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%  Author Package
%% Translation via Omnimark script a2l, April 29, 1996 (all in one day!)
%\controldates{16-JUL-1996,16-JUL-1996,16-JUL-1996,23-JUL-1996}
\documentclass{era-l}
\usepackage{graphicx}
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%\issueinfo{2}{1}{}{1996}

\hyphenation{para-me-trised para-me-trise}

%% Declarations:

\theoremstyle{plain}
\newtheorem*{theorem1}{Theorem {1.6} \cite{Po,Uhl}}
\newtheorem*{theorem2}{Theorem (The Equivalence)}
\newtheorem*{theorem3}{Corollary (Wood's Conjecture)}
\newtheorem*{theorem4}{Theorem (Monad representation)}
\newtheorem*{theorem5}{Theorem (Closed Form)}
\newtheorem*{theorem6}{Conjecture (Uniton Number)}
\newtheorem*{theorem7}{Corollary (Finite Gap)}
\newtheorem*{theorem8}{Proposition (Moduli Topology)}

\theoremstyle{remark}
\newtheorem*{remark0}{Remark on metrics}
\newtheorem*{remark1}{Remark}
\newtheorem*{remark2}{Remark}

\theoremstyle{definition}
\newtheorem*{definition1}{Definition (Uniton Bundles)}

\renewcommand{\qed}{}

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\newcommand{\T}{\widetilde{T\mathbb{P}}{}^1}
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\begin{document}

\title{Unitons and their moduli }
\author{Christopher Kumar Anand}
%\address{University of Warwick}
\address{Mathematics Research Centre, 
  University of Warwick, Coventry CV4 7AL, UK}
\email{anand@maths.warwick.ac.uk}
\date{September 19, 1995}
\thanks{Research supported by NSERC and FCAR scholarships.}
\keywords{Uniton, harmonic map, chiral field, sigma model}
\subjclass{Primary 58E20, 58D27, 58G37}
\commby{Eugenio Calabi}
\begin{abstract}We sketch the proof that unitons (harmonic spheres in 
$\operatorname{U}(N)$) 
correspond to holomorphic `uniton bundles', and that these admit
monad representations analogous to Donaldson's representation of
instanton bundles.  We also give a closed-form expression for the 
unitons involving only matrix operations, a finite-gap result 
(two-unitons have energy $\ge 4$), computations of fundamental 
groups of
energy $\le 4$ components, new methods of proving discreteness of the
energy spectrum and of Wood's Rationality Conjecture, a discussion of
the maps into complex Grassmannians and some open problems.
\end{abstract}
\maketitle

\section*{1. Harmonic maps}

Harmonic maps between Riemannian manifolds $M$ and $N$ 
are critical values of an energy functional
\begin{equation*}\text{energy}(S:M\to N)=\frac{1}{2}\int _{M}|dS|^{2}.\end{equation*}
They  generalise geodesics, which are the local 
minima of the length functional.  (Length is energy with respect to the 
induced metric.)
  Harmonic maps of surfaces are likewise branched minimal immersions 
(locally least-area surfaces with $z\mapsto z^{k}$ type singularities).

In the case of surfaces in a matrix group, with 
the standard (bi-invariant) metric, the energy takes the form
\begin{equation*}\text{energy}(S)=\frac{1}{2}\int _{\mathbb{R}^{2}}
\left (|S^{-1}\dd {x} S|^{2}+|S^{-1}\dd {y} S|^{2}\right ) dx\wedge dy.\tag{{1.1}}\end{equation*}

{\em Unitons} are harmonic maps  $S:{\mathbb{S}}^{2}\to \operatorname{U}(N)$.  
Some authors call them multi-unitons.  Since the energy is 
conformally invariant in the case of surfaces, it is natural to stick to 
coordinates $x$ and $y$ on $\mathbb{R}^{2}$ and derive the Euler-Lagrange 
 equations,
\begin{equation*}\dd {x} (S^{-1}\dd {x} S)+\dd {y} (S^{-1}\dd {y} S)=0,\tag{{1.2}}\end{equation*}
which we will refer to as the uniton equations.
{}From \cite[Theorem 3.6]{{SaUhl}}, we know that  
harmonic maps from $\mathbb{R}^{2}\to \operatorname{U}(N)$ extend to 
${\mathbb{S}}^{2}$ iff they 
have 
finite 
energy, and that such maps are always smooth.
So working in terms of coordinates 
$x$ and $y$ on $\mathbb{R}^{2}$ or $z\in \mathbb{C}$ poses no real limitation.

\begin{remark0} Since harmonic maps $\mathbb{S}^{2}\to\operatorname{U}(N)$
have constant determinant, and $\operatorname{SU}(N)$ has an essentially
bi-invariant metric, nothing is gained by considering other bi-invariant
metrics on $\operatorname{U}(N)$. On the other hand, bi-invariance of the
metric is essential in what follows. \end{remark0}

\subsection*{{1.3} Based unitons} Unitons are determined by the pullback of 
the Maurer-Cartan form on 
$\operatorname{U}(N)$, 
\begin{equation*}
A\deq \frac{1}{2}S^{-1}d S=A_{z}dz+A_{\bar {z}} d\bar {z}\tag{{1.4}}
\end{equation*}
and a choice of a basepoint, $S(\infty )\in \operatorname{U}(N)$, 
as we can see by thinking 
of $d+2A$ as a flat connection and $S$ as a gauge transformation.
We thus have a decomposition of the
 $\operatorname{U}(N)$-unitons,
\begin{equation*}\operatorname{Harm}(\mathbb{S}^{2}, 
 \operatorname{U}(N))=\operatorname{U}(N)\times \operatorname{Harm}^{*}(\mathbb{S}^{2}, 
 \operatorname{U}(N)),\tag{{1.5}}\end{equation*}
into initial values $S_{0}\in \operatorname{U}(N)$ and {\em based} unitons 
(which we take to have $S(\infty )=\one $).

Since 
$\operatorname{Harm}^{*}(\mathbb{S}^{2}, \operatorname{U}(N))=\left \{A=dS:
S\in \operatorname{Harm}(\mathbb{S}^{2},\operatorname{U}(N))\right \}$,
 it is useful to 
have equations for $A$ as well.
Two $\operatorname{U}(N)$-valued maps $A_{x},A_{y}$ come from a map 
$S:\mathbb{R}^{2}\to \operatorname{U}(N)$ in this 
way iff $d+2A$ has zero curvature.
They come from a {\em harmonic\/} map if, in addition, $d^{*}A=0$. 
This has a zero-curvature formulation:
\begin{theorem1}  
Let $\Omega \subset \mathbb{S}^{2}$ 
be a simply connected neighbourhood and 
$A:\Omega \to T^{*}(\Omega )\otimes \operatorname{U}(N)$.
Then $2A=S^{-1}d S$, with $S$ harmonic iff the curvature of the
connection \begin{equation*}\mathcal{D}_{\lambda }=(\dd {\bar {z}}+(1+\lambda )A_{\bar {z}},
\dd {z} +(1+\lambda ^{-1})A_{z})\tag{{1.7}}\end{equation*}
vanishes for all $\lambda \in \mathbb{C}^{*}$.\end{theorem1}
After Uhlenbeck, we will write 
$E_{\lambda }:\Omega \times \mathbb{C}^{*}\to \operatorname{Gl}(N)$ 
for flat sections of $\mathcal{D}_{\lambda }$
and call them extended solutions.
In the setting of \cite{BuRa}, they are the twistor lifts associated to the
harmonic maps.

\section*{2. Holomorphic methods }

In investigating unitons and their moduli, we will be using an  idea 
that goes back to Weierstra\ss ' description of minimal surfaces 
in $\mathbb{R}^{3}$ via a pair of holomorphic functions and their 
derivatives.  His starting point was Enneper's closed form expression for 
these minimal surfaces in terms of analytic functions and quadratures.

Uhlenbeck proved \cite{Uhl}  that  
all unitons can  be constructed via quadratures (i.e. B\"{a}cklund 
transformations, written explicitly as quadratures by Wood \cite{Wo}),
and our results giving an equivalence between unitons and holomorphic
bundles are analogous to 
Weierstra\ss ' description, but the uniton story is more complicated than 
the history of minimal surfaces in $\mathbb{R}^{3}$.

The first\footnote{The referee has pointed out that Bochner did work in this
directon; see Trans. Amer. Math. Soc. {\bf 47} (1940), 146--164.} 
modern use of complex methods, and the starting point for 
twistor methods in harmonic map theory, is Calabi's association to 
minimal immersions 
$\mathbb{S}^{2}\to \mathbb{S}^{2n}=\operatorname{SO}(2n+1)/\operatorname{SO}(2n)$
 of a 
holomorphic curve in
$\operatorname{SO}(2n+1)/\operatorname{U}(n)$ 
(a K\"{a}hler 
manifold).  Analysis of the holomorphic curves led him to a 
classification of the minimal spheres and the discovery that the area 
spectrum was discrete.  Calabi's work was followed by work of physicists 
and mathematicians on maps into complex projective spaces and 
Grassmannians, and then Uhlenbeck's work on maps into $\operatorname{U}(N)$.  
Since 
Grassmannians are symmetric spaces in $\operatorname{U}(N)$ (they are 
the components of $\{S\in \operatorname{U}(N):S^{2}=\one \}$)
 and are totally geodesically embedded via the Cartan embedding 
$(V\subset \mathbb{C}^{N}\mapsto \pi _{V}-\pi _{V}^{\perp }; 
\pi _{V}=\text{projection onto } V),$ Uhlenbeck's results subsume the 
previous work.  Finally, symmetric spaces contain 
$\operatorname{U}(N)$ as a symmetric space of 
$\operatorname{U}(N)\times \operatorname{U}(N)$ and unitons
 can be treated in 
this most general context (see \cite{BuRa}), although we will not do so
now.

\section*{3. Uniton bundles }

The present work rests on the observation of Ward and Uhlenbeck  
that the harmonic map equations for maps 
$\mathbb{R}^{2}\to \operatorname{U}(N)$ are dimensionally reduced Bogomolny 
equations on $\mathbb{R}^{2+1}$.  Ward (\cite{Wa}) 
additionally observed that Hitchin's 
(\cite{Hi}) equivalence of Bogomolny 
solutions on $\mathbb{R}^{3}$ and holomorphic bundles on 
$T\mathbb{P}^{1}=T\mathbb{S}^{2}=\{\text{oriented lines in } \mathbb{R}^{3}\}$ 
applied to maps $\mathbb{R}^{2}\to \operatorname{SU}(N)$, and showed how this 
construction naturally `compactified' for maps 
$\mathbb{S}\to \operatorname{SU}(2)$.  
In \cite{An2} we show that this equivalence can be `compactified', 
identifying finite-energy harmonic maps 
$\mathbb{S}^{2}\to \operatorname{U}(N)$ with bundles which are the 
restriction of bundles on the fibrewise compactification of 
$T\mathbb{P}^{1}$, a Hirzebruch surface.

The uniton bundles are bundles on $\widetilde{T\mathbb{P}}{}^{1} 
\deq \mathbb{P}{}(\mathcal{O}{}\oplus \mathcal{O}{}(2))$,
the fibrewise compactification of the tangent bundle $T\mathbb{P}{}^{1}$ of 
the 
{\em complex} 
projective line.

Let $(\lambda ,\eta )$  and 
$(\hat {\lambda }=1/\lambda ,\hat {\eta }=\eta /\lambda ^{2})$ be coordinates 
on $T\mathbb{P}^{1}\cong \mathcal{O}_{\mathbb{P}^{1}} (2)$, 
where $\lambda $ is the usual coordinate on $\mathbb{P}^{1}$ and $\eta $ 
is the coordinate associated to $d/d\lambda $.
Meromorphic sections ($s$) of $T\mathbb{P}^{1}$ give all the holomorphic 
sections of $\widetilde {T\mathbb{P}}{}^{1}$ ($[s,1]$ in projective 
coordinates on 
$\widetilde {T\mathbb{P}}{}^{1})$, save one.  We fix notation for the lines on 
$\T $:
\begin{equation*}\begin{split}
P_{\lambda }&=\pi ^{-1}(\lambda )=\text{a pfibre 
(silent p)},\\ {G}_{a,b,c}&=\left \{(\lambda ,[a-2b\lambda -c\lambda ^{2},2]\right \},\text{ for } 
(a,b,c)\in \mathbb{C}^{3}\cong H^{0}(\mathbb{P}^{1},T\mathbb{P}^{1}),\\
{G}_{\infty }&=\{(\lambda ,[1,0])\}=\text{infinity section of 
$\widetilde {T\mathbb{P}}{}^{1}$}.\end{split}\tag{{3.1}}\end{equation*}

To encode unitarity, we need the real structure
\begin{equation*}\sigma ^{*}(\lambda ,\eta )=(1/\bar {\lambda } ,-\bar {\lambda } 
^{-2}\bar {\eta })\tag{{3.2}}\end{equation*}
acting on $\T $ and compatibly on 
$\mathbb{C}^{3}\cong H^{0}(\mathbb{P}^{1},\mathcal{O}(2))$, the space of finite 
sections:
\begin{equation*}\sigma ^{*}(a,b,c)=(\bar c,-\bar b,\bar a).\tag{{3.3}}\end{equation*}
We similarly define time translation
\begin{align*}
\delta _{t}:\ & (\lambda ,\eta )\mapsto (\lambda ,\eta -2t\lambda ),\\ & 
(a,b,c)\mapsto (a,b+t,c).\tag{{3.4}}\end{align*}

\begin{definition1} \ 
A rank $N$, or $\operatorname{U}(N)$, {\em uniton bundle}, 
$\mathcal{V}$, 
is a holomorphic rank $N$ bundle on $\widetilde {T\mathbb{P}}{}^{1}$ which is 
a) trivial when restricted 
to the following curves 
in $\widetilde {T\mathbb{P}}{}^{1}$:
\begin{enumerate}
\item the section at infinity ($G_{\infty }$),
\item nonpolar fibres (i.e.\ $P_{\lambda }$ for $\lambda \in \mathbb{C}^{*}$),
\item real sections of $T\mathbb{P}^{1}$ 
(sections invariant under $\sigma $, $G_{(z,it,\bar {z})}$);
\end{enumerate}b) is equipped with bundle lifts
\begin{equation*}\begin{CD}
\mathcal{V}@>\tilde \delta _{t}>>\mathcal{V}\\@VVV 
@VVV\\\widetilde {T\mathbb{P}}{}^{1}@>\delta _{t}>>\widetilde {T\mathbb{P}}{}^{1}
\end{CD}\quad \text{and}\quad \begin{CD}
\mathcal{V}@>\tilde \sigma >>\mathcal{V}^{*}\\@VVV @VVV\\
\widetilde {T\mathbb{P}}{}^{1}
@>\sigma >>\widetilde {T\mathbb{P}}{}^{1}\end{CD}\end{equation*}
\begin{enumerate}
\item $\tilde \delta _{t}$ a one-parameter family of holomorphic 
transformations fixing $\mathcal{V}$ above the section at infinity and
lifting $\delta _{t}$,  and 
\item $\tilde \sigma $ a norm-preserving, antiholomorphic lift of
 $\sigma $ such that the induced 
hermitian metric on $\mathcal{V}$ restricted to a fixed point of $\sigma $ 
is positive definite.  Equivalently, the induced lift to  the 
principal bundle of frames  acts on fibres of fixed points of $\sigma $ by 
$X\mapsto X^{*-1}$; and\end{enumerate}c) has a framing, 
$\phi \in H^{0}(P_{-1},Fr(\mathcal{V}))$, of the bundle 
$\mathcal{V}$ restricted 
to the fibre $P_{-1}=\{\lambda =-1\}\subset \widetilde {T\mathbb{P}}{}^{1}$ 
such that $\tilde \sigma (\phi )=\phi $.
\end{definition1}
\begin{remark1}  An argument similar to the one in 
\cite{ADHM} shows that real triviality is implied by the other bundle 
properties.  The proof depends essentially on the monad construction
in \S 5.\end{remark1}
\begin{theorem2} \ 
The space of based unitons, 
$\operatorname{Harm}^{*}(\mathbb{S}^{2},\operatorname{U}(N))$, is 
isomorphic to the 
space of rank $N$ uniton bundles, with energy corresponding to the second 
Chern class.\end{theorem2}


\begin{proof}[Sketch of Proof]
The elements of Hitchin's bundles are solutions of a linear differential 
operator on the corresponding line of $\mathbb{R}^{3}$.  
A $\bar {\partial }$-operator 
defines a Koszul-Malgrange holomorphic structure on the bundle.  
In adapting the construction one must show that a singularity in this 
$\bar {\partial }$-operator along $G_{\infty }$ can be removed by a 
continuous gauge change.  Away from $\lambda =0,\infty $, any extended 
solution lifts to give a trivialisation of 
$\mathcal{V}|_{\widetilde {T\mathbb{C}}^{*}}$, so the analytic difficulties are only 
at two points.

Going the other way, Hitchin reconstitutes the Bogomolny system via 
`null' connections on lines in $\mathbb{R}^{3}$ corresponding to the set of 
sections of $T\mathbb{P}^{1}$ mutually tangent at a fixed point.  
(Evaluation at the point gives a flat trivialisation along the line.)  
The extension involves
\begin{enumerate}
\item pushing the holomorphic bundle down to the one-point 
compactification of $T\mathbb{P}^{1}$,
\item realising this as a quadric in $\mathbb{P}^{3}$,
\item showing that pulling and pushing (taking a direct image of) the bundle 
gives a bundle on $\mathbb{R}P^{2}\times \mathbb{R}\subset 
\mathbb{P}^{3\text{dual}}$, and
\item showing that the reconstituted solution on $\mathbb{R}P^{2}\times \mathbb{R}$ gives a harmonic map from $\mathbb{S}^{2}$ (this involves 
calculating the energy of the map on the big open cell 
$\mathbb{R}^{2}\hookrightarrow \mathbb{R}P^{2}$ in well-chosen affine 
coordinates on $\mathbb{R}P^{2}$, and using Sacks' and Uhlenbeck's result on 
extendibility of finite energy maps on $\mathbb{R}^{2}$).
\end{enumerate}Finally, one must check that the extra conditions on the bundle are 
necessary and sufficient.  We explain the energy calculation below.
\end{proof}


\section*{4. Monodromy construction }

The above constructions depend heavily on the twistor correspondence
\begin{equation*}
\mathbb{R}^{3} %@<<<
\leftarrow\mathbb{S}^{2}\times 
\mathbb{R}^{3}\cong \mathbb{R}\oplus T\mathbb{S}^{2}
%@>>>
\rightarrow T\mathbb{P}^{1}\tag{{4.1}}\end{equation*}
which relates points on $\mathbb{R}^{3}$ to sections of $T\mathbb{P}^{1}$ and 
points of $T\mathbb{P}^{1}$ to null lines of 
$\mathbb{C}^{3}=\mathbb{R}^{3}_{\mathbb{C}}$.  We use it to construct flat 
sections or 
trivialisations of the bundles on $\mathbb{R}^{3}$ and $T\mathbb{P}^{1}$, and 
their compact counterparts.  

The most important example of this is the $G_{\infty }$-trivialisation of 
$\mathcal{V}|_{\widetilde {T\mathbb{C}}^{*}}$, which can be intrinsically 
defined using the trivialisation of $\mathcal{V}|_{G_{\infty }}$.  (Since 
$\mathcal{V}|_{G_{\infty }}$ is holomorphically trivial, 
$\mathcal{V}|_{pt}\cong H^{0}(G_{\infty },\mathcal{V})$, giving a notion 
of parallel translation along holomorphically trivial curves in $\T $, 
$G_{\infty }$, $G_{\text{real}}$, $P_{\lambda }:\lambda \in \mathbb{C}^{*}$ for 
example.)

The uniton bundle $\mathcal{V} \to \widetilde {T\mathbb{P}}{}^{1}$ is constructed 
generically as the kernel of a differential operator on $\mathbb{R}^{3} 
\times \mathbb{S}^{2}$;  $\mathcal{D}_{\lambda }$ is the projection of 
that operator from 
 $\mathbb{R}^{2}\times \mathbb{C}^{*}$.  The extended solution spans the kernel
of  $\mathcal{D}_{\lambda }$.  Pulling 
$E_{\lambda }$ back to $\mathbb{R}^{3}\times \mathbb{C}^{*}$ gives a trivialisation 
of $\mathcal{V}$ restricted to the open set $\{\lambda \in \mathbb{C}^{*}\}$. 
Intrinsically, the $G_{\infty }$-infinity trivialisation is
the trivialisation on each fibre $P_{\lambda }$, 
$\lambda \in \mathbb{C}^{*}$, which agrees with a trivialisation of $\mathcal{V} 
|_{G_{\infty }}$.  We `evaluate' this trivialisation and get the extended 
solution in terms of the constant trivialisation of $\mathbb{C}^{N}\times \mathbb{S}^{2}$, by comparing it to the trivialisation of $\mathcal{V}$ on the real 
sections.  We obtain
\begin{equation*}E_{\lambda }(z,\bar {z})\in \operatorname{Hom}(\mathbb{C}^{N}) = 
\operatorname{Hom}(H^{0}(P_{-1},\mathcal{V}))\tag{{4.2}}\end{equation*}
by composing the cycle of maps
\begin{equation*}\begin{CD}
\mathcal{V}_{\lambda ,\infty } @<\text{restr}<< H^{0}({G}_{\infty },\mathcal{V}) 
@>\text{restr}>> 
  \mathcal{V}_{-1,\infty }\\
@A\text{restr}AA @. @AA\text{restr}A\\
H^{0}(P_{\lambda },\mathcal{V}) @. @. H^{0}(P_{-1},\mathcal{V})\\
@V\text{restr}VV @. @VV\text{restr}V\\
\mathcal{V}_{(\lambda ,z/2-t\lambda -\lambda ^{2}\bar {z}/2)} @<\text{restr}<< 
H^{0}({G}_{(z,\bar {z},t)},\mathcal{V}) 
  @>\text{restr}>> \mathcal{V}_{(-1,z/2-t\lambda -\bar {z}/2)}\end{CD}\tag{{4.3}}\end{equation*}
counterclockwise.  The existence of the bundle isomorphism 
$\tilde {\delta }_{t}$ (time translation) ensures that the result does not 
depend on $t$.  Finiteness, i.e.\ extension to $\mathbb{S}^{2}$, follows from 
the compactness of $\T $.

Since any extended solution encodes the uniton as $S=E_{-1}^{-1}E_{1}$, 
this gives a construction of the uniton which does not involve 
quadratures.  If we define the bundle as a collection of transition 
functions, it reduces to solving the Riemann-Hilbert problem.  This is 
Ward's construction.  (We illustrate 
this for a two-uniton in \cite{An2}.)  While solving the Riemann-Hilbert 
problem is not as routine as taking derivatives, 
the monodromy construction is quite useful as a theoretical tool.  
We can use it to affirm Wood's conjecture:
\begin{theorem3}\ 
If $S:{\mathbb{S}}^{2}\to \operatorname{U}(N)$ is a uniton, 
then the composition with 
$\operatorname{U}(N)\hookrightarrow \operatorname{Gl}(N)$
is rational, i.e.\ the functions in $x$ and $y$ which make up the matrix 
$S\in \operatorname{U}(N)$ are rational.
\end{theorem3}


We use it to show energy$(S)=c_{2}(\mathcal{V})$ (and hence that the 
energy spectrum is discrete) by computing the Chern-Weil integral
for $c_{2}$ for a constructed connection.  The connection is pieced
together with a partition of unity from connections defined in terms
of flat frames:  take the
$G_{\infty }$-trivialisation/extended solution to be flat away from 
$\lambda =0,\infty $, and the standard (nonholomorphic!) frame flat near 
$\lambda =0 \text{ or } \infty $.  The integral is then reduced
to the energy integral by using the zero-curvature 
equations for $E_{\lambda }$ and integration by parts.

Previous proofs of energy discreteness and Wood's conjecture  
\cite{Va1}, \cite{Va2}  are technically quite different
and do not have interpretations in terms of the uniton bundles.

\subsection*{4.4 Grassmannian solutions }The union of Grassmannians in $\mathbb{C}^{N}$ is realised as 
$\{S\in \operatorname{U}(N):S^{2}=\one \}$.  We will call 
$S\in \operatorname{Harm}(\mathbb{S}^{2},\operatorname{U}(N))$ a {\em Grassmannian solution} if there exists a $Q\in \operatorname{U}(N)$ 
such that $(QS)^{2}=\one $.  Let 
$E_{\lambda }$ now be an extended solution for $S$ and assume 
$E_{\lambda }(\infty )$ is conjugate to 
$\begin{pmatrix}\one &\\&-\lambda \one \end{pmatrix}
$.  
Uhlenbeck shows 
$\widetilde {E}_{\lambda }=E_{-\lambda }E_{1}^{-1}$ 
is an extended solution for $S^{-1}$.

Now define an involution by 
\begin{equation*}\mu ^{*}\lambda =-\lambda ,\quad \mu ^{*}\eta =\eta \quad (\text{equivalently }\mu ^{*}z=z),\end{equation*}
and fix generators $\pi _{1}(G_{\infty }\cup P_{-1}\cup G_{z}
 \cup P_{\lambda })$
above which we calculate the monodromies which determine $E_{\lambda }$:
\begin{equation*}
E_{\lambda }=
\begin{matrix}\includegraphics[scale=.68]{era2e-fig-1}\end{matrix}
%\vskip 2in
% \kern -85pt 
\hspace{-85pt}
 \begin{matrix}&&&\\ 
& & &  G_{\infty } \\ 
&&&\\ 
&&&\\
  & & & G_{z} \\ 
&&&\\
  P_{\lambda }& \ \ \ \ \ \ \ \ &P_{-1}&  \end{matrix}
\end{equation*}
Then the formula for $\widetilde {E}_{\lambda }$ has the interpretation
\begin{align*}
E_{-\lambda }E_{1}^{-1}&=
\begin{matrix}\includegraphics[scale=.54]{era2e-fig-1}\end{matrix}
%\vskip \kern -85pt 
\hspace{-75pt}
 \begin{matrix}&&&\\
&&&  \\ 
&&&\\ 
&&&\\   
&&&\\ 
&&&\\
P_{-\lambda }& \ \ \ \ \ \ &P_{-1}&  \end{matrix}
\circ \begin{matrix}\includegraphics[scale=.54]{era2e-fig-3}\end{matrix}
%\vskip 2in
% \kern -85pt 
\hspace{-75pt}
 \begin{matrix}&&&\\  
&&&\\ 
&&&\\ 
&&&\\   
&&&\\ 
&&&\\
P_{1}& \ \ \ \ \ \ \  \ &P_{-1}&  \end{matrix}
=\begin{matrix}\includegraphics[scale=.48]{era2e-fig-2}\end{matrix}
%\vskip 2in
% \kern -120pt 
\hspace{-85pt}
 \begin{matrix}&&&\\  
&&&\\ 
&&&\\ 
&&&\\   
&&&\\
&&& \\
 P_{\lambda }&  \    &P_{-1}& \ &P_{1} \end{matrix}
\\
&=\begin{matrix}\includegraphics[scale=.54]{era2e-fig-1}\end{matrix}
%\vskip 2in
% \kern -85pt 
\hspace{-75pt}
 \begin{matrix}&&&\\  
&&&\\ 
&&&\\ 
&&&\\   
&&&\\ 
&&&\\
P_{-\lambda }& \ \ \ \ \ \ \ &P_{1}&  \end{matrix}
=\mu ^{*}\begin{pmatrix}\includegraphics[scale=.53]{era2e-fig-1}\end{pmatrix}
%\vskip 2in
% \kern -95pt 
\hspace{-80pt}
 \begin{matrix}&&&\\  
&&&\\ 
&&&\\ 
&&&\\   
&&&\\ 
&&&\\
P_{\lambda }& \ \ \ \ \ \ \ &P_{-1}&  \end{matrix}
=\mu ^{*}E_{\lambda }.\end{align*}
Since $\widetilde {E}_{\lambda }$ determines the uniton bundle 
and vice versa,
the uniton bundle for $S^{-1}$ is the $\mu $-pullback of the bundle for 
$S$, up to the choice of framing.  To see the effect on the framing, note 
that when $E_{\lambda }(\infty )= \begin{pmatrix}\one &\\&-\lambda \one \end{pmatrix}
$ (the difference between the chosen framing of 
$\mathcal{V}|_{G_{\infty }}$ and the canonical one) $\mu $ carries a frame 
at $P_{-1}\cap G_{\infty }$ to $\begin{pmatrix}\one &\\&-\one \end{pmatrix}
$ times 
itself (after `transporting' it back using evaluation of 
the canonical frame).

So uniton bundles ($\mathcal{V},\Phi $) correspond to Grassmannian 
solutions iff $\mu $ lifts to $\tilde {\mu }:\mathcal{V}\to \mathcal{V}$ and 
the signature of $\mu ^{*}\phi \phi ^{-1}:\mathbb{C}^{N}\to \mathbb{C}^{N}$ 
determines the component (rank of 
the image Grassmannian).

\section*{5. Horrocks' monad construction }

Atiyah et al (\cite{ADHM}) used Horrocks' monad representation for 
holomorphic bundles on projective spaces  to 
represent Yang-Mills' instantons.  Donaldson \cite{Do} then used the 
monads themselves  to equate the real instanton bundles on 
$\mathbb{P}^{3}$ with bundles on $\mathbb{P}^{2}$.  
See \cite{OSS} for a general account of such representations.  For our 
purposes, it is enough to know that holomorphic bundles on $\mathbb{P}^{2}$ 
which are trivial on generic lines can be represented as
\begin{equation*}\underline{\operatorname{ker}K}/
\underline{\operatorname{im}J},\end{equation*}
where 
\begin{equation*}
\begin{CD}
\mathcal{O}(-1)^{k}@>J>>\mathcal{O}^{2k+N}@>K>>\mathcal{O}(1)^{k},
\end{CD}
\end{equation*}
i.e.\ by two $k\times (2k+N)$ matrices of degree $1$ homogenous polynomials 
in three variables.  By blowing up a point on $\T $ and blowing down two 
exceptional divisors, one arrives at a birational equivalence of $\T $ and 
$\mathbb{P}^{2}$ along which one can push and pull uniton bundles.  A few 
technical difficulties aside, we obtain
\begin{theorem4}
The space of based unitons \linebreak $\operatorname{Harm}^{*}
(\mathbb{S}^{2},\operatorname{U}(N))$ is isomorphic to the set of monad 
data
\begin{gather*}
\gamma , \alpha '_{1}, \delta \in \operatorname{gl}(k/2),
\quad \gamma \text{ nilpotent},\\
a'\in \operatorname{M}_{k/2+N,k/2},\quad b'\in \operatorname{M} _{k/2,k/2+N}\end{gather*}
satisfying
\begin{gather*}
\operatorname{rank} \begin{pmatrix}\gamma \\ \alpha '_{1}+z\\ a' 
\end{pmatrix}
=\operatorname{rank}\begin{pmatrix}\gamma &\alpha '_{1}+z&b'
\end{pmatrix}
=k/2\qquad \forall z\in \mathbb{C},\tag{{\text{nondegeneracy}}} \\
[\gamma ,\alpha '_{1}]+b'a'=0,\tag{{\text{monad equation}}} \end{gather*}
\begin{equation*}
\begin{gathered}[c]
[\delta,\gamma]=0,\\
[\delta ,\alpha _{1}']=\gamma, \\
a'\delta =0,\\
\delta b'=0,\end{gathered}\tag{{\text{time invariance}}}\end{equation*}
quotiented by the action 
of $g\in \operatorname{Gl}(k/2)$:
\begin{gather*}
\gamma \mapsto g\gamma g^{-1},
\quad \alpha '_{1}\mapsto g\alpha '_{1} g^{-1},
\quad \delta \mapsto g\delta g^{-1},\\
a'\mapsto a' g^{-1},\quad b'\mapsto gb'.\end{gather*}
\end{theorem4}


Reinterpreting the monodromy construction of $E_{\lambda }$, we obtain
\begin{theorem5} \ 
Based rank N unitons of energy $8\pi k$ are all of 
 the form
 \begin{equation*}S=\mathbb{I} +a\alpha _{2}^{-1}(\alpha _{1}-2(x+iy\alpha _{2}))^{-1}b.\end{equation*}
 (Multiplication is matrix 
 multiplication.)
 
The primed monad data determine the uniton bundle over a hemisphere.  Reality 
determines it over the other hemisphere.  The unprimed data 
describing the whole bundle and appearing in the closed form are
\begin{equation*}\begin{matrix}\alpha _{1}=2\begin{pmatrix}\alpha _{1}^{\prime *}& \phi _{1}\\
\phi _{2}&\alpha '_{1}
\end{pmatrix},
&
\alpha _{2}=\begin{pmatrix}-\one -2\gamma ^{*}&\\&\one +2\gamma \end{pmatrix},
\\
a=2\begin{pmatrix}b^{\prime *}&a'
\end{pmatrix},
&
b=2\begin{pmatrix}a^{\prime *}\\b' 
\end{pmatrix},
\end{matrix}
\tag{{\text{reality}}}\end{equation*}
where $\phi _{1}$ and $\phi _{2}$ are functions of $\gamma $, $a'$ and 
$b'$ determined by the big monad equation 
$[\alpha _{1},\alpha _{2}]+ba=0$. 
\end{theorem5}
\begin{remark2}  
Because homogenous monads give simple bases of sections above lines in 
$\mathbb{P}^{2}$, computing the monodromy expression for $E_{\lambda }$ is
much easier on $\mathbb{P}^{2}$.  Such bases are very difficult to 
compute for $\T $ monads (see \cite{An1}).\end{remark2}


The monads representing uniton bundles are 
stratified by
the order of the nilpotent matrix $\gamma $.  When it is zero, the 
resulting solutions factor through a holomorphic map, via a translation 
of the Cartan
embedding of a Grassmannian into $\operatorname{U}(N)$.  (Putting
the  $\operatorname{U}(2)$ monads in a normal form  
recovers the poles and principal parts description of rational maps.)
Examination of the corresponding monad data reveals that  
 the $\gamma =0$ maps have $\mathbb{N}$ components given by energy, so the map
\begin{equation*}\operatorname{Harm}(\mathbb{S}^{2},\operatorname{Gr}(\mathbb{C}^{N}))
\hookrightarrow \operatorname{Harm}(\mathbb{S}^{2},\operatorname{U}(N))\end{equation*}
is not injective, as it maps degree components onto each other.
When $\gamma $ is not zero, on the other hand, the
existence of time translation is obstructed and generic $\gamma $'s do not
occur.

That $\gamma =0$ maps are $1$-unitons (i.e. the extended solutions 
can be written as degree $1$ polynomials,  \cite{Uhl}) 
and that the extended solution produced from 
the monad data never has terms of degree (as a Laurent polynomial) 
greater than the degree of $\gamma $, lead us to
\begin{theorem6} 
The uniton number is the smallest $n\in \mathbb{Z}$ such that $\gamma ^{n}=0$.
\end{theorem6}
In particular, this is the case when the uniton bundle
has a section in a neighbourhood of $\lambda =0$ which has a zero 
on the $n$th formal neighbourhood (i.e. as a truncated $n$th order
polynomial).

\section*{6. Applications }

\begin{theorem7} 
Maps with energy $\leq 3$ are $1$-unitons; therefore, \linebreak 
$\operatorname{Harm}_{k}^{*}(\mathbb{S}^{2},\!\operatorname{U}(N))$, $k\le 3$ 
(the first three nontrivial energy components) are connected.
\end{theorem7}
This bound on the uniton number is interesting in that it marks a 
distinction between 
unitons and instantons.  To the author's knowledge, it was not predicted.
The bound is sharp as the $\operatorname{U}(3)$ uniton from \cite{An2} 
has energy $4$ and is a $2$-uniton.

Finally, to demonstrate the potential for naive approaches to moduli 
topology we calculate
\begin{theorem8} 
\begin{equation*}\pi _{0}
\left (\operatorname{Harm}_{4}^{*}(\mathbb{S}^{2},\operatorname{U}(N))\right )=0,
\quad \pi _{1}
\left (\operatorname{Harm}_{4}^{*}(\mathbb{S}^{2},\operatorname{U}(N))\right )=0,
\end{equation*}
for $N\geq 3$.
\end{theorem8}
Martin Guest is also able to make these computations by extending
the ideas of \cite{FGKO}.  

\section*{7. Prospects }One should be able 
 to calculate the homology of low energy  components using the 
L-stratification spectral sequence \cite{BHMM} or GIT techniques.  
One could also try to compute the relative invariants of 
$\operatorname{Harm}(\mathbb{S}^{2},\operatorname{Gr}_{k}(\mathbb{C}^{N})) 
\hookrightarrow \operatorname{Harm}(\mathbb{S}^{2},\operatorname{U}(N))$ 
(based and unbased) to understand the relationship between these spaces, 
and  get information about the Grassmannian maps.  These 
questions are the subject of joint enquiry with A.\  Crawford.

Related to the conjecture on the uniton number are the problems of trying 
to represent the bundle transformations which correspond to
\begin{enumerate}
\item composing the harmonic map with $z\mapsto 1/z$ (since translations 
act nicely on the monad, this would give us an action of 
$\operatorname{Conf}(\mathbb{S}^{2},\mathbb{S}^{2})$ on the bundles) and
\item adding a uniton/performing a flag transformation.
\end{enumerate}These operations should be interesting in their own right and understanding
the second 
is a first step toward possible generalisations to maps 
into the symmetric spaces as in \cite{BuRa}.  

\bibliographystyle{amsalpha}
\begin{thebibliography}{MMMM}


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\end{thebibliography}



\end{document}