are only a subclass of Witten's.
The situation here is similar to the comparison between a four-dimensional
$BF$ theory to a gravitational theory in the first order formalism.
In order to connect the above two theories one has to require that the
$B$ field is the ``square" of the vierbein \cite{js}. In the situation
described
in this paper, in order to connect our $BF$ theory with Witten
monopole equation, one has to assume the validity  of \rf{spinor}.

However,
we want to stress that the results we present in the following of this
paper are completely independent on the above
relation between our and Witten's monopole
equations.

Let us now look for an explicit  solution of~\rf{nuova}.
Requiring that, in the limit $g\to 0$, we have non trivial
solutions of \rf{ventotto}
implies $\sin({g\over 2}\phi_m)\ne 0$.
In the path integral the effect of the cosmological-like term
\begin{equation}\label{cosm}
\Delta S=-{g^2\over 4}\int_X {\,\hbox{\rm Tr}\,}(B\wedge B)
\end{equation}
with $g\approx 0$ but $g\ne 0$
is to select
the minimal energy solutions, obtained
when $|\sin({g\over 2}\phi_m)|=1$, thus giving flux quantization.

We can choose coordinates $(x,y,u,v)$ on $X$ locally in a
neighborhood of the base point $\bar x^\mu$ so that the
surface $\Sigma$ is given by the equations $x=y=0$.
For the Poincar\'e dual form $\omega$ restricted to $\Sigma$ we can
take~\cite{dk}
\begin{equation}\label{trentacinquez}
\omega(x,y)=\psi(x,y)dx\wedge dy
\end{equation}
where $\psi$ is a bump function on ${\bf R}^2$, supported
near $\vec{\bar x}=(x,y)=(0,0)$ and with integral $1$.
We can choose
$\psi(x,y)=\delta^{(2)}(x,y)$, with $\delta^{(2)}(x,y)$
the two-dimensional delta function.
Therefore we may think that the Maxwell's field strength $f$
is not zero only on $\Sigma$ by choosing
$\alpha=\alpha_\mu dx^\mu=\alpha_1 dx+\alpha_2 dy$,
i.e. $\alpha_3=\alpha_4=0$, so that
$f=d\alpha=f_{12}dx\wedge dy$,
where
$f_{12}\equiv(\partial_1\alpha_2-\partial_2\alpha_1)$,
$\partial_i\equiv\partial/\partial x^i$ ($i=1,2$),
$\vec x\equiv\{x^i\}\equiv(x,y)$.
Then~\rf{nuova} through~\rf{trentacinquez} becomes at order $g^2$
\begin{equation}\label{trentaseiz}
f_{12}(x,y)=\mp 2\pi q_mg
\delta^{(2)}(x,y)
\end{equation}
The Maxwell potential $\alpha_i$ which solves~\rf{trentaseiz}
can be written as\footnote{
Dropping the assumption of minimal energy, the general solution
is
$\alpha_i(\vec x)=cg(\epsilon_{ij}+i\delta_{ij})
\partial^j\log|\vec x-\vec {\bar x}|$
with $c/q_m=\sin (\pi g c)$.
}
\begin{equation}
\alpha_i(\vec x)=\pm  q_mg(\epsilon_{ij}+i\delta_{ij})
\partial^j\log|\vec x-\vec {\bar x}|
\label{trentasettez}
\end{equation}
This is the Kohno connection \cite{Koh} that is obtained in a
similar contest also in \cite{msz}.
Indeed one has that
\begin{equation}\nonumber
\partial_i\partial^i\log|\vec x-\vec{\bar x}|
=2\pi\delta^{(2)}(\vec x-\vec{\bar x})
\end{equation}
and locally around $\vec{\bar x}$ the monopole magnetic flux
of $\alpha/g$ is
\begin{equation}\label{phim}
\phi_m=\pm 2\pi q_mQ(\Sigma,\Sigma)
\end{equation}
where
\begin{equation}\label{intersec}
Q(\Sigma,\Sigma')=\int_X\omega[\Sigma]\wedge\omega[\Sigma']
\end{equation}
denotes the {\it algebraic intersection number}~\cite{dk}
of the oriented surfaces $\Sigma$ and $\Sigma'$.
Notice that~\rf{intersec} is well defined even when
$\Sigma'=\Sigma$.
For the minimal energy solution one has $\sin (g\pi q_m)=1$,
implying
\begin{equation}\label{Dirac}
2g q_m Q(\Sigma,\Sigma)\equiv 1\quad({\rm mod}\, 4)
\end{equation}
which corresponds to the {\it Dirac quantization condition}.
{}From~\rf{phim} it follows that the operator~\rf{ventuno}, when expressed
in terms of the fields $(\tilde A$, $\tilde B)$ is related to
the exponential of the ``magnetic flux".

Summarizing,  we   have   shown   that   the   inclusion  of   the
gauge-invariant non-local operator
$M(\Sigma)$ into the expectation value~\rf{tredici}  given
by   the   non-abelian   ${\rm    SU}(2)$    $BF^+$-theory
reduces  this theory   to   an  { effective   abelian} one {
with monopoles} and
{\it without} the $M(\Sigma)$ operator.
The resulting quantum field theory, which is ``dual" to the original one,
is coupled to monopoles and is
written in terms of an abelian connection $\alpha$
on  a  holomorphic
line bundle $L_\Sigma$ (the monopole line bundle),
and  a
self-dual
two-form
$\tilde B$.
This result proves~\rf{quindici}
in the  semiclassical  approximation.

At  this  perturbative  order
the underlying moduli space is the classical one
and coincides with the moduli
space $\tilde {\cal M}$.

Thus the partition function given by the path-integral written  in
the r.h.s. of~\rf{quindici}  (or  equivalently  the  ``one-point''
function described by the l.h.s. of~\rf{quindici}) is equal to the
number of points in our moduli-space of the fields
$(\tilde A, \tilde B)$,
counted with a $\pm$ sign
\begin{equation}\label{trentadue}
\left\langle 1\right\rangle_{(\tilde{\cal M},L_\Sigma)}\equiv\#{\tilde{\cal M}}
(\tilde A,\tilde B; X,L_\Sigma)\quad   .
\end{equation}
Equation~\rf{trentadue} is  the same as  in
Witten~\cite{w1,w3}.
Equation ~\rf{trentadue} follows from the fact  that
the integration over $\tilde B$, gives
a delta-functional contribution,
i.e. $\delta(d^+\tilde A +2\pi q_mg\omega{\rm Tr}\{\hol{}{}{C}
t^3\})$,
which in turn gives a counting measure (with signs) on
the space $\cal M$.

Observe that the $\pm$ signs are provided by the ratio
between the functional determinant coming from the ghosts\footnote{
The gauge fixing \rf{singular} requires by \rf{topo}
a Faddeev--Popov term $\bar q \wedge
d_Aq$, where $\bar q$ is the anti-self-dual
anti-ghost associated to the 1-form ghost $q$.
}
and
the Jacobian associated with the delta-functional.

\section{Donaldson's polynomial invariants}
\label{sezsei}

In this section we show how the original Donaldson polynomial
invariants~\cite{dk,w2} naturally come in the
framework of our theory after the introduction of Wilson-line
operators
\begin{equation}\label{wilson}
W(C)={\,\hbox{\rm Tr}\,} P\exp ( \oint_C A)
\end{equation}
Let us now consider a closed oriented surface
$\hat\Sigma$. We open a small contractible disk $\Sigma''$ so that
$\hat\Sigma=\Sigma'\cup\Sigma''$
and compute the following expectation value:
\begin{equation}\label{ev}
\left\langle M(\Sigma)W(C)\right\rangle
\end{equation}
where $C=\partial\Sigma'=-\partial \Sigma''$.
Since the Wilson loop does not depend on the $B$-field,
we can perform the $B$ integration exactly as in the previous
section.
Then the resulting delta-functional restricts the $A$-integration
to the reducible connections $\tilde A$  and this allows to write
the Wilson loop as
\begin{equation}\label{wilsonuno}
W(C)={\,\hbox{\rm Tr}\,}\exp (\int_{\Sigma'} f \, t_3)
\end{equation}
Since in this case one has
$f=\mp 2\pi q_mg\omega[\Sigma]$
and
$2gq_m Q(\Sigma,\Sigma)=1+4n$,~\rf{ev} gives
\begin{equation}\label{QQ}
\left\langle M(\Sigma)W(C)\right\rangle=\left\langle M(\Sigma)\right\rangle
2\cos\left[
{\pi\over 2}(1+4n){Q(\Sigma,\hat\Sigma)\over Q(\Sigma,\Sigma)}
\right]
\end{equation}
If the monopole has the fundamental charge ($n=0$),~\rf{QQ}
gives the algebraic intersection number $Q(\Sigma,\hat\Sigma)$
modulo $2Q(\Sigma,\Sigma)$. Notice that the insertion of the operator
$M(\Sigma)$ creates a monopole whose flux through $\Sigma'$ measures
the intersection number.

We now move to the general case of $d+\hat d$ classes
$[\Sigma_1],\ldots,[\Sigma_d]$
and $[\hat\Sigma_1],\ldots,[\hat\Sigma_{\hat d}]$
in $H_2(X,{\bf Z})$.
We can associate as before $\hat d$ loops $C_1,\ldots,C_{\hat d}$
to the $[\hat\Sigma_b]$ and compute the expectation value
\begin{equation}\label{eevv}
\left\langle \prod_{a=1}^d M(\Sigma_a)\prod_{b=1}^{\hat d}
W(C_b)\right\rangle
\end{equation}
which turns out to be proportional to
\begin{equation}\label{vvee}
2^{\hat d}\prod_{a=1}^d\prod_{b=1}^{\hat d}
\cos\left(
{\pi\over 2}(1+4n_a){Q(\Sigma_a,\hat\Sigma_b)\over Q_0(\Sigma_.)}
\right)
\end{equation}
where
$Q_0(\Sigma_.)={\rm G.C.D. } \{Q(\Sigma_a,\Sigma_a); a=1,\ldots,d\}$.

Moreover, choosing $d=\hat d$ and $\Sigma_a=\hat\Sigma_a$,
$a=1,\ldots ,d$, one gets a symmetrized version of~\rf{vvee},
which is a generating functional for the Donaldson polynomial invariants
associated to $X$,
$q_X([\Sigma_1],\ldots,[\Sigma_d])$
\cite{w2}.

{}From a physical point of view the Wilson-loops
detect the monopole flux generated by
the insertion of the operators  $M(\Sigma)$ associated to surfaces.
Monopoles of the same charge on the two sides of the
surfaces give opposite contributions to the
flux\rlap,\footnote{
This picture, usually understood as a
``double monopole layer'', has been suggested to us by A.~Sagnotti
and has been firstly considered by A.~Polyakov~\cite{pol}
in 3D-QED and more recently in Ref.~\cite{msz} in
4D-QED.
}
which turn out to be proportional to the algebraic intersection numbers.

\section{Conclusions}
In this work we have studied
the self-dual $BF$ theory as the first order version
of pure Yang--Mills theory with theta-term.
In this framework
we have the Wilson loop operator $W(C)$, which
is the ``order operator" for ordinary Yang--Mills theory,
and an operator $M(\Sigma)$ playing the r\^ole of
a ``disorder parameter'' in the sense
of Ref.~\cite{th1}. Indeed
the abelian reduction \rf{ventinovea} maps the highly non-trivial relations
among the ``Weyl group operators"\footnote{
Notice that by \rf{BF} in $3+1$ dimensions the spatial components
$\epsilon_{ijk}B_{jk}$ are canonically conjugated to $A_i$.
}
$\rm Hol$ and $M$ into the 't~Hooft commutation relations.

{}From a physical point of view, the insertion of $M(\Sigma)$
plays the r\^ole of the standard ``abelian projection'' in pure QCD
without Higgs mechanism~\cite{th2}.

The  expectation  values  of
$M(\Sigma)$ alone and of the combination $M(\Sigma)W(C)$ are related
to the Donaldson--Witten invariants of
a K\"ahler manifold $X$
\cite{dk,w3,w2,w1,t,km}.
However, these invariants arise here in a different setting with respect to
the one considered by Witten.

Yet the computation of the invariants relies on the
existence of monopole equations
both here and in Witten's approach.

One of the differences between our framework and Witten's one,
is that our observables introduce singularities, so our theory
appears to be non-trivial, even if we work with ${\bf R}^4$.
For a related approach see also \cite{ans}.

The observable $M(\Sigma)$ (in the case when $\Sigma$ is a torus)
can be given a geometrical interpretation as parallel transport
of one of the fundamental cycles along the other. This suggests that $BF$
theory can be described as a ``gauge theory of loops"
\cite{ccmr}.
Our formulation gives an effective field-theoretical
description of the string picture for QCD introduced in Ref.~\cite{p}.

All our calculations have been done in
the limit $g\to 0$. In this limit both the topological
$BF$ theory with the
``cosmological term" \rf{BF} in 4-dimensions  and the Yang--Mills theory with
theta-term appear to detect 4-manifolds invariants. For these two theories
to be equivalent (on shell),  the choice of the ``gauge" $B^- =0$
(that breaks the larger ``topological invariance" of \rf{BF}) is
a key ingredient.

If we instead consider a ``pure" $BF$ theory (i.e.\ we take $g=0$) and do
not impose the constraint of self-duality $B^-=0$,
we can study a different topological field theory,
as discussed in \cite{cm,c,ccfm}.
Perturbation theory in  $\kappa=q_mg$
detects 2-knots, i.e. embedded (or immersed) 2-surfaces, up to
diffeomorphisms of the 4-manifolds.

\begin{center}
{\bf Acknowledgements}
\end{center}
\noindent
The authors acknowledge  fruitful  discussions  with J.~Baez,
M.~Bianchi, J.~Fr\"ohlich,  F.~Fucito,  G.~Rossi,
A.~Sagnotti and M.~Zeni.
In particular we thank R.~Catenacci for some critical remarks.