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% Anhado dos mas:

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%\begin{document}

\preprint{FTUAM-98-13; IFT-UAM/CSIC-98-10} 

%\begin{flushright}
%FTUAM-98-13

%IFT-UAM/CSIC-98-10
%\end{flushright}

%\vskip 0.2cm

%{\Large
%\centerline{{\bf Nahm's Transformation on the Lattice}}
%\vskip 0.5cm

\title{\bf Nahm's Transformation on the Lattice}

\author{ A. Gonz\'alez-Arroyo\thanks{also at: \  Instituto de F\'{\i}sica
Te\'orica C-XVI, Universidad Aut\'onoma de Madrid, Cantoblanco, Madrid~28049,
 SPAIN.} \ and C. Pena \\ Departamento de F\'{\i}sica Te\'orica C-XI\\
Universidad   Aut\'onoma de Madrid\\Cantoblanco, Madrid 28049, SPAIN.}
%\\
%\\Departamento de F\'{\i}sica Te\'orica C-XI\\
%Universidad   Aut\'onoma de Madrid\\Cantoblanco, Madrid 28049, SPAIN.}



%\author{ C. Pena \\Departamento de F\'{\i}sica Te\'orica C-XI\\Universidad
%Aut\'onoma de Madrid\\Cantoblanco, Madrid 28049, SPAIN.}
%\centerline{\qquad \ \ 
%  A. Gonz\'alez-Arroyo{$^\dag$}{$^\ddag$} and C. Pena{$^\dag$} \\ }}
%\vskip 0.5cm

%\centerline{{$^\dag$}Departamento de F\'{\i}sica Te\'orica C-XI,}
%\centerline{Universidad Aut\'onoma de Madrid,}
%\centerline{Cantoblanco, Madrid 28049, SPAIN.}
%\vskip 10pt
%\centerline{{$^\ddag$}Instituto de F\'{\i}sica Te\'orica C-XVI,}
%\centerline{Universidad Aut\'onoma de Madrid,}
%\centerline{Cantoblanco, Madrid 28049, SPAIN.}

%\vskip 1.0cm

%\begin{center}
%{\bf ABSTRACT}
%\end{center}
\abstract{ By studying zero modes of the Dirac operator on the lattice, 
 we explicitly construct the Nahm transform of some topologically 
 non-trivial gauge field configurations on the torus.}

%\vskip 1.5 cm
%\begin{flushleft}
%PACS: 11.15.-q, 11.15.Ha
\keywords{Nahm transform, Fermion zero modes, Instanton solutions, Lattice
gauge theory.}

%Keywords: Nahm transform, Fermion zero modes, Instanton solutions, Lattice gauge theory.
%\end{flushleft}

\begin{document}



%\newpage

\input{section1.tex} 


\input{section2.tex}


\input{section3.tex}


\input{section4.tex}


\input{section5.tex}


\input{acknow.tex}


\newpage



\begin{thebibliography}{10}


\bibitem{Polyakov1}
A.M. Polyakov,
\newblock {\em Phys. Lett. }  59B (1975) 82.


\bibitem{thooft1}
G.~'t~Hooft,
\newblock {\em Phys. Rev. Lett.} 37 (1976) 8.


\bibitem{BPST}
A.~A.~Belavin, A.~M.~Polyakov, A.~S.~Schwartz and Yu.~S.~Tyupkin,
\newblock {\em Phys. Lett. }  59B (1975) 85.


\bibitem{ADHM}
M.~F.~Atiyah, N.~J.~Hitchin, V.~G.~Drinfeld and Y.~I.~Manin,
\newblock {\em Phys. Lett. } 65A (1978) 185.


\bibitem{saviddy}
G.~K.~Saviddy,
\newblock {\em Phys. Lett. } 71B (1977) 133. 


\bibitem{copen}
J.~Ambj\o rn and P.~Olesen,
\newblock {\em Nuc. Phys. } B170 (1977) 60.


\bibitem{torusrev}
A.~Gonzalez-Arroyo,
\newblock {Proceedings of the 1997 Pe\~n\'{\i}scola Advanced School in Non--Perturbative Quantum Physics,} World Scientific (1998);
hep-th/9807108.


\bibitem{twist}
G.~'t~Hooft,
\newblock {\em Nucl. Phys.}  B153 (1979)  141.


\bibitem{stsds}
G.~'t~Hooft,
\newblock {\em Commun. Math. Phys. } 81 (1981) 267.


\bibitem{vanbaal6}
P.~van~Baal,
\newblock {\em   Commun. Math. Phys.} 94 (1984) 397.


\bibitem{gpg-as}
M.~Garc\'{\i}a~P\'erez, A.~Gonz\'alez-Arroyo and B.~S\"oderberg,
\newblock {\em    Phys. Lett. } 235B (1990) 117.


\bibitem{gpg-a}
M.~ Garc\'{\i}a~ P\'erez and A.~Gonz\'alez-Arroyo,
\newblock {\em   J. Phys. A. } 26 (1993) 2667.


\bibitem{investigating}
A.~Gonz\'alez-Arroyo and P. Mart\'{\i}nez,
\newblock{\em Nucl. Phys.} B459 (1996) 337.


\bibitem{nahm}
W.~Nahm,
\newblock {\em Phys. Lett. } 90B (1980) 413.


\bibitem{pbpvb}
P.~J.~Braam and P.~van~Baal,
\newblock {\em   Commun. Math. Phys.} 122 (1989) 267.


\bibitem{vanbaalnew}
Th.~C.~Kraan and P.~van~Baal,
\newblock hep-th/9805168.

\bibitem{vanbaalNT}
P.~van~Baal,
\newblock {\em Nucl. Phys. B (Proc. Suppl.)}  49 (1996) 238.


\bibitem{Proc97}
M.~Garc\'{\i}a P\'erez, A.~Gonz\'alez-Arroyo, A.~Montero and C.~Pena,
\newblock {\em Nucl. Phys. B (Proc. Suppl.)} 63A-C (1998) 501. 


\bibitem{FermionZM}
A.~Gonz\'alez-Arroyo, A.~Montero and C.~Pena,
\newblock In preparation.


\bibitem{KalkCG}
T.~Kalkreuter and H.~Simma,
\newblock {\em Comput. Phys. Commun.} 93 (1996) 33.



\bibitem{BiCGorig}
H.~van der Vorst,
\newblock {\em SIAM J. Sc. Stat. Comp.} 13 (1992) 631.


\bibitem{BiCG}
A.~Frommer, V.~Hanemann, B.~N\"ockel, Th.~Lippert and K.~Schilling,
\newblock {\em Int. J. Mod. Phys.} C5 (1994) 1073.


\bibitem{pvb5}
P.~van~Baal,
\newblock {\em Nucl. Phys. (Proc. Suppl.)}  49 (1996) 238.


\bibitem{CohenGomez}
E.~Cohen and C.~G\'omez,
\newblock {\em Phys. Rev. Lett.}  52 (1984) 237.




\end{thebibliography}




\newpage


\begin{figure}[!h]

\caption{Action density for the original lattice gauge configuration in the plane of the four absolute maxima.
The total action has been normalized to~1.}

\vbox{ \hbox{ \hskip40pt \epsfxsize=300pt \hbox{\epsffile{gauge_dens.eps} } }   
     }

\label{OrigConf}

\end{figure}


\newpage


\begin{figure}[!h]

\caption{Invariant densities for the two orthogonal zero modes chosen for the original lattice configuration,
in the plane of the four absolute maxima. Both have norm 1 in the continuum.}

\vbox{ \hbox{ \hskip70pt \epsfxsize=250pt \hbox{\epsffile{zm.1.eps} } }   
     }

\vbox{ \hbox{ \hskip70pt \epsfxsize=250pt \hbox{\epsffile{zm.2.eps} } }   
     }

\label{ZMs}

\end{figure}


\newpage


\begin{figure}[!h]

\caption{Action density for the transformed gauge configuration in the $(z_0=1/2,z_2=1/2)$ plane. The normalization
is the same as in the original lattice configuration. The center in this plane has been displaced to the
origin of coordinates for clarity, and
an interpolating procedure has been used to obtain a smooth surface. 
}

\vbox{ \hbox{ \hskip40pt \epsfxsize=300pt \hbox{\epsffile{z_2_z_4_plane.eps} } }   
     }

\label{zPlane}

\end{figure}


\newpage


\begin{figure}[!h]

\caption{Comparison of action densities for the Nahm transformed field (dots) and the original lattice field (solid line)
along a line with two absolute maxima. The line has been extracted from a fit to the lattice points densities. The
center has been displaced to the origin of coordinates for clarity.}

\vbox{ \hbox{ \hskip75pt \epsfxsize=300pt \hbox{\epsffile{doble_maximo.eps} } }   
     }

\label{Compara}

\end{figure}



\newpage


\begin{table}[!h]

\label{tb:TabSd}


\begin{center}
\caption{Values for the transformed electric and magnetic gauge fields in some selected points of the
$z$-space are given. The number of significative figures is dictated by the deviations from self-duality
appearing in the original lattice configuration. The first point corresponds to an absolute maximum;
the second, to the maximum in a plane in which $E_1=E_3=0$; the third, to a point with no special
character.}

\vspace{1cm}

\input{tabla_campo.tex}

\end{center}

\end{table}



\begin{table}[!h]

\label{tb:TabCorr}


\begin{center}
\caption{Values for the traces of products of electric field components for corresponding $x_0$ and $z_0$ points:
$x_0=(0.312,0.584,0.596,0.766)$, $z_0=(0.584,0.688,0.766,0.404)$.
$x_0$ corresponds to a lattice point, selected such that clear hierarchies among the invariants are established.}

\vspace{1cm}


\input{tabla_corr.tex}

\end{center}

\end{table}





\end{document}















\section{The Nahm transformation}
Let us consider a  4 dimensional torus of size $l_0 \times l_1 \times l_2
\times l_3$, and let $\hat{l}_{\mu}$ represent the vector $(0,\ldots,
l_{\mu},\ldots,0)$, whose only non-zero component is the  $\mu^{th}$ component.
Now consider a self-dual gauge field configuration  $A_{\mu}(x)$ 
defined on this torus. It satisfies:
\be
\label{tbc}
A_{\nu}(x + \hat{l}_{\mu}) = \Omega_{\mu}(x)\, A_{\nu}(x)\, \Omega^+_{\mu}(x) +
\imath\ \Omega_{\mu}(x)\,  \partial_{\nu} \Omega^+_{\mu}(x)\ \ ,
\ee
where $\Omega_{\mu}(x)$ are the twist matrices. For $SU(N)$, these matrices must fulfill the consistency
condition:
\be
\Omega_{\mu}(x+\nu) \Omega_{\nu}(x) = Z_{\mu \nu}\ \Omega_{\nu}(x+\mu) \Omega_{\mu}(x)\ \ .
\ee
When fields transforming in the fundamental representation of the gauge group appear, 
all the constants $Z_{\mu \nu}$ must be equal to 1; otherwise, they are in general
elements of the center of the group $\ZN$, which can be
parametrized as $Z_{\mu \nu}=\exp(2 \pi \imath n_{\mu \nu}/N)$. The twist tensor $n_{\mu \nu}$
is antisymmetric, and its elements are integers defined modulo $N$.

Let $Q$ stand for the
topological charge of the gauge field configuration. As is well-known,
the Atiyah-Singer index theorem implies that the difference between the
number of positive chirality and negative chirality solutions of the Dirac
equation for fermion fields transforming in the fundamental representation of
the gauge group is given by $Q$. Let us assume that for our gauge configuration
there are no  negative chirality
solutions.  Then,  there are exactly $Q$ positive chirality solutions, which we
will label $\Psi^{\alpha}(x)$, with $\alpha=1,\ldots ,Q$. They  satisfy:
\be
\hat{\bar{D}}\Psi^{\alpha}(x)=0\ \ ,
\ee
where $\hat{\bar{D}} \equiv D_{\mu} \bar{\Gamma}_{\mu}$ is the positive chirality Weyl
operator and $D_{\mu} = \partial_{\mu} - \imath A_{\mu}$. In the Weyl basis we have:
\be
\Dsl = \pmatrix{ 0 & \hat{D} \cr  \hat{\bar{D}} & 0 }
\hspace{0.5 cm}
\gamma_5 = \pmatrix{ 1 & 0 \cr  0 & -1 }  \quad ,
\ee
where $\Gamma_{\mu}=(I, -\imath \vec{\sigma})$ and
$\bar{\Gamma}_{\mu}=\Gamma^+_{\mu}$. Furthermore, the solutions satisfy the
following boundary conditions:
\be
\label{bc}
\Psi^{\alpha}(x + \hat{l}_{\mu}) =  \Omega_{\mu}(x) \Psi^{\alpha}(x)\ \ .
\ee

Now  consider the family of gauge fields:
\be
A_{\mu}(x,z)=A_{\mu}(x)+ 2 \pi z_{\mu}\, I ,
\ee
where $z_{\mu}$ are 4 real numbers. For all $z$, the field strength $F_{\mu
\nu}$ is the same, and hence they are all self-dual and have the same
topological charge. Therefore, we obtain a family $\Psi^{\alpha}(x,z)$ 
of positive chirality
solutions of the Dirac equation:
\be
\label{equD}
\hat{\bar{D}}_z \Psi^{\alpha}(x,z) = (\hat{\bar{D}}- 2 \pi \imath \hat{\bar{z}}\,
I) \Psi^{\alpha}(x,z) = 0 \ \ ,
\ee
satisfying the
boundary condition Eq.~\ref{bc} and normalized as follows:
\be
\label{normalization}
\int d^4x\ \left(\Psi^{\alpha}(x,z)\right)^+\,  \Psi^{ \beta}(x,z) = \delta_{\alpha \beta}
\ \ .
\ee
Now notice that  $\exp(- 2 \pi \imath \tilde{z}_{\mu} x_{\mu})\,
\Psi^{\alpha}(x,z+\tilde{z})$ satisfies the same equation than
$\Psi^{\alpha}(x,z)$. However, in general, the boundary conditions are
different, since 
 the right-hand side of Eq.~\ref{bc} gets multiplied by $\exp (-  2 \pi
\imath \tilde{z}_{\mu} l_{\mu})$. This  new factor becomes simply unity if
$\tilde{z}_{\mu}$ is an integer multiple of $1/l_{\mu}$. Hence, defining  the
vector $\hat{\tilde{l}}_{\mu} = (0,\ldots, \frac{1}{l_{\mu}},\ldots,0)$ we can
write:
\be
\label{tbcfors}
\Psi^{\alpha}(x,z+\hat{\tilde{l}}_{\mu}) = \Psi^{\beta}(x,z) (\Omega'^{+}_{\mu}(z))_{\beta \alpha} \exp(2 \pi \imath x_{\mu}/l_{\mu})\ .
\ee
This is so because any solution can be written as a linear combination of the
basis  functions $\Psi^{\beta}(x,z)$. The coefficients
$(\Omega'^{+}_{\mu}(z))_{\beta \alpha}$ cannot in general be chosen equal to
1, if we insist in $\Psi^{\beta}(x,z)$ being continuous in $z$.

Now let us construct the Nahm transform of the gauge field $A_{\mu}(x)$. 
It is given by:
\be
\label{NT}
(\hat{A}_{\mu}(z))_{\alpha \beta} = \imath\, \int d^4x\ \left(\Psi^{
\alpha}(x,z)\right)^+\ \frac{\partial}{\partial z_{\mu}}\Psi^{\beta}(x,z) \ \  .  
\ee
This is a $U(Q)$ gauge field defined on the dual torus (of size
$\frac{1}{l_0} \times \frac{1}{l_1} \times \frac{1}{l_2} \times
\frac{1}{l_3}$). Using Eq.~\ref{tbcfors} one finds that $\hat{A}_{\mu}(z)$
satisfies a relation analogous to Eq.~\ref{tbc} (exchanging the roles of x and z)
in terms of $\Omega'_{\mu}$. 
Now, one can in terms of this field construct the
field strength tensor $\hat{F}_{\mu \nu}$. The Nahm-transformed gauge 
field has the following properties:
\begin{itemize}
\item $\hat{F}_{\mu \nu}$ is again self-dual.
\item The first and second Chern classes and the ranks for the original
and transformed gauge fields are related through:
{\setlength \arraycolsep{2pt}
\bea
rk(\hat{F}) &=& c_2(F)-\frac{1}{2}c_1^2(F) \\
c_1(\hat{F}) &=& -\int_{T^4}(dz_{\mu} \wedge dx_{\mu})^2 \wedge c_1(F) \\
c_2(\hat{F}) &=& rk(F)+\frac{1}{2}c_1^2(F) \quad .
\eea
}
Thus, in particular,  one sees that the roles of the rank of the group 
and the topological charge are exchanged by the Nahm transformation when
the first Chern class vanishes.
\item From the previous statement it follows that if we start with $SU(N)$
gauge fields (with no twist $n_{\mu \nu}= 0 \bmod N$), then the Nahm
transform is in  $SU(Q)$.
\item The Nahm transformation  is an involution: if we apply it twice we go back to 
the original gauge field. Thus, it can be considered a duality transformation. 
\item If we start with a family of gauge fields depending on some 
parameters, then the Nahm transformation  generates a new set of self-dual gauge 
fields depending on those parameters. Hence, we have induced a mapping between 
the moduli spaces of the gauge field and its transform. 
This mapping is an isometry with respect to the natural metric of these 
moduli spaces. 
\end{itemize}
For a proof of these properties, see Ref.~\cite{pbpvb}.

Now, let us consider the vicinity of a point $z$ in the dual torus. 
We can make a Taylor expansion of the positive chirality solutions
of the Dirac equation in the vicinity of this point:
\be
\Psi^{\alpha}(x,z+\Delta z)= \Psi^{\alpha}(x,z) + \Delta z_{\mu}
\Psi_{\mu}^{\alpha}(x,z)+ \Delta z_{\mu} \Delta z_{\nu}
\Psi_{\mu \nu}^{\alpha}(x,z)+ \ldots
\ee
By plugging this equation into the Dirac equation for $z+\Delta z$ and
equating powers of $\Delta z_{\mu}$ on both sides, we obtain for the first two orders:
{\setlength\arraycolsep{2pt}
\begin{eqnarray}
\label{normaleq}
\hat{\bar{D}}_z \Psi^{\alpha}(x,z) & = & 0 \\
\label{main}
\hat{\bar{D}}_z \Psi_{\mu}^{\alpha}(x,z) & = & 2 \pi \imath\,
\bar{\Gamma}_{\mu} \Psi^{\alpha}(x,z) 
\end{eqnarray}}
Now, in terms of these functions, and defining:
{\setlength\arraycolsep{2pt}
\bea
P^{\alpha \beta}_{\mu}(z) & \equiv & \langle \Psi^{\alpha} | \Psi_{\mu}^{\beta} \rangle 
\equiv \int d^4x \left(\Psi^{\alpha}(x,z)\right)^+ \Psi_{\mu}^{\beta}(x,z) \nonumber \\
Q^{\alpha \beta}_{\mu \nu}(z) & \equiv & \langle \Psi_{\mu}^{\alpha} | \Psi_{\nu}^{\beta} \rangle 
\label{prodesc}
\eea}
one can write the vector potential and
the field-strength tensor coming out of the Nahm transformation as follows:
{\setlength\arraycolsep{2pt}
\begin{eqnarray}
\hat{A}_{\mu}(z) & = & \imath P_{\mu}(z) \\ 
\hat{F}_{\mu \nu}(z) & = & \imath \left(Q_{\mu \nu}(z)- Q_{\mu \nu}^{+}(z)+\lbrack P_{\mu}(z),P_{\nu}(z) \rbrack \right)
\label{fmunu}
\end{eqnarray}}
The normalization conditions  imply that $\hat{A}_{\mu}(z)$ is hermitian.
Henceforth, to obtain  the Nahm-transformed gauge field at one point $z$ of the
dual torus, one has simply to solve Eqs.~\ref{normaleq}-\ref{main}.
None of the two equations has a unique solution. Choosing a  solution
within both sets of equations amounts  to a choice of gauge for the 
Nahm-transformed fields. More specifically, any normalized set of solutions  of
Eq.~\ref{normaleq} is  related by a unitary transformation
$\Omega'_{\alpha \beta}(z)$ to any other. Furthermore, once this
choice is made, it can be easily shown that selecting a particular solution
$\Psi_{\mu}^{\alpha}(x,z)$ of  Eq.~\ref{main} amounts to the choice of a
gauge in the neighbourhood of $z$. One can obtain any  solution of
Eq.~\ref{main} by adding to $\Psi_{\mu}^{\alpha}(x,z)$ a general solution of
 Eq.~\ref{normaleq}:  $\Psi^{\alpha}_{\mu}(x,z)+ S_{\mu}^{\beta \alpha}\
\Psi^{\beta}(x,z)$, where $S_{\mu}$ must be antihermitian due to the
normalization conditions. This produces a change in the vector potential
$\hat{A}_{\mu} \longrightarrow \hat{A}_{\mu} + \imath \, S_{\mu}$, but as
can be readily verified from the equations above, $\hat{F}_{\mu \nu}$ is left invariant.
It is in principle possible to impose some set of conditions on the
solutions in order to select a particular gauge for $\hat{A}_{\mu}$.
However, in this paper we will 
concentrate on gauge invariant quantities and, hence, any solution will do. 

In the following sections we will describe how we have been able to 
numerically construct the Nahm transform of a given self-dual gauge field
configuration on the torus by  finding the solutions of
Eqs.~\ref{normaleq}, \ref{main} using the lattice formulation of the theory. 
In the next section we will describe the numerical technique and in the
following we will apply our construction to some explicit examples. 




\begin{tabular}{|c|c|c|}
\hline 
{\em z}--space point &  $E_1^a,E_2^a,E_3^a$ & $B_1^a,B_2^a,B_3^a$ \\   \hline \hline
 
$(1/2,1/4,1/2,1/4)$ & 
$(-0.049,0.012,-1.262)$ & 
$(-0.049,0.012,-1.262)$ \\
 &
$(-1.204,0.444,0.051)$ & 
$(-1.212,0.447,0.051)$  \\
 &
$(-0.436,-1.182,0.005)$ & 
$(-0.436,-1,182,0.005)$ \\

\hline

$(1/2,1/2,1/2,1/2)$ & 
$(0.000,0.000,0.000)$ & 
$(0.000,0.000,0.000)$  \\
 &
$(0.218,-0.705,0.000)$ & 
$(0.225,-0.725,0.000)$  \\
 &
$(0.000,0.000,0.000)$ & 
$(0.000,0.000,0.000)$  \\

\hline

%$(1/2,3/8,1/2,1/4)$ & 
%$(-0.082,0.002,-1.088)$ & 
%$(-0.082,0.002,-1.088)$  \\
% &
%$(-1.129,0.179,0.086)$ & 
%$(-1.140,0.180,0.087)$  \\
% &
%$(-0.169,-1.075,0.011)$ & 
%$(-0.169,-1.075,0.011)$  \\


$(3/4,1/2,1/2,2/3)$ & 
$(0.031,-0.001,-0.262)$ & 
$(0.032,-0.002,-0.262)$  \\
 &
$(-0.311,-0.166,-0.038)$ & 
$(-0.316,-0.169,-0.037)$  \\
 &
$(0.076,-0.287,0.010)$ & 
$(0.076,-0.287,0.010)$  \\





\hline


\end{tabular}




\begin{tabular}{|c|c|c|}
\hline 
 &  $x_0$ & $z_0$ \\   \hline \hline
$Tr(E_1(x)\,E_1(x))$ & 0.418  & 0.460  \\
$Tr(E_2(x)\,E_2(x))$ & 0.465  & 0.488  \\
$Tr(E_3(x)\,E_3(x))$ & 0.383  & 0.412  \\
$Tr(E_1(x)\,E_2(x))$ & 0.033  & 0.031  \\
$Tr(E_1(x)\,E_3(x))$ & -0.027  & -0.027  \\
$Tr(E_2(x)\,E_3(x))$ & 0.024  &  0.025 \\

\hline

\end{tabular}






