Get Your Free Pool Price Estimates Now!
Find and compare reputable swimming pool companies operating within the Knysna area and listed on PoolBuilders.co.za.
BayLine Pools has more than 18 years of experience in the design and construction of swimming pools. Their repair and renovation team is able to transform you existing pool into something new and exciting. The companies offices are situated in Jeffreys Bay and they are able to operate throughout the Eastern Cape and parts of the Western Cape. These areas include: Plettenberg Bay and Knysna, Port Elizabeth, Jeffreys Bay and St Francis.
What pool designs appeal to you?
Where will the pool be built?
How can we get the estimates to you?
Website Design © VisAbility Joomla Website Developers. All Rights Reserved.
Why not enter your details here and we will provide you with up to 5 personalised pool price estimates? Quick, easy and obligation free.
Complete this simple form and get up to five separate and professional safety quotations.
This must be done to prove that you are a real person!


text:
\section{#1}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}

\def\rf#1{(\ref{eq:#1})}
\def\lab#1{\label{eq:#1}}
\def\nonumber{\nonumber}
\def\begin{eqnarray}{\begin{eqnarray}}
\def\end{eqnarray}{\end{eqnarray}}
\def\begin{equation}{\begin{equation}}
\def\end{equation}{\end{equation}}
\def\!\!\!\! &=& \!\!\!\! {\!\!\!\! &=& \!\!\!\! }
\def\foot#1{\footnotemark\footnotetext{#1}}
\def\lbrack{\lbrack}
\def\rbrack{\rbrack}
\def\left\langle{\left\langle}
\def\right\rangle{\right\rangle}
\def\Bigl\langle{\Bigl\langle}
\def\Bigr\rangle{\Bigr\rangle}
\def\left\lbrack{\left\lbrack}
\def\right\rbrack{\right\rbrack}
\def\left\{{\left\{}
\def\right\}{\right\}}
\def\({\left(}
\def\){\right)}
\newcommand{\noindent}{\noindent}
\newcommand{\ct}[1]{\cite{#1}}
\newcommand{\bi}[1]{\bibitem{#1}}
\def\vskip\baselineskip\vskip-\parskip\noindent{\vskip\baselineskip\vskip-\parskip\noindent}
\relax

\def\mathop{\rm tr}{\mathop{\rm tr}}
\def\mathop{\rm Tr}{\mathop{\rm Tr}}
\def\widehat{\rm Tr}{\widehat{\rm Tr}}
\def\vert{\vert}
\def\bigm\vert{\bigm\vert}
\def\;\Bigg\vert{\;\Bigg\vert}
\def\bigg\vert{\bigg\vert}
\newcommand\partder[2]{{{\partial {#1}}\over{\partial {#2}}}}
\newcommand\funcder[2]{{{\delta {#1}}\over{\delta {#2}}}}
\newcommand\Bil[2]{\Bigl\langle {#1} \Bigg\vert {#2} \Bigr\rangle}
\newcommand\bil[2]{\left\langle {#1} \bigg\vert {#2} \right\rangle}
\newcommand\me[2]{\left\langle {#1}\bigm\vert {#2} \right\rangle}
\newcommand\sbr[2]{\left\lbrack\,{#1}\, ,\,{#2}\,\right\rbrack}
\newcommand\pbr[2]{\{\,{#1}\, ,\,{#2}\,\}}
\newcommand\pbbr[2]{\left\{\,{#1}\, ,\,{#2}\,\right\}}

\def\ket#1{\mid {#1} \rangle}
\def\bra#1{\langle {#1} \mid}
\newcommand{\braket}[2]{\langle {#1} \mid {#2}\rangle}
\def\alpha{\alpha}
\def{\tilde A}^R{{\tilde A}^R}
\def{\tilde {\cal A}}^R{{\tilde {\cal A}}^R}
\def\atcm#1{{\tilde {\cal A}}^{(R,#1)}}
\def\beta{\beta}
\def{\cal D}{{\cal D}}
\def\delta{\delta}
\def\Delta{\Delta}
\def\epsilon{\epsilon}
\def\varepsilon{\varepsilon}
\def\gamma{\gamma}
\def\Gamma{\Gamma}
\def\nabla{\nabla}
\def{1\over 2}{{1\over 2}}
\def\lambda{\lambda}
\def\Lambda{\Lambda}
\def\mu{\mu}
\def\nu{\nu}
\def\over{\over}
\def\omega{\omega}
\def\Omega{\Omega}
\def\phi{\phi}
\def\Phi{\Phi}
\def\partial{\partial}
\def\prime{\prime}
\def{\tilde \Phi}{{\tilde \Phi}}
\defQ_{\bf s}{Q_{\bf s}}
\def\rightarrow{\rightarrow}
\def\sigma{\sigma}
\def\Sigma{\Sigma}
\def\tau{\tau}
\def\theta{\theta}
\def\Theta{\Theta}
\def\Theta_{+}{\Theta_{+}}
\def\Theta_{-}{\Theta_{-}}
\def\Theta_{+}^{>}{\Theta_{+}^{>}}
\def\Theta_{-}^{<}{\Theta_{-}^{<}}
\def\Theta_{+}^{(0)}{\Theta_{+}^{(0)}}
\def\Theta_{-}^{(0)}{\Theta_{-}^{(0)}}
\def\tilde{\tilde}
\def\widetilde{\widetilde}
\defJ^C{J^C}
\def{\bar J}{{\bar J}}
\def{\jmath}{{\jmath}}
\def{\bar \jmath}{{\bar \jmath}}
\def{\bar \p}{{\bar \phi}}
\def\varphi{\varphi}
\def\varepsilon{\varepsilon}
\def{\tilde \varphi}{{\tilde \varphi}}
\defFa\'a di Bruno~{Fa\'a di Bruno~}
\def{\cal A}{{\cal A}}
\def{\cal B}{{\cal B}}
\def{\cal E}{{\cal E}}
\def{\cal G}{{\cal G}}
\def{\hat {\cal G}}{{\hat {\cal G}}}
\def{\cal H}{{\cal H}}
\def{\hat {\cal H}}{{\hat {\cal H}}}
\def{\cal L}{{\cal L}}
\def{\cal M}{{\cal M}}
\def{\cal N}{{\cal N}}
\def\mid u\mid^2{\mid u\mid^2}
\def{\bar u}{{\bar u}}
\def\mid z\mid^2{\mid z\mid^2}
\def{\bar z}{{\bar z}}
\def\mid w\mid^2{\mid w\mid^2}
\def{\bar w}{{\bar w}}
\newcommand\sumi[1]{\sum_{#1}^{\infty}}
\newcommand\fourmat[4]{\left(\begin{array}{cc}
{#1} & {#2} \\ {#3} & {#4} \end{array} \right)}

\def{\cal G}{{\cal G}}
\def{\hat{\cal G}}{{\hat{\cal G}}}
\def{\cal G}^{\ast}{{\cal G}^{\ast}}
\def{\widetilde \lie}{{\widetilde {\cal G}}}
\def{\elie}^{\ast}{{{\widetilde \lie}}^{\ast}}
\def{\cal H}{{\cal H}}
\def{\cal F}{{\cal F}}
\def{\widetilde \lie}{{\widetilde {\cal G}}}
\def\f#1#2#3 {f^{#1#2}_{#3}}
\def{\sf w_\infty}{{\sf w_\infty}}
\def{\sf w_{1+\infty}}{{\sf w_{1+\infty}}}
\def{\sf {\hat w}_{\infty}}{{\sf {\hat w}_{\infty}}}
\def{\sf W_\infty}{{\sf W_\infty}}
\def{\sf W_{1+\infty}}{{\sf W_{1+\infty}}}
\def{\sf {\hat W}_{\infty}}{{\sf {\hat W}_{\infty}}}
\def\Rm#1#2{r(\vec{#1},\vec{#2})}
\def\OR#1{{\cal O}(R_{#1})}
\def\ORti{{\cal O}({\widetilde R})}
\def\AdR#1{Ad_{R_{#1}}}
\def\dAdR#1{Ad_{R_{#1}^{\ast}}}
\def\adR#1{ad_{R_{#1}^{\ast}}}
\def\KP{${\rm \, KP\,}$}
\def\KPl{${\rm \,KP}_{\ell}\,$}
\def\KPo{${\rm \,KP}_{\ell = 0}\,$}
\def\mKPa{${\rm \,KP}_{\ell = 1}\,$}
\def\mKPb{${\rm \,KP}_{\ell = 2}\,$}
\def\relax\leavevmode{\relax\leavevmode}
\def\vrule height1.5ex width.4pt depth0pt{\vrule height1.5ex width.4pt depth0pt}
\def\rlx\hbox{\sf Z\kern-.4em Z}{\relax\leavevmode\hbox{\sf Z\kern-.4em Z}}
\def\rlx\hbox{\rm I\kern-.18em R}{\relax\leavevmode\hbox{\rm I\kern-.18em R}}
\def\rlx\hbox{\,$\inbar\kern-.3em{\rm C}$}{\relax\leavevmode\hbox{\,$\vrule height1.5ex width.4pt depth0pt\kern-.3em{\rm C}$}}
\def\rlx\hbox{\rm I\kern-.18em N}{\relax\leavevmode\hbox{\rm I\kern-.18em N}}
\def\rlx\hbox{\,$\inbar\kern-.3em{\rm O}$}{\relax\leavevmode\hbox{\,$\vrule height1.5ex width.4pt depth0pt\kern-.3em{\rm O}$}}
\def\rlx\hbox{\rm I\kern-.18em P}{\relax\leavevmode\hbox{\rm I\kern-.18em P}}
\def\rlx\hbox{\,$\inbar\kern-.3em{\rm Q}$}{\relax\leavevmode\hbox{\,$\vrule height1.5ex width.4pt depth0pt\kern-.3em{\rm Q}$}}
\def\rlx\hbox{\rm I\kern-.18em F}{\relax\leavevmode\hbox{\rm I\kern-.18em F}}
\def\rlx\hbox{\,$\inbar\kern-.3em{\rm G}$}{\relax\leavevmode\hbox{\,$\vrule height1.5ex width.4pt depth0pt\kern-.3em{\rm G}$}}
\def\rlx\hbox{\rm I\kern-.18em H}{\relax\leavevmode\hbox{\rm I\kern-.18em H}}
\def\rlx\hbox{\rm I\kern-.18em I}{\relax\leavevmode\hbox{\rm I\kern-.18em I}}
\def\rlx\hbox{\rm I\kern-.18em K}{\relax\leavevmode\hbox{\rm I\kern-.18em K}}
\def\rlx\hbox{\rm I\kern-.18em L}{\relax\leavevmode\hbox{\rm I\kern-.18em L}}
\def\hbox{{1}\kern-.25em\hbox{l}}{\hbox{{1}\kern-.25em\hbox{l}}}
\def\0#1{\relax\ifmmode\mathaccent"7017{#1}%
B        \else\accent23#1\relax\fi}
\def\0 \omega{\0 \omega}
\def\mathrel{\vrule height1ex}\joinrel\mathrel\times{\mathrel{\vrule height1ex}\joinrel\mathrel\times}
\def\mathrel\times\joinrel\mathrel{\vrule height1ex}{\mathrel\times\joinrel\mathrel{\vrule height1ex}}
\def\noindent{\bf Remark.}\quad{\noindent{\bf Remark.}\quad}
\def\noindent{\bf Proposition.}\quad{\noindent{\bf Proposition.}\quad}
\def\noindent{\bf Theorem.}\quad{\noindent{\bf Theorem.}\quad}
\def\noindent{\bf Definition.}\quad{\noindent{\bf Definition.}\quad}
\def\noindent{\bf Example.}\quad{\noindent{\bf Example.}\quad}
\def\noindent{\bf Proof.}\quad{\noindent{\bf Proof.}\quad}

\begin{titlepage}
\vspace*{-1cm}

\vskip 3cm

\vspace{.2in}
\begin{center}
{\large\bf  Self-Duality in the Context of the  Skyrme Model }
\end{center}

\vspace{.5cm}

\begin{center}
L. A. Ferreira\footnote{laf@ifsc.usp.br} and L. R. Livramento\footnote{leandrorl@ifsc.usp.br}

\vspace{.3 in}
\small

\par \vskip .2in \noindent
Instituto de F\'\i sica de S\~ao Carlos; IFSC/USP;\\
Universidade de S\~ao Paulo, USP  \\
Caixa Postal 369, CEP 13560-970, S\~ao Carlos-SP, Brazil\\

\normalsize
\end{center}

\vspace{.5in}

\begin{abstract}

 We study a recently proposed modification of the Skyrme model that possesses an exact self-dual sector leading to an infinity of exact Skyrmion solutions with arbitrary topological (baryon) charge.  The self-dual sector is made possible by the introduction, in addition to the usual  three $SU(2)$ Skyrme fields, of six scalar fields assembled in a symmetric and invertible three dimensional matrix $h$. The action presents   quadratic and quartic terms in derivatives of the Skyrme fields, but instead of the group indices being contracted by the $SU(2)$  Killing form, they are contracted with the $h$-matrix in the quadratic term, and by its inverse on the quartic term. Due to these extra fields the static version of the model, as well as its self-duality equations, are conformally invariant on the three dimensional space $\rlx\hbox{\rm I\kern-.18em R}^3$.  We show that the static and self-dual sectors of such a theory are equivalent, and so the only non-self-dual solution must be time dependent. We also show that for any configuration of the Skyrme $SU(2)$ fields, the $h$-fields adjust themselves to satisfy the self-duality equations, and so the theory has plenty of non-trivial topological solutions. We present explicit exact solutions using a holomorphic rational ansatz, as  well as a toroidal ansatz based on the conformal symmetry. We point to possible extensions of the model that break the conformal symmetry as well as the self-dual sector, and that can perhaps lead to interesting physical applications.

\end{abstract}
\end{titlepage}

\section{Introduction}
\label{sec:intro}
\setcounter{equation}{0}

The concept of self-duality plays a crucial role in the study of classical solutions of a great variety of field theories, from  kinks in $(1+1)$-dimensions to instantons and magnetic monopoles in four dimensional non-abelian gauge theories. The appearance of self-dual solutions depends crucially on the existence in the theory, of a topological charge admitting an integral representation, i.e. there must exists a local density of topological charge. The self-dual solutions present two main properties: first, they are solutions of self-duality equations which are first order (partial) differential equations that imply the (static) second order Euler-Lagrange equations of the full theory, and second, they saturate a lower bound of the static energy (or Euclidean action) which is related to the modulus of the topological charge. This second property makes the self-dual solutions very stable as they have the lowest  energy in any topological sector of the theory. The fact that the construction of self-dual solutions involve one integration less than the case of the ordinary solutions of the Euler-Lagrange equations, is not related to the use of dynamical conservation laws, but a consequence of the topological structures of the theory. Indeed, the invariance of the topological charge under any smooth (homotopic) variation of the fields leads,  with its integral representation,  to local differential identities that together with the self-duality equations imply the Euler-Lagrange equations \cite{selfdual}.

In this paper we explore the concept of self-duality in the context of the Skyrme model \cite{skyrme}, a $(3+1)$-dimensional field theory with fields taking values on the group $SU(2)$ and presenting a lot of interesting physical applications \cite{mantonbook,shnirbook}. Despite the fact that the Skyrme model does have a topological charge admitting an integral representation, it does not possess a non-trivial self-dual sector \cite{ruback}. The study of the properties of its topological solitons, the Skyrmions, has therefore to rely on numerical methods. Even though that has not prevented a large number of physical applications, the existence of a self-dual sector could shed light on many of the structures of the model.  In the last few years there has appeared some modifications of the Skyrme model that do admit  self-dual sectors. There is the approach, based of self-dual Yang-Mills, that amounts to the coupling an infinite tower of meson fields to the Skyrme model \cite{sutcliffetower} and that has obtained very interesting results for the spectrum of light nuclei \cite{naya1,naya2}. There is the so-called BPS Skyrme model \cite{adam1,adam2} that instead of the usual quadratic and quartic Skyrme terms in derivatives of the fields, it has in its action a term with six derivatives of the fields (the square of the topological current) and a potential term. The Skyrmions in such a model are constructed analytically and are compacton-like solutions.  Some applications in nuclear physics and neutron stars were achieved through such a model \cite{adamprl,adamneutron}. Then there is a modification of the Skyrme model \cite{ua845}, using the ideas of \cite{selfdual}, that  possesses an exact self-dual sector, and when coupled to an extra field presents an infinite number of analytical self-dual solutions   \cite{shnir} with arbitrary values of the Skyrme topological charge.

The model we consider in this paper is the one proposed in \cite{laf}, possessing an exact self-dual sector, and also constructed based on the ideas of \cite{selfdual}. Besides the usual $SU(2)$ group valued fields $U$ of the Skyrme model, it has six scalar fields assembled in a symmetric and invertible matrix $h_{ab}$, $a,b=1,2,3$. The action is similar to that of the original Skyrme model, in the sense it has  quadratic and quartic terms in derivatives of the $U$-fields, but the group indices are not contracted with the $SU(2)$ Killing (trace) form, but with the matrix  $h_{ab}$ in the quadratic term, and its inverse in the quartic term. The model is defined by the action
\begin{equation}
S= \int d^4x\left[ \frac{m_0^2}{2}\, h_{ab}\,R^a_{\mu}\,R^{b\,,\, \mu}-\frac{1}{4\,e_0^2}\, h^{-1}_{ab}\,H^a_{\mu\nu}\,H^{b\,,\,\mu\nu}\right]
\lab{modelintro}
\end{equation}
where, like in the usual Skyrme model, $R^a_{\mu}$ are the components of the Maurer-Cartan form, i.e. $i\,\partial_{\mu}U\,U^{\dagger}\equiv R^a_{\mu}\,T_a$, with $T_a$ being a basis of the $SU(2)$ Lie algebra, and $H^a_{\mu\nu}$ is the curl of that form, i.e. $H^a_{\mu\nu}\equiv \partial_{\mu} R^a_{\nu}-\partial_{\nu} R^a_{\mu}$, and $m_0$ and $e_0$ are coupling constants. Of course, in order to keep the energy of the theory \rf{modelintro} positive definite, we shall restrict the matrix $h_{ab}$ to the cases where its eigenvalues   are positive. If the fields $h_{ab}$ are considered as independent fields, then their presence does not destroy, as we discuss in section \ref{sec:model}, the usual global left and right symmetry $SU(2)_L\otimes SU(2)_R$ of the original Skyrme model \cite{skyrme}. The introduction of the extra fields $h_{ab}$ is motivated by the methods of \cite{selfdual} of constructing theories with self-dual sectors. The method involves the splitting of the density of the topological charge into two pieces and the static energy is built by adding the squares of these quantities. In the splitting process  one is free to attach a matrix to one of the pieces and its inverse to the other piece. The self-duality equations is obtained by imposing the equality, up to a sign, of these two pieces. In the present case the topological charge used in such a  process is the usual Skyrme topological charge associated   to the mappings $\rlx\hbox{\rm I\kern-.18em R}^3 \rightarrow SU(2)$, with the spatial infinity identified to a point, and so with $\rlx\hbox{\rm I\kern-.18em R}^3$ compactified to the $3$-sphere $S^3$. The corresponding self-duality equations are given by
\begin{equation}
m_0\,e_0 \,h_{ab}\,R^b_i=\pm \frac{1}{2}\,\varepsilon_{ijk}\,H^a_{jk}
\lab{selfdualeqsintro}
\end{equation}
with the indices $i,j,k=1,2,3$, corresponding to the spatial Cartesian coordinates $x^i$. Any solution of the nine first order partial differential equations \rf{selfdualeqsintro} also solves not only the static (second order) Euler-Lagrange equations associated to the $SU(2)$ $U$-fields, but also the static Euler-Lagrange equations associated to the extra fields $h_{ab}$. Another interesting point about the presence of the fields $h_{ab}$ is that they render the static energy, associated to the theory \rf{modelintro}, and the self-duality equations \rf{selfdualeqsintro}, invariant under conformal transformations in the three dimensional spatial sub-manifold $\rlx\hbox{\rm I\kern-.18em R}^3$. As a consequence, the self-dual Skyrmions do not have a fixed size, but can be re-scaled without changing its energy and topological charge. However, the equality of the contributions to the static energy coming from the quadratic and quartic terms, which in the original Skyrme model is implied by Derrick's scaling argument  \cite{derrick}, here is a consequence of the static Euler-Lagrange equations associated to the fields $h_{ab}$, as we discuss in section \ref{sec:model}.

In \cite{laf} it was constructed analytical self-dual Skyrmions of unity topological charge for the theory \rf{modelintro}, in the case where the matrix $h_{ab}$ is proportional to the unity matrix, and so one has just one extra field. In this paper we analyse the properties of the theory \rf{modelintro} in its full generality. The first two important results that we obtain are the following. The static sector of the theory \rf{modelintro} is the same as its self-dual sector, i.e. any static solution of \rf{modelintro} is a solution of the self-duality equations \rf{selfdualeqsintro}. The static Euler-Lagrangian equations associated to the fields $h_{ab}$ play a crucial role in establishing such a result, and in fact they are in some sense equivalent to the self-duality equations \rf{selfdualeqsintro}. In addition, the fields $h_{ab}$ are just spectators in the self-dual or static sector, in the sense that given any configuration for the $SU(2)$ $U$-fields, the fields $h_{ab}$ adjust themselves to satisfy all the nine self-duality equations \rf{selfdualeqsintro}. In fact, for any self-dual solution,    the fields $h_{ab}$ can be written as
\begin{equation}
h=\frac{\sqrt{{\rm det}\,\tau}}{\mid m_0\,e_0\mid}\; \tau^{-1}
\lab{goodrelhtauintro}
\end{equation}
where $\tau$ depends only on the $U$-fields and it is given by  $\tau_{ab}= R_{i}^a\,R_{i}^b$.  Note that $\tau$ is similar to the strain tensor of the Skyrme model given by $D_{ij}=R_i^a\,R_j^a$, and indeed we show that their eigenvalues are the same. Therefore, the self-duality equations \rf{selfdualeqsintro} has plenty of analytical solutions, and we analyse two types of such solutions. We construct explicitly self-dual solutions for any value of the topological charge using the holomorphic rational ansatz for the $SU(2)$ $U$-fields \cite{mantonbook}. For those  configurations we construct the matrix $h_{ab}$, in particular its eigenvalues $\varphi_a$. It turns out that the first two eigenvalues $\varphi_1$ and $\varphi_2$, are equal and spherically symmetric. They have its maximum value at the origin and decay to zero at infinity. The third eigenvalue $\varphi_3$ decomposes into the product of two pieces, one  depending on the radial distance only, and the other on the  polar and azimuthal angles. The radial part resembles very much the form of $\varphi_1$ and $\varphi_2$, and the angular part is proportional to the Wronskian of the two holomorphic polynomials entering in the rational map. As usual the topological charge of the solution is determined by the degrees of these polynomials.

We then use the conformal symmetry of the static version of the theory \rf{modelintro}, to construct an ansatz involving the toroidal coordinates in $\rlx\hbox{\rm I\kern-.18em R}^3$. The symmetries of the ansatz are such that the dependence upon the two toroidal angles are explicitly given and the matrix $h_{ab}$ is given in terms of a profile function of the third toroidal coordinate, which is  left undetermined. Again the eigenvalues of $h$ have a maximum at the origin and decay to zero at spatial infinity. The topological charge is equal to the product of two integers associated in the ansatz, with the two toroidal angles. We have an infinity of analytical solutions with arbitrary topological charges.

The construction of the model \rf{modelintro}, based on the  ideas of \cite{selfdual}, is such that the scalar fields $h_{ab}$ are introduced without a kinetic term, i.e. they are not propagating fields. In addition, it makes the static version of the theory \rf{modelintro} conformally invariant which is not very suitable for physical applications. Therefore, in order to make the model \rf{modelintro} more realistic we have to introduce a kinetic term for the fields $h_{ab}$ and break the conformal symmetry. The addition of the kinetic term alone is sufficient to break that symmetry explicitly, but one can also add  mass and  potential terms for those fields as well. The self-dual sector is also lost with the addition of such terms, but that might be  desirable for some physical applications. For instance, in applications to nuclear physics, the topological charge of the Skyrme model is interpreted as the baryon number and the Skyrmions as nuclei. In a self-dual theory the static energy of the Skyrmion is proportional  to the topological charge, and so there is no binding energy.   We have checked that the addition of the kinetic, mass and potential terms to the model \rf{modelintro} brings a positive binding energy, at least for small barion numbers. Therefore,  such modifications of the theory \rf{modelintro} seems to be promissing in applications to nuclear physics and we shall report elsewhere \cite{tocome} the numerical results we have obtained on those lines.

The paper is organised as follows. In section \ref{sec:model} we discuss in details the properties of the model \rf{modelintro}, its symmetries, the static and self-dual sectors. In section \ref{sec:ansatzholom} we present the self-dual solutions using the rational map, and in section \ref{sec:ansatztoro} the self-dual solution in a toroidal ansatz based on the conformal symmetry. The appendix \ref{app:conformal} contains the proof of the conformal invariance of the static version of the theory \rf{modelintro}, and in appendix \ref{app:proof} we prove some results important in  establishing the equivalence of the static and self-dual sectors. The conclusions are presented in section  \ref{sec:conclusion}.

\section{The description of the model}
\label{sec:model}
\setcounter{equation}{0}

We consider in this paper the Skyrme-type model proposed in \cite{laf}, on a four dimensional Minkowski space-time, with the metric signature as $ds^2=dx_0^2-dx_i^2$, $i=1,2,3$, defined by the action
\begin{equation}
S= \int d^4x\left[ \frac{m_0^2}{2}\, h_{ab}\,R^a_{\mu}\,R^{b\,,\, \mu}-\frac{1}{4\,e_0^2}\, h^{-1}_{ab}\,H^a_{\mu\nu}\,H^{b\,,\,\mu\nu}\right]
\lab{model}
\end{equation}
where $m_0$ (of dimension $\({\rm length}\)^{-1}$) and $e_0$ (dimensionless) are coupling constants, and  it is based on the $SU(2)$ Lie algebra with
generators $T_a$, $a=1,2,3$,  satisfying
\begin{equation}
\sbr{T_a}{T_b}=i\,\varepsilon_{abc}\,T_c \; ;\; \qquad\qquad\qquad\qquad
{\rm Tr}\(T_a\,T_b\)=\kappa\, \delta_{ab}
\lab{su2killing}
\end{equation}
where $\kappa$ is a constant that depends upon the representation ($\kappa=1/2$ for the spinor representation, and $\kappa=2$ for the triplet (adjoint) representation). We shall use a normalized trace, independent of the representation, as follows
\begin{equation}
\widehat{\rm Tr}\(T_a\,T_b\)=\frac{1}{\kappa}\,{\rm Tr}\(T_a\,T_b\)=\delta_{ab}
\lab{normtrace}
\end{equation}
In addition,
$R^a_{\mu}$, $\mu=0,1,2,3$, are the components of the $SU(2)$ Maurer-Cartan form given by
\begin{equation}
R_{\mu} \equiv i\,\partial_{\mu}U\,U^{\dagger}\equiv R^a_{\mu}\,T_a\;; \qquad\qquad\qquad
R^a_{\mu}=i\, \widehat{\rm Tr}\(\partial_{\mu}U\,U^{\dagger}\,T_a\)
\lab{rdef}
\end{equation}
with $U$ being an element of the group $SU(2)$. The quantities $H_{\mu\nu}^a$ correspond to the curl of  $R_{\mu}$, and since these satisfy the Maurer-Cartan equation,  i.e.
\begin{equation}
\partial_{\mu}R_{\nu}-\partial_{\nu}R_{\mu}+i\,\sbr{R_{\mu}}{R_{\nu}}=0
\lab{maurercartaneq}
\end{equation}
we  have that
\begin{equation}
H^a_{\mu\nu}\equiv \partial_{\mu} R^a_{\nu}-\partial_{\nu} R^a_{\mu}
=-i\, \widehat{\rm Tr}\(\sbr{R_{\mu}}{R_{\nu}}\,T_a\)=\varepsilon_{abc}\,R_{\mu}^b\,R_{\nu}^c
\lab{hdef}
\end{equation}
The theory \rf{model} differs from the original Skyrme model  \cite{skyrme,mantonbook} by the fact that the group indices are not contracted by the $SU(2)$ Killing form but instead by the symmetric matrix $h_{ab}$ on the quadratic term in derivatives and by its inverse on the quartic term. We shall consider the six entries of  that symmetric and invertible matrix  as extra fields of the theory, that  transform as scalars fields under the Lorentz  group. Clearly, we shall be concerned with  the cases where the eigenvalues of the matrix $h_{ab}$ are positive definite in order for the energy of the theory \rf{model} to be positive definite.

Since the theory \rf{model} does not have local gauge symmetries,  in order to have finite energy solutions the fields have to go to a constant vacuum configuration at spatial infinity. Then as long as topological considerations are concerned we can compactify the space $\rlx\hbox{\rm I\kern-.18em R}^3$ into the three-sphere $S^3$, and the fields $U$ define mappings $S^3 \rightarrow SU(2)\equiv S^3$. The corresponding topological charge (winding number)  is the same as in the usual Skyrme theory, i.e.
\begin{equation}
 Q= \frac{i}{48\,\pi^2}\int d^3x\; \varepsilon_{ijk}\,\widehat{\rm Tr}\(R_i\,R_j\,R_k\)
 \lab{topcharge}
 \end{equation}

The Euler-Lagrange equations, obtained from \rf{model},  associated to the $SU(2)$ $U$-fields,  are given by
\begin{equation}
\partial_{\mu}\left[m_0^2\,e_0^2\,h_{ab}\,R^{b,\mu}-\varepsilon_{abc}\,R^b_{\nu}\,h_{cd}^{-1}\,H^{d,\mu\nu}\right]=\varepsilon_{abc}\left[m_0^2\,e_0^2\,R_{\mu}^b\,h_{cd}\,R^{d,\mu}+\frac{1}{2}\,H_{\mu\nu}^c\,h_{bd}^{-1}\,H^{d,\mu\nu}\right]
\lab{elforu}
\end{equation}
and the Euler-Lagrange equations associated to the fields $h_{ab}$ are given by
\begin{equation}
m_0^2\,e_0^2\,R_{\mu}^a\,R^{b,\mu}+\frac{1}{2}\,h_{ac}^{-1}\,H_{\mu\nu}^c\,h_{bd}^{-1}\,H^{d,\mu\nu}=0
\lab{elforh}
\end{equation}
Note that by contracting \rf{elforh} with $h_{ca}$ one gets  that the r.h.s. of \rf{elforu} must vanish. Therefore, one gets three conserved currents ($a=1,2,3$)
\begin{equation}
\partial_{\mu}J^{\mu}_a=0\;;\qquad \qquad {\rm with} \qquad \qquad
J^{\mu}_a=m_0^2\,e_0^2\,h_{ab}\,R^{b,\mu}-\varepsilon_{abc}\,R^b_{\nu}\,h_{cd}^{-1}\,H^{d,\mu\nu}
\lab{conscurr}
\end{equation}

The action \rf{model} is invariant under the global symmetry $SU(2)_L\otimes SU(2)_R$ defined by the transformations
\begin{equation}
U\rightarrow g_L\,U\;;\qquad\qquad R_{\mu}^a\rightarrow d_{ab}\(g_L\)\,R_{\mu}^b \;;\qquad\qquad
h_{ab}\rightarrow d_{ac}\(g_L\)\, h_{cd}\,d_{db}^T\(g_L\)
\lab{leftsymm}
\end{equation}
and
\begin{equation}
U\rightarrow U\,g_R\;;\qquad\qquad\qquad R_{\mu}^a\rightarrow R_{\mu}^a \;;\qquad\qquad \qquad
h_{ab}\rightarrow h_{ab}
\lab{rightsymm}
\end{equation}
with $g_L\,,\,g_R\in SU(2)$, and where $d\(g\)$ is the $3\times 3$ matrix for the group element $g$ in the adjoint representation of $SU(2)$, i.e.
\begin{equation}
g\,T_a\, g^{-1}=T_b\,d_{ba}\(g\)\;;\qquad\qquad\qquad d\(g_1\)\,d\(g_2\)=d\(g_1\,g_2\)
\lab{adjoint}
\end{equation}
The Noether currents associated to the left symmetry \rf{leftsymm} are those given in \rf{conscurr}, and the Noether currents associated to the right symmetry \rf{rightsymm} are given by
\begin{equation}
{\tilde J}_{\mu}^a= J_{\mu}^b\,d_{ba}\(U\) ;\qquad\qquad\qquad {\rm with}\qquad\qquad
\partial^{\mu}{\tilde J}_{\mu}^a=0
\lab{rightcurr}
\end{equation}
The conservation of the left and right currents, given in \rf{conscurr} and \rf{rightcurr} respectively, imply that
\begin{equation}
J^{\mu}_b\,\partial_{\mu}d_{ba}\(U\)=0
\lab{funnyrel}
\end{equation}
The relation \rf{funnyrel} is a consequence of the Euler-Lagrange equations \rf{elforh} for the fields $h_{ab}$. Indeed, using the fact that the adjoint representation is unitary and real and so the matrices $d_{ab}$ are orthogonal, i.e. $d^T\(U\)=d\(U^{\dagger}\)$, one can show that
\begin{eqnarray}
J^{\mu}_b\,\partial_{\mu}d_{ba}\(U\)\,T_a&=&
-i\, U^{\dagger}\sbr{J^{\mu}_a\, T_a}{R_{\mu}}\,U
\\
&=& \varepsilon_{abc}\,U^{\dagger}\,T_c\,U\left[ m_0^2\,e_0^2\,h_{ad}\,R^{d\,,\,\mu}\,R^b_{\mu}+
\frac{1}{2}h^{-1}_{bd}H^{d\,,\,\mu\nu}\,H^a_{\mu\nu}\right]
\nonumber
\end{eqnarray}
and that vanishes as a consequence of \rf{elforh}.

Note that the introduction of the fields $h_{ab}$ in the action \rf{model}, does not destroy the target space symmetries of  the original Skyrme model, namely $SU(2)_L\otimes SU(2)_R$. One still has six (left and right) conserved currents, and one can say, like in the original Skyrme model, that the equations of motion are equivalent to the conservation of these currents. Indeed, \rf{elforu} and \rf{elforh} follow from \rf{conscurr} and \rf{rightcurr}.  The introduction of the fields $h_{ab}$, however, brings two new structures to the  the theory \rf{model} as compared to the original Skyrme model. It allows the existence of an exact self-dual (BPS) sector, as we explain below in section \ref{subsec:bps}, and it renders the theory \rf{model} conformally invariant in the three dimensional space, as we explain in the appendix  \ref{app:conformal}. Indeed, one can check that the static energy associated to \rf{model}, given by
\begin{equation}
 E=\int d^3 x \left[ \frac{m_0^2}{2}\, h_{ab}\,R^a_{i}\,R^{b}_{i}+\frac{1}{4\,e_0^2}\, h^{-1}_{ab}\,H^a_{ij}\,H^{b}_{ij}\right]
 \lab{staticenergy}
\end{equation}
is invariant under the conformal transformations in three dimensions (see \rf{densitiestransf}). In addition, the static version of the equations of motion \rf{elforu} and \rf{elforh}, are also conformally invariant (see \rf{eqofmotionconformal}).

Note that by contracting the Euler-Lagrange equation \rf{elforh} with $h_{ab}$, one gets that the two terms of the Lagrangian density in \rf{model} must be equal on-shell, i.e.
\begin{equation}
\frac{m_0^2}{2}\, h_{ab}\,R^a_{\mu}\,R^{b\,,\, \mu}=-\frac{1}{4\,e_0^2}\, h^{-1}_{ab}\,H^a_{\mu\nu}\,H^{b\,,\,\mu\nu}
\end{equation}
For static configurations that implies that the two terms in the energy density in \rf{staticenergy} are also equal
\begin{equation}
 \frac{m_0^2}{2}\, h_{ab}\,R^a_{i}\,R^{b}_{i}=\frac{1}{4\,e_0^2}\, h^{-1}_{ab}\,H^a_{ij}\,H^{b}_{ij}
 \lab{fakederrick}
 \end{equation}
 In the original Skyrme model, where $h_{ab}$ is the identity matrix, a relation equivalent to
 \rf{fakederrick} is obtained by Derrick's scaling argument \cite{derrick}.   The static energy \rf{staticenergy} is invariant under the scaling of the coordinates $x_i\rightarrow \alpha\, x_i$, due to the conformal symmetry. So, the static solutions of the theory \rf{model} do not have a fixed size, as the original Skyrmions do. However, the balance between the quadratic and quartic terms in space derivatives of the energy density of the theory \rf{model}, is provided by the Euler-Lagrange equations for the fields $h_{ab}$ as shown in \rf{fakederrick}.

 \subsection{The static sector}
\label{subsec:static}

Another important role of the static Euler-Lagrange equations  for $h$ \rf{elforh} is to relate  $h$ to a real and symmetric matrix given by
\begin{equation}
\tau_{ab}\equiv R_{i}^a\,R_{i}^b
\lab{ourstraintensor}
\end{equation}
where the sum is over  the space index $i=1,2,3$, only. That matrix   is similar to the usual strain tensor of the Skyrme model defined by \cite{mantonstrain,mantonbook}
\begin{equation}
D_{ij}=R_i^a\,R_j^a
\lab{mantonstrain}
\end{equation}
Note that
\begin{equation}
{\rm det}\,\tau={\rm det}\,D\;;\qquad\qquad\qquad\qquad {\rm Tr}\,\tau^n={\rm Tr}\,D^n
\end{equation}
and so the eigenvalues of $\tau$ and $D$ are the same. Now if $v_a$ is an arbitrary real vector then
\begin{equation}
v^T\, \tau\, v= \sum_{i=1}^3\(v_a\,R_i^a\)^2 \geq 0
\lab{positivetau}
\end{equation}
Then $\tau$ is a positive semidefinite matrix and so its eigenvalues are all non-negative. At the same time the eigenvalues of the matrix $h$ can not vanish as it has to be invertible, and they have to be all positive for the energy of the theory \rf{model} to be positive definite. As a consequence of the static Euler-Lagrange equations  for $h$ \rf{elforh}, we then get the following results. Consider a domain ${\cal D}$ in the three dimensional space $\rlx\hbox{\rm I\kern-.18em R}^3$. If the matrix $\tau$ is singular in ${\cal D}$, then the $SU(2)$ fields $U$ have to be constant in ${\cal D}$, i.e.
\begin{equation}
{\rm det}\,\tau=0 \qquad\qquad \rightarrow \qquad\qquad U={\rm constant}\qquad\qquad {\rm for}\qquad x^i\in {\cal D}
\lab{uconstantresult}
\end{equation}
and so the whole matrix $\tau$ vanishes (see \rf{rdef} and \rf{ourstraintensor}).
On the other hand if none of the eigenvalues of $\tau$ vanish in ${\cal D}$, i.e. ${\rm det}\,\tau\neq 0$, then  it follows that the matrices $h$ and $\tau$  can be diagonalised by the same orthogonal matrix $M$ on that domain, i.e.
\begin{equation}
\(M^T\,h\,M\)_{ab}= \varphi_a\,\delta_{ab}\;;\qquad\qquad \qquad \qquad \(M^T\,\tau\,M\)_{ab}= \omega_a\,\delta_{ab}
\lab{eigenvalueshtau}
\end{equation}
and so they commute in ${\cal D}$, i.e. $\sbr{h}{\tau}=0$.
In addition, their eigenvalues are related by
\begin{equation}
\omega_1=m_0^2\,e_0^2 \,\varphi_2\,\varphi_3\;;\qquad\qquad
\omega_2=m_0^2\,e_0^2 \,\varphi_1\,\varphi_3\;;\qquad\qquad
\omega_3=m_0^2\,e_0^2 \,\varphi_1\,\varphi_2
\lab{eigenvalueshtaurel}
\end{equation}
The proof of the relations \rf{uconstantresult}, \rf{eigenvalueshtau} and  \rf{eigenvalueshtaurel} is given in Appendix \ref{app:proof}.

Note that the matrix $\tau$, defined in \rf{ourstraintensor}, transforms under the action of the group $SU(2)_L\otimes SU(2)_R$, given in \rf{leftsymm} and \rf{rightsymm}, in the same way as the matrix $h$. Therefore, the eigenvalues of $h$ and $\tau$ are invariant under the group $SU(2)_L\otimes SU(2)_R$. Indeed, if $\ket{v}$ is an eigenvector of $h$  with eigenvalue $\varphi$, i.e. $h\ket{v}=\varphi\ket{v}$, then $d\(g_L\)\ket{v}$ is an eigenvector of the transformed matrix $d\(g_L\)h\,d\(g_L\)^T$, with the same eigenvalue $\varphi$. The same applies to the matrix $\tau$.

The relations \rf{eigenvalueshtau} and  \rf{eigenvalueshtaurel} have some important consequences for the structure of the static solutions of the model \rf{model}. Using \rf{eigenvalueshtau} and  \rf{eigenvalueshtaurel} we can write that
\begin{equation}
\(M^T\,\tau^{-1}\,M\)_{ab}=\frac{1}{m_0^2\,e_0^2 \,\varphi_1\,\varphi_2\,\varphi_3}\; \varphi_a\,\delta_{ab}
\end{equation}
and so  the matrices $h$ and $\tau^{-1}$ are proportional in ${\cal D}$
\begin{equation}
h=m_0^2\,e_0^2\, \({\rm det}\,h\)\; \tau^{-1}=\frac{\sqrt{{\rm det}\,\tau}}{\mid m_0\,e_0\mid}\; \tau^{-1}
\lab{goodrelhtau}
\end{equation}
From \rf{rdef} and \rf{ourstraintensor} we observe that the matrix $\tau$ is a functional of the $SU(2)$  fields $U$ and their first derivatives only. Therefore, what the relation \rf{goodrelhtau} is telling us is that for static solutions of the theory \rf{model}, and in regions where the matrix $\tau$ is non-singular, the fields $h_{ab}$ are explicitly written in terms of the $U$ fields and their first derivatives without any integration needed. Note that this is a consequence of only the static Euler-Lagrange equations for the fields $h$ \rf{elforh}, and the positivity of the eigenvalues of $h$ and $\tau$. In the cases where the matrix $\tau$ is singular, it is also possible to find a matrix $h$ that solves the equations \rf{elforh} (see subsection \ref{subsec:tausing} of appendix \ref{app:proof}). We have shown that by using the Euler-Lagrange equations for $h$ \rf{elforh}, one can write the Euler-Lagrange equation for $U$ \rf{elforu} as in \rf{conscurr}. Then using \rf{goodrelhtau} one can express, in regions where $\tau$ is non-singular, the static version of \rf{conscurr} only in terms of the $U$ fields as
\begin{equation}
\partial_{i}J_{i}^a=0\;;\qquad \qquad {\rm with} \qquad \qquad
J_{i}^a=\sqrt{{\rm det}\,\tau}\, \tau^{-1}_{ab}\,R_i^{b}+\frac{m_0^2\,e_0^2}{\sqrt{{\rm det}\,\tau}}\;\varepsilon_{abc}\,R^b_{j}\,\tau_{cd}\,H^{d}_{ij}
\lab{conscurrstatic}
\end{equation}

An interesting consequence of the equation \rf{goodrelhtau}  is that it implies that the determinant of the matrix $h_{ab}$ is proportional to the density of topological charge. In order to see that, let us treat the quantities $R_i^a$, introduced in \rf{rdef}, as a $3\times 3$ matrix with the following ordering of rows and columns $R_i^a\equiv \(R\)_{ia}$, $i=1,2,3$ and $a=1,2,3$.  We then have that
 \begin{equation}
 \varepsilon_{ijk}\,R_i^a\,R_j^b\,R_k^c=\varepsilon_{abc}\,\varepsilon_{ijk}\,R_{i1}\,R_{j2}\,R_{k3}=\varepsilon_{abc}\,{\rm det} R
 \lab{nicedet}
 \end{equation}
 Using \rf{su2killing} and \rf{nicedet} one then gets that
 \begin{equation}
 \varepsilon_{ijk}\,\widehat{\rm Tr}\(R_i\,R_j\,R_k\)=i\, 3\, {\rm det} R
 \lab{detrtopcharge}
 \end{equation}
 Then from \rf{topcharge} one gets
 \begin{equation}
 Q= -\frac{1}{16\,\pi^2}\int d^3x\; {\rm det} R
 \lab{topcharger}
 \end{equation}
 From  \rf{goodrelhtau} it follows that, in regions where $\tau$ is non-singular,  ${\rm det}\,\tau= \(m_0^2\,e_0^2\)^3\,\({\rm det}\, h\)^2$, and \rf{ourstraintensor} implies ${\rm det}\,\tau=\({\rm det}\,R\)^2$. Therefore
 \begin{equation}
 {\rm det}\,h={\rm det}\,R/\lambda^3\qquad\qquad\qquad {\rm with}\qquad\qquad \lambda=\pm m_0\,e_0
 \lab{nicerhrel}
 \end{equation}
 In order for the energy of the theory \rf{model} to be positive definite we need the eigenvalues of $h$ to be all positive, i.e.   ${\rm det}\,h>0$. Therefore, we conclude that
 \begin{equation}
 {\rm sign}\(\lambda\, {\rm det}\,R\)=+1
 \lab{signlambdadetr}
 \end{equation}
 and so ${\rm det}\,R$ has to have the same sign in all points in $\rlx\hbox{\rm I\kern-.18em R}^3$.
Therefore, in regions where $\tau$ is non-singular, from \rf{detrtopcharge} one gets that
\begin{equation}
{\rm det}\,h=-\frac{i}{3\,\lambda^3}\,\varepsilon_{ijk}\,\widehat{\rm Tr}\(R_i\,R_j\,R_k\)
\lab{dethtopchargerel}
\end{equation}
If $\tau$ is singular only in zero measure sets, it follows that the topological charge \rf{topcharge} is given by
\begin{equation}
Q=-\frac{\lambda^3}{16\,\pi^2}\,\int d^3x\, {\rm det}\,h
\lab{topchargedeth}
\end{equation}
Since we need ${\rm det}\,h>0$,  for the energy of the theory \rf{model} to be positive definite, we need
 \begin{equation}
 {\rm sign}\(\lambda\,Q\)=-1
 \lab{signlambdaq}
 \end{equation}

The relation \rf{goodrelhtau} between the matrices $h$ and $\tau$, valid in regions where $\tau$ is non-singular,  shows that the matrix $h$ does give a measure of the strain of the map from the compactfied three dimensional space $S^3$ to $SU(2)$.
In addition, we can write the quadratic and quartic terms of the static energy \rf{staticenergy} as functions of these eigenvalues, Indeed, from \rf{fakederrick} and \rf{eigenvalueshtaurel} we get that
\begin{equation}
 h_{ab}\,R^a_{i}\,R^{b}_{i}=\frac{1}{2\,\lambda^2}\, h^{-1}_{ab}\,H^a_{ij}\,H^{b}_{ij}={\rm Tr}\(h\,\tau\)
 =3\,\lambda^2\,\varphi_1\,\varphi_2\,\varphi_3
 \lab{bpsdensity}
  \end{equation}
 Therefore, the static energy density in \rf{staticenergy} evaluated on a given static solution of the theory \rf{model}  is
\begin{equation}
{\cal E}_{{\rm static}}\equiv \frac{m_0^2}{2}\, h_{ab}\,R^a_{i}\,R^{b}_{i}+\frac{1}{4\,e_0^2}\, h^{-1}_{ab}\,H^a_{ij}\,H^{b}_{ij}=m_0^2\,\lambda^2\;3\,\varphi_1\,\varphi_2\,\varphi_3=\frac{3\,m_0^2}{\mid\lambda\mid}\,\sqrt{\omega_1\,\omega_2\,\omega_3}
\end{equation}
On the other hand, using \rf{hdef}, \rf{mantonstrain}, \rf{eigenvalueshtau} and \rf{eigenvalueshtaurel}, one gets that the static energy for the usual Skyrme, evaluated on a static solution of the theory \rf{model},  is given by
\begin{eqnarray}
{\cal E}_{{\rm Skyrme}}&\equiv& \frac{m_0^2}{2}\,R^a_{i}\,R^{a}_{i}+\frac{1}{4\,e_0^2}\, \,H^a_{ij}\,H^{a}_{ij}=\frac{m_0^2}{2}\,\left[{\rm Tr} D +\frac{1}{2\,\lambda^2}\(\({\rm Tr}D\)^2-{\rm Tr}\,D^2\)\right]
\nonumber\\
&=&\frac{m_0^2}{2}\,\left[\omega_1+\omega_2+\omega_3+\frac{1}{\lambda^2}\(\omega_1\,\omega_2+\omega_2\,\omega_3+\omega_1\,\omega_3\)\right]
\nonumber\\
&=&\frac{m_0^2\,\lambda^2}{2}\,\left[\varphi_1\,\varphi_2+\varphi_2\,\varphi_3+\varphi_1\,\varphi_3+\varphi_1\,\varphi_2\,\varphi_3\,\(\varphi_1+\varphi_2+\varphi_3\)\right]
\lab{skyrmedensity}
\end{eqnarray}
where in the last equality we have assumed the relation \rf{eigenvalueshtaurel}, i.e. that the field configuration satisfy the static Euler-Lagrange equations for $h$ \rf{elforh}. Therefore, we can write that
\begin{eqnarray}
\Delta {\cal E}&\equiv& {\cal E}_{{\rm Skyrme}}-{\cal E}_{{\rm static}}
\nonumber\\
&=&\frac{m_0^2}{2}\,\left[
\(\sqrt{\omega_1}-\frac{\sqrt{\omega_2\,\omega_3}}{\mid\lambda\mid}\)^2
+\(\sqrt{\omega_2}-\frac{\sqrt{\omega_1\,\omega_3}}{\mid\lambda\mid}\)^2
+\(\sqrt{\omega_3}-\frac{\sqrt{\omega_1\,\omega_2}}{\mid\lambda\mid}\)^2
\right]
\nonumber\\
&=&\frac{m_0^2\,\lambda^2}{2}\,\left[\(1-\varphi_1\)^2\,\varphi_2\,\varphi_3+
\(1-\varphi_2\)^2\,\varphi_1\,\varphi_3+\(1-\varphi_3\)^2\,\varphi_1\,\varphi_2\right]
\lab{deltaenergy}
\end{eqnarray}
where again  in the last equality we have assumed the relation \rf{eigenvalueshtaurel}.
Consequently, $\Delta {\cal E}$ is positive if the eigenvalues of the matrix $h$ are positive, and so if the energy of the theory \rf{model} is positive definite. On the domain where all $\varphi_a$'s are positive we have three extrema for $\Delta{\cal E}$. The points $\varphi_a=0$ and $\varphi_a=1$, for $a=1,2,3$, are minima and the point $\varphi_a=1/2$, $a=1,2,3$, is a saddle point.

\subsection{The self-dual sector}
\label{subsec:bps}

As explained in \cite{laf}, the introduction of the matrix $h_{ab}$ allows the existence of an exact self-dual sector, and makes the theory conformally invariant in three dimensional spatial sub manifold. The self-duality equations, defining that exact-self-dual sector are given by
\begin{equation}
\lambda \,h_{ab}\,R^b_i=\frac{1}{2}\,\varepsilon_{ijk}\,H^a_{jk}\;;\qquad\qquad\qquad \qquad\lambda=\pm\,m_0\,e_0
\lab{selfdualeqs}
\end{equation}
Note that the self-duality equations \rf{selfdualeqs} can be written in $3$-vector space notation as
\begin{equation}
{\vec \nabla}\wedge {\vec R}_a =\lambda\, h_{ab}\,{\vec R}_b
\lab{genforcefree}
\end{equation}
which is a generalization, to several vectors, of the well known force-free equation for magnetic fields in solar and plasma physics \cite{marsh,chandra,ua845,shnir}.

The solutions of the first order equations \rf{selfdualeqs} not only solve  the static second order Euler-Lagrange equations \rf{elforu},   associated to the $SU(2)$ $U$-fields, but also solve those associated to the six extra fields $h_{ab}$, given in \rf{elforh}. Note that  the static version of \rf{elforh} can be written as
\begin{equation}
\left[\mid \lambda\mid \,R_i^a-\frac{1}{2}\,h^{-1}_{ac}\,\varepsilon_{ijk}\,H_{jk}^c\right]\,
\left[\mid \lambda\mid \,R_i^b+\frac{1}{2}\,h^{-1}_{bd}\,\varepsilon_{ilm}\,H_{lm}^d\right]=0
\lab{factorstaticeqomh}
\end{equation}
as the crossed terms cancel each other. Indeed, using \rf{nicedet} and \rf{hdef} it follows that, $h^{-1}_{ac}\,\varepsilon_{ijk}\,H_{jk}^c\,R_i^b=h^{-1}_{ac}\,\varepsilon_{cde}\,\varepsilon_{ijk}\,R_i^b\,R_j^d\,R_k^e=2\, h^{-1}_{ab}\, {\rm det}\,R$. Similarly, $h^{-1}_{bd}\varepsilon_{ilm}H_{lm}^d R_i^a=2h^{-1}_{ab} {\rm det}R$, and so they indeed cancel each other. Then \rf{factorstaticeqomh} is the same as the static version of \rf{elforh}, and \rf{selfdualeqs} does imply \rf{factorstaticeqomh}.
 Using \rf{selfdualeqs} twice, one gets that
 \begin{equation}
 \varepsilon_{abc}\,\lambda^2\, R_{i}^b\,h_{cd}\,R^{d}_{i}= \frac{1}{4}\, \varepsilon_{abc}\,\varepsilon_{ijk}\,h_{bd}^{-1}\,H^d_{jk}\,\varepsilon_{ilm}\,H^c_{lm}=\frac{1}{2}\, \varepsilon_{abc}\,h_{bd}^{-1}\,H^d_{jk}\,H^c_{jk}
 \end{equation}
  and that implies that the static version of r.h.s. of \rf{elforu} vanishes. Now contract the static version of the l.h.s. of \rf{elforu} with $T_a$, and use \rf{selfdualeqs} to get
\begin{eqnarray}
&\partial_i&\left[\lambda^2\, h_{ab}\,R^{b}_i+\varepsilon_{abc}\,R^b_{j}\,h_{cd}^{-1}\,H^{d}_{ij}\right]\,T_a=\frac{\lambda}{2}\,\varepsilon_{ijk}\,\partial_i\left[H^a_{jk}+2\,\varepsilon_{abc}\,R^b_{j}\,R^c_k\right]\,T_a
\nonumber\\
&=&-i\,\frac{3\,\lambda}{2}\,\varepsilon_{ijk}\,\partial_i \sbr{R_j}{R_k}=\frac{3\,\lambda}{2}\,\varepsilon_{ijk}\,\sbr{\sbr{R_i}{R_j}}{R_k}=0
\end{eqnarray}
where we have used the fact that $R_i$ satisfies the Maurer-Cartan equation \rf{maurercartaneq}, and in the last equality we have used the Jacobi identity. So, \rf{selfdualeqs} does imply the static versions of \rf{elforu} and \rf{elforh}.

The solutions of \rf{selfdualeqs}  saturate a Bogomolny-type bound of the static energy associated to \rf{model}.  Indeed, one can write the  static energy  \rf{staticenergy} as
 \begin{eqnarray}
E =\frac{1}{2\,e_0^2}\,\int d^3 x \left[ \lambda\, R_i^b\,k_{ba}-\frac{1}{2}\,k^{-1}_{ab}\,\varepsilon_{ijk}\,H_{jk}^b\right]^2-{\rm sign}\(\lambda\)\,\frac{\mid m_0\mid}{\mid e_0\mid}\,48\,\pi^2\,Q
\lab{prebpsenergy}
 \end{eqnarray}
where $Q$ is the topological charge, given in \rf{topcharge},  and where we have written the matrix $h$ in terms of a real and invertible matrix $k$ as
 \begin{equation}
 h=k\,k^T
 \lab{haskk}
 \end{equation}
 Note that if $O$ is an orthogonal $3\times 3$ matrix ($O\,O^T=\hbox{{1}\kern-.25em\hbox{l}}$), then $k$ and $k\,O$ give the same $h$. Such a freedom accounts for the three extra entries of $k$ as compared with those of $h$, since $O$ has three independent entries. In fact, the theory \rf{model} was constructed in \cite{laf} using arguments of \cite{selfdual} based on a splitting of the density of the topological charge \rf{topcharge}. That splitting procedure implies that the matrix $h$ has the form \rf{haskk}.     Therefore, using \rf{signlambdaq} we get that the self-dual solutions saturate the bound of the static energy, and in that case it is given by
 \begin{eqnarray}
 E_{{\rm BPS}}=m_0^2\, \int d^3 x \,  h_{ab}\,R^a_{i}\,R^{b}_{i}=
\frac{1}{2\,e_0^2}\,\int d^3 x\,  h^{-1}_{ab}\,H^a_{ij}\,H^{b}_{ij}=\frac{\mid m_0\mid}{\mid e_0\mid}\,48\,\pi^2\,\mid Q\mid
\lab{bpsenergy}
 \end{eqnarray}
 We now want to show that the static Euler-Lagrange equations for the fields $h_{ab}$ \rf{elforh} imply the self-duality equations \rf{selfdualeqs}. Let us denote
 \begin{equation}
 S_i^{(\pm),a}\equiv \mid \lambda\mid \,R_i^a\pm \frac{1}{2}\,h^{-1}_{ac}\,\varepsilon_{ijk}\,H_{jk}^c
 \lab{siapmdef}
 \end{equation}
 Then the static Euler-Lagrange equations for the fields $h_{ab}$, as written in \rf{factorstaticeqomh}, can be expressed as
 \begin{equation}
 S_i^{(+),a}\,S_i^{(-),b}=0
 \lab{factorstaticeqomhfors}
 \end{equation}
 From \rf{siapmdef} we have
 \begin{equation}
 S_i^{(+),a}+S_i^{(-),a}= 2\, \mid \lambda\mid \,R_i^a
 \lab{sumsiapm}
 \end{equation}
Contracting \rf{sumsiapm}  with  $S_i^{(\pm),b}$ and using \rf{factorstaticeqomhfors} one gets
\begin{eqnarray}
S_i^{(\pm),a}\,S_i^{(\pm),b}&=&2\, \mid \lambda\mid \,R_i^a\, S_i^{(\pm),b}=2\,\mid \lambda\mid\left[\mid \lambda\mid\,\tau_{ab}\pm \({\rm det}\,R\)\,h^{-1}_{ab}\right]
\nonumber\\
&=&2\,\mid \lambda\mid^2\left[\tau_{ab}\pm \({\rm sign}\,\lambda\)\,\mid \lambda\mid^2\,\({\rm det}\,h\)\,h^{-1}_{ab}\right]
\nonumber\\
&=&2\,\mid \lambda\mid^2\,\tau_{ab}\left[1\pm \({\rm sign}\,\lambda\)\right]
 \end{eqnarray}
 where we have used \rf{ourstraintensor}, \rf{nicedet}, \rf{nicerhrel} and \rf{goodrelhtau}. Consequently
 \begin{eqnarray}
\qquad\qquad {\rm sign}\,\lambda =\pm 1\qquad\qquad
 \rightarrow \qquad\qquad S_i^{(\mp),a}=0
 \end{eqnarray}
 But the vanishing of $S_i^{(\pm),a}$, as given in \rf{siapmdef}, is equivalent to the self-duality equations \rf{selfdualeqs}. Therefore, we come to a very important conclusion: the static Euler-Lagrange equations for the fields $h_{ab}$ \rf{elforh} imply the self-duality equations \rf{selfdualeqs}, and these in their turn imply the static Euler-Lagrange equations for the $SU(2)$ fields $U$ \rf{elforu}. So, the static and self-dual sectors of the theory \rf{model} coincide. Consequently, the only non-self-dual solutions of the theory \rf{model} must necessarily be time dependent.

 Note in addition that in the relation \rf{goodrelhtau}, which is a direct consequence of the static Euler-Lagrange equations for the fields $h_{ab}$ \rf{elforh} , the matrix $h$ is expressed entirely in terms of the matrix $\tau$,  which in its turn is expressed in terms of the $SU(2)$ fields $U$ and their first derivatives. Therefore, by choosing any static field configuration for the $U$-fields one gets from \rf{goodrelhtau} the matrix $h$ that satisfies the static Euler-Lagrange equations for the fields $h_{ab}$ \rf{elforh}, and so both $U$ and $h$ are static solutions of the theory \rf{model}. Note that in the cases where the matrix $\tau$ is singular, the relation \rf{goodrelhtau} does not hold true. However, as we show in subsection \ref{subsec:tausing} of appendix \ref{app:proof}, it is possible to find a matrix $h$ that solves the self-duality equations \rf{selfdualeqs} even when $\tau$ is singular. Consequently, \rf{model} has plenty of static solutions. We now analyse some special types of these static solutions.

\section{The Holomorphic Ansatz}
\label{sec:ansatzholom}
\setcounter{equation}{0}

In this section we construct exact self-dual Skyrmion solutions for the self-duality equations \rf{selfdualeqs} using the so-called rational map ansatz \cite{rational1,rational2,mantonbook}.  We parameterize the $SU(2)$ group elements $U$, with a real scalar field $f$ and a complex scalar field $u$, together with its complex conjugate ${\bar u}$,  as \cite{skyrmejoaq,laf}
\begin{equation}
U= W^{\dagger}\, e^{i\,f\, T_3}\,W
\lab{udecompdef}
\end{equation}
with $W$ having the following form in the spinor representation of $SU(2)$
\begin{eqnarray}
W=\frac{1}{\sqrt{1+\mid u\mid^2}}\(
\begin{array}{cc}
1& i\,u\\
i\, {\bar u}&1
\end{array}\)
\end{eqnarray}
Therefore we have that
\begin{equation}
R_{i} = i\,\partial_{i}U\,U^{\dagger}=- V\,\Sigma_i\,V^{\dagger}\;;\qquad\qquad\qquad\qquad
V\equiv W^{\dagger}\,e^{i\,f\,T_3/2}
\end{equation}
and
\begin{equation}
\Sigma_i\equiv \partial_if\,T_3+\frac{i\,2\,\sin\(f/2\)}{1+\mid u\mid^2}\(\partial_i u\,T_{+}-\partial_i{\bar u}\,T_{-}\)
\end{equation}
with $T_{\pm}=T_1\pm i\,T_2$. The self-duality equations \rf{selfdualeqs} become
\begin{equation}
\lambda\, {\rm Tr}\( \Sigma_i\, T_b\){\tilde h}_{ba}=\frac{i}{2}\,\varepsilon_{ijk}\,{\rm Tr}\(\sbr{\Sigma_j}{\Sigma_k}\,T_a\)
\lab{selfdualeqssigma}
\end{equation}
where we have introduced the matrix ${\tilde h}_{ab}$ as
\begin{equation}
h_{ab}\equiv d_{ac}\(V\)\,{\tilde h}_{cd}\,d^T_{db}\(V\)
\lab{htildedef}
\end{equation}
with $d\(V\)$ being the adjoint representation matrix for the group element $V=W^{\dagger}\,e^{i\,f\,T_3/2}$ (see \rf{adjoint}). The adjoint representation (triplet) is real and unitary and so  $d$ is a $3\times 3$ orthogonal matrix. Therefore, since $h$ is symmetric, so is the matrix  ${\tilde h}$.

We now use spherical coordinates, but instead of using the polar and azimuthal angles we stereographic project the two sphere on a plane and parameterize that plane by a complex coordinate $w$, together with its complex conjugate ${\bar w}$. So, we have the coordinate transformation
\begin{equation}
x_1=r\;\frac{-i\(w-{\bar w}\)}{1+\mid w\mid^2}\;;\qquad \qquad x_2=r\;\frac{\(w+{\bar w}\)}{1+\mid w\mid^2}\;;\qquad \qquad
x_3=r\;\frac{\mid w\mid^2-1}{1+\mid w\mid^2}
\lab{wcartesiandef}
\end{equation}
where $r$ is the radial distance. The  Euclidean space metric becomes
\begin{equation}
ds^2=dr^2+\frac{4\,r^2}{\(1+\mid w\mid^2\)^2}\,dw\,d{\bar w}
\lab{holometric}
\end{equation}
The self-duality equations \rf{selfdualeqssigma} become
\begin{eqnarray}
\lambda\, {\rm Tr}\( \Sigma_r\, T_b\){\tilde h}_{ba}&=&
\frac{\(1+\mid w\mid^2\)^2}{2\,r^2}\,{\rm Tr}\(\sbr{\Sigma_w}{\Sigma_{{\bar w}}}\,T_a\)
\nonumber\\
\lambda\, {\rm Tr}\( \Sigma_{{\bar w}}\, T_b\){\tilde h}_{ba}&=&
{\rm Tr}\(\sbr{\Sigma_{{\bar w}}}{\Sigma_{r}}\,T_a\)
\lab{selfdualeqsrzzb}\\
\lambda\, {\rm Tr}\( \Sigma_{w}\, T_b\){\tilde h}_{ba}&=&
{\rm Tr}\(\sbr{\Sigma_{r}}{\Sigma_{w}}\,T_a\)
\nonumber
\end{eqnarray}
We now use the holomorphic ansatz
\begin{equation}
f\equiv f\(r\)\;;\qquad\qquad \qquad u\equiv u\(w\)\;;\qquad\qquad \qquad {\bar u}\equiv {\bar u}\({\bar w}\)
\lab{holoansatz}
\end{equation}
and so
\begin{equation}
\Sigma_r=\partial_r f\,T_3\;;\qquad\qquad
\Sigma_w=\frac{i\,2\,\sin\(f/2\)}{1+\mid u\mid^2}\,\partial_w u\,T_{+}\;;\qquad\qquad
\Sigma_{{\bar w}}=-\frac{i\,2\,\sin\(f/2\)}{1+\mid u\mid^2}\,\partial_{{\bar w}} {\bar u}\,T_{-}
\end{equation}
Therefore, we get from \rf{selfdualeqsrzzb} that ${\tilde h}_{ab}$ is diagonal
\begin{eqnarray}
{\tilde h}_{11}={\tilde h}_{22}=\frac{\partial_rf}{\lambda}\;;\qquad
{\tilde h}_{33}=\frac{4\,\sin^2\(f/2\)}{\lambda\,r^2\,\partial_r f}\,\frac{\(1+\mid w\mid^2\)^2}{\(1+\mid u\mid^2\)^2}\,\partial_w u\,\partial_{{\bar w}}{\bar u}\;;\qquad {\tilde h}_{12}={\tilde h}_{13}={\tilde h}_{23}=0
\nonumber\\
\lab{htildeholo}
\end{eqnarray}
Consequently, in order to have the eigenvalues of the matrix $h$ positive, we have to impose that (see \rf{signlambdaq})
\begin{equation}
{\rm sign}\(\lambda\,\partial_rf\)=+1
\lab{signlambdaf}
\end{equation}
and so, $f$ has to be a monotonic function of $r$, monotonically increasing for $\lambda>0$ and monotonically decreasing for $\lambda<0$.

The matrix $h$ can be obtained from \rf{htildedef} with $d\(V\)=d\(W^{\dagger}\)\,d\(e^{i\,f\,T_3/2}\)$, and
\begin{eqnarray}
d\(e^{i\,f\,T_3/2}\)=\left(
\begin{array}{ccc}
 \cos \frac{f}{2} & \sin \frac{f}{2} & 0 \\
 -\sin \frac{f}{2} & \cos \frac{f}{2} & 0 \\
 0 & 0 & 1 \\
\end{array}
\right)
\end{eqnarray}
and
\begin{eqnarray}
d\(W^{\dagger}\)=\frac{1}{1+\mid u\mid^2}\,\left(
\begin{array}{ccc}
 \frac{1}{2} \left(2+u^2+{\bar u}^2\right) & \frac{1}{2} i \left(u^2-{\bar u}^2\right) & i (u-{\bar u}) \\
 \frac{1}{2} i \left(u^2-{\bar u}^2\right) & \frac{1}{2} \left(2-u^2-{\bar u}^2\right) & -(u+{\bar u} ) \\
 -i (u-{\bar u}) & u+{\bar u} & 1-\mid u\mid^2  \\
\end{array}
\right)
\end{eqnarray}
We then have that
\begin{equation}
{\rm det}\,h={\rm det}\,{\tilde h}=
\frac{4}{\lambda^3}\,\frac{\partial_r f\,\sin^2\(f/2\)}{r^2}\,\frac{\(1+\mid w\mid^2\)^2}{\(1+\mid u\mid^2\)^2}\,\partial_w u\,\partial_{{\bar w}}{\bar u}
\end{equation}
Therefore, from \rf{topchargedeth} we have that the topological charges of these configurations are
\begin{equation}
Q=\frac{i}{4\,\pi^2}\int dw\wedge d{\bar w}\,\frac{\partial_w u\,\partial_{{\bar w}}{\bar u}}{\(1+\mid u\mid^2\)^2}\;\left[f-\sin f\right]\mid_{r=0}^{r=\infty}
\lab{pretopcharge}
\end{equation}
where we have used the fact that the volume element is $d^3 x=-\frac{i\,2\,r^2}{\(1+\mid w\mid^2\)^2}\,dr\wedge dw\wedge d{\bar w}$ (see \rf{holometric}).

Note that the self-duality equations \rf{selfdualeqs} do not determine the functions $f\(r\)$, $u\( w\)$ and ${\bar u}\({\bar w}\)$. It only determines the matrix $h$ in terms of these functions. Therefore, any holomorphic ansatz configuration \rf{holoansatz} leads to a solution of the self-duality equations \rf{selfdualeqs}.  However, for the function $u\(w\)$ to be a well defined map between two-spheres it has to be a ratio of two polynomials $p$ and $q$, i.e. the so-called rational map
\begin{equation}
u\(w\)=\frac{p\(w\)}{q\(w\)}
\lab{rationalmap}
\end{equation}
The degree $n$ of the map between two-spheres defined by \rf{rationalmap} is the highest power of $w$ in either of the polynomials $p$ or $q$, and it is equal to \cite{mantonbook}
\begin{equation}
n=\frac{i}{2\,\pi}\int dw\wedge d{\bar w}\,\frac{\mid\,q\,\partial_w p-p\,\partial_w q\mid^2}{\(\mid p\mid^2+\mid q\mid^2\)^2}
\end{equation}
In addition, for the topological charge \rf{pretopcharge} to be non-trivial the profile function $f\(r\)$ has to be such that the quantity $\left[f-\sin f\right]\mid_{r=0}^{r=\infty}$ does not vanish.  For a given integer $m$  one has that $e^{i\,2\,\pi\,m\,T_3}=\pm \hbox{{1}\kern-.25em\hbox{l}}$, depending if the representation used is of integer (+) or half-integer (-) spin. Therefore, from \rf{udecompdef} we have that if $f=2\,\pi\,m$ then $U=\pm \hbox{{1}\kern-.25em\hbox{l}}$. Consequently, we shall consider boundary conditions such that $f\(0\)= 2\,\pi\,m$, and $f\(\infty\)=0$, and so $U\(\infty\)=\hbox{{1}\kern-.25em\hbox{l}}$ and $U\(0\)=\pm \hbox{{1}\kern-.25em\hbox{l}}$, depending if the representation used for $U$ is of integer (+) or half-integer (-) spin. The topological charge \rf{pretopcharge} then becomes
\begin{equation}
Q=-m\,n
\end{equation}
Clearly, if we swap the boundary conditions for $f$ at the origin and at spatial infinity, the topological charge changes sign.

Note that alternatively we could have chosen the boundary condition for $f$ such that $f\(0\)= 2\,\pi$, and $f\(\infty\)=0$, leading to a solution of topological charge $(-n)$. We could then use  the product ansatz  \cite{mantonbook} to construct a  Skyrmion of charge $(-m\,n)$ by taking the $SU(2)$ field $U$ as  $U_{(-m\,n)}= U_{(-n)}^m$, where $U_{(-n)}$ is the field $U$ for the Skyrmion of charge $(-n)$, i.e. using  \rf{udecompdef} one has  $U_{(-n)}= W^{\dagger}\, e^{i\,f_{(1)}\, T_3}\,W$, with  $f_{(1)}\(0\)= 2\,\pi$, and $f_{(1)}\(\infty\)=0$. Therefore,  $U_{(-m\,n)}= W^{\dagger}\, e^{i\,m\,f_{(1)}\, T_3}\,W$.  Consequently, for a given profile function $f_{(1)}$, the number of ways one can construct  a self-dual solution of topological charge $N$ is given by the number of partitions of $N$ into the product of two integers. As we have said before, any configuration for the $SU(2)$ field $U$ leads to a self-dual solution, since the $h$-fields act as spectators (see \rf{goodrelhtau}). So, one can change the profile function smoothly without changing its boundary values and the self-dual solution obtained has the same topological charge. So, the self-dual solutions are infinitely degenerated.

In order to illustrate the kind of solutions we get for the matrix $h$ we  shall consider the simple profile function
\begin{equation}
f\(r\)=4\,{\rm ArcTan}\(\frac{1}{\zeta}\)\qquad\qquad\qquad \qquad \zeta=\frac{r}{a}
\lab{chosenf}
\end{equation}
which implies
\begin{equation}
\frac{\partial_rf}{\lambda}=
\frac{4\,\sin^2\(f/2\)}{\lambda\,r^2\,\partial_r f}=-\frac{1}{\lambda\, a}\,\frac{4}{\(1+\zeta^2\)}
\end{equation}
with $a$ being an arbitrary positive parameter of dimension of length. Then, from \rf{htildeholo}, we get that the eigenvalues of the matrix $h$ are
\begin{equation}
\(\varphi_1\,,\,\varphi_2\,,\,\varphi_3\)=\frac{1}{\mid\lambda\mid\, a}\,\frac{4}{\(1+\zeta^2\)}\(1\,,\,1\,,\,
\frac{\(1+\mid w\mid^2\)^2}{\(1+\mid u\mid^2\)^2}\,\partial_w u\,\partial_{{\bar w}}{\bar u}\)
\lab{eigenvaluesconfig}
\end{equation}
where we have used \rf{signlambdaf} and the fact that the profile function \rf{chosenf} is monotonically decreasing.

\section{The Toroidal Ansatz}
\label{sec:ansatztoro}
\setcounter{equation}{0}

It was shown in \cite{laf} (see \rf{trickone}) that the self-duality equations \rf{selfdualeqs} are invariant under conformal transformations on the three dimensional spatial sub manifold. We now use  that symmetry together with the target space symmetries \rf{leftsymm} and \rf{rightsymm} to build an ansatz based on the toroidal coordinates \cite{babelon,shnir}. In order to implement that we parameterize the $SU(2)$ group elements as
\begin{eqnarray}
U= \(
\begin{array}{cc}
Z_2&i\,Z_1\\
i\,{\bar Z}_1& {\bar Z}_2
\end{array}\)\;;\qquad\qquad\qquad \mid Z_1\mid^2+\mid Z_2\mid^2=1
\lab{torou}
\end{eqnarray}
We now select two commuting $U(1)$ subgroups of the target space symmetry group $SU(2)_L\otimes SU(2)_R$, given in \rf{leftsymm} and \rf{rightsymm}, as follows
\begin{eqnarray}
U\rightarrow e^{i\,\alpha\,T_3}\,U\,e^{-i\,\alpha\,T_3}\;;\qquad\qquad {\rm or}\qquad \qquad
Z_1\rightarrow e^{i\,\alpha}\,Z_1\;;\qquad Z_2\rightarrow Z_2
\lab{firstu1}
\end{eqnarray}
and
\begin{eqnarray}
U\rightarrow e^{i\,\beta\,T_3}\,U\,e^{i\,\beta\,T_3}\;;\qquad\qquad {\rm or}\qquad \qquad
Z_1\rightarrow Z_1\;;\qquad Z_2\rightarrow e^{i\,\beta}\,Z_2
\lab{secondu1}
\end{eqnarray}
We also select two commuting $U(1)$ subgroups of the conformal group in three dimensions defined by the vector fields \cite{babelon,shnir}
\begin{eqnarray}
\partial_{\phi}&\equiv& x^2\,\partial_1-x^1\,\partial_2
\lab{phitransf}
\\
\partial_{\xi}&\equiv&\frac{x^3}{a}\(x^1\,\partial_1+x^2\,\partial_2\)+\frac{1}{2\,a}\(a^2+x_3^2-x_1^2-x_2^2\)\,\partial_3
\lab{xitransf}
\end{eqnarray}
where $a$ is a free  parameter with dimension of length, and where we have introduced the angles $\phi$ and $\xi$, such that the vector fields above generate rotations along these angular directions. The transformation \rf{phitransf} corresponds to rotations on the plane $x_1\,x_2$, and \rf{xitransf} to a linear combination of a special conformal transformation and a translation along the $x_3$ direction. It turns out that $\phi$ and $\xi$ correspond in fact to the two angles of the toroidal coordinates in three dimensions defined by
\begin{eqnarray}
x^1&=& \frac{a}{p}\,\sqrt{z}\,\cos\phi\;; \qquad\qquad\qquad \qquad p=1-\sqrt{1-z}\,\cos\xi
\nonumber\\
x^2&=& \frac{a}{p}\,\sqrt{z}\,\sin\phi\;; \qquad\qquad\qquad \qquad 0\leq z\leq 1
\lab{torocoord}\\
x^3&=& \frac{a}{p}\,\sqrt{1-z}\,\sin \xi\;; \qquad\qquad\qquad\;\; 0\leq \phi\,,\,\xi\leq 2\,\pi
\nonumber
\end{eqnarray}
where the Euclidean metric becomes
\begin{equation}
ds^2=\frac{a^2}{p^2}\left[\frac{dz^2}{4\,z\(1-z\)}+\(1-z\)\,d\xi^2+z\,d\phi^2\right]
\lab{torometric}
\end{equation}
We build an ansatz that is invariant under the joint action of the  $U(1)$  subgroups \rf{firstu1} and \rf{phitransf},  and also invariant under the joint action of the subgroups \rf{secondu1} and \rf{xitransf}, leading to
\begin{equation}
Z_1=\sqrt{F\( z\)}\,e^{i\,n\,\phi}\;;\qquad\qquad\qquad Z_2=\sqrt{1-F\( z\)}\,e^{i\,m\,\xi}\;;\qquad\qquad m\,,\,n \in \rlx\hbox{\sf Z\kern-.4em Z}
\lab{toroansatz}
\end{equation}
with $0\leq F\leq 1$. From \rf{torou} and \rf{toroansatz} we get that
\begin{eqnarray}
R_z&=&i\,\partial_zU\,U^{\dagger}=\frac{F^{\prime}}{\sqrt{F\(1-F\)}}\,\left[-\cos\(m\xi+n\phi\)\,T_1+\sin\(m\xi+n\phi\)\,T_2\right]
\lab{toror}\\
R_{\xi}&=&i\,\partial_{\xi}U\,U^{\dagger}=-2m\left[\sqrt{F\(1-F\)}\left[\sin\(m\xi+n\phi\)\,T_1+\cos\(m\xi+n\phi\)\,T_2\right]+\(1-F\)\,T_3\right]
\nonumber
\\
R_{\phi}&=&i\,\partial_{\phi}U\,U^{\dagger}=2n\left[\sqrt{F\(1-F\)}\left[\sin\(m\xi+n\phi\)\,T_1+\cos\(m\xi+n\phi\)\,T_2\right]-F\,T_3\right]
\nonumber
\end{eqnarray}
and these quantities satisfy the commutation relations
\begin{eqnarray}
\sbr{R_z}{R_{\xi}}&=&-i\,\frac{m}{n}\,\frac{F^{\prime}}{F}\, R_{\phi}
\nonumber\\
\sbr{R_{\phi}}{R_{z}}&=&-i\,\frac{n}{m}\,\frac{F^{\prime}}{\(1-F\)}\, R_{\xi}
\lab{rcommrel}\\
\sbr{R_{\xi}}{R_{\phi}}&=&-i\,4\,m\,n\,\frac{F\(1-F\)}{F^{\prime}}\, R_{z}
\nonumber
\end{eqnarray}
Then the  self-duality equations \rf{selfdualeqs} become
\begin{eqnarray}
\lambda\,\frac{a}{p}\, R_z^b\,h_{ba}&=&-2\,m\,n\,\frac{1}{F^{\prime}}\,\frac{F\(1-F\)}{z\(1-z\)}\, R_z^a
\nonumber\\
\lambda\,\frac{a}{p}\, R_{\xi}^b\,h_{ba}&=&-2\,\frac{n}{m}\, F^{\prime}\,\frac{\(1-z\)}{\(1-F\)}\, R_{\xi}^a
\lab{toroselfdualeqs}\\
\lambda\,\frac{a}{p}\, R_{\phi}^b\,h_{ba}&=&-2\,\frac{m}{n}\, F^{\prime}\,\frac{z}{F}\, R_{\phi}^a
\nonumber
\end{eqnarray}
So, the self-duality equations \rf{toroselfdualeqs} imply that the three  quantities $R_z^a$, $R_{\xi}^a$ and $R_{\phi}^a$, are eigenvectors of the matrix $h_{ab}$. Since eigenvectors and eigenvalues are known, one can reconstruct that matrix, obtaining
\begin{equation}
h= M\, h_D\, M^T\;;\qquad\qquad\qquad\qquad\qquad M\,M^T=\hbox{{1}\kern-.25em\hbox{l}}
\lab{hdiagonaltoroidaldef}
\end{equation}
with
\begin{equation}
h_D=-\frac{2}{\lambda}\,\frac{p}{a}\,
\mbox{\rm diag.}\,\left(m\,n\,\frac{1}{F^{\prime}}\,\frac{F\(1-F\)}{z\(1-z\)}\;,\; \frac{n}{m}\,
F^{\prime}\,\frac{\(1-z\)}{\(1-F\)}\;,\; \frac{m}{n}\,F^{\prime}\,\frac{z}{F}\right)
\equiv \mbox{\rm diag.}\,\left(\varphi_1\,,\,\varphi_2\,,\,\varphi_3\right)
\lab{hdiagonaltoroidalfinal}
\end{equation}
and
\begin{eqnarray}
M=\(
\begin{array}{ccc}
\cos\(m\xi+n\phi\)&\sqrt{F}\,\sin\(m\xi+n\phi\)&\sqrt{1-F}\,\sin\(m\xi+n\phi\)\\
-\sin\(m\xi+n\phi\)&\sqrt{F}\,\cos\(m\xi+n\phi\)&\sqrt{1-F}\,\cos\(m\xi+n\phi\)\\
0&\sqrt{1-F}&-\sqrt{F}
\end{array}\)
\lab{mtoroidaldef}
\end{eqnarray}
Therefore
\begin{equation}
{\rm det}\,h={\rm det}\,h^{({\rm diag.})}=-m\,n\,\left[\frac{2}{\lambda}\,\frac{p}{a}\right]^3\, F^{\prime}
\end{equation}
Consequently, from \rf{topchargedeth} one gets that the topological charges for these configurations are
\begin{equation}
Q=m\,n\,\left[ F\(1\)-F\(0\)\right]
\end{equation}
where we have used the fact that the volume element is $d^3x=\frac{1}{2}\,\frac{a^3}{p^3}\,dz\,d\xi\,d\phi$ (see \rf{torometric}).

Note that the self-duality equations do not determine the profile function $F\(z\)$. However, for the matrix $h$ to be invertible, one has to have $m , n \neq 0$, and $F^{\prime}\neq 0$. Note in addition that in order for the eigenvalues $\varphi_a$ to be positive, one observes from \rf{hdiagonaltoroidalfinal}, that one must have
\begin{equation}
{\rm sign}\(\lambda\,m\,n\,F^{\prime}\)=-1
\end{equation}
Consequently, $F$ has to be a monotonic function of $z$, increasing for $\lambda\,m\,n<0$, and decreasing for $\lambda\,m\,n>0$. That is the equivalent of relation \rf{signlambdaf} in the holomorphic ansatz.

The $\tau$-matrix, defined in \rf{ourstraintensor}, in the case of this toroidal ansatz, is given by
\begin{equation}
\tau_{ab}=\frac{p^2}{a^2}\left[ 4\,z\,\(1-z\)\, R_z^a\,R_z^b+\frac{1}{1-z}\,R_{\xi}^a\,R_{\xi}^b+\frac{1}{z}\, R_{\phi}^a\,R_{\phi}^b\right]
\lab{tautoroidal}
\end{equation}
and one can write it as
\begin{equation}
\tau=M\,\tau_D\,M^T
\lab{diagonalizetautoroidal}
\end{equation}
with $M$ given by \rf{mtoroidaldef}, and
\begin{equation}
\tau_D=4\, \frac{p^2}{a^2}\,{\rm diag.}\,\({F^{\prime}}^2\,\frac{z\(1-z\)}{F\(1-F\)}\,,\, m^2\,\frac{\(1-F\)}{\(1-z\)}\,,\, n^2\, \frac{F}{z}
\)\equiv {\rm diag.}\,\(\omega_1\,,\,\omega_2\,,\, \omega_3\)
\lab{tautoroidaldiag}
\end{equation}
From \rf{hdiagonaltoroidaldef} and \rf{diagonalizetautoroidal} one sees that the matrices $h$ and $\tau$ are indeed diagonalised by the same orthogonal matrix $M$, given in \rf{mtoroidaldef}, and such a fact is compatible with \rf{eigenvalueshtau}. In addition, comparing  \rf{hdiagonaltoroidalfinal} and  \rf{tautoroidaldiag}  one observes their  eigenvalues  do satisfy the relations \rf{eigenvalueshtaurel}.

\section{Conclusions}
\label{sec:conclusion}
\setcounter{equation}{0}

We have studied the properties of a modified Skyrme model, originally proposed in \cite{laf}, that possesses an exact self-dual sector. The novelty of such a modification is the introduction of six scalar fields assembled into a symmetric and invertible matrix $h_{ab}$, that not only makes the existence of the self-dual sector possible, but also renders it conformally invariant in the spatial sub-manifold $\rlx\hbox{\rm I\kern-.18em R}^3$. We have shown that the static and self-dual sector  are in fact equivalent, in the sense that any static solution also satisfies the self-duality equations. In addition, the fields $h_{ab}$ are spectators in the self-dual sector since  for any configuration for the $SU(2)$ $U$-fields, the fields $h_{ab}$ adjust themselves to satisfy the self-duality equations. Consequently, the model possesses an infinity of analytical self-dual solutions. We have construct explicitly two classes of such solutions: one based on the holomorphic rational ansatz, and another based on a toroidal ansatz constructed from the conformal symmetry of the  self-dual sector.

The construction of the self-dual sector is such that it does not leave room for a kinetic term for the fields $h_{ab}$.  The addition of a kinetic term, as well as of a mass and potential terms, break the conformal symmetry of the theory in $\rlx\hbox{\rm I\kern-.18em R}^3$, and also destroys the self-dual sector. However, that is desirable for physical applications, since the exact self-dual solutions have their energy proportional to the topological charge and so they do not have an interaction when at rest relative to each other.  We have verified that the addition of the kinetic, mass and potential terms can lead to a positive binding energy in some range of the coupling constants. That can be useful in application of the model to nuclear physics. The numerical results we have obtained on those lines will be presented elsewhere \cite{tocome}.

\vspace{2cm}

\noindent {\bf Acknowledgements:} The authors are grateful to Carlos Naya for many helpful discussions. LAF is partially supported by Conselho Nacional de Desenvolvimento Cient\'ifico e Tecnol\'ogico - CNPq (contract 308894/2018-9) and Funda\c c\~ao de Amparo \`a Pesquisa do Estado de S\~ao Paulo - FAPESP (contract 2018/01290-6). LRL is supported by CAPES.

\vspace{2cm}

redpajama_set_name: RedPajamaArXiv


Have you seen any coffee table that has a design like this before ? Not for me!
I named this "transformer coffee table" because this table design consist of an elevated table top and a side drawer. Elevated table top provides options of using it as writing or eating table. So cool right?
Last Saturday I woke up early to to check out Stanzo Collection official launch in 1 Mont Kiara. I am always on the hunt for new furniture for our home that was renovated last year. The above transformer table is one of my favorite piece that I really like and the price is surprisingly reasonable considering it's design.
Stanzo" or "Stanza" is an Italian term, which literally means "rooms". Stanzo Collection concept store currently houses two tasteful brands, elegantly names "Songdream" and "Prostoria" that offer the best of East and West.
"We chose "Stanzo" because its translation as "rooms" in English very aptly befits the property showroom style we seek to adopt for our Stanzo Collection concept store. ," said Ken Low, Group Managing Director of Stanzo Collection.
Ken emphasizes that the Stanzo Collection is aimed at discerning, affluent home owners who are passionate about designing their home interiors by themselves. To this end, he believes that this premium Collection is versatile enough to meet even the most stringent of requirements, no matter how diverse one's tastes may be. Put simply, he believes that the Stanzo Collection is the perfect ingredient to create the most exclusive look for any home or workplace.
And I do like how furniture are displayed in room concept because it's easier for me to see how to mix & match the furniture back home and from there I can make reference to recreate my own furniture arrange.
The "Songdream" collection will surely appeal to those seeking a contemplative environment, with its modern, contemporary furniture selection providing a distinct Zen-like charm. It comes with complete range of sofas, coffee tables, room dividers, chairs, bookcases, dining tables and wardrobes.
Prostoria Collection bases its design along clean lines working in tandem with different shades of calm palettes, accentuated by bright shades such as passionate red and lively green. With a versatile collection of wood, steel and fabric furniture and furnishings, it is unsurprising the "Prostoria's" multifunctional sofa model "Uplift" recently won the German Design Council's prestigious Interior Innovation Award.
Let me take your for a virtual tour here and show you which is my favourite piece!
I love this bookcase design and colour!!! I think instead of filling up with books, I would prefer to display with souvenirs I collected from my travel adventure. RM5299 for this... hmm....should I get this ?
It also comes in white and walnut colour!
Very simple & sleek work place set up!
I love this living room settle up by Designer AfroditiKrassa in collaboration with Stancco Songdream Collection.
I think is is my favoueite bedframe and can add in side table to have breakfast on bed!
I spotted this unique dressing table where it comes with a lift-up mirror. All made of medium-density fibreboard (MDF) with walnut veneer and leather interior! I would totally get this if the surface is bigger because I like to display more like show off and admire my beauty stuff on my table!
Spotted some colourful bean chair.. not only it looks in style but it's really comfortable too!
Most of the furniture showcased in Stanzo Collection looks really cosy and modern to me! I'm sure you will be thinking that these furniture piece show must be really expensive considering the location of this store is in Mont Kiara. I was really surprised to find out that the pricing were more reasonable than I thought as each of Stanzo's collection is sourced from the highest quality when it comes to design.
I am now thinking if I should get the "transformer coffee table" or not.. it's RM2419 if I remembered correctly!
It's nice to catch up with blogger friends at the launching! Took all the pictures here using HTC One M8 camera... the pictures quality turns out to be quite nice that I can actually leave my dlsr camera at home for a while.


Quickly, off the top of your head – what's the most important factor in your daily corrosion industry safety work life?
It's Y-O-U and Captain Obvious—what a Dynamic Duo!
Safety managers train you. Regional managers establish procedures and rules. Clients require certain clothing. Your supervisor warns you. But the bottom line is: No program, sign or protective equipment will work if YOU elect not to make safety an intentional part of your daily work life.
Make no mistake, every job must finish safely. As soon as it doesn't, safety becomes the MOST IMPORTANT measure of the job; and you become the yardstick.
Let's review some simple ways for you to practice safety at every job every day.
Check your work area frequently.
Look around and listen up! Use your senses to prevent a safety event (and you already do it all day every day). What do you see, hear or smell in your work area? Mobile equipment in use? Materials being moved? Walking area slippery? Cords or ropes in your path? Trash not discarded correctly? Machine making a 'funny' noise? Do you smell hydraulic fluid?
Observe how your work area changes during the day. Let those working around you know about the changes. Take an active role in making your job site safer by helping to make others aware of the little changes. Sure, you might sound like Captain Obvious. So what? When it helps everyone go home at the end of the day with nothing but a fatter wallet… then THAT'S a good day!
All MATCOR employees abide by our Stop Work Obligation. Every employee at our job sites has an obligation to intervene and stop work when a situation gets identified that could break one of MATCOR's Life Saving Rules or cause injury or illness.
Electricity and equipment. Both start with "E" and both kill.
Here's another one from Captain Obvious. Check your electrical equipment every time you use it. Drills, grinders, reeling machines and irons. If it has to be plugged in or charged, there's the potential for a cracked case, frayed or cut cords or a short somewhere. Make a quick inspection for wear and tear, and intact strain relief and connections.
For larger equipment, like Pipeline Current Mappers, generators or interrupters, take the time to read the Operator's/Owner's Manual at least once before you running the machine. If the manual isn't available, ask someone who has used the machine for a quick rundown.
Wide open and burning hot.
Many times when we hear about a workplace fatality, it involves some kind of fall…in a hole, from heights, or something falling on a worker. Considering the kind of work we do, every worker needs to be aware of 1) what he or she might fall over/into; and 2) what might fall onto him or her. Falls aren't always fatal, but often workplace fatalities involve falls. Here comes Captain Obvious, again. Be aware and make others aware of anything that might make you fall; or that might fall on you.
Another situation that could burn us on a job site is fire (sorry for the bad pun there). Anywhere there's machinery, combustible chemicals, or cad welding, the possibility of fire exists. Open flame, especially on a pipeline right-of-way could turn an ordinary day into a hot time that no one wants. Knowing what to do in the event of a fire at EVERY job site makes sense. Where's the extinguisher? Do you know how to use it? What about the escape route from the site? Yes, you and your crew should create and review the emergency plan at each job site. After all, knowing the best direction to run IS an emergency action plan.
Where IS that first aid box?
Calling Captain Obvious… Do you know where the first aid kit is in your work area? Can you get to it quickly and easily? When was the last time you checked to see if it needed to be restocked? Though you may think that's someone else's job, it will matter most to you when you need to remove a splinter, bandage up a cut or treat a burn. Remember, we're talking about how Y-O-U have the most important part of job site safety. If the first aid kit isn't ready and available when you need it, the problem isn't someone else's. Check the first aid kit.
Corrosion Industry Safety: The last line of defense.
If we can't engineer a hazard away, we write a procedure to protect you from it. When engineering or administration doesn't quite eliminate a hazard, you get personal protective equipment (PPE). Here's the rub. If you choose not to wear PPE, it doesn't protect you (OMG, Captain Obvious snuck right in there). Hard hats, safety glasses, gloves, or earplugs might be uncomfortable. Then again, so are skull fractures, blindness, amputations and deafness.
PPE may be a last line of defense when it comes to corrosion industry safety, but that makes it no less important. Make sure you don it every time on every job…and speak up if your co-workers need to be reminded.
Workers need to take care at all times and at all job sites. No matter how many machines have guards, or safety policies the company writes, Y-O-U have the biggest impact on your safety. Awareness of your work space, inspecting your tools, knowing how to get out in an emergency or where to find the first aid kit are all factors in YOUR control.
You make decisions every day that impact how, or if, you and your crew go home. Will it be with a fatter wallet? Or will it be with stitches, or in a cast or worse? Captain Obvious knows.
MATCOR takes corrosion industry safety very seriously and maintains an excellent safety record.
Learn more about MATCOR Safety Programs or contact us for additional information.
Pipeline cathodic protection industry received a boost this week: proposed high-volume natural gas pipeline construction across 3 Southeastern states. Spectra, the company behind this jolt of economic opportunity, has dubbed it the "Renaissance Project". The natural gas pipeline proposes several lines branching off the main pipeline to potential customers along the route.
The pipeline is a complex project and will require a number of business services such as pipeline cathodic protection. MATCOR's pipeline protection program uses a number ISO 9001:2008-certified solutions to protect such a project. For example, MATCOR's SPL™-FBR linear Anode and Durammo™ Deep Anode System would help lower total cost of ownership on the pipeline.
The proposed pipeline is almost 300 miles with three different pipeline diameters. The natural gas pipeline will feature two compressor stations to maintain line pressures, according to officials. It was stated the line will have a capacity of 1 billion cubic feet per day, and can be expanded to over 1.5 billion cubic feet per day.
Spectra plans for the pipeline to run from the Chattanooga, Tennessee area, through Alabama and towards the Atlanta, Georgia area. "We are continuing to work with multiple potential customers to design a project to fit their supply demand needs," Grover said in a statement on the project.
Furthermore, Spectra executed letter of intent with the AGL, the parent company of Chattanooga Gas Co., and Atlanta Gas Light Co.to explore a joint arrangement for local distribution. Sources close to the matter state the "Renaissance Project" could be up and running by mid-2017.
"From a project kickoff standpoint, we continue to reach out to federal, state and local public officials informing them of the project," she said. "We'll send letters and start contacting landowners along our proposed study corridor pending further market feedback." Grover stated Spectra the Renaissance Project study corridor map is in the final stages. Currently, the map highlights 15 counties across Tennessee, Alabama and Georgia.
Economic groups across these states laud the move as one that will stimulate the economy and bring jobs to the region. "It's going to be a good project and the infrastructure for natural gas is such that, industry-wide, there's a great demand for it," one source familiar with the matter said. Experts familiar with the project say it is a crucial move to support industrial expansion and business growth in the region.
The Renaissance pipeline is indeed poetically named. The Southeastern region has recently seen improvements to its economic state and industrial competitiveness. However, key investments such as the natural gas pipeline and other infrastructure must be put in place to attract jobs, manufacturers and families into these committees. That said, the pipeline is a key cog to the continued rebirth of this region. MATCOR and other service providers will be stewards of this bright future, protecting valuable assets that power communities.
MATCOR is a leading provider of ISO 9001:2008-certified cathodic protection for pipelines and cathodic protection management. Our team maintains the highest quality standards of cathodic protection for systems for storage tanks and other products that let you focus on your business operations.
When calling 811, homeowners and contractors are connected to their local one-call center, which notifies the appropriate utility companies of their intent to dig. Professional locators are then sent to the requested digging site to mark the approximate locations of underground lines with flags, spray paint or both.
Striking a single line can cause injury, repair costs, fines and inconvenient outages. Every digging project, no matter how large or small, warrants a call to 811. Installing a mailbox, building a deck and planting a trees and are all examples of digging projects that need a call to 811 before starting.
811 is a free, FCC-designated national one-call number that connects a caller from anywhere in the country to the appropriate local one-call center. The one-call center will then alert the appropriate underground facility owners so they can dispatch locators to mark the approximate location of their lines with paint or flags.
How can you celebrate 8-11? Learn more about 811. Make a promise, and make a difference. Visit http://www.the811promise.com for further details.


Dance Coloring Pages are the best way to help your kids show some love about their dance activity. Maybe their favorite activity? Dancing is definitely a passion for many many people, children included. Whether it be Jazz, Tap, Hip-Hop, Ballet or another form of dance, get your kids saying I love Dance! and excited to get to dance class. Our coloring pages and dance pictures can help them get there. Print them all for free.


Are you a Product Design Graduate? Are you looking for a long term career with a stable and well-established lighting manufacturer that has a consistent track record of success?
My client are a lighting manufacturer, with a strong reputation in the industry for exceptional products and extremely high standards. They are currently undergoing a period of intense growth, and are looking to capitalise on this by expanding their design team further.
You are a recent grad in product design, or similar, and are looking to join a company that will invest heavily in your future. You are ambitious and committed to working hard and developing extremely high standards in a professional working environment.
You will be based out of the office in the Hampshire area.
If you feel you match the description above, and are looking to join a company in which you can really challenge yourself and develop your skillset with a growing company, then you should email arthur.rees@panacea-selection.co.uk or call Arthur Rees on 0117 973 8090 for a confidential discussion.


Getting rid of your stuff is supposed to make you happier.
That's the main idea fueling Marie Kondo's book, "The Life-Changing Magic of Tidying Up." When it came out a few years ago, it fueled the minimalist movement and encouraged us to throw out those sweaters from The Limited that we loved once upon a time in the '80s.
This book also gave me permission me to purge other random items in a way that I'd never done before. Her approach of "thanking things for their service," is an effective one for someone like me who is emotionally attached to 'things'. I do this exercise about twice a year and always feel lighter and happier afterwards. My closets and my mind are clearer and the things I keep, as Kondo says, truly spark joy.
So why should getting rid of a stroller and the last of the baby gear make me so miserable?
Sitting on the cold tile of our basement floor, I'm in nearly in tears. Boxes of dusty baby gear and deconstructed stroller pieces are spread out around me. The seat and cover have been scrubbed and the wheels greased. The aluminum frame gleams, having been wiped clean of fingerprints and dried yogurt and who knows what else.
I'm agonized with memories of each little toy, each bright plastic what-not and then I channel my inner KonMarie: What do we need? What are we using? What sparks joy? I am making good headway, sorting things to donate, things to sell, things to just throw out and then I find Adventure Bear.
Adventure Bear is a small, brown, stuffed animal with a green pacifier attached. It was my older son's constant companion. I clipped it on the shoulder strap before our first outing in our new stroller. It was a glorious spring day in Chicago, one of those rare early warm days when we emerge, pale and blinking at the sunlight like hibernating animals from their winter dens.
My baby was strapped in his five-point-harness, blue eyes following my every move, clutching Adventure Bear close. It was the first of countless outings as mother and son with that stroller, bear in hand.
Now I rubbed his soft, slightly-pilled ears and inhaled the lingering scent of Dreft. I wondered how many miles he'd logged with us. I started to put him in the Donate bin but I can't do it. I set him aside with a sigh.
Just then my son wandered into the room. He saw the stroller and the piles of accessories and gear, and he turned to me, his eyes brimming with tears.
"Mom!" he said, "Are you getting rid of that stroller? My stroller?" His voice quivered.
Yes, I said. It's going to a family with a little boy and they will make more memories just like we did.
He started to cry. I was not prepared for this. I expected him to want to keep special toys and other artifacts from baby days, but a stroller? He fell into my arms, tears falling on my shoulder, his body shaking as he sobbed.
He pulled away, wiping his face with the back of his hands, his lovely blue eyes now red with tears.
He still sobbed. I held him, patted his back, wiped my own tears. Adventure Bear looked up at me from the table where I'd placed him in indecision.
He turned the bear over in his hands, and hugged it to his face, stroking its head softly. He sniffed, and managed a smile. He hugged me again, and I held him tight in my arms, struck with how tall and lean he'd become in the past year. By far a boy, no longer my baby.
"I love you, Mommy. Can we go to the park?" he asked, breaking the spell.
Of course we can. I said. Get your brother.
He handed the bear back, and skip-hopped up the stairs, yelling for his sibling.
I carried the toy upstairs and tucked it into in a box in the back of my closet. It contains just a few precious things from those early days: A tiny Chicago Cubs jersey, a special rattle, and now Adventure Bear.
Downstairs, the boys were wrestling on the couch. Maybe someday they would have their own babies, I thought with a smile, and Adventure Bear will come along for the ride.
And I finally felt that spark of joy.
This essay originally appeared in Fete Lifestyle Magazine, October 2018.


text:
Q: Pydantic generate json-schema enforcing field requirements I have the following model, where the field b is a list whose length is additionally enforced based on the value of field a.
from pydantic import BaseModel, validator
from enum import Enum

class A(BaseModel):
    a: int
    b: list[int]

    @validator("b")
    def check_b_length(cls, v, values):
        assert len(v) == values["a"]

a = A(a=1, b=[1])

A.schema_json()

Generating the following JSON-schema:
{
    "title": "A",
    "type": "object",
    "properties": {
        "a": {
            "title": "A",
            "type": "integer"
        },
        "b": {
            "title": "B",
            "type": "array",
            "items": {
                "type": "string"
            }
        }
    },
    "required": [
        "a",
        "b"
    ]
}

Is there any way to enfore the additional validation being done by pydantic in the check_b_length function in JSON-schema? I.e. this could be done possibly via oneOf, as in this question: JSON Schema - specify field is required based on value of another field ... So its possible to enfore in JSON-schema, and possible to enforce in pydantic, but how can you map them?

redpajama_set_name: RedPajamaStackExchange


1 These things have I spoken unto you, that ye should not be offended. 2 They shall put you out of the synagogues: yea, the time cometh, that whosoever killeth you will think that he doeth God service. 3 And these things will they do unto you, because they have not known the Father, nor me. 4 But these things have I told you, that when the time shall come, ye may remember that I told you of them. And these things I said not unto you at the beginning, because I was with you.
Satanists surround Street Preacher & mock the Crucifixion!
Arrested for reading Bible in public!


Greg Smith appears at Fantasium Comics & Games on Saturday, August 25 at 3 PM to sign JUNIOR BRAVES OF THE APOCALYPSE.
It's dangerous to go alone! Take this invitation and join us in fighting the good fight!
All proceeds benefit the Breast Cancer Research Foundation and the Pink Ink Fund.
The adventure begins at 7PM !
- Earn gold and prizes by completing quests and playing mini games set up throughout the marketplace!
- Visit the Ta-Ta Tavern for the most refreshing Elixers!
- Make your way to Skooder's Hot Dog Co. and feast upon their delectable eats!
- Amazing local vendors with unique items exclusive to this event!
Vol. 1 collects the first half of the epic comic series Legend.
Already read it? Haven't had time? Just curious to find out more?
JOIN OUR CLUB FOR DISCUSSION ON TUESDAY, APRIL 26th @ 6PM!


Planes make the world go round but what makes the planes go around? That question has kept me intrigued and involved in aviation for years. In my aviation management program I learnt about the challenges and opportunities behind pesky industry acronyms like ASK, FTL and FOD. I get excited by the chance to increase airline schedule capacity (or Available Seat Kilometres), make sure pilots don't fly too many hours (Flight Time Limitations) and rostering ground staff (who should never leave behind Foreign Object Debris). That's all in a day's work for an Aviation Solutions Consultant.
The faster airlines react, the less passengers will be inconvenienced. How can airline operations keep their plans from descending into hell?
What has to be planned to make tomorrow's flights go without a hitch? Solving the equation to complex airport operations is not as simple.


A diverse portfolio of services, suitable for all organizations, in most industries and varied business segments. Your progress and development is in the hands of professionals.
* Training subjects beyond the ones listed in the published catalogues may be developed on request. The lists of subjects are amended and updated on regular time intervals.
a couple of core principles that we apply.
If you haven't found a specific title-topic, in the Listing of Programs and Training Courses which is provided, a customized design is potentially the solution after performing a Training Needs Analysis (TNA) to access the actual training requirements and the missing gap.
Do contact us to discuss!
2-plus: A cost effective solution!
You may batch together a number of individual courses from the listing of training titles that is provided and make up a unique training program, suitable to your in-house needs. This is not only an advanced and comprehensive approach to learning and development but also a cost effective solution due to significant discounts that apply for programs made up of 2-courses-plus in a row.
Do contact us for the ideal solution!
Value for money and measurable Results – a Win-Win collaboration and a Seamless experience!


text:
Q: Customizing Android app UI with more than one setContentView on top of camera view? I'm a beginner Android developer working on a prototype app. I'm using a proprietary barcode scanner inside my app, and I want to overlay a small menu at the bottom of the screen while my barcode scanner is running. Ideally this menu would be sized roughly 1/5th of the screen and contain a few buttons. I've thought about using fragments to achieve this UI, but I'm having a hard time converting my scanner's SDK tutorial to use fragments.
Right now, my activity produces a full screen camera view with the barcode picker overlayed in the middle with the following code:
getWindow().setFlags(WindowManager.LayoutParams.FLAG_FULLSCREEN,
        WindowManager.LayoutParams.FLAG_FULLSCREEN);
requestWindowFeature(Window.FEATURE_NO_TITLE);

ScanditSDKAutoAdjustingBarcodePicker picker = new
ScanditSDKAutoAdjustingBarcodePicker(
    this, sScanditSdkAppKey,ScanditSDKAutoAdjustingBarcodePicker.CAMERA_FACING_BACK);

setContentView(picker);
mBarcodePicker = picker;

How would I go about overlaying a menu that I can describe in an XML file with this setup? Invoking an XML layout via another setContentView() covers the whole screen, even when I set its background to transparent (or  null).
Apologies in advance if this question is worded poorly.

A: I've checked documentation for this ScanditSDKAutoAdjustingBarcodePicker and found next link:
http://docs.scandit.com/stable/android/android-scanview-options.html
As the BarcodePicker is a normal RelativeLayout you can add it to your own layout like any other view:
// Set app key. This will only have to be done once
ScanditLicense.setAppKey("yourAppKey");
// Create the scan view. Note that you will have to configure the settings in your app
// since the default settings have all symbologies disabled and you won't be able to
// scan anything.
BarcodePicker barcodePicker = new BarcodePicker(context, ScanSettings.create());
// Add the scan view to the root view.
rootView.addView(barcodePicker);

So you can have "ordinary" activity with layout, defined in XML. You can use RelativeLayout. There you can put container (FrameLayout) for picker and your menu, aligned to bottom of RelativeLayout. You can find container by id and put picker into that container with addView().


"At the time of my first consultation with Dr. Hodges in late October, 2009, I was in constant, excruciating pain due to an incorrectly diagnosed/improperly treated problem in the lumbar region of my spine. Standing and walking, even with a cane, were both difficult; sleep was elusive. An overwhelming sense of loss and grief due to the death in September of my beloved brother exacerbated the effects of the physical problems.
Dr. Hodges immediately arranged for an x-ray of my lower back, enabling the correct diagnosis of the severe curvature/twisting of the lumbar region. She was able to work me into her schedule eight times over the next three weeks, and we began to see slow but sure progress in dealing with the problem and the pain. I credit Dr. Hodges with making it possible for me to travel to my brother's memorial service in Southern California in mid-November.
From the beginning, I was impressed with Dr. Hodges' care and concern for the entirety of the patient's well being. In addition to checking on the status of my back at each consultation, she also asks about how life is going in general – if/how my back problems are affecting my ability to enjoy/participate in activities that bring me pleasure (like gardening), etc. The concern for my well being clearly goes beyond the status of the bones that need adjusting.
I should note the feeling of care and concern in Dr. Hodges' office starts when the patient meets Gayle Baker, the receptionist. The warm, sincere welcome, which extends to dealing with mundane things like paying a bill or making the next appointment, adds to my appreciation for the care provided.


← Outsourcing ESIC's recruitment process to private agencies: Is it valid in law?
Time spent traveling to and from work is "work"- European Court of Justice !
Opposite is in India, run by businessmen-controlled-politicians, notwithstanding Sec. 51-E, which was a definite progress, in spite of procedural difficulties.
"Requiring them to bear the burden of their employer's choice would be contrary to the objective of protecting the safety and health of workers pursued by the directive, which includes the necessity of guaranteeing workers a minimum rest period." – says the European Court of Justice.
Increasing the spread-over period to 12 hours is shown as the a safety and health measure in India.
There is no necessity to guarantee the workers any "minimum rest period".
Mayday revolution resulted in 8+8+8 hours, i.e., 8 hours for work, 8 hours for forest, recreation with family and 8 hours to sleep. But, already the official 8 hours work has become 12 hours including the time taken for commuting to and fro workspot, in India. When the spread-over period is 12 hours, and the journey between the residence and workspot takes another 4 hours, where is the time for rest, family and sleep?
We are creating a society that would be poor in health and would need more medical attendance resulting in more expenditure on medical side by the ESIC and the State.
We do not care for any respectable slot in the Human Development Index or Global Prosperity Index.
Those who can change things, do not do their bit. But, they work for extricating themselves from that situation personally.
When are we going to usher in a civilised society free from exploitation?