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Home six things six things: Douglas Robertson
six things: Douglas Robertson
Kona Macphee 23rd April 2011 23rd April 2011 six things
Douglas Robertson is an artist, who divides his time between his home in Hampshire, and exhibiting and researching his artwork in his native Scotland.
He has worked on numerous collaborations with some of Scotland's most respected poets including Valerie Gillies, Kenneth White, Donald S. Murray, Jen Hadfield and Andrew Philip, and his work has been exhibited widely throughout the country.
"Douglas Robertson is a landscape artist in the best sense of that description, committed to a fundamental exploration of the relationship of human beings to their natural environment.
Robertson's work is steeped in poetry and oral tradition. But he makes use of folklore and poetry rather than illustrating it. Instead the references in his work become themselves part of a living folklore, a poetical visualisation which takes one closer to nature through its spirits."
— Murdo Macdonald, The Scotsman
Douglas's website: www.douglasrobertson.co.uk
• One thing that's always worth getting out of bed for
Journeys, and the people you meet on the way.
One thing I have loved about working as an artist, is that it has given me the opportunity to travel around the country, visiting a wide variety of interesting places, and work with some of the country's finest poets and writers.
Many of the journeys I have taken have brought me to some of the most beautiful islands of Scotland, experiencing their colour and customs. The people of these islands have played a major part in the shaping of my work with their language, customs, folklore and traditions.
I hope to be getting out of bed for many journeys to come!
• One thing about myself that often obstructs me
Procrastination.
My studio, or should I call it 'the library', is evidence of this – three large pencil drawings on the board almost close to completion, several assemblages sitting on the bench in various stages of production, and sketchbooks with numerous ideas and plans for new pieces, and exhibitions looming close. And what am I doing… reading poetry!
• One thing I've learned the hard way
That art is not easy, or should I say, creating art is not.
Any maker will probably agree that the process of creating is a love/hate relationship. Long periods of solitude and intense working in the studio are often the norm, but they are rewarded by the pleasure of seeing and hearing audiences enjoying the finished work.
• One thing that gets under my skin
Man's inhumanity to man.
It worries me that with all the other pressing needs we have in this world in terms of human suffering and the plight of many nations, we still spend vast amounts of time and money on weaponry and warfare. Will we never learn?
• One thing I'd love to change
(Apart from the previous one thing, and my obvious procrastination problem.)
On a personal level, I'd love to relocate my studio to somewhere on the west coast of Scotland, and have more time to spend on my work and with my family.
On a wider level, I'd like to see a day where people have the opportunity to pursue their ambitions and dreams in a much more egalitarian world.
• One thing I hope for
Good health, strength and imagination to keep me working, for many years to come.
I've too many things on my list that I still want to do, see and experience. It all comes back to that first question of what's worth getting out of bed for!
About six things
'six things' is a series of micro-interviews with interesting and creative people, in which they're asked to respond to a standard set of six prompts. A new 'six things' is published on the site each Saturday.
artist, sixthings. Bookmark.
sunday best #13: The Poetics of Space
sunday best #14: The Wilton Diptych
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text:
\subsection*{$\text{(1)} = \text{(2)}$}

Recall that periodicity condition (P1), namely that $\Theta $ and $\Psi $ have periods lying in $ 2\pi\sqrt {-1}\mathbb{ Z} $,  guaranteed that the spectral data corresponded to a harmonic section of an $ SU (2) $-principal bundle over a 2-torus, to obtain a harmonic map we required also periodicity condition (P2). This can also be expressed in terms of integrality of periods but to do so we must pull $\Theta $ and $\Psi $ back to the singular curve $ \hat X $
defined by
\[
y ^ 2 = (\lambda +1) ^ 2 (\lambda -1) ^ 2 a (\lambda)
\]
Otherwise said, $\hat X$ is the curve obtained from $X$ by identifying the two points $p_1, q_1$ in $\lambda^{-1} (1)$ together to form an ordinary double point, and doing likewise with the two points $p_{-1}, q_{-1}$ in $\lambda^ {- 1} (-1)$.

 A line bundle on $\hat X$ can be thought of as a line bundle on $X$, together with $e ^ {c_j}\in\mathbb{ C} ^\times $ giving the identification of the points over $p_j, q_j,$ for $j = 1, -1$. As described above, the eigenline bundles $\mathcal E_ z$ are naturally specified with respect to an open cover consisting  of $X_A = X -\lambda^ {- 1}\{0,\infty \}$ and neighbourhoods of the points in $\lambda^{-1}\{0,\infty \}$. These neighbourhoods are taken sufficiently small so that they do not contain any of the points $p_j, q_j$ and so we obtain a linear flow of line bundles $\hat{\mathcal E}_z $ on $\hat X$ by employing the same transition functions.

Periodicity conditions (P1) and (P2) together are now exactly the requirement that (P1) holds for $ \hat X $. Of the various equalities proven in Theorem~\ref{theorem:periodicity}, only for those involving $\Pi_\Theta,\Pi_\Psi $ did our proof utilise the assumption that $ X $ is smooth. We now explain how to modify our interpretation of the periods of $\Theta $ and $\Psi $ as elements of the dual of the space of regular differentials for the case when our curve has a pair of ordinary double points.

We begin by supplementing our normalised homology basis $ A_j, B_j $ for $ X $ by additional curves which we push forward under the normalisation map
\begin {align*}
\iota:  X &\rightarrow\hat X\\
 (\lambda, y) &\mapsto (\lambda, (\lambda +1) (\lambda -1) y)
\end {align*}
 to yield a homology basis for $\hat X $. Recall that we represented our basis by curves emanating from a single point $ x_0\in X $ and so that $\rho_*(A_j) \sim - A_j $, $\rho_*(B_j) \sim B_j\;\mbox{mod}\;\langle A_1,\ldots, A_g \rangle $.
For $ k =\pm  1 $ choose an embedded curve $\gamma_k $ from  $ p_k $ to $ q_k $ not intersecting any $ A_j, B_j $ and such that $\rho_*\gamma_k \sim-\gamma_k $.

The regular differentials on $\hat X $ correspond to holomorphic differentials on $ X $ together with meromorphic differentials whose only singularities are simple poles  at the points $ p_1, q_1 $ or $ p_{-1}, q_{-1} $ or both and satisfying
\
\mathrm{Res}_{p_k}\omega = -\mathrm{Res}_{q_k}\omega\quad\text { for } k = 1, 2.
\]
We take a normalised basis $ H ^ 0 (X,\mathcal K) $ represented by the differentials $\omega ^ 1,\ldots,\omega ^ g $ satisfying $\int_{A _ j}\omega ^ i =\delta ^ i_j $. Define $\eta ^ k $ for $ k =\pm 1 $ to be the meromorphic differential whose only singularities are simple poles at $ p_k, q_k $ such that $\int_{A_j}\eta ^ k = 0 $ for $ j = 1,\ldots, g $ and with residue $ \frac {1} {2\pi\sqrt {- 1}} $ and $ - \frac {1} {2\pi\sqrt {- 1}}  $ at $ p_k $ and $ q_k $ respectively. Assume that $\Theta_0 $, $\Psi_0 $ are normalised as above, that is that their $ A_j $-periods vanish. Then define $\hat\Pi_\Theta,\hat\Pi_\Psi\in H ^ 0 (\hat X,\mathcal K) ^\vee $ by
\begin{align*}
\hat\Pi_\Theta (\omega ^ j) &=\int_{B_j}\Theta_0,\quad\hat\Pi_{\Theta_0} (\eta ^ k) =\int_{\gamma_k }\Theta_0\text { and }\\
\hat\Pi_\Psi(\omega ^ j) &=\int_{B_j}\Psi_0,\quad\hat\Pi_\Psi(\eta ^ k) =\int_{\gamma_k }\Psi_0\quad\text { for } j = 1,\ldots, g, k =\pm1.
\end{align*}
\begin {theorem}\label {theorem:singular}
Let $ (X,\lambda,\Theta,\Psi)$ be smooth spectral data satisfying (\ref{item:first})--(\ref{item:last}) of Theorem ~\ref {theorem:Hitchin} and such that $\Theta $ and $\Psi $ have purely imaginary periods. Let $\hat X $ be the curve obtained from $X$ by identifying the two points $p_1, q_1$ in $\lambda^{-1} (1)$ together to form an ordinary double point, and doing likewise with the two points $p_{-1}, q_{-1}$ in $\lambda^ {- 1} (-1)$. Then the statement of Theorem ~\ref {theorem:periodicity} holds also for $\hat X $, and periodicity condition (P2) is equivalent to the requirement that this pair of elements of $ H ^ 1 (\hat X,\mathcal O) $ are integral with respect to the lattice $ H_1 (\hat X, 2\pi\sqrt {-1}\mathbb{ Z}) $.\end {theorem}

\begin {proof}
As noted above, the equalities not involving $\hat\Pi_\Theta $ and $\hat\Pi_\Psi $ were already established in the proof of Theorem~\ref {theorem:periodicity}. We argue now that these are given by the Abel-Jacobi derivatives stated above for the singular curve $\hat X $. As before we assume that $\lambda $ is not branched over $ 0 $ and $\infty $.

Denote by $\Delta $ the simply connected region formed by cutting $ X $ along the homology basis $ A_1,\ldots, A_g, B_1, \ldots , B_g $
 specified above. Then
exactly as in the proof of Theorem ~\ref {theorem:periodicity}, reciprocity yields that
\[
\hat\Pi_\Theta (\omega ^ j) =\int_{B_j}\Theta = 4\pi\sqrt {-1}\left (\bar c   {
\left.  \dfrac {d\mathcal A_{p_\infty } } {d\lambda^ {- 1}} (\omega ^ j)\right|_{\lambda = \infty }}-c\left. \dfrac {d\mathcal A_{p_0} } {d\lambda} (\omega ^ j)
\right|_{\lambda = 0}\right)
\]
Furthermore, set $\hat\Delta_k =\Delta\setminus\gamma_k $,  fix $ x_0\in\hat\Delta_k $ and define $ l _k (x) =\int_{x_0} ^ x\eta ^k $.
Then with $\gamma_k ^ + $ and $\gamma_k ^ - $ denoting either side of the split left by the deletion of $\gamma_k $ as shown in Figure~\ref {figure:cuts},
\begin{figure}[htb]
\centering
\includegraphic
{pants2.pdf}
\caption{Polygon with cut}
\label{figure:cuts}
\end{figure}

\begin {align*}
2\pi\sqrt {-1}\sum_{p\in\hat \Delta_k}\mathrm{Res}_p (l_k\Theta) & =
\int_{\partial\hat\Delta}  l _k\Theta \\
&=\sum_{j = 1} ^ g\left (\int_{A _ j}\eta ^ k\int_{B_j}\Theta -\int_{B_j}\eta ^ k\int_{A_j}\Theta\right) +
\int_{\gamma_k ^ +} l_k\Theta -\int_{\gamma_k ^ -} l_k \Theta\\
& =\int_{\gamma_k ^ +} l_k\Theta -\int_{\gamma_k ^ -} l_k \Theta
\end {align*}
since $\eta ^ k $ and $\Theta $ are both normalised to have vanishing $ A $-periods. But for $p ^ +\in\gamma_k ^ + $ and the corresponding $ p ^ -\in\gamma_k ^ - $,
\[
l_k (p ^ +) - l_k (p ^ -) = 2\pi\sqrt {-1}\mathrm{Res}_{p_k}\eta ^ k = 1,
\]
so
\begin {align*}
\Pi_\Theta (\eta ^ k) =\int_\gamma\Theta & = -2\pi\sqrt {-1}\sum_{p\in\hat\Delta_k}\mathrm{Res}_{p} (l_k\Theta)\\
& = 4\pi\sqrt {-1}\left (\bar c   {
\left.  \dfrac {\partial\mathcal A_{p_\infty } } {\partial\lambda^ {- 1}} (\eta ^ k)\right|_{\lambda = \infty }} -c\left. \dfrac {d\mathcal A_{p_0} } {d\lambda} (\eta ^ k)
\right|_{\lambda = 0}\right).
\end {align*}
Similarly,
\[
\Pi_\Psi = 4\pi \sqrt {-1}\left (- c\tau\left. \dfrac {d\mathcal A_{p_0} } {d\lambda}\right|_{\lambda = 0}  + \bar c\bar\tau {\left. \dfrac {d\mathcal A_{p_\infty } } {d\lambda^ {- 1}}\right|_{\lambda = \infty }}\right).\qedhere
\]
\end{proof}

\section {Spectral curves and their applications}
\label {section:Lax}

\subsection {Finite-type harmonic maps from a surface to a symmetric space}
We begin by extending the zero-curvature description of harmonic maps into symmetric spaces, as these arise more frequently in geometric applications then do Lie groups. We then introduce the notion of harmonic maps of the plane of {\it finite-type} and explain how a spectral curve construction can be given for these. Certainly not all harmonic maps of the plane are finite-type, but in many situations it has been shown that all doubly-periodic such maps are either totally isotropic (and  given by holomorphic data in terms of a Weierstrass-type representation) or of finite-type and expressible in terms of spectral curve data.
Harmonic maps into Lie groups and symmetric spaces can be studied in terms of one another due to the fact that there is a natural totally geodesic immersion from a symmetric space into the corresponding group, namely the Cartan immersion. As such we could discuss finite-type solutions at either the Lie group or symmetric space level, we choose to do  the latter due to the aforementioned prevalence of symmetric spaces in geometric applications and also because one can always regard a Lie group $ G $ as a symmetric space $ (G\times G)/G $.

A homogeneous space $ G/H $ is a \emph{symmetric space} if there exists an involution $\sigma: G\rightarrow G $ such that
\[
(G ^\sigma)_0\subset H\subset G ^\sigma
\] where $ G ^\sigma $ denotes the fixed point set of $\sigma $, and $ (G ^\sigma)_0 $ the identity component of $ G ^\sigma $.

Recall that a map $ \iota: N\rightarrow P $ between pseudo-Riemannian manifolds is \emph{totally geodesic} when it sends geodesics to geodesics, or equivalently when
\[
\text {second fundamental form of $ \iota $}= (\nabla (d\iota)) = 0.
\]
 The Cartan map of a symmetric space is given by
\begin {align*}
\iota:\;\; & G/H\rightarrow G\\
& gH\mapsto \sigma (g ) g^{-1}.
\end {align*}
The following result is standard when $ G $ is compact and hence has a bi-invariant Riemannian metric, and is not difficult to extend to the pseudo-Riemannian case (see \cite {CT:11} for details).
\begin {theorem}
Let $ G $ be a semisimple  Lie group with bi-invariant (pseudo)-metric $\langle \cdot,\cdot \rangle $ and $ G/H $ a symmetric space with respect to the involution $\sigma: G\rightarrow G $.
Then $\iota: gH\mapsto \sigma (g)g ^ {- 1} $ is a totally geodesic immersion $ G/H\rightarrow G $.
If $ H = G ^\sigma $, then $\iota$ is additionally an embedding.
\end {theorem}
Since the Cartan immersion $\iota: G/H\rightarrow G $ is totally geodesic, a smooth map $ f: M\rightarrow G/H $ is harmonic if and only if the composition $\tilde{f} = \iota\circ f: M\rightarrow G $ is harmonic. Writing $\Phi = \tilde {f} ^ {- 1} d\tilde {f} $, this is equivalent to
\[
d*\Phi = 0.
\]
It is more useful to phrase this in terms of a lift $ F: U\subset M\rightarrow G $ of $ f $ on a simply connected open set $ U $.
The involution $\sigma:\mathfrak{g}\rightarrow\mathfrak{g} $ gives a splitting
\[
\mathfrak{g} =\mathfrak h\oplus \mathfrak m
\]
into the $ (+1) $- and $ (-1) $-eigenspaces spaces of $\sigma $.
Let  $\varphi : =  F^ {- 1} d  F  $ and  then $\varphi  =\varphi _{\mathfrak{h}} +\varphi _{\mathfrak{m}}$ is the decomposition of $\varphi $
 into the eigenspaces of $\sigma $.

Since $ \tilde { f} =\sigma ( F)  F^ {- 1} $, we have
\[
\Phi =\tilde f^ {- 1} d\tilde f = F\left (\sigma (F)^ {- 1} d (\sigma (F)) - F^{-1}d F \right) F^ {- 1} = -2\mathrm{Ad}_F (\varphi_{\mathfrak{m}}).
\]

It is now straightforward to verify that a harmonic map $ f $ from a simply connected surface into $ G/H $ is equivalent to an $S ^ 1 $-family of $\mathfrak{g} $-valued 1-forms
\[
\varphi _\lambda =\lambda\varphi'_\mathfrak{m} +\varphi_\mathfrak{h} +\lambda^ {- 1}{\varphi''}_\mathfrak{m}
\]
each satisfying the Maurer-Cartan equation
\begin {equation}\label {eq:symmetric}
d\varphi_\lambda +\frac 12 [\varphi_\lambda\wedge\varphi_\lambda ] = 0,
\end {equation}
where $\varphi = F^ {- 1} dF $ for $ F $ a lift of $ f $ into $ G $.

Some solutions to \eqref{eq:symmetric} may be obtained merely by solving a pair of commuting ordinary differential equations on a finite dimensional loop algebra. It is upon these {\em finite-type} solutions that we shall focus and we begin by explaining how harmonic maps into symmetric spaces can be described in this straightforward way.

Let $\Omega\mathfrak{g} $ be the loop algebra
\[
\Omega\mathfrak{g}: =\{\xi: S ^ 1\rightarrow\mathfrak{g}\mid \xi\text { is smooth.}  \}.
\]
For studying maps into symmetric spaces it is helpful to consider the twisted loop group
\[
\Omega ^\sigma G =\{\gamma: S ^ 1\rightarrow G:\gamma (-\lambda)\} =\sigma (\gamma (\lambda))\}
\]
 and corresponding twisted loop algebra $\Omega ^\sigma\mathfrak{g} $.
The (possibly doubly infinite) Laurent expansion
\[
\xi(\lambda) =\sum_{j} \xi_j \lambda  ^ j,\quad\xi_{\text even}\in\mathfrak h ^\mathbb{ C},\quad\xi_{\text odd}\in\mathfrak m ^\mathbb{ C},\quad\xi_{- j} =\bar\xi_j
\]
allows us to filtrate $\Omega ^\sigma\mathfrak{g} ^\mathbb{ C} $ by finite-dimensional subspaces
\[
\Omega ^\sigma_d =\{\xi\in\Omega\mathfrak{g}\mid \xi_j = 0\text { whenever }\left|j\right| >d\}.
\]

Choose a Cartan subalgebra $\mathfrak t$ of $\mathfrak{g} $ such that $\mathfrak t\subset\mathfrak h $
and recall that a non-zero $\alpha \in (\mathfrak{t}^\mathbb{ C})^*$ is a {\em root}  with corresponding {\em root space} $\mathcal{G}^{\alpha}\subset\mathfrak{g} ^\mathbb{ C} $
 if  $[X_1, X_2 ]=\alpha(X_1) X_2 $ for all $X_1\in \mathfrak{t}$ and $ X_2\in \mathcal{G}^{\alpha}$.
Fix also a set of {\em simple roots}, that is roots $\alpha_1, \ldots, \alpha_N $  such that every root $\alpha $ can be written uniquely as
$$\alpha=\sum_{j=1}^{N} m_j \alpha_j,$$ where the $m_j$ are either all positive integers or all negative integers.

The roots act also on $\mathfrak k ^\mathbb{ C} $ and writing $ \mathfrak n $ for the positive root spaces of $\mathfrak h ^\mathbb{ C} $ with respect to this choice of simple roots, we obtain an Iwasawa decomposition
\[
\mathfrak{h}^\mathbb{ C} =\mathfrak{n}\oplus\mathfrak{t}^\mathbb{ C}\oplus\bar{\mathfrak {n}}.
\]
Define $ r:\mathfrak{h} ^\mathbb{ C}\rightarrow\mathfrak{h} ^\mathbb{ C} $ by
\[
r (\eta) =\eta_{\bar{\mathfrak{n}}} +\frac 12\eta_{\mathfrak k}
\]
and note that
\[
(\eta dz)_{\mathfrak h} = r (\eta) dz +\overline {r (\eta)} d\bar z.
\]
The following result appears in \cite[Theorem 2.5] {BP:94}; in this context the reader is also referred to  \cite {OU:02}.
\begin {theorem}
\label {theorem:finite}
Suppose $ G $ is a semisimple Lie group with a bi-invariant pseudo-metric and $ G/H $ is a symmetric space. If $ d $ is a positive odd integer and $\xi:\mathbb{ R} ^ 2\rightarrow\Omega ^\sigma_d $ satisfies the Lax pair
\[
\dfrac {\partial\xi} {\partial z} = [\xi,\lambda\xi_d + r (\xi_{d -1})]
\]
then
there exists  $ F:\mathbb{ R} ^ 2\rightarrow G $ unique up to left translation, such that the map  $ f:\mathbb{ R} ^ 2\rightarrow G/H $
framed by $ F $ is harmonic and $\varphi = F^ {- 1} dF $ satisfies
\begin {equation}\label {eq:pair}
\varphi'_{\mathfrak m} =\xi_d,\quad\varphi'_{\mathfrak h} =\xi_{d -1}.
\end {equation}
If $ G $ is compact then global solutions of the Lax pair exist for any choice of initial condition.
\end {theorem}

\newcommand{\mathfrak{p}}{\mathfrak{p}}
\begin {proof}
Suppose that $\xi:\mathbb{ R} ^ 2\rightarrow\Omega_d $ satisfies  \eqref {eq:pair} and hence also the conjugate equation
\begin {equation}\label {eq:conjugate}
\dfrac {\partial\xi} {\partial\bar z} = - [\xi, \lambda^ {- 1}\xi_{- d} +\overline {r (\xi_{d -1})} ].
\end {equation}
Then
\[
d\xi = [\xi, \phi_\lambda ],
\]
where $\varphi_\lambda = (\lambda\xi_d + r (\xi_{d -1})) d z + (\lambda^ {- 1}\xi_{- d} +\overline {r (\xi_{d -1})}) d\bar z$.
Writing
\begin {align*}
\varphi & = ( \xi_d + r (\xi_{d-1})) dz + (\xi_{- d} +\overline {r (\xi_{d -1})}) d\bar z\\
& = (\varphi '_{\mathfrak p} +\varphi'_{\mathfrak k}) + (\varphi ''_{\mathfrak p} +\varphi''_{\mathfrak k}),
\end {align*}
 equations \eqref {eq:pair} and \eqref {eq:conjugate} yield
\begin {align}
d\varphi'_\mathfrak{p} + [\varphi _{\mathfrak {k}}\wedge\varphi '_\mathfrak{p}] & = 0\label {eq:MCg1}\\
d\varphi_{\mathfrak {k}} + \frac 12 [\varphi _{\mathfrak {k}}\wedge\varphi_{\mathfrak {k}}] + [\varphi '_\mathfrak{p}\wedge\varphi''_\mathfrak{p}] & = 0\nonumber\\
d\varphi''_\mathfrak{p} + [\varphi _{\mathfrak {k}}\wedge\varphi ''_\mathfrak{p}] & = 0.\nonumber
\end {align}
which are the various components of the Maurer-Cartan equation  \eqref {eq:symmetric}.

Now suppose that $ G $ is compact and note that since the coefficient of $\lambda ^ {d +1} $ on the right hand side of
\[
Z (\xi) = \frac 12 ( X  (\xi) - i Y  (\xi)) = \lambda\xi_d + r (\xi_{d -1})
\]
 vanishes, this equation defines defines vector fields $ X $, $ Y $ and $Z $  on $\Omega_d $. Taking $\bar Z $ to be the vector field conjugate to $ Z $, then $ X, Y $ commute if and only if  $ [  Z,\bar Z] =  0 $. This follows from a straightforward but tedious computation, using the Jacobi identity as well as the inclusions $ [\mathfrak t ^\mathbb{ C}, \mathfrak n]\subset\mathfrak n ,   [\bar{\mathfrak n}, \bar{\mathfrak n}]\subset\bar{\mathfrak n} $, $ [\mathfrak n,\mathfrak n]\subset\mathfrak n $ which one sees for example by taking a Chevalley basis for $\mathfrak k ^\mathbb{ C} $.

The flows of the vector fields $ X, Y $ are given by Lax equations $\frac {d\xi} {dx} = [\xi,  \hat X (\xi) ]$ and $\frac {d\xi} {d y} = [\xi, \hat Y (\xi) ]$ for $\hat X,\hat Y :\Omega_d\rightarrow\Omega_d $ and the $  L ^ 2 $ inner product on $\Omega\mathfrak{g} $ is ad-invariant. Thus
\begin {align*}
\frac {d} {d x}\langle\xi,\xi\rangle & = 2\langle [\xi,\hat X (\xi) ],\xi\rangle \\
& = -2\langle [\xi,\xi ],\hat X (\xi)\rangle = 0
\end {align*}
and similarly for $ Y $, so the flows evolve on spheres and hence are complete whenever $ G $ is compact.  Therefore this pair of complete commuting vector fields defines an action of $\mathbb{ R} ^ 2 $ on $\Omega_d$ via
\[
(x, y)\cdot \xi = X ^ {x}_1\circ X ^ {y}_2 (\xi).
\]
We see that  for any $\psi_0\in\Omega_d$, we may define $\xi:\mathbb{ R} ^ 2\rightarrow\Omega_d$ by
\[
\xi (x, y) = (x, y)\cdot\xi_0
\]
and the $\xi $ so defined satisfies \eqref{eq:pair}.\end {proof}

Given a harmonic map $ f:\mathbb{ R} ^ 2\rightarrow G/H $, we may define a family of flat connections $\phi_\lambda $ as above and then a solution $\xi: S ^1\rightarrow\Omega_d $ to the Lax pair
\[
\frac {\partial\xi} {\partial z} = [\xi,\phi_\lambda']
\]
is called a  {\em polynomial Killing field} and if furthermore $ \xi_d + r (\xi_{d -1}) =\phi_{\bar z} $ then the polynomial Killing field is said to be {\em adapted}.  As Theorem~\ref{theorem:finite} demonstrates, the harmonic map $ f $ can be recovered from an adapted polynomial killing field.
\begin {definition}
Harmonic maps $ f:\mathbb{ R} ^ 2\rightarrow G/H $ arising from the above construction, or equivalently those possessing an adapted polynomial Killing field are said to be of \emph{finite-type}.
\end {definition}
Harmonic maps into groups can be analysed analogously or by considering the group as a symmetric space.
Harmonic maps of finite type can thus be constructed by remarkably more simple means than general harmonic maps. The obvious question of course is how special are these maps? Certainly by no means all harmonic maps of the plane are of finite type but if we restrict our attention to maps which are periodic with respect to a rank-two lattice $\Lambda\subset\mathbb{ R} ^ 2 $ then the compactness of the domain $\mathbb{ R} ^ 2/\Lambda $ makes it reasonable to ask whether all such doubly periodic maps are of finite type. The first finite-type results were contained in the work of Hitchin on harmonic maps of a 2-torus $\mathbb{ R} ^ 2/\Lambda $ to the 3-sphere \cite {Hitchin:90} and in the study by Pinkall and Sterling \cite {PS:89} of constant mean curvature immersions of the plane in Euclidean-3 space. As described in the previous section, Hitchin gave a complete characterisation of harmonic $\mathbb{ R} ^ 2/\Lambda\rightarrow S ^ 3\cong SU (2) $ in terms of a spectral curve. He showed that with the exception of conformal harmonic maps into a totally geodesic $ S ^ 2\subset S ^ 3 $ (the so-called totally isotropic maps), all harmonic maps $\mathbb{ R} ^ 2/\Lambda\rightarrow  S ^ 3 $ are of finite type (the totally isotropic  are dealt with by separate means, \cite {Calabi:67}). His approach used the holonomy of the family of flat connections rather than polynomial Killing fields and this result came down to an application of the fact that an elliptic operator on a compact domain (the 2-torus) has but a finite dimensional kernel.

The approach by Pinkall and Sterling was the first in a series of papers \cite {FPPS:92,BPW:95, BFPP:93} showing a large classes of harmonic maps of tori are of finite type. Indeed a major advantage of the polynomial killing field approach is that it has been more amenable to proving finite-type results. In particular this approach yielded the following quite general two theorems.
\begin {theorem}[\cite {BFPP:93}]
Let $ f:T ^ 2 =\mathbb{ R} ^ 2/\Lambda\rightarrow G $ be a semi-simple adapted harmonic map into a compact semi-simple Lie group. Then $ f $ is of finite type.
\end {theorem}
From this point of view the importance of the double-periodicity condition comes from the fact that the 1-form $ 4i\xi_d = f ^*\omega\left ( \frac {\partial} {\partial z}\right) $ is holomorphic, since the harmonic map equation may be expressed as the condition that
\[
f^{-1}\nabla_{\frac {\partial} {\partial\bar z}} ^ Gf_*\frac {\partial} {\partial z} = 0.
\]
On a genus-one surface the only holomorphic differentials are constant.

\begin {theorem}[\cite {BFPP:93}]
 Suppose $ G $ is compact and the symmetric space $ G/H $ has rank one. A harmonic map $ f:\mathbb{ R} ^ 2/\Lambda\rightarrow G/H $ is of finite type if and only if it is non-conformal.
 \end {theorem}
The rank is the maximum dimension of a subspace of the tangent space (to any point) on which the sectional curvature is identically zero. Rank one symmetric spaces include spheres and projective spaces.

Of course the most geometrically interesting harmonic maps into symmetric spaces are the conformal ones, and when the target is a sphere or complete projective space, Burstall  \cite {Burstall:95} showed that all but the totally isotropic harmonic maps of 2-tori have lifts into an appropriate flag manifold which are of finite type (this involves expanding the notion somewhat to primitive maps into $ k $-symmetric spaces). The compactness assumption also excludes a number of geometrically interesting situations, but recently Turner and the author have shown \cite {CT:11, CT:12} that maps of 2-tori into $G/T$ possessing a Toda frame are necessarily of finite-type, for $ G $ a simple Lie group with bi-invariant pseudo-metric and $ T $ a Cartan subgroup. In particular then superconformal harmonic maps of 2-tori to de-Sitter spheres $S ^ {2n}_1 $ whose harmonic sequence is everywhere defined lift to maps of finite type. Since Willmore surfaces in $ S ^ 3 $ without umbilic points are characterised by the property that their conformal Gauss maps, which take values in $ S ^ 4_1 $, are harmonic this result yields a simple proof that Willmore tori without umbilic points are all of finite-type. Using the multiplier curve described below, the Willmore result is proven in \cite {Schmidt:02, Bohle:08} without the umbilic assumption.

\subsection {A comparison of spectral curve constructions}

As with Hitchin's holonomy construction for harmonic 2-tori in $ S ^ 3 $, one would like to use polynomial Killing fields to build spectral curves for other harmonic maps. The reason to not necessarily use holonomy directly is simply that beyond the case $ G = SU (2) $ there has not been success in proving in this way that the resulting spectral curve is actually algebraic, that is has finite genus. A map being of finite-type means exactly that the resulting spectral curve has finite genus.  The polynomial Killing fields are solutions to a Lax pair, and there is a long and rich history of spectral curve constructions in the study of solutions to Lax pair equations (see for example \cite {AM:80:1, AM:80:2}). However when studying classical examples such as the Toda lattice, or geodesics on an ellipsoid, or more modern examples such as Higgs bundles, one is concerned with a single solution to the Lax equation. The natural approach is then to take the characteristic polynomial of this solution. When considering harmonic maps, one is presented with an entire algebra of solutions. These solutions for example have differing degrees, and hence the characteristic polynomials clearly yield algebraic curves of different genus. The genus has geometric meaning, since the dimension of the space on which one can choose the eigenline bundle determines the dimension of the family in which the harmonic map lies. The eigenline bundle can usually be chosen from an appropriate Prym variety or perhaps a Prym-Tjurin subvariety of the Jacobian and the dimension of this variety can be computed in terms of the spectral genus.

In   \cite {McIntosh:95, McIntosh:96}, McIntosh considers the entire algebra of polynomial Killing fields for harmonic 2-tori in $\C\P ^ n $ by taking the spectrum of a maximal abelian subalgebra, then proving that the resulting curve is independent of the choice of maximal abelian subalgebra. Another construction which is somewhat simpler is to take the ``eigenline'' curve of (a maximal abelian subalgebra) of the polynomial Killing fields, as in  \cite {FPPS:92}. However McIntosh and Romon have given an example in which the spectral curve obtained by this construction is different to that obtained by using the spectrum \cite {MR:10} (and it is possible to reconstruct the map from the latter but not the former)
and so some caution is required with the eigenline approach.

The spectral curve clearly offers a powerful tool in studying the moduli space of harmonic maps and as such has been instrumental in a number of recent approaches on various geometric conjectures. In   \cite {KSS:10}, the authors present a detailed analysis of the moduli space of  equivariant constant mean curvature tori in $ S ^ 3 $ and show that the spectral curve of any embedded equivariant minimal tori of positive (arithmetic) spectral genus can be deformed through a family of spectral curves of constant mean curvature tori to a curve of geometric genus zero and arithmetic genus one,  which is known not to be the spectral curve of any constant mean curvature torus. This contradiction implies that any embedded equivariant minimal torus in $ S ^ 3 $ must in fact have (arithmetic) spectral genus zero and hence be the Clifford torus,  proving the equivariant case of Lawson's conjecture that the only embedded minimal torus in $ S ^ 3 $ is the Clifford torus. A similar approach was employed in   the preprint \cite{KS:08} to yield the Pinkall-Sterling conjecture that the only embedded constant mean curvature tori in $ S ^ 3 $ are those of revolution, a fortiori as yielding the full Lawson conjecture. An analogous approach to the Willmore conjecture shows promise, with partial results  established in \cite {Schmidt:02} (in this context we mention also the recent announcement of a proof of the Willmore conjecture using rather different methods \cite{MN:12}.

More recently, another spectral curve construction has come into vogue, namely the multiplier, or Fermi curve  \cite {Taimanov:97, Schmidt:02, BLPP:07}. In a sense this has its roots in the holonomy construction, but it is more general in that it does not rely upon the existence of a family of flat connections and so applies to maps which are not necessarily harmonic. Instead one considers maps $ f $ from a Riemann surface $\Sigma $ into $ S ^ 4$ which are merely conformal or equivalently, (quaternionic) holomorphic as maps into $\H\P ^ 1\cong S ^ 4 $. As with holomorphic maps into complex projective spaces, such $ f $ correspond to quaternionic line sub-bundles of the trivial rank two quaternionic vector bundle $ V $ on $\Sigma $. Geometrically, the multiplier curve encodes a subspace of the space of Darboux transforms of the original map. These are a natural generalisation of classical Darboux transforms, where two surfaces in $\mathbb{ R} ^ 3 $ are classical Darboux transforms of one another if they share a common sphere congruence. More generally, given a surface $\Sigma $ and conformal immersion  $ f: \Sigma\rightarrow S ^ 4 $, a {\em Darboux transform} of $ f $ is a conformal map $\hat f: \Sigma\rightarrow S ^ 4 $ such that
for each $ p\in \Sigma $, $f (p)\neq \hat f (p) $, and there is a smooth oriented sphere congruence $ S: M\rightarrow\{\text {oriented round 2-spheres in } S ^ 4\} $ such that $ S  $ left-envelopes $\hat f $ and $ S  $ both left- and right-envelopes $ f $.

To say that $ S $ left-envelopes $\hat f $ means that for all $p\in M,\,\hat f (p)\in S (p) $, and the oriented great circles in $ S ^ 3 $ corresponding to the tangent planes of $\hat f (M) $ and $ S (p) $ at $\hat f (p) $ differ by left translation in $ S ^ 3\cong S U (2) $. Right-enveloping is defined analogously. Alternatively, considering the oriented Grassmannian of 2-planes in $\mathbb{ R} ^ 4 $ as $ S ^ 2\times S ^ 2 $, one can think of having a pair of Gauss maps given by the left and right normals and we require that the left normal of $\hat f $ matches that of $ S $ whilst both the left and right normals of $ f $ match those of $ S $. Darboux transforms of $ f $ are exactly the maps defined away from isolated points by holomorphic sections of the pull-back $\widetilde {V/L} $ of the quotient bundle $ V/L $ to the universal cover of $\Sigma $. The multiplier spectral curve is then the the space of holonomies realised by holomorphic sections of $\widetilde {V/L} $.

For the fundamental case of constant mean curvature tori in $\mathbb{ R} ^ 3 $, we have in this quaternionic line bundle also a family of flat connections, gauge equivalent to the family of flat connections described in section~\ref {section:content}. The holomorphic structure on $\widetilde {V/L} $ is precisely the $ (0, 1) $ part of these connections (which is independent of the spectral parameter $\lambda $). Thus sections which are actually parallel with respect to some connection $\nabla ^\lambda $ are in particular holomorphic sections, and the corresponding Darboux transforms are termed $\lambda $-Darboux transforms. These maps are included in Hitchin's study of harmonic maps into $ S ^ 3 $, as the Gauss map of the constant mean curvature surface is harmonic and vice versa. In terms of Hitchin's spectral curve, the fact that the map corresponds to a constant mean curvature torus means precisely that the curve is unbranched over $ 0 $ and $\infty $ and it possesses a holomorphic involution covering $\lambda \mapsto -\lambda $. If we term the quotient of Hitchin's curve by this involution the {\it eigenline spectral curve}, we have the following  \cite {CLP:09}.
\begin {theorem} [  \cite {CLP:09}]
The eigenline and multiplier curves of a constant mean curvature torus in $\mathbb{ R} ^ 3 $ are not birational, however they have the same normalisation. The multiplier curve is always singular whereas the eigenline curve is generically smooth. \end{theorem}
This point of view is a particularly natural way of recovering the original constant mean curvature immersion.
\begin {theorem}[  \cite {CLP:09}]
The original constant mean curvature immersion $ f: T ^ 2\rightarrow\mathbb{ R} ^ 3 $ is given by the limit of the $\lambda $-Darboux transforms as $\lambda $ tends toward $ 0 $ or $\infty $.
\end {theorem}

Clearly it would be highly desirable to be able to extend spectral curve methods to harmonic maps of surfaces of genus higher than one. The multiplier curve offers an approach here, as by considering only the holomorphic structure (so ``half" the connection $\nabla_\lambda $), one is not so tightly tied to the assumption that the fundamental group must be abelian but can rather consider abelian representations of the holonomy of the more general holomorphic sections.

\bibliographystyle{alpha}

\def$'${$'$}

redpajama_set_name: RedPajamaArXiv


I'm using this thread as a channel for communication about the BuddyFriendGuy Pickup server, in addition to IRC.
The configuration is taken from packer (thanks, packer!) with the map collections being mine.
The primary motivation of this server is to provide lower ping for North American pickup players. Thanks to SPLAT for his suggestion.
This server is listed in IRC channel #xonotic.pickup of quakenet, but it's not listed in the in-game public list.
I reverted this server's code to git 82e38560842fa7ac81336d4788de7e3fe51a4cc4 before a recent sound bug. So far it's running the same version as packer's server. Thanks SPLAT for feedback on this one.
djstrikes3 kindly provided his map server to run a pilot project -- to test the idea of a centralized but speed-optimized map server. That is, players will download the maps from his server instead of mine. This should help lower the load and traffic on my server, and in turn improving players' experience.
Please let me know if you experience any problems!
Considering the low usage of this server, I'm allocating the server resource to my BuddyFriendGuy Fun Server, i.e. this server is taken down for now, and the Fun server is now up.


text:
Q: Adding __declspec(dllexport) to struct causes compilation error I have a struct in a DLL which, so far, has been used within that DLL, only.
I wanted to use the struct in another DLL, via pointer. So, I added __declspec(dllexport) to the struct definition. Simply adding that dll export statement is now causing the struct to fail to compile (not just the regular DLL export warnings). This is the first time that I've seen this behavior. I'm using Visual Studio 2019. I don't know what the underlying problem is.
What could be a reason that simply adding '__declspec(dllexport)' to a struct/class would make it fail compilation?
struct __declspec(dllexport) PipelineManager {
// ...
private:
    std::unordered_map<std::string, std::string>            mPathToName;
    std::unordered_map<std::string, vko::PipelineLayout>    mLayouts;
    std::unordered_map<std::string, vko::GraphicsPipeline>  mPipelines;
    std::unordered_map<std::string, std::unique_ptr<GraphicsPipelineDef>> mDefs;
};

A chunk of the errors are below:
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\xmemory0(820,1): error C2280:  'std::pair<const _Kty,_Ty>::pair(const std::pair<const _Kty,_Ty> &)': attempting to reference a deleted function
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\xmemory0(820,1): error C2280:         with
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\xmemory0(820,1): error C2280:         [
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\xmemory0(820,1): error C2280:             _Kty=std::string,
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\xmemory0(820,1): error C2280:             _Ty=std::unique_ptr<ngen::rendo::GraphicsPipelineDef,std::default_delete<ngen::rendo::GraphicsPipelineDef>>
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\xmemory0(820,1): error C2280:         ] (compiling source file Z:\NGEN\Source\Renderer\GpuProgramManager.cpp)
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133): message :  see declaration of 'std::pair<const _Kty,_Ty>::pair'
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133): message :         with
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133): message :         [
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133): message :             _Kty=std::string,
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133): message :             _Ty=std::unique_ptr<ngen::rendo::GraphicsPipelineDef,std::default_delete<ngen::rendo::GraphicsPipelineDef>>
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133): message :         ] (compiling source file Z:\NGEN\Source\Renderer\GpuProgramManager.cpp)
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133,5): message :  'std::pair<const _Kty,_Ty>::pair(const std::pair<const _Kty,_Ty> &)': function was implicitly deleted because a data member invokes a deleted or inaccessible function 'std::unique_ptr<ngen::rendo::GraphicsPipelineDef,std::default_delete<_Ty>>::unique_ptr(const std::unique_ptr<_Ty,std::default_delete<_Ty>> &)'
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133,5): message :         with
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133,5): message :         [
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133,5): message :             _Kty=std::string,
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133,5): message :             _Ty=std::unique_ptr<ngen::rendo::GraphicsPipelineDef,std::default_delete<ngen::rendo::GraphicsPipelineDef>>
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133,5): message :         ]
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133,5): message :         and
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133,5): message :         [
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133,5): message :             _Ty=ngen::rendo::GraphicsPipelineDef
1>C:\Program Files (x86)\Microsoft Visual Studio\2019\Preview\VC\Tools\MSVC\14.21.27619\include\utility(133,5): message :         ] (compiling source file Z:\NGEN\Source\Renderer\GpuProgramManager.cpp)

// ...

1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(67): message :  see reference to function template instantiation 'std::unordered_map<std::string,std::unique_ptr<ngen::rendo::GraphicsPipelineDef,std::default_delete<_Ty>>,std::hash<_Kty>,std::equal_to<_Kty>,std::allocator<std::pair<const _Kty,std::unique_ptr<_Ty,std::default_delete<_Ty>>>>>::unordered_map(const std::unordered_map<_Kty,std::unique_ptr<_Ty,std::default_delete<_Ty>>,std::hash<_Kty>,std::equal_to<_Kty>,std::allocator<std::pair<const _Kty,std::unique_ptr<_Ty,std::default_delete<_Ty>>>>> &)' being compiled
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(67): message :         with
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(67): message :         [
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(67): message :             _Ty=ngen::rendo::GraphicsPipelineDef,
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(67): message :             _Kty=std::string
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(67): message :         ] (compiling source file Z:\NGEN\Source\Renderer\GpuVM.cpp)
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(66): message :  see reference to class template instantiation 'std::unordered_map<std::string,std::unique_ptr<ngen::rendo::GraphicsPipelineDef,std::default_delete<_Ty>>,std::hash<_Kty>,std::equal_to<_Kty>,std::allocator<std::pair<const _Kty,std::unique_ptr<_Ty,std::default_delete<_Ty>>>>>' being compiled
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(66): message :         with
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(66): message :         [
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(66): message :             _Ty=ngen::rendo::GraphicsPipelineDef,
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(66): message :             _Kty=std::string
1>Z:\NGEN\Source\Renderer/PipelineManager.hpp(66): message :         ] (compiling source file Z:\NGEN\Source\Renderer\GpuVM.cpp)

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