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Curating art in churches
The Rev'd Nigel di Castaglione, Team Vicar of All Saints and St Andrew, Kingston, wearing one of the chasubles made for 'A World to Come' by Bettina Furnée. Photograph by Torrie Smith
Emma Smith with visitors to Euphonia at St Michael's & All Angels Walthamstow
Victoria Burgher, Crown, 2019 at All Saints Leyton
Naomi Maxwell, The Intercessions – Worth, 2019
Installed as part of Praxis and Promiximty at St Edmund's Chingford
Hannah Whittaker, Parquet Picioare, 2019 installed at St Margaret's Leytonstone
This two-year initiative focuses on forming partnerships to deliver contemporary art exhibitions in Anglican churches and cathedrals. Recent case studies and guidelines for curating temporary exhibitions have also been drawn up.
A WORLD TO COME
All Saints and St Andrew, Kingston, Cambridge CB23 2NG
13 July – 1 October. Open daily 10am – 6pm
An installation by Bettina Furnée responding to medieval wall paintings at All Saints and St Andrew, Kingston
Bettina Furnée's newly commissioned installation imagines the Wheel of the Seven Acts of Corporal Mercy, one of a number of scenes of judgement and moral conflict depicted in the church's medieval wall paintings, being 'turned again' for a contemporary audience. Presenting a cast of seven symbolic creatures sewn into robes inspired by church vestments, the work playfully considers morality beyond good and evil, and the capacity of art to hold together contradictory positions.
In connection with Wysing Arts Centre's Open Studios Day, Project Curator Laura Purseglove will lead a Q&A session with the artist Bettina Furnée on 17 August at 2pm. No booking required, just turn up.
DIRECTIONS FROM CAMBRIDGE
Take the Barton road, followed by the B1O46 through Comberton. The church is accessed via a left turn (Church Lane) which comes up just before you get to Bourn.
Kindly supported by Allchurches Trust, the Jerusalem Trust and the Headley Trust.
Four artists respond to the theme of fellowship in a series of installations for churches in Waltham Forest
Emma Smith, Euphonia
St Michael's & All Angels, Northcote Rd, Walthamstow E17 6PQ 22 June – 24 July
Holy Trinity, Leytonstone 26 July – 12 August
Euphonia is an interactive sound installation based on artist Emma Smith's research into the musicality of human communication. The work invites visitors to become part of the installation, to add their own voices to the space and, in doing so, to author the score itself.
St Michaels and All Angels walk-through events:
Thursday 18 July 10am-5pm
Working with her unique 5Hz language, these events will demonstrate that we are all musical, regardless of training and experience, by guiding participants through a simple process resulting in a collaborative composition. Open to all. Free, no need to book.
Holy Trinity opening times:
Tuesdays and Wednesdays 1.15 - 4pm
Sundays 12.30 - 5pm
Naomi Maxwell, Praxis and Proximity
St Edmund's, Larkswood Rd, Chingford E4 9DS 8 July – 1 September
Documentary photographs celebrating acts of fellowship in Waltham Forest, structured around the liturgy of the Eucharist.
Thursdays 9.30am - 12pm
Saturdays 10am - 11am
Sundays 9am - 12noon
St Edmunds is also open on Sundays for services at 8am and 10am and Thursdays for morning prayer at 8.30am and a service at 12.30pm with lunch afterwards. Visitors are welcome to join in with services and view the artwork around these times.
Victoria Burgher, Crown
Exterior of All Saints, Melbourne Rd, Leyton E10 7HF 10 June – 1 November
Crown is a public artwork in the form of a wreath of individually made metal-foil chrysanthemums produced by the artist Victoria Burgher in collaboration with school groups and the local community. As the work is installed on the exterior of the church, it can be seen at all times.
Hannah Whittaker, Parquet Picioare
St Margaret's, Woodhouse Rd, Leytonstone E11 3NG 6 July – 1 September
Inspired by the church's lively Romanian Orthodox congregation, Hannah Whittaker presents an installation of tessellating tiles which sit on top of the existing church floor and are personalised by members of the community.
Coinciding with Leytonstone Art Trail
8 – 12 July 10am - 3pm
15 – 16 July 10am - 3pm
St Margarets is also open for services on Sundays at 10.30am, on Wednesdays at 7.30pm, and Thursdays at 10am. Visitors are welcome to join in with services and view the artwork around these times.
Kindly supported by London Over the Border, Allchurches Trust and the Jerusalem Trust.
Case studies of temporary visual arts projects in churches and cathedrals are available here:
All Saints, West Dulwich: a long duration, participatory residency
St Michael's, Discoed: an annual, themed, multi-artist exhibition and fundraiser
St John on Bethnal Green: short term rolling exhibitions with a community focus
St Paul's Bow Common: site-specific art with professional installation and community impact
Art in Romney Marsh: an annual, site-specific festival across multiple rural churches
Coventry Cathedral: performance piece
Salisbury Cathedral: an installation with a focus on community participation and education
Sheffield Cathedral: an exhibition in a citywide event showcasing works from private collections
All Saints, Skeyton: an exhibition in a rural area showcasing local art for sale
Commissioning New Art for Churches: a Guide for Parishes and Artists
A pdf to download with details guidance on commissioning new permanent works of art as well as temporary exhibitions. Published in partnership with ChurchCare.
The first Art in Churches project was a site-responsive installation hosted by St John the Baptist church in Newcastle. Ritournelle (once more, from the top) by Katia Kameli is the result of a short residency in Summer 2017, programmed in collaboration with Platforma, an arts and refugee network. This site-specific art work filled the small diamond panes of glass with coloured film, in patterns that recalled design and architecture motifs native to some of the primary countries from which refugees and migrants relocate to Newcastle. Countries and designs referenced include Punjabi Phulkari and Congolese Kuba textiles and Iraqi mosaic design.
Wanting also to create a sound work, Kameli invited migrants and refugees from the city, as well as members of the church choir, to sing a song which reminds them of home. These songs were recorded in situ at St John the Baptist and were available to listen to in the church.
The residency, production and installation of the work was managed by Laura Purseglove, Project Curator for Art and Christianity and the Church Buildings Division of the Church of England.
Kindly supported by Allchurches Trust, Arts Council of England, the Jerusalem Trust, the Headley Trust and the Pea Green Boat Community First Fund at the Community Foundation Tyne & Wear and Northumberland.
'A Light Shines in the Darkness', a touring exhibition of artists' films organised by Paul Bayley with Film and Video Umbrella. The programme included:
Suki Chan, Still Point
Alexander and Susan Maris, Uriel
Melanie Manchot, LEAP after the Great Ecstasy
Graham Ellard and Stephen Johnstone, Everything Made Bronze
Kathleen Herbert, Stable
Nick Crowe and Ian Rawlinson, The Carrier's Prayer
Christian partner in the Three Faiths Forum Urban Dialogues exhibition.
All Hallows on the Wall: Host to Jeffrey Charles Henry Peacock evening event.
13 October – 3 Decmber 2010
Wallspace gallery: 'Commission' an exhibition of contemporary art in British Churches.
26 June – 13 July 2006
City of London Festival: three works by Yoko Ono at St Paul's Cathedral.
City of London Festival: INSIGHT included works by Nick Crowe and Ian Rawlinson, Adele Prince, Victoria Jones, John Newling, and Rebecca Horn.
St Edmund the King & Martyr, Lombard Street: Tom Gidley Your actual
St Stephen's Walbrook: Simon Hitchens In the Presence of Absence
Banner image: Katia Kameli, Ritournelle (once more, from the top), 2017


Many writers believe that the only or best literary agents are located on the East Coast, but West Coast agents beg to differ. The major publishing houses may still reside in and around New York City, but major authors live throughout the world, and Pacific Coast agents have found that literary representation outside New York may actually be to an author's advantage. Join this panel of West Coast pros to learn how they navigate a publishing world without borders.
Aimee Liu is author of the novels Flash House, Cloud Mountain, and Face. Nonfiction works include Gaining and Solitaire. Her short stories and essays have appeared in more than a dozen anthologies, magazines and literary journals. She teaches in Goddard College's MFA Program in Creative Writing.
Betsy Amster is a literary agent and president of Betsy Amster Literary Enterprises. Before opening her agency, she spent ten years as an editor at Random House. Her clients include bestselling novelists María Amparo Escandón and Joy Nicholson, MacArthur Fellow Will Allen, and psychologist Elaine Aron.
Angela Rinaldi is president of the Angela Rinaldi Literary Agency which was founded in 1994. She held managerial and editorial positions at Penguin Random House and Simon & Schuster. She founded the book publishing program at the Los Angeles Times. She represents both fiction and nonfiction.
Elise Capron is an agent at the Dijkstra Literary Agency, which was established over thirty years ago and represents a wide range of genres. She has been with SDLA for more than a decade, and she is most interested in representing adult literary fiction and narrative nonfiction, particularly history.
Rebecca Friedman has been an agent for twelve years. A graduate of Barnard College, she is interested in commercial and literary fiction with a focus on literary suspense, women's fiction, romance, young adult, and journalistic nonfiction and memoir. She is looking for great stories told in strong voices.


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Arrest made in fatal shooting of Maryland rapper Tre Da Kid
ANNAPOLIS, Md. (AP) — Police say they've charged a Baltimore man with murder in the fatal shooting of popular Annapolis rapper Tre Da Kid.
News outlets report 30-year-old James Esau Davis III was arrested Thursday. Investigators say they think he shot the 32-year-old rapper, whose real name was Edward Montre Seay, earlier this month.
Police say Davis shot Seay while driving through Annapolis. Seay's car then crashed and he died of his injuries.
The Capital reported in 2016 that Seay won a contest seeking the country's best freestyle lyricist. His prize was $10,000 and the opportunity to tour with a hip hop label's recording artists.
Police say Davis is being held without bond.
Information from: The Capital, http://www.capitalgazette.com/

redpajama_set_name: RedPajamaCommonCrawl


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>>BACK
Contact: Joe Sherk, jlsherk@aol.com/360-981-4806, mobile
Kilgore's Steve Torrence excited to be racing at the Texas Motorplex
DALLAS – The time has come for Steve Torrence to return to NHRA racing after taking a self-imposed four-month hiatus to start a new team and build a new race car.
"I couldn't be more excited for myself and my family because we will debut our new Torrence Racing Top Fuel dragster at our home track, the Texas Motorplex in Ennis," said Torrence, of Kilgore, Texas – less than a two-hour drive from Ennis. "I consider this the high point of my racing career. It feels real good to be here to race."
Torrence will take his first 1,000-foot ride in the sleek 8,000-horsepower dragster Friday when qualifying begins for the 26th annual AAA Texas NHRA Fall Nationals.
"Our (crew) guys were loading the car into the trailer Tuesday and were heading for Dallas to get ready to race," he continued. "We have surrounded ourselves with the most capable people and they have all worked hard to prepare for this race. It was a big achievement to start a new team and get it ready in four months."
Crew chief Richard Hogan and Bryan (Speedy) Shipman were the lone crewmen at the team's shop in Brownsburg, Ind. during the team's infancy. Orders for the myriad of tools, nuts bolts, car parts, engine blocks and assorted pieces were sent and the stuff eventually made its way to the team. A flurry of activity in the last few weeks got the job done.
Torrence made two starting-line launches at Indianapolis last week and Hogan was satisfied with the data . . . then it was full speed ahead.
This is the first of four races the team plans to enter before the end of the season. Other stops will be at Phoenix (Oct. 14-16), Las Vegas (Oct. 28-30) and Pomona, Calif. (Nov. 10-13).
"We want to put on a good show for the Torrences – Steve and his parents, Billy and Kay," said Hogan. "We'd like to make four good qualifying runs at Dallas and race on Sunday, but when you are starting a race team from scratch, you're not really sure which way it is going to go. We have a fairly proven combination, but it takes time to get everything going in the right direction."
"This has been a huge undertaking . . . and it turned into a full-blown race team," Torrence added. "There is a sense of accomplishment, but we still have a lot to do. The biggest accomplishment will be when this car runs well and we get into a final round or get a race win.
"Everything is a stepping stone. We'll set small goals and try to achieve them first and see where it goes."
The team's Friday qualifying sessions are scheduled at 2:15 and 5:15 p.m. and Saturday's sessions will be at 11 a.m. and 1:30. Eliminations begin Sunday at 11 a.m.
download release as .doc file


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Thank you for visiting my personal website. I have been in lending for almost 30 years in residential home loans, wholesale and loan servicing. In that time, I have worked with thousands of households from first time buyers through high-end repeat buyers as well. At Mortgage Network, Inc., we have a variety of traditional and sophisticated mortgage programs that provide the best financing options available in today's competitive market. Each situation has a solution and I can help you choose, with confidence and understanding, the best one for your family and personal needs.
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In fairness to other patients and in order to provide safe treatment to engaged patients who are most likely to benefit, repeated no-shows or late cancellations may be cause to discontinue treatment at our clinic.
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Q: library not found for -lProtobuf in Xcode I'm using Xcode 10.2.1 and objective-c in my project. after using this commond in terminal:
git checkout "specific commit"

I'm bring back my codes to 4 month ago. after using this command in terminal I must change Bundle Identifier (because I'm no longer have access to that developer account).
after that I'm getting this error in Xcode (before that I'm getting error about account, No account for team "*****".):
library not found for -lProtobuf

I tried to delete pod file and reinstall pods by run this command in terminal:
pod install

but the problem not solve.

A: Try to open your project from its .xcworkspace file instead of .xcodeproj file.

redpajama_set_name: RedPajamaStackExchange


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NIHL North Round Up - Weekend of 11th & 12th January 2014
Blackburn Hawks continued their impressive run of form with two more victories at the weekend, beating bottom club Telford Titans 10-6 in the Moralee Conference on Saturday night and then despatching Manchester Minotaurs 6-1 in the Northern Cup on Sunday evening.
The win on Saturday moved the Hawks up into 2nd place in the table overnight but they were then displaced by both Billingham Stars and Sutton Sting who picked up wins over Whitley Warriors and Telford Titans respectively on Sunday evening.
Billingham's 6-2 win in the North East derby will have been especially welcome to their fans as it brought to an end a terrible run of results where they lost 4 games in a row, scoring only 4 goals in the process.
Solway Sharks remain top of the Moralee Conference with a 3 point cushion over the Stars. They also have three games in hand, having played no league games this weekend.
In the Laidler Conference (North 2), 2nd placed Deeside Dragons closed the gap with leaders Solihull Barons to 3 points with a 2-1 win at home to Nottingham Lions on Sunday night. They missed the chance to move to within 1 point of top spot as they lost their game on Saturday 3-2 away to Sheffield Senators.
Lions are now level on points with Coventry Blaze in what looks as if it will be a two horse battle for 3rd place.
In the other games of the weekend, Widnes Wild won 1-5 away to Blackburn Eagles and then lost 3-5 at home to Sheffield Senators.
Photo: Outstanding young Blackburn player James Riddoch was voted MVP for his performance for the Hawks against Manchester on Sunday. James had previously been playing for Blackburn Eagles in the Laidler Conference and has only just stepped up to play for the N1 side. (Photo by John Milton - www.togger.co.uk)
NIHL North: One In & One Out At Whitley Warriors
Chelmsford Chieftains - Shut Out Kings!
NIHL North Moralee Conference Round Up - Weekend o...
NIHL North Laidler Conference Round Up - Weekend o...
Blackpool Seagulls - Next Match 8th February
NIHL North 1 News: Hawks' Dunford To Sit Out Seaso...
New Blackpool Rink Looking Good
NIHL North Round Up - Weekend of 11th & 12th Janua...
September 2013: 2012/13 Season Review - NIHL Nort...
July 2013: Deeside Dragons v Liverpool Leopards L...
April 2013: Bruce Sims Souvenir Programme
2011/12 North 2 - First Ever Edition - Back Copies...


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\section{Introduction and Model}

Nonlinear sigma models have had great success in describing low energy
QCD phenomenology.  However, they have not captured the hallmark
feature of QCD: confinement, where the potential between $q\bar{q}$
pairs grows linearly with separation.  Such linear potentials can
arise as tension-carrying flux tubes --- between $q$ and $\bar{q}$,
for Yang-Mills QCD, or between any unscreened sources which persist in
low energy effective theories. The conventional Skyrme model, induced
by the global flavor symmetry breaking $SU({N_f})_L \times SU({N_f})_R
\rightarrow SU({N_f})_{diag}$, supports neither unscreened sources nor
flux tubes. Global flux tubes are classified by $\pi_2 (G/H)$, due to
their constant vev at spatial infinity. This necessarily vanishes
whenever the vacuum manifold $G/H$ is itself a Lie group, as occurs in
the Skyrme model.

Witten \cite{wittenon} noted that an $SO(N_c)$ gauge theory of QCD
induces a different Skyrme model, whose topology {\em can} support
flux tubes. This theory has ${N_f}$ lefthanded quarks $q_L$, which
transform as (real) fundamentals under color $SO(N_c)$.  $SO(N_c)$ has
no gauge anomaly, and thus requires no right-handed quarks; however,
its $U(1)_B$ anomaly breaks baryon number to $Z_2$.  The high energy
theory thus has the full symmetry group $Z_2 \times SO(N_c) \times
SU({N_f})$. At low energies, flavor $SU({N_f})$ breaks to its $SO({N_f})$
subgroup, due to formation of a quark condensate $\langle
q_{Li}^\alpha \,\, \Sigma_{ij} \,\, q_{Lj}^\alpha\rangle$ (where
$\alpha$ and $i$ are color and flavor indices respectively). This
condensate interacts with quark excitations in the theory, inducing
an effective Majorana mass for the quarks
\begin{equation}
\label{LMaj}
{\cal L}_m =         - \mu \left( {\overline{q_L}}  \Sigma  q_{Lc}
         +             {\overline{q_{Lc}}} {\Sigma}^\dagger  q_{L} \right)\ ,
\end{equation}
which displays the $SU({N_f})\rightarrow SO({N_f})$ flavor symmetry breaking
explicitly.  The Goldstone modes $\Sigma$ are described by a
$SU({N_f}) / SO({N_f})$ nonlinear sigma model, with skyrmions
($\pi_3(G/H)$ = $Z_2$, for ${N_f}
\ge 4$) and flux tubes ($\pi_2(G/H)$ = $Z_2$, for ${N_f}
\ge 3$). This defect classification makes physical sense: for baryons,
identified with antibaryons because real quarks are identified with
antiquarks; and for flux tubes, whose $Z_2$ structure emerges as a
response to external spinor sources, which can be screened by
fundamental quarks only in even combinations.

In this paper, we elaborate on our recent work with Manohar
\cite{shortskyrme}, constructing the flux tubes in this theory and showing
that their interactions with skyrmions and spinor sources obey
heuristic expectations.  Confinement of spinor sources in an $SO(N_c)$
gauge theory can thus manifest itself in the low energy sigma model,
through relic phenomenology in the presence of unscreened sources.

We organize our results as follows: section II derives the unique flux
tube form with minimal energy; section III examines its classical
stability and dynamics; section IV its quantum stability and spectrum;
and section V its interactions and relationships with other
fundamental objects in the theory. The early sections reveal that
minimal flux tubes lie in a planar subspace of the vacuum manifold
$SU({N_f})/ SO({N_f})$; while section V discusses why our flux tubes are
Alice strings, despite ambiguities in defining parallel transport
around them; why they carry skyrmion number when twisted; and how
they can be viewed as mediating the confinement of spinor sources.

\section{Finding Nontrivial Flux Tubes}

To construct topological defects, we seek configurations where the
condensate $\Sigma$ varies spatially in both nonsingular and
nontrivial ways. Nonsingular variations assume a form dictated by the
transformation properties of $\Sigma$. Demonstrating their
nontriviality, however, is complicated in our modified Skyrme model.  The
difficulty stems from the same source as the flux tubes themselves:
the fact that the global vacuum manifold $G/H$ is not itself a Lie
group, but only a quotient space.  Thus, unlike skyrmions in the
standard Skyrme model, winding numbers for $\Sigma$ cannot be
obtained from a group structure on $G/H$ alone, as indices
dependent only on $\Sigma$. Instead they must be determined by
homotopy arguments from the embedding of $\Sigma$ in $G$.

The $SU(2)_L \times SU(2)_R$ Skyrme model presents a simpler case.
Here symmetry group transformations $(g_L, g_R)$ act on $G$ by left
multiplication and leave $\Sigma$ invariant under the diagonal
subgroup.  $\Sigma$ can thus be constructed from underlying group
transformations: $\Sigma = g_L {g_R}^\dagger,$ transforming as $\Sigma
\rightarrow L \Sigma {R}^\dagger$ under the group element $(L,R)$. When
$\Sigma = {{\mbox{1\hskip-0.22em\relax l}}\,} ,$ $L=R$ gives the diagonal
unbroken subgroup. For
$\Sigma \ne {{\mbox{1\hskip-0.22em\relax l}}\,},$ however, the embedding $H
\subset G$ changes and
$L=R=g$ rotates $\Sigma$ on the vacuum manifold, $\Sigma \rightarrow g
\Sigma {g}^\dagger.$ Using these transformation laws, and imposing
spherical symmetry to minimize energy, we may write an arbitrary
$\pi_3$ defect as $\Sigma = L(\Omega) \Sigma_o(r) {R}^\dagger
(\Omega)$. We choose our basis to diagonalize $\Sigma_o (r) = \exp
{\{iF(r)\tau_z\} }$, with $F(r \rightarrow \infty) = 0.$ $\Sigma$ then has
finite energy only when $L=R,$ keeping $\Sigma$ constant at spatial
infinity, and is singular at the origin unless $F(0)= n\pi$.

Establishing this nonsingular form $\Sigma = g(\Omega) \exp
{\{i F(r)\tau_z\}} {g}^\dagger (\Omega)$ requires only the quotient
space structure $G/H = \left( SU(2)_L \times SU(2)_R \right)/
SU(2)_{diag}.$ We require the full Lie group structure of $G/H$,
however, to show that a particular $g(\Omega)$ induces nontrivial
$\pi_3$, or equivalently, to reduce the Pontryagin index to a function
of $\Sigma.$ Since $\pi_3(G/H) = \pi_3(SU(2)) = Z,$ the homotopy
index $\pi_3$ simply measures the number of times $\Sigma(r,\Omega )$
covers $SU(2).$ This is easily determined by group integration or
inspection. For the skyrmion, $g(\Omega) = \exp{\{-i\theta \tau_z/2\}}
\exp{\{-i\phi\tau_y/2\}}$ rotates $\tau_z$ to an
arbitrary Lie algebra element $\hat{r}\cdot\hat{\tau}.$ Thus $\Sigma =
\exp{\{i F(r)\hat{r}\cdot\hat{\tau} \}}$ consists of the exponentiation of
the full Lie algebra from $0$ to $n\pi$ --- by definition, covering the
Lie group $n$ times, for a $\pi_3$ winding $n$.

Our model yields a nonsingular ansatz for $\Sigma$ as directly as the
standard one. $G = SU({N_f}),$ acting on itself by left multiplication,
leaves $\Sigma$ invariant under the orthogonal subgroup. We can thus
write $\Sigma$ as $g {g}^T,$ transforming as $\Sigma \rightarrow a
\Sigma{a}^T$ under the group element $a$. When $\Sigma =
{{\mbox{1\hskip-0.22em\relax l}}\,} ,
a \in SO({N_f})$ gives the unbroken subgroup $H$; elsewhere, the
embedding $ H\subset G$ is parallel transported to $g H {g}^\dagger$.
Again, a cylindrically symmetric defect can be written as an
$r$-dependent vev, with angle-dependent group rotation: $\Sigma =
g(\theta) \Sigma_o(r){g}^T (\theta).$ Choosing $\Sigma_o =
{{\mbox{1\hskip-0.22em\relax l}}\,} $ at spatial infinity restricts $g(\theta)$
to lie in the unbroken
$SO(n)$ subgroup, and to commute with $\Sigma_o(r=0),$ if $\Sigma$ is
to be nonsingular with finite energy.

We can further characterize $\Sigma $ in terms of $su({N_f})$ basis
generators. These specify rotations in all 2-dimensional subplanes
$(jk)$, and are usually taken as follows (with Cartan norm ${\mbox{tr}\,} T_a
T_b = {{\textstyle\frac{1}{2}}} \delta_{ab}$). The rank ${N_f}-1$ Cartan
subalgebra has as
basis diagonal matrices $T_d$,
\begin{equation}
\label{defTd}
T_d = ( 2d(d +1)\, )^{-1/2}\ {\rm diag} (1,\ldots,1, -d,0,
\ldots,0),
\end{equation}
with ones in the first $d$ entries and $d = 1$ to ${N_f} -1$.  We can
overspecify this basis, sacrificing the Cartan norm, by the set $\{
{{\textstyle\frac{1}{2}}}\tau_{z\,(jk)\,}\}$ in all subplanes of the ${N_f}$
dimensional
vector space.  Off-diagonal generators $\{
{{\textstyle\frac{1}{2}}}\tau_{x\,(jk)\,},
{{\textstyle\frac{1}{2}}}\tau_{y\,(jk)\,}\}$ complete the $su({N_f})$ basis.
Flavor
symmetry breaking divides the basis into two sets: antisymmetric
matrices $\{T_h\} \equiv\{{{\textstyle\frac{1}{2}}} \tau_{y\,(jk)\,}\}$,
generating $H =
SO({N_f});$ and symmetric matrices $\{T_b\} \equiv\{
{{\textstyle\frac{1}{2}}}\tau_{x\,(jk)\,},
{{\textstyle\frac{1}{2}}}\tau_{z\,(jk)\,} \}$, generating no
symmetries.  $\Sigma_o(r),$ a unitary symmetric matrix, can be written
$\exp {\{i F_b(r) T_b\} }.$ Thus $\Sigma$ assumes the most general
nonsingular form $\Sigma = h(\theta) \exp {\{ i F_b(r) T_b)\} }
h^{-1}(\theta),$ for $h(\theta) \in H$ and $F_b(r)$ ranging from zero
at infinity to $ 2\pi n \,\,\delta_{bb'}$ (for some fixed direction
$b\, '$) at the origin.

Having obtained nonsingular configurations $\Sigma$, we must
demonstrate their nontriviality. Unlike the conventional Skyrme
model, this requires an understanding of how $\Sigma = g
{g}^T$ arises from the underlying group mapping $g$.
For flux tubes, we must construct $\Sigma$ from the exact sequence
\begin{eqnarray*}
\pi_2\left( SU({N_f})\right) = 0  \quad \rightarrow \quad&&
\pi_2\left( SU({N_f})/SO({N_f})\right) \\
&&\quad\quad \rightarrow \quad
\pi_1\left( SO({N_f})\right) = Z_2 \quad \rightarrow \quad
\pi_1\left( SU({N_f})\right) = 0 \quad .
\end{eqnarray*}
That is, $g g^T$ gives a nontrivial $\Sigma$ only if $g$ corresponds
to some mapping from the plane to $SU({N_f}),$ with boundary values in
the $SO({N_f})$ subgroup. Furthermore, when parametrized as a family of loops,
$g$ must start at the identity and end on a nontrivial loop in $SO({N_f})$.
Taking $\left (\alpha\in[0,2\pi], \beta\in[0,\pi]\right) $ as our
coordinates on the plane, these criteria become
\begin{equation}
\label{gbc}
g(\alpha, \beta)  = \left\{
\begin{array}{ll}
{{\mbox{1\hskip-0.22em\relax l}}\,} & \mbox{when $\alpha = 0$, $\alpha = 2\pi$,
\ or $\beta = 0$}\\
h^2(\alpha) & \mbox{when $\beta = \pi$},
\end{array}
\right.  \end{equation}
where $h^2(\alpha)$ is a nontrivial loop in $SO({N_f}).$ Such nontrivial
loops can be written $h^2(\alpha) = \exp {\{ i \alpha \, 2n_h T_h\}
  },$ where $T_h$ is the set $\{{{\textstyle\frac{1}{2}}} \tau_{y(jk)} \}$
introduced
above and $n_h T_h$ generates rotations in a single plane. (Of
course, these loops can be deformed, but deformations from
geodesic form lengthen the loop and induce additional gradient
energy in $\Sigma.$ We discard them, to focus on mappings that
produce minimal gradient energy.)

We now show that the embeddings $g(\alpha, \beta)$ have minimal
gradient energy only when they induce flux tubes $\Sigma$ of a unique
form. We show this by imposing consistency and minimal energy
conditions on the most general trivialization $g(\alpha,
\beta)$. We then relate the topological coordinates $(\alpha, \beta)$ to
cylindrical coordinates $(r,\theta)$, to obtain the physical flux tube
$\Sigma(r,\theta)$. Finally we consider the low-lying deformations of
$\Sigma(r,\theta)$ which can be favored by potential energy terms. We
thus obtain a family of non-trivial flux tubes, among whom a minimal
representative is selected dynamically.

We construct the most general trivialization $g(\alpha,\beta)$ as follows.
Start, at $\beta = 0,$ by left multiplying $h(\alpha)$ by its inverse:
$g( \beta = 0) = h^{-1}(\alpha)\, h(\alpha) ={{\mbox{1\hskip-0.22em\relax
l}}\,}.$ As $\beta$ varies,
allow the left multiplier $h^{-1}(\alpha)$ to vary over the full group
$G$: $g(\alpha, \beta) = g_1(\beta)\, h^{-1}(\alpha) \, g_2(\beta) \,
h(\alpha) $.  The $\alpha=0$ condition and uniqueness of the inverse
imply $g_2(\beta) = g_1^{-1}(\beta).$ To probe structure on $G/H$
(where $a\rightarrow a a^T$ identifies cosets $\{gH\}$),  write
$g_1(\beta)= b(\beta)\,\tilde{h}(\beta),$ for $\tilde{h}(\beta)\in H$ and
$b$ generated by broken generators $T_b$. (Specifically, since
$g_1(\beta) \, g_1^T(\beta) =\exp {\{ i G_b(\beta)T_b\}}$, we can write
$g_1(\beta) = \exp {\{ i G_b(\beta)T_b/2)\}}\,\,\tilde{h}(\beta)$.) Thus
we may write the most general trivialization of the loop $h^2(\alpha)$
in $G$:
\begin{equation}
\label{gformgen}
g(\alpha, \beta) =
b(\beta) \ \tilde{h}(\beta) \,\,h^{-1}(\alpha)\,\, \tilde{h}^{-1}(\beta)
                        \ b^{-1}(\beta) \ h(\alpha).
\end{equation}
This induces minimal energy only in geodesic form
\begin{eqnarray}
\label{gformgeod}
b(\beta) &=& \exp{\{i l \beta \,\, n_b T_b\}} \nonumber\\
\tilde{h}(\beta)  &=& \exp{\{i m \beta\,\, \tilde{n}_h T_h\}} \nonumber\\
h(\alpha)  &=& \exp{\{i \alpha \,\, n_h T_h\}} ,
\end{eqnarray}
where $n_b$ and $n_h,\tilde{n}_h$ are unit vectors over the range of
${T_b}, {T_h}$ respectively.

We now show that consistency reduces the distinct choices of
$b,h,$ and $\tilde{h}$.  For simplicity, take $\tilde{h} =
{{\mbox{1\hskip-0.22em\relax l}}\,}$ at
first.  Choose a basis in $su({N_f})$ so that $n_b T_b =
{{\textstyle\frac{1}{2}}}\tau_{z\,(12)\,}$.  Then, under conjugation by
$b(\beta)$, $2n_h T_h =
 n_{(jk)}\,\, \tau_{y(jk)}$ rotates as follows:
\begin{eqnarray}
\label{rotTh}
b(\beta)\ 2 n_h T_h\ b^{-1}(\beta)&=&
n_{(12)}\ \left(\cos l\beta \,\tau_{y(12)} + \sin l\beta \,\tau_{x(12)}\right)
+
{\sum_{2<j<k}} n_{(jk)}\,\,\tau_{y(jk)}\nonumber\\
&&+ {\sum_{\stackrel{j=1,2}{k>2}}} \, n_{(jk)} \
\left(\cos(l\beta/2) \,\tau_{y(jk)}
- (-1)^j \, \sin(l\beta/2) \,\tau_{x(jk)}\right)  .
\end{eqnarray}
To obtain $g(\alpha,\beta=\pi) = h^2(\alpha)$, this conjugated
generator must give $-2 n_h T_h$ at $\beta = \pi$. This can occur for
only two lowest winding possibilities: $l$ can be 1, with $n_{(jk)} =
e_{(12)}$ vanishing outside the $(12)$ plane; or $l$ can be $2$, with
$n_{(jk)}$ describing a plane that intersects $(12)$ in a single line.
The second possibility is further constrained by the boundary
condition at $\alpha =2\pi$, which requires $h(2\pi)$ to commute
with $n_b T_b$. This occurs only when ${\hbox{\bf [} n_b T_b, (n_h T_h)^2
\hbox{\bf ]} }$
vanishes --- that is, when the plane $n_{(jk)}$ intersects $(12)$
along a coordinate axis.  Thus only three distinct candidates arise
for the pair $b,h$: $l=1, n_{(ij)} = e_{(12)}$; $l=2, n_{(ij)} =
e_{(13)}$; and $l=2, n_{(ij)} = e_{(23)}$.

Allowing nontrivial $\tilde{h}(\beta)$ produces no additional flux tubes
of minimal energy, as we demonstrate explicitly in an appendix.
Thus the form \ref{gformgen} for the flux
tubes's embedding reduces to two candidates:
\begin{equation}
\label{gform}
g(\alpha, \beta) =
b(\beta) \ h^{-1}(\alpha)
                        \ b^{-1}(\beta) \ h(\alpha),
\end{equation}
with
\begin{equation}
\label{defbh}
h(\alpha)  = \exp{\{i \alpha \,\, T_{h\Box}\}}  \quad
b(\beta) =
\left\{ \begin{array}{c}
 \exp{\{i \beta \,\, T_{b\Box}\}}\\
 \exp{\{i 2\beta \,\, T_{b\Box '}\}}
 \end{array}\right.,
 \end{equation}
where $\Box$ and $\Box '$ denote planes intersecting along a
coordinate axis, and $T_b, T_h$ are ${{\textstyle\frac{1}{2}}}\tau_z ,
{{\textstyle\frac{1}{2}}}\tau_y$ in
the indicated planes. Written in this form, we see that the two
candidates are in fact the same. From equation \ref{rotTh}, both
choices for $b(\beta)$ induce the same rotated group element $b(\beta)
\ h^{-1}(\alpha)\ b^{-1}(\beta)$ (modulo sign redefinition of $\beta$
).  We can thus choose a single deformation $g(\alpha, \beta)$,
exploring only a planar $SU(2)$ subgroup of $SU({N_f})$, to produce the
flux tube of minimal gradient energy associated with the $Z_2$ loop
$h^2(\alpha)$. Note that $g$ indeed gives a $Z_2$ object, with
$g(\alpha, \beta = 2\pi) = {{\mbox{1\hskip-0.22em\relax l}}\,}$.  (This $Z_2$
structure appears in the
$\beta \rightarrow -\beta$ equivalences above, as well).

 From this form we construct the flux tube
$\Sigma = g g^T:$
\begin{equation}
\label{Sigbh}
\Sigma = b(\beta) \ h^{-1}(\alpha)
                        \ b^{-2}(\beta) \ h(\alpha) \ b(\beta).
\end{equation}
This gives
\begin{equation}
\label{Sigtpexp}
\Sigma(\alpha, \beta) =
{{\mbox{1\hskip-0.22em\relax l}}\,} - 2 \sin^2{(\alpha /2)} \sin^2{\beta}\
{{\mbox{1\hskip-0.22em\relax l}}\,}_\Box
+  i \sin^2{(\alpha /2)} \sin{(2\beta)}\ \tau_{z\Box}
-  i \sin{\alpha } \sin{\beta}\ \tau_{x\Box} ,
\end{equation}
where ${{\mbox{1\hskip-0.22em\relax l}}\,}_\Box$ gives the identity in the
plane $\Box$ and vanishes
outside it, and ${{\mbox{1\hskip-0.22em\relax l}}\,}$ is the usual $SU({N_f})$
identity.  Of course,
this form for $\Sigma$ can deform while
remaining a nontrivial flux tube. In particular, for $\Sigma$ on
$G/H$, the variables $\alpha$ and $\beta$ give coordinates on the
physical plane $R^2$, identified to $S^2$ by the condition $\Sigma
\rightarrow {{\mbox{1\hskip-0.22em\relax l}}\,}$ at spatial infinity ({\em
i.~e. } on the boundary $\alpha \in \{0,
2\pi\}$; $\beta \in\{ 0,\pi\})$. We can deform $\Sigma$ to a radially
symmetric form by identifying $r=\infty$ with this boundary, and
$r=0$ with the center $\alpha = \pi, \beta =
\pi/2$ of the deformation $g$ producing $\Sigma$. This yields
\begin{equation}\begin{array}{c}
\label{Sigralform}
\Sigma(r, \theta) = h(\theta) \ b(r) \ h^{-1}(\theta),
\ \ \mbox{for }\nonumber\\[3pt]
 h(\theta) = \exp{\{i \theta \,\, T_{h\Box}\}},
\ b(r) = \exp{\{i F(r) \,\, T_{b\Box}\}},
\end{array}\end{equation}
with $T_{h\Box}, T_{b\Box}$ as
in our definition \ref{defbh} for $g(\alpha, \beta)$.

We derive this radially symmetric form for $\Sigma$ and fix
$F(r=0)$ as follows. We expand equation \ref{Sigralform} to obtain
\begin{equation}
\label{Sigrexp}
\Sigma(r, \theta) =
{{\mbox{1\hskip-0.22em\relax l}}\,} + \left(\,\,\cos (F/2) - 1 \,\,\right)\
{{\mbox{1\hskip-0.22em\relax l}}\,}_\Box
+ i \sin{(F/2 )} \left(\,\, \cos{\theta}\ \tau_{z\Box} -
\sin{\theta}\ \tau_{x\Box} \,\,\right).
\end{equation}
which can be identified with equation \ref{Sigtpexp}, term by term. This
gives an undeformed relation between the topological description $g(\alpha,
\beta)$ and the spatial form $F(r),\theta$:
\begin{eqnarray}
F(r) &=& 4 \tan^{-1} \left( \,\,[\sin(\alpha/2) \sin\beta]^{-2} - 1\,\,
                 \right)^{-1/2}\nonumber\\
\theta &=& 2 \tan^{-1} \left( \,\cot(\alpha/2)
                \left[ \cos\beta  + \sin\beta
                \left( \,\, [\sin(\alpha/2) \sin\beta]^{-2} - 1\,\,
                 \right)^{1/2}\right]^{-1}\,\right),
\end{eqnarray}
where $\tan^{-1}$ gives values in a single period $[-\pi/2,\pi/2]$.
The arguments to $\tan^{-1}$ range from $0$ to $\pm\infty$ for
$\theta$, and from $0$ to $\infty$ for $F(r)$. Thus $\theta$ covers a
range $[-\pi,\pi]$, suitable for a polar angle, and $F(r)$ ranges from
$0$ to $2\pi$. $F(r)$ assumes the extremes of its range at the
boundary $r=\infty$, where $F(r)=0$, and the center $r=0$, where
$F(r)=2\pi$. These boundary conditions comprise the only relevant
feature of the mapping, the homotopy invariant $F(r=0) - F(r=\infty).$
They establish boundary conditions for any single-winding flux tube of
form \ref{Sigralform}: $F(0) = 2\pi, F(r \rightarrow \infty) = 0$.

Finally, we note that while form \ref{Sigralform} for $\Sigma$ indeed
minimizes gradient energy, potential terms for $\Sigma$ --- like a quark
mass term --- can favor a vev other than $\exp{(i F(r) \,\,
T_{b\Box})}\ $. Potential terms consistent with unbroken $SO({N_f})$
symmetry give a vev we can always diagonalize.  We thus consider
the extending $\Sigma$ so that $F(r) \, T_{b\Box}\ \rightarrow F_d(r)
\,T_d,$ where $F_1 \equiv F$ and $T_d$
varies over the Cartan subalgebra generators \ref{defTd}.
The nonplanar $T_{d>1}$ commute with both $T_{b\Box}$ and $T_{h\Box}$; that is,
they commute with the full embedding $g(\alpha, \beta)$. We may thus
obtain our extension by the simple modification
$
g(\alpha, \beta) \rightarrow g(\alpha, \beta)\ \exp{\{i F_{d>1}(r) \,
T_{d>1}/2\}}.
$
Taking $F_{d>1}(r)$ to vanish as $r \rightarrow \infty$, this does not affect
the behavior of $g$ on the boundary --- hence it leaves the topology
unchanged.  It changes the resulting flux tube only by the desired
overall multiplication $\Sigma \rightarrow \Sigma \ \exp{\{i F_{d>1}(r)\,
T_{d>1}}\}
.$ Thus the true minimum energy flux tube is one of a family of
nontrivial configurations:
\begin{equation}\begin{array}{c}
\label{Sigfinform} \Sigma(r, \theta)=h(\theta) \ b(r) \
h^{-1}(\theta),\ \ \mbox{with }\\[3pt]
h(\theta)= \exp{\{i \theta
\,\, T_{h\Box}\}},\ b(r) =  \exp{\{i F_d(r) T_d\, \}},\\[3pt]
\mbox{and } F_1(r=0) = 2\pi; \ \ F_d(r\rightarrow\infty) = 0.
\end{array}\end{equation}
Which of these candidates is realized remains a question
of dynamics.

\section{Flux Tube Stability and Dynamics}

Studying flux tube dynamics begins with the
question of stability. As minimal model for the Goldstone field
 $\Sigma$, with stable skyrmions, we have
the Skyrme lagrangian
\begin{equation}
\label{SSkyrme}
{\cal L}_0= \frac{F_\pi^2}{16} \,\,{\mbox{tr}\,} \,\partial_\mu \Sigma
\partial^\mu {\Sigma}^\dagger
        +\frac{1}{32e^2} \,\,{\mbox{tr}\,}\,
  {\hbox{\bf [}  {\Sigma }^\dagger \partial_\mu \Sigma,{\Sigma }^\dagger
\partial_\nu \Sigma  \hbox{\bf ]} }^2
\ .
\end{equation}
Unaltered, this lagrangian implies a size instability for all flux tubes.
For under the rescaling $\Sigma(r,\theta) \rightarrow \Sigma (\lambda
r,\theta)$, the tension of a finite flux tube can always decrease.
Specifically, its quadratic contribution stays invariant, while its
quartic rescales by a factor $\lambda^2$ --- leading to an energy
minimized when $\lambda = 0.$ Physically, this corresponds to flux
tubes which diffuse to infinite size to lower their energy.

To stabilize the flux tubes, we must consider modifications of the
minimal forms for $\Sigma$ and ${\cal L}_o$. First, we note that adding
higher derivative gradient terms to ${\cal L}_o$, analogous to the Skyrme
term, cannot both stabilize $\Sigma$ and produce a positive definite
Hamiltonian.  This holds because such terms can be at most second
order in time derivatives, and, by Lorentz symmetry, $r$ and $\theta$
derivatives; thus they can never give tension contributions scaling
more strongly than $\lambda^2$. This leaves only two options for
stabilizing $\Sigma$. First, we can consider a potential for
$\Sigma$, giving a tension component that scales as $\lambda^{-2}$.
Second, we can allow the flux tube to vary along its axis --- giving
$\Sigma$ some $z$-dependence --- to obtain a quadratic contribution to
the energy scaling as $\lambda^{-2}$. The second option, implemented
by exciting zero modes along the flux tube's $z$-axis, has a
structure paralleling that which arises in quantizing the flux tube.
We thus defer a study of its dynamics until the next section
on quantization, and focus on stabilization by a potential. Ultimately
both approaches give similar results: both give as minimal flux tubes
representatives from the family \ref{Sigfinform} with vanishing
$F_{d>1}(r)$. Stabilization by $z$-rotation has a richer
dynamical structure, however --- partly because it introduces
unconstrained rotational parameters into the problem.

A stabilizing potential, on the other hand, arises naturally in our
theory.  By giving the quarks bare Majorana masses, we explicitly
break the $SU({N_f})$ flavor symmetry of our original QCD gauge theory.
This induces a pion mass term,
\begin{equation}
\label{SMass}
{\cal L}_m =       \frac{F_\pi^2 \, m_\pi^2}{16} {\mbox{tr}\,}
        \left(  \Sigma  + {\Sigma}^\dagger - 2\cdot
{{\mbox{1\hskip-0.22em\relax l}}\,} \right)
\ ,
\end{equation}
in the limit of degenerate quark masses. The mass term stabilizes the
flux tube while affecting its dynamics in a simple way, as we discuss
below.

To study dynamics, we calculate the energy density of flux tubes of the
general form \ref{Sigfinform}. The Skyrme action \ref{SSkyrme} gives
a gradient contribution
\begin{equation}
\label{Egradgen}
\rho_{0}= \frac{F_\pi^2}{16}
        {\mbox{tr}\,} \left \{
        \left(\,F'_d\, T_d\, \right)^2
        +\frac{1}{r^2} \,\, \left( \tilde{T}_\Box^2 -
        {\hbox{\bf [} \,F'_d\, T_d\, ,\tilde{T}_\Box  \hbox{\bf ]} }^2 \right)
        \right \} \ ,
\end{equation}
where
\begin{equation}
\label{defTtilde}
{\tilde{T}_\Box} \equiv
b^{-1} (r) \ {\hbox{\bf [} \ T_{h\Box}\ ,\ b(r)\  \hbox{\bf ]} }
\ .
\end{equation}
and $r$ has been rescaled into the dimensionless units $eF_\pi \,
r_{phys}$.  This form for $\rho_0$ follows algebraically, using no
information about the generators $T_d$ and $T_{h\Box}$.
${\tilde{T}_\Box}$ measures the noncommutativity of the radial
generator $ F_d(r)\, T_d$ with its angular counterpart $T_{h\Box}$.
The nonplanar $T_{d>1}$ commute with all generators $T_{h\Box}, T_d$
--- making ${\tilde{T}_\Box}$ and ${\hbox{\bf [} \,F'_d\, T_d\,
,{\tilde{T}_\Box} \hbox{\bf ]} }$
independent of $F_{d>1}$.  This has two consequences: first, nonplanar
terms contribute only the positive definite sum $\sum_{d>1}
{F'_d}^2/2$ to the gradient energy density. Second, the
${\tilde{T}_\Box}$-dependent terms span only a planar $SU(2)$
subgroup, where $T_{d=1} = {{\textstyle\frac{1}{2}}} \tau_{z\Box}$ and
$T_{h\Box} = {{\textstyle\frac{1}{2}}}
\tau_{y\Box}$. They are thus calculable, giving the flux tube gradient
energy density
\begin{equation}
\label{Egrad}
\rho_{0}= \frac{F_\pi^2}{16}
        \left \{ {{\textstyle\frac{1}{2}}}  {\textstyle \sum_{d}} {F'_d}^2
        +\frac{1}{r^2} \left( 1- \cos F_1 \right) \left( 1 + {F_1'}^2 \right)
        \right \} \ .
\end{equation}

The pion mass term \ref{SMass} also contributes to the energy.
Form \ref{Sigfinform} for the flux tube yields its contribution in closed form:
\begin{equation}
\label{EpotN}
\rho_m = \frac{m_\pi^2}{16e^2}
        \left \{ 2 {N_f} - 4 \cos \left(\sum_{d>1}\omega_d/2\right)
\cos (F_1/2) - 2 \sum_{d>1}\cos\omega_d
        \right \} \ ,
\end{equation}
with
\begin{equation}
\label{defomegad}
\omega_d \equiv \frac{- d \,\, F_d}{\sqrt{2d(d+1)}}
 + \sum_{i>d}\frac{F_i}{\sqrt{2i(i+1)}} \, ,
\end{equation}
where $d = 2, \ldots,{N_f}-1$ and $r$ has again been rescaled to $eF_\pi
\, r_{phys}$.

The minima of this potential fix boundary conditions for
$\Sigma$ at spatial infinity.  Such minima have
$F_{1} = 2\pi m$, consonant with our a priori
boundary condition $F_1(r\rightarrow\infty) = 0$. To minimize with respect to
the nonplanar $\omega_d$, we set
\begin{equation}
\label{omegavac}
\frac{\partial\rho_m}{\partial \omega_d} \sim
 2\sin\left(\sum_{d>1}\omega_d/2\right) \cos (F_1/2)
+ 2 \sin \omega_d = 0\ ,
\end{equation}
with positive second derivatives ${\partial^2 \rho_m} /{\partial
  \omega_d^2}, {\partial^2 \rho_m}/ {\partial F_1^2}$.  This gives the
degenerate family $\omega_d = 2\pi j_d$, for any integers $j_d$
summing to the same parity as $m$. As the simplest representative, we
take $\omega_2 = 2m\pi$ and $\omega_d = 0$ for $d>2$. This
completes our boundary conditions at infinite radius: $F_1 \rightarrow 0$,
$\omega_d \rightarrow 0$, for all $d$; corresponding to $F_d \rightarrow 0$ for
all $d$, in our original variables.

Away from spatial infinity, we must propagate these boundary
conditions inward to obtain flux tubes obeying the full equations of
motion. However, given our boundary conditions $\omega_d(r\rightarrow\infty)
= 0$, these equations trivialize for the nonplanar variables
$F_{d>1}$.  They have independent gradient energy terms \ref{Egrad},
which combine with the potential \ref{EpotN} to give equations of
motion
\begin{equation}
\label{eomnonplan}
\frac{F_\pi^2}{16}\ \frac{1}{r}\,\,\partial_r \left( \, rF'_d \right) =
\frac{\partial \omega_e}{\partial F_d}\
\frac{\partial \rho_m}{\partial \omega_e}
\quad\mbox{for $d>1$.}
\end{equation}
 From equation \ref{omegavac}, the vacuum values $\omega_e =2\pi j_e$
cause the source terms to vanish on the right hand side --- even when
the planar field $F_1$ departs from its vacuum value.  Thus our
boundary conditions, setting $F_d=\omega_d=0$ at infinite radius,
induce only constant solutions as we propagate equation \ref
{eomnonplan} inward to the origin. This gives $F_d$ that vanish
identically for nonplanar $d>1$.

This implies that the minimum energy flux tube for any ${N_f}$ varies
only over the planar $SU(2)$ subgroup. It takes the simple form
\ref{Sigralform}, with $F(r) \equiv F_1(r)$, and has energy density
\begin{equation}
\label{Eflux}
\rho= \frac{F_\pi^2}{16}
        \left \{ {{\textstyle\frac{1}{2}}} {F'} ^2
        +\frac{1}{r^2} \left( 1- \cos F \right) \left( 1 + {F'}^2 \right) +
         \lambda^2 \left( 1 - \cos (F/2)\right) \right \} \ ,
\end{equation}
where $\lambda = 2 m_\pi/e\,F_\pi.$ This determines a nonlinear
equation of motion for $F$:
\begin{eqnarray}
\label{eomplan}
\lefteqn{
\left( \,\, 1 +\frac{4}{r^2} \sin^2 (F/2)\, \right)\
\frac{1}{r}\,\,\partial_r \left( \, rF' \right) =}&&\nonumber\\
&&\sin (F/2)\left\{ \ \frac{2}{r^2}\,\cos(F/2) \
\left( \,\, 1-{F'}^2 \right) \ + \
\frac{8F'}{r^3} \sin(F/2) \ + \ \frac{\lambda^2}{2}
\right\} \ .
\end{eqnarray}

Numerical solutions to equation \ref{eomplan} are shown in Figure 1
for different values of $\lambda$, including the physical
$\lambda_0=0.236$ ($e=2\pi$, $m_{\pi}=138$ Mev and $F_{\pi}=186$ Mev).
Increasing $\lambda$ raises the flux tube's energy density while
shrinking its core size. The asymptotic regimes agree with limits to
equation \ref{eomplan}. Inside the core, $F$ falls linearly from its
boundary value of $2\pi$ at the origin; outside, it scales as the
hyperbolic Bessel function $F \sim x^{-1/2}\,\exp{\{-x\}},$ for $x =
\lambda r/2$. The parameter $2\lambda^{-1}$ thus sets the flux tube's
core size in dimensionless units. In physical units, this gives core
size $m_\pi^{-1}$, sensible for a bound state of Goldstone modes
$\Sigma$.  It carries tension proportional to $F_\pi^2$ --- which is
proportional to the number of colors $N_c$ in a large $N_c$ QCD limit.
This supports its interpretation as a mediator of the confining force
between spinor sources, whose scales as the spinor Casimir $N_c$.
Numerically we find tension $4.6 F_\pi^2$ when $\lambda=\lambda_0$.
As $\lambda$ varies, the tension varies as shown in figure 2.

\section{Quantizing the Flux Tube}

\subsection{ Semiclassical Zero Modes and Quantum Stability}

Under quantization, the flux tube samples not only the ground state
above, but also its zero modes. Recall, from section II, $\Sigma$'s
transformation laws:
$\Sigma \rightarrow a\Sigma a^T$ under
global rotations $a$, giving the residual symmetry $H
= SO({N_f})$ at infinite radius where $\Sigma = {{\mbox{1\hskip-0.22em\relax
l}}\,}$. However, because $\Sigma$
varies nontrivially, global $H$ rotations are not
symmetries within the core. Instead they produce distinct
 degenerate configurations, coincident at spatial infinity. The
zero modes of $\Sigma$ explore these configurations:
\begin{equation}
\label{defSigt}
\Sigma(t, r, \theta) = A(t) \ \Sigma(r,\theta)\ A^{-1} (t) \ ,
\end{equation}
where $A(t) \equiv \exp{\{ieF_\pi\,\omega\, t \, \,n_h T_h\}}$ rotates
in $H$ with dimensionless frequency $\omega$. \footnote{We neglect
  other excitations, such as bending modes.} These zero modes have
rotational energy confined to the string core; an energy that can be
made arbitrarily small, classically, by taking $\omega\rightarrow 0$.

Classically, we calculate this rotational energy from the Skyrme
action \ref{SSkyrme}:
\begin{equation}
\label{Ezms}
\rho_{\omega} = \frac{F_\pi^2 \,\,\omega^2}{16}
{\mbox{tr}\,} \left ( \hat{T}^2 -
        {\hbox{\bf [} \,F'_d\, T_d\, ,\hat{T} \hbox{\bf ]} }^2 -
        \frac{1}{r^2}   {\hbox{\bf [} \,{\tilde{T}_\Box}\, ,\hat{T} \hbox{\bf
]} }^2
        \right) \ .
\end{equation}
Here ${\tilde{T}_\Box},$ from equation \ref{defTtilde},
measures the noncommutativity of radial and angular generators
$F_d(r) T_d$ and $T_{h\Box}$, while
\begin{equation}
\label{defThat}
\hat{T} \equiv b^{-1}(r) \ {\hbox{\bf [} \ h^{-1 } (\theta)\ n_h T_h\ h
  (\theta)\ ,\ b(r)\ \hbox{\bf ]} }
\end{equation}
measures the noncommutativity of $A(t)$ with $\Sigma(r,\theta)$.  We
have retreated to the general form \ref{Sigfinform} --- including
nonplanar contributions --- for $\Sigma(r,\theta)$; thus we retain
complete expressions \ref{Egrad} and \ref{EpotN} for its gradient and
potential energy.  We do this for two reasons: first, we must insure
that the planar vacuum $F_{d>1} =0$ --- favored by the weak pion mass
term --- remains minimal despite quantum fluctuations due to zero
modes; and second, we wish to explore stabilization by $z$-rotation,
where $A = A(z)$.

Calculating the rotational energy \ref{Ezms} is straightforward.  We
expand $2n_h T_h$ as $ n_{(jk)} \,\tau_{y(jk)}$; then note that the
tilde operation, defined in equation \ref{defTtilde}, acts on basis
elements as follows:
\begin{equation}
\label{Ttildebasis}
2 \tilde{T}_{(jk)}  =
\left(  \begin{array}{rr}
&-i{(\pi_{jk}-1)}\\ i(\pi^*_{jk}-1)&
\end{array} \right)_{(jk)} =
( \cos\omega_{jk}-1) \,\,\tau_{y(jk)} +
\sin \omega_{jk}  \,\,\tau_{x(jk)} \ .
\end{equation}
Here $\pi_{jk}$ is given by
\begin{equation}
\label{defomjk}
\pi_{jk} \equiv \exp{\{i\omega_{jk}\}}
\equiv
\exp{\{i F_d (T_{d,\,\,k} - T_{d,\,\,j})\}} \ ,
\end{equation}
where $T_{d,\,\,k}$ denotes the $k$-th diagonal entry in the Cartan
generator $T_d$, from equation \ref{defTd}.
The commutator terms  act on the relevant basis elements $\tau_{x(jk)},
\,\,\tau_{y(jk)}$ to give
\begin{eqnarray}
{\hbox{\bf [} \ T_d\ , \ \tau_{a(jk)}\   \hbox{\bf ]} } &=& -i (T_{d,\,\,k} -
T_{d,\,\,j})\
 \epsilon_{ab}\,\,\tau_{b(jk)} \quad \forall j,k\nonumber\\
{\hbox{\bf [} \ 2\,\tilde{T}_\Box\ , \ \tau_{a\Box'} \   \hbox{\bf ]} } &= &
i p\,(\cos{\omega_\Box} -1 ) \,\,\tau_{a\Box''}
+ i \, \sin{\omega_\Box}\,\,  \epsilon_{ab}\,\,\tau_{b\Box''}
\ , \end{eqnarray}
where $\epsilon_{ab}$ is the two dimensional permutation matrix
$\epsilon_{xy}.$ The second commutator contributes only when
planes $\Box$ and $\Box'$ intersect, with ${\hbox{\bf [} \tau_{y\Box} ,
  \tau_{y\Box'}\hbox{\bf ]} } = ip\,\, \tau_{y\Box''}$ defining $\Box''$ and $p
=
\pm 1$.  Altogether, these expressions imply the rotational energy
\begin{equation}
\label{Erot}
\rho_\omega = \frac{F_\pi^2\omega^2}{16}
\left\{\begin{array}{c}
n^2_{(12)}   \left( 1- \cos F_1 \right) \left( 1 + {F'_1}^2 \right) +
{\displaystyle \sum_{k>j>2}} n^2_{(jk)}
 \left( 1- \cos \omega_{jk} \right) \left( 1 + {\omega'_{jk}}^2
\right) +\\[0.3ex]
{{\textstyle\frac{1}{2}}} {\displaystyle \sum_{k>2}} ( n^2_{(1k)} +  n^2_{(2k)}
)
\left(\begin{array}{r}
 \left( 1- \cos \omega_{1k} \right) \left( 1 + {\omega'_{1k}}^2
        + \frac{1}{2r^2} \left( 1- \cos F_1 \right)  \right) \\ +
 \left( 1- \cos \omega_{2k} \right) \left( 1 + {\omega'_{2k}}^2
        + \frac{1}{2r^2} \left( 1- \cos F_1 \right)  \right)
\end{array}\right)
\end{array}\right\} \ .
\end{equation}

This form for the rotational energy depends on the planar $F_1$, both
explicitly and through the variables $\omega_{jk}$. To extract its
$F_1$ dependence, we note that
\begin{equation}
\omega_{jk} = \omega_{k-1} - \omega_{j-1} ,
\end{equation}
where
\begin{equation}
\omega_0 \equiv  - {{\textstyle\frac{1}{2}}}
\left({\displaystyle\sum_{d>1} \omega_d} \right) \ + \ F_1/2 \ ,\quad
\omega_1 \equiv - {{\textstyle\frac{1}{2}}}
\left({\displaystyle\sum_{d>1} \omega_d}  \right) \ - \ F_1/2 \ .
\end{equation}
and the nonplanar $\omega_{d>1}$ are given by eq.~\ref{defomegad}.
We may thus rewrite the rotational energy as
\begin{equation}
\label{Erotfin}
\rho_\omega = \frac{F_\pi^2\omega^2}{16}
\left\{\begin{array}{c}
n^2_{(12)}   \left( 1- \cos F_1 \right) \left( 1 + {F'_1}^2 \right) +
{\displaystyle \sum_{k>j>2}} n^2_{(jk)}
 \left( 1- \cos \omega_{jk} \right) \left( 1 + {\omega'_{jk}}^2
\right) \\[0.5ex] +
{{\textstyle\frac{1}{2}}} {\displaystyle \sum_{k>2}} ( n^2_{(1k)} +  n^2_{(2k)}
)
\left(\begin{array}{c}
\left( 1 + {S'_{k-1}}^2
        + {{F'_1}^2}/{4} + \frac{1}{2r^2} \left( 1- \cos F_1 \right)
\right) \\
\cdot \left( 1- \cos S_{k-1} \cos(F_1/2)\right) \\
+ F'_1\ S'_{k-1}
\sin S_{k-1}\sin(F_1/2)
\end{array}\right)
\end{array}\right\} \ ,
\end{equation}
where $ S_{k} \equiv \omega_{k} + {{\textstyle\frac{1}{2}}} \sum_{d>1}
\omega_d$.
This depends only on $F_1$ (explicitly) and the nonplanar
$\omega_{d>1}$ (explicitly and through $S_{k-1}$, $\omega_{jk}$).

Neglecting the pion mass term for now, our boundary conditions require
$F_1$ and $\omega_{d>1}$ to approach a minimum of the potential \ref
{Erotfin} at spatial infinity. As in the previous section, these
minima occur when $F_{1} = 2\pi m$ and $\omega_{d>1} = 2\pi
j_d$, for any integers $j_d$ whose sum has the same parity as $m$.  We
take as the simplest representative
$\omega_2 = 2m\pi$ and $\omega_{d>2} = 0$. (The specific case
${N_f} = 3$ has additional degenerate vacua, non-coincident with
those of the pion mass term, given by $\omega_2 = 2m\pi/3$; however,
these reduce to the pion mass vacua above when more families
are added.) Thus our boundary condition $F_1(r\rightarrow\infty) = 0$
induces the full boundary conditions at infinite radius: $F_1 \rightarrow
0$, $\omega_{d>1} \rightarrow 0$; corresponding to $F_d \rightarrow 0$
for all $d$, in our original variables.

The rotational energy $\rho_\omega$ generates spatially-dependent
corrections to the equations of motion for $F_d$. However, these
corrections trivialize for the nonplanar variables $F_{d>1}$, given our
boundary conditions $\omega_d = 0$. We show this as follows: the
rotational energy \ref{Erotfin} introduces an additional term
\begin{equation}
\Delta  = \frac{\partial \omega_e}{\partial F_d}\
\left(\frac{\partial \rho_\omega}{\partial \omega_e}
 - \frac{1}{r}\,\,\partial_r \left(
        \,\, r \,\,\frac{\partial \rho_\omega}{\partial \omega'_e}\right)
\right)
\end{equation}
into the right hand side of the equation of motion \ref{eomnonplan}
for $F_d$. This term is generally quite complicated;
however, when the nonplanar $\omega_{d>1}$ assume their asymptotic
values of zero, it simplifies. For a single $\omega_e$ (before
multiplying by the Jacobian), it becomes
\begin{eqnarray}
\label{eompt}
\Delta_e& = &\omega^2 \left\{
E_{e+1} + {{\textstyle\frac{1}{2}}} {\displaystyle \sum_{k>2}}  E_k\right\}\ ,
\quad
\mbox{with}\nonumber\\
E_k &=&
-  F'_1\ S'_{k-1} \ \sin(F_1/2) - {{\textstyle\frac{1}{r}}}\, \partial_r
\,(\,2r S'_{k-1})\
\left( 1- \cos(F_1/2)\right) \ .
\end{eqnarray}
Asymptotically, where $F_1 =0$, all $\Delta_e$ vanish, and
the full equation of motion \ref {eomnonplan} is again sourceless.
This gives asymptotic solutions for the nonplanar $F_d$ which are
again Bessel functions of order zero --- implying, for finite energy,
that $F_{d>1}$ and all its derivatives (or $\omega_{d>1}$ and all its
derivatives) vanish at infinite radius.

However, given nonplanar terms that obey $\omega_{d>1} = \omega'_{d>1} =
\omega''_{d>1} = 0,$ the perturbations $\Delta_e$
vanish, regardless of the planar field $F_1(r)$. So, as for pion mass
stabilization, equation \ref {eomnonplan} remains sourceless as we
propagate it inward to the origin, giving $F_{d>1}$ that vanish
identically.

Thus the planar vacuum $F_{d>1}=0$ survives quantum fluctuations due
to zero modes. Its static limit retains the form \ref{Sigralform},
with $F(r) \equiv F_1(r)$ varying only over a planar $SU(2)$ subgroup.
It has classical energy, from equation \ref{Erotfin},
\begin{equation}
\label{Ezmfin}
\rho_{tot} = \frac{F_\pi^2}{16}
        \left \{
\begin{array}{c}
         {{\textstyle\frac{1}{2}}} {F'} ^2
        +\left( r^{-2} + \omega^2 n^2_\Box \right)
 \left( 1- \cos F \right) \left( 1 + {F'}^2 \right)
        \ + \lambda^2
        \left( 1 - \cos (F/2)\right)
         \\
+  \omega^2 n^2_{\Box'}\ \left( 1- \cos(F/2)\right)
\left( 1 + {{F'}^2}/{4} + \frac{1}{2r^2} \left( 1- \cos F \right)
\right)
\end{array} \right \}
\end{equation}
where
\begin{equation}
n^2_\Box = n^2_{(12)} \,\, ,\quad  n^2_{\Box'} =
 {\scriptstyle \sum_{k>2}}  n^2_{(1k)} +  n^2_{(2k)}  \ ,
\end{equation}
describe the orientation of the zero mode rotation $A(t)$ relative to
$\Sigma(r, \theta)$.

For a flux tube stabilized by $z$-rotation, where $A = A(z)$, the
rotational terms in $\rho_{tot}$ deform the nonlinear equation of motion for
$F$. Classically, we can rescale $\omega$ to set
$n^2_{\Box} + n^2_{\Box'} = 1$ (a rescaling unnecessary when ${N_f} =
3$).  This gives energy as a monotonic function of $n^2_{\Box}$
--- which measures the component of $\Sigma\,\mbox{'s}$ $z$-dependence
due to slip dislocation, where $\Sigma(z,r,\theta)$ rotates about its
core in a single internal space plane $\Box$, but starts at the
$x$-axis with changing offset angle $\theta_0(z)$.  Its converse,
$n^2_{\Box'}$, measures the twist dislocation in $\Sigma(z,r,\theta)$:
the extent to which $A(z)$ rotates the internal space plane, $\Box
\rightarrow R_{ab}\,(z) \Box$, in which $\Sigma(z,r,\theta)$ cycles about
its core.

Numerical solutions for the $z$-stabilized $F$ are shown in Figures 3
and 4.  Their
asymptotic behavior agrees with analytic limits: inside the core, $F$
falls linearly from its boundary value of $2\pi$ at the origin; while
outside, it scales as the hyperbolic Bessel function $F \sim
x^{-1/2}\,\exp{\{-x\}},$ for $x = \left( \lambda^2 + \omega^2 (1 + 3
n^2_{\Box}) \right)^{1/2} r/2$. That is, the flux tube becomes more
compact as either $\omega$ or $n_\Box$ grows. As in the mass
stabilization case, shrinking core sizes correlate with growing
tensions. Thus we see, from Figure 3, that the value $n_{\Box}=0$ is
favored, and twisting dislocations cost less energy than their
slip-offset counterparts. Figure 4 confirms the flux tube's tendency
both to shrink and to gain energy with increasing rotational frequency, and
shows how significant the rotational deformation of $F(r)$ is. Thus the
 tension not only acquires rotational energy terms,
\begin{equation}
\label{rotten}
\tau
\rightarrow\ \tau + {{\textstyle\frac{1}{2}}} \omega^2
(\,n^2_{\Box}\,\Lambda_{\Box} + n^2_{\Box'}\,\Lambda_{\Box'}) ) \ ;
\end{equation}
it also contains hidden rotational dependence through deformations of
the ground state tension $\tau$ and moments of inertia
$\Lambda_{\Box}, \Lambda_{\Box'}$. We explore these deformations in
Figure 5, showing that $\tau$ grows linearly with $\omega$, due to
flux tube compression. The moments of inertia instead fall rapidly,
dropping by a factor of five as $\omega$ grows from $0$ to $2$, before
stabilizing at roughly $\Lambda_{\Box} = 10 \ F_\pi^2,
\Lambda_{\Box'} = 4 F_\pi^2$ for $\omega \ge 2\,$ . We show below that
this range $\omega \ge 2$ is typical of quantum rotational excitations
of the flux tube; thus we are justified in neglecting rotational
deformation of $\Lambda_{\Box}, \Lambda_{\Box'}$ over this range.

Finally, we note that the geodesic parametrization assumed
above for the rotation $A(t)$, while useful for discussing the flux
tube's classical limit, does not restrict our analysis. Any function
$A(t)$ over $H= SO({N_f})$ induces the energy \ref{Ezmfin}, with $\omega
\, n_{(jk)}$ defined by
\begin{equation}
\label{defoms}
\omega\, n_{(jk)} \equiv {-i}{(eF_\pi )}^{-1}\,\,
        {\mbox{tr}\,} \ {A}^\dagger \dot{A} \,\, \tau_{y(jk)} \ .
\end{equation}

\subsection{The Quantized Spectrum}

 From the previous section, there are two ways to stabilize the flux
tube: by exploiting the pion mass term in the Lagrangian or by
$z$-rotation.  These give the two classical solutions
$\Sigma(r,\theta)$ (equation \ref{Sigfinform}) and
$\Sigma(r,\theta,z)$ (equation \ref{defSigt}, with $t\rightarrow z$).  Both
have rotational zero modes induced by $A(t)\in H$. We now
quantize the spectrum of these zero modes to find the flux tube's quantum
numbers and lowlying excitations \cite{Balach}.

Quantizing $\Sigma(r,\theta,z)$ is a complicated task, because it
involves two rotations, the $z$ and $t$ rotations. Furthermore,
there is no physical input for the $z$-rotation frequency $\omega$.
Therefore, we restrict ourselves to quantizing the mass-stabilized solution
$\Sigma(r,\theta)$.

Equations \ref{rotten} and \ref{defoms} determine
a two-dimensional Lagrangian for the flux tube:
\begin{equation}
\label{MQSkyrme}
L({\tilde z},t) = -\tilde{\tau}
-{\tilde{\Lambda}_{\Box}\over 2}({\mbox{tr}\,}{A}^\dagger{\dot A}T_{h\Box})^2
 -{\tilde{\Lambda}_{\Box'}\over 2} \sum_{\Box'}\
({\mbox{tr}\,}{A}^\dagger{\dot A}T_{h\Box'})^2\ ,
\end{equation}
where $\tilde{\tau}$ is a ground state tension and
${\tilde z}=eF_{\pi}z$ measures dimensionless length
along the flux tube.
The moments of inertia $\tilde{\Lambda}_{\Box}$ and $\tilde{\Lambda}_{\Box'}$
come from equation \ref{Ezmfin}:
\begin{eqnarray}
\tilde{\Lambda}_{\Box} &=& {\pi\over {e^3F_{\pi}}}
\int\,rdr\ (1+ F^{\prime 2})(1-\cos F)
\nonumber\\
\tilde{\Lambda}_{\Box'} &=& {\pi\over {e^3F_{\pi}}}
\int\,rdr\ \left(1+{F^{\prime 2}/ 4}+ {{\textstyle\frac{1}{2r^2}}}
(1-\cos F) \right)\left(1-\cos(F/ 2)\right)\ .
\end{eqnarray}

We note that form {\ref{MQSkyrme}} for the Lagrangian comes from
only two facts: from the confinement of $\Sigma$
to the planar $SU(2)_{\Box}$ subgroup in internal space;
and from cylindrical symmetry in physical
space. The skyrmion in this theory, which we discuss in the
following section, shares these characteristics.
Thus its one-dimensional Lagrangian $L(t)$ also has form
{\ref{MQSkyrme}}, with $\tilde{\tau}$, $\tilde{\Lambda}_{\Box}$ and
$\tilde{\Lambda}_{\Box'}$ dependent upon skyrmion dynamics.

To quantize such a Lagrangian, we must do two things:
first, we must understand the transformation properties
of $\Sigma$ under the symmetries of $L$;
and second, we must write $L$ in terms of invariants of the
quantized Noether charges of those symmetries.
The first task is facilitated by considering the most
general form of $\Sigma$ on the vacuum manifold:
\begin{eqnarray}
\label{GSigma}
\Sigma(x_i,t)\ &=&\ A(t)\ e^{iF(x_i)n_b(x_i)T_b}\ A^{-1}(t)\nonumber\\
\ \ \ \  &=&\ e^{iF(x_i)n_b(x_i)R_{b'b}T_{b'}}\ ,
\end{eqnarray}
where $R_{b'b}(t)$ is an orthogonal matrix encoding the isospin rotation
due to $A$-conjugation:
\begin{equation}
\label{Rotation}
A^{\dagger}T_iA\ =\ R_{ij}T_j \quad\Rightarrow\quad
R_{ij}(t)\ =\ 2\ {\mbox{tr}\,}(\,\, {{A}^\dagger}(t)\,\, T_{i}\,\, A(t)\,\,
T_j\,\, )\ .
\end{equation}
While $R_{ij}$ is defined for all $SU(N)$ generators $T_i$,
it breaks into block diagonal form, $R_{hh'}\bigoplus R_{bb'}$,
over the space of unbroken and broken generators respectively.

For the general form $\Sigma(x_i,t)$, the Lagrangian
{\ref{MQSkyrme}} involves a general sum
$-{\tilde{\Lambda}_h\over 2}({\mbox{tr}\,}{A}^\dagger {\dot A}T_h)^2$.
This Lagrangian is invariant under global $h\in SO({N_f})$,
implemented by either
left or right multiplications $A\to hA$ or $A\to Ah$.
For infinitesimal transformations
$h_{\epsilon}=1-2i\epsilon_hT_h$, $\Sigma(x_i,t)$
transforms as:
\begin{eqnarray}
\label{DA}
\delta_LA &=& (-i2\epsilon_hT_h)A
\ \ \Longleftrightarrow\ \
\delta_L\Sigma(x_i,t)=-i\epsilon_h\ [\ 2T_h\ ,\
\Sigma(x_i,t)\ ]\nonumber\\
\delta_RA &=& A(-i2\epsilon_hT_h)
\ \ \Longleftrightarrow\ \
\delta_R\Sigma(x_i,t)=-i\epsilon_h\ A\
[\ 2T_h\ ,\ \Sigma(x_i)\ ]\ {{A}^\dagger}\ .
\end{eqnarray}
Therefore the left transformation corresponds to $SO({N_f})$
isospin rotation, and its Noether
charge ${\cal I}_h$ satisfies
an $so({N_f})$ algebra after quantization. The right transformation corresponds
to $SO({N_f})$ isospin rotation in the body fixed frame, with
Noether charge ${\cal I}_h'$.
To understand the physical interpretation of ${\cal I}_h'$,
we calculate the commutator in the second line of equation {\ref{DA}},
obtaining
\begin{equation}
\label{DRA}
A[2T_h,\Sigma(x_i)]{A}^\dagger =
2i f_{hbb'}n_{b}\partial_{n_{b'}}\Sigma(x_i,t)\ .
\end{equation}

For the specific cases of flux tube and skyrmion, $n_b (x_i)$ has
two special properties: $n_b = n_{b\Box}$, lying only in an $SU(2)$
subplane of $SU({N_f})$; and $n_b (x_i)$ is spatially axisymmetric,
depending linearly on spatial direction components $u_x$ and $u_y$.
These properties
identify planar isospin rotations with spatial
$z$-rotations. We see this in two steps: the planarity of $n_b$
reduces equations \ref{DA} and \ref{DRA}, when $h = h\Box$, to
\begin{equation}
\delta_R \,\, \Sigma (r, n_b(r, u_i) )
= -2\epsilon \,\, f_{h\Box\,b\Box\,b'\Box}\,\,
n_{b\Box} \,\, \partial_{n_{b'\Box}} \ \Sigma (r, n_b'(r, u_i) ) \ ,
\end{equation}
which the Jacobian relating $n_b$ to $u_i$,
at fixed $u_z$, equates with spatial $z$-rotation :
\begin{equation}
\delta_R \,\, \Sigma (r,  u_i)
= -2\epsilon \,\, \epsilon_{zij}\,\,
u_i \,\, \partial_{u_{j}} \ \Sigma (r, u_i)  \ .
\end{equation}
Thus ${\cal I}'_{\Box}$, which generates planar body-centered isospin
rotations $R$, must equal the defect's spin operator $J_z$, generating
spatial $z$-rotations. Other isospin rotations, which destroy the
planarity of $n_b$, are distinct from other spatial rotations, which
change the defect's axis of axisymmetry.

Under right and left infinitesimal transformations of $A(t)$, the
rotations $R_{b'b}$, which give physical coordinates for $\Sigma(x_i,t)$
on the vacuum manifold, transform as
\begin{eqnarray}
\label{LRR}
\delta_LR_{b'b}\ &=&\ -2\epsilon_hf_{hb'c}R_{cb}\nonumber\\
\delta_RR_{b'b}\ &=&\ -2\epsilon_hR_{b'c}f_{hcb}\ .
\end{eqnarray}
That is, $R_{b'b}$ is left or right multiplied
by ${({\tilde h}_{\epsilon})}_{bc}=1+i\epsilon_h(2if_{hbc})$.
This shows how isospin rotation acting directly on the space
of broken generators
can obey an $so({N_f})$ algebra --- specifically the $so({N_f})$
algebra which occurs as a subalgebra
of $su({N_f})$ in the adjoint representation,
restricted to the block-diagonal component $2if_{hbc}$.

To find ${\cal I}_h$ and ${\cal I}_h'$,
we compute the variation of the Lagrangian under
the two infinitesimal transformations:
\begin{equation}
\label{DL}
\delta_LL = {\dot{\epsilon_h}}{\cal I}_h\ \ \ \ \ \ \
{\hbox{and}}\ \ \ \ \ \
\delta_RL = {\dot{\epsilon_h}}{\cal I}_h'\ .
\end{equation}
This gives
\begin{equation}
\label{LR}
{\cal I}_h' = i\tilde{\Lambda}_h {\mbox{tr}\,}({A}^\dagger {\dot A}T_h)\ \
{\hbox{(no sum),}}\
\ \ \ {\hbox{and}}\ \ {\cal I}_h=R_{hh'}{\cal I}_{h'}'\ .
\end{equation}

We can now write the defect Hamiltonian in terms of the physical
quantized Noether charges: the isospin ${\cal I}$ and angular
momentum ${\cal J}$.  Combining the Lagrangian {\ref{MQSkyrme}} with
Noether charges {\ref{LR}} yields a Hamiltonian dependent on ${\cal
I}_{\Box}'^2$ and ${\cal I}_{\Box'}'^2$, where $\Box'$ are planes
intersecting $\Box$ in only one line. However, the defect
$\Sigma(r, u_i )$ is invariant under a subgroup
$SO({N_f}-2)$ of $SO({N_f})$, acting on the subspace $\Box_{\bot}$
orthogonal to $\Box$.  This implies two things: ${\cal I}'^2 =
{\cal I}_{\Box}'^2 + \sum_{\Box'} {\cal I}_{\Box'}'^2$; and
the defect's allowed quantum states ${\cal I}'$ must contain a singlet
under the $SO({N_f}-2)$ subgroup. For ${N_f} > 3$, this excludes spinor
representations --- constraining the skyrmions and flux tubes to
be bosonic.  Lastly, because ${\cal I}$ is simply a rotation of ${\cal
I}'$, and ${\cal I}_{\Box}'= {\cal J}_z$, we write the Hamiltonian as
\begin{equation}
\label{finalH}
H (\tilde{z}, t) =\tilde{\tau}+
{1\over{2{\tilde\Lambda}_{\Box}}} {\cal J}_{z}^2
+{1\over {2{\tilde\Lambda}_{\Box'}}} (\ {\cal I}^2 - {\cal J}_{z}^2\ ) \ ,
\end{equation}
subject to the constraints above: ${\cal I}_{\Box}'= {\cal J}_z$,
with bosonic defects for ${N_f} > 3$.

Using the numerical results of Section IVA, we may be more explicit
about the flux tube's quantized energy levels. Relating the rotational
energies of equation \ref{finalH} to their counterparts in the tension
\ref{rotten} (for dimensionful $z$) gives
\begin{equation}
\omega^2 = (eF_\pi)^4 \
( \,\, {{\cal J}_z}^2/4{\Lambda_{\Box}}^2 + ( \,\, {\cal I}^2 -
{{\cal J}_z}^2\,\, )/4{\Lambda_{\Box'}}^2\,\, ) \approx 4{{\cal
J}_z}^2 +25 ( \,\, {\cal I}^2 - {{\cal J}_z}^2\,\, )\ .
\end{equation}
Thus $\omega$ assumes values $\omega \ge 2$ for quantized zero modes.
Over this range, $\tilde{\Lambda}_{\Box} $ and
$\tilde{\Lambda}_{\Box'}$ vary little but $\tilde{\tau}$ is described
by the linear fit $\tilde{\tau} = (9.0 + 3.7 (\omega - 2))\,\,
F_\pi/e$, as shown in Figure 5.  Thus our model predicts
\begin{equation}
H (\tilde{z}, t) = \frac{F_\pi}{2\pi} \ \left (1.6 + 3.7\,\, \sqrt{
4{{\cal J}_z}^2 +25 ( \,\, {\cal I}^2 - {{\cal J}_z}^2\,\, )} +
20 {{\cal J}_z}^2 + 50 ( \,\, {\cal I}^2 - {{\cal J}_z}^2\,\, )
\right )
\end{equation}
for the allowed excited states $({\cal I}, {\cal J}_z)$, and
\begin{equation}
H (\tilde{z}, t) = 4.6 \ \frac{F_\pi}{2\pi}
\end{equation}
for the ground state of the flux tube.

\section{Flux Tube Interactions}

We now consider how flux tubes interact with fundamental objects in
our low energy theory. First, we establish that flux tubes can be
Alice strings --- forcing some charges and particle wave functions to
become double-valued. Like gauge theories with Alice strings and
monopoles, our sigma model has twisted flux loops which form point
defects.  We construct these defects --- the skyrmions --- and
demonstrate their nontrivial topology. We then consider how they
interact with the flux tubes, showing that only topologically trivial
combinations of two flux tubes can end on skyrmions. This suggests the
physical interpretation that, while the baryons in this theory are not
confined, the spinor sources which combine to form them are, with
confinement mediated by the $Z_2$ flux tubes joining them.

First we consider twisted flux tubes. We note that our flux tubes may
share the defining property of Alice strings, which arise in the
symmetry-breaking of certain gauge theories. \cite{Alice} Alice
strings have unbroken symmetries which preserve a local vev but cannot
be extended globally, since they become multivalued when parallel
transported around the spatially varying vev of the string.  This
``Alice'' nature is not topologically invariant, but depends on the
specific Wilson line integral producing the string's asymptotic
winding.  Whether our flux tubes have this trait is similarly
ambiguous: using the physical embedding \ref{Sigralform} of the flux
tube, we can take $g(r,\theta) = h(\theta)\,\, b(F(r)/2)$. This gives
parallel transported unbroken generators $g\,\, T_{h\Box'} \,\,
g^{-1}$ that are double-valued in $\theta$ for each radius.  However,
all generators can be made single-valued by choosing instead
$g(r,\theta) = h(\theta)\,\, b(F(r)/2)\,\, h(\theta),$ which induces the same
flux tube $\Sigma$.

Because $\Sigma$ can be viewed as an Alice string, we might expect it
to share a property of gauged Alice strings in models with monopoles:
that twisted string loops support monopole charge. For gauged Alice
strings, this is necessary, as monopoles alter their charge when
passing through string loops; moreover, specific examples indicate
that twisted Alice loops can carry a single unit of monopole charge.
\cite{Almon} For global Alice strings, the suggestion arises by
analogy.  In our case, it gains further support from the hidden gauge
character of the low energy global theory. Specifically, we can recast
the flavor-dependent quark mass $\Sigma$ in the usual way, as an
interaction between flavor gauge fields $g^{-1} (r,\theta)
\,\,\partial_\mu \,\, g(r,\theta)$ and shifted quarks $Q_{L} =
g(r,\theta) \,\, q_{L}.$ \cite{Balach} This accomplishes two goals: it
equates skyrmion number with anomalous baryon current, while
identifying the flux tube with a gauged string, whose Wilson
line integral is $ {\rm P}\exp{ \left ( \oint\vec{A} \cdot d\vec{l}\
\right ) }.$ However, the Alice ambiguity persists: choosing $g(r,\theta)$
double-valued gives a gauge theory with Wilson loop $U(2\pi)= {
  {\mbox{1\hskip-0.22em\relax l}}\,} - 2 \,\,{
  {\mbox{1\hskip-0.22em\relax l}}\,}_\Box$, making both the quarks
$Q_{L}$ and the generators $T_{h\Box'}$ double-valued.  A
single-valued choice for $g(r,\theta)$ produces instead single-valued
quarks and generators, with Wilson loop $U(2\pi)= {
  {\mbox{1\hskip-0.22em\relax l}}\,}$.

This ambiguity can be resolved physically, by considering adiabatic
transport of quarks around the flux tube. \cite{wilczek} Under such
transport, the quarks remain in their mass eigenstates. These are
governed by two terms: a flavor-independent bare Majorana mass $M$
(breaking $SU(N_f) \rightarrow SO(N_f)$ explicitly) and the
flavor-dependent Majorana mass $\mu\Sigma$, where $\mu$ is the
$SU(N_f)$-breaking vev. The two determine mass eigenstates that are
double-valued in $\theta$. However, the states have mass splitting
$\Delta m^2 = -2i M (\mu - \mu^*) \sin(F/2)$, where we have chosen $M$
real. Thus the quarks are degenerate, and unaffected by transport
around the string, unless the vev $\mu$ is misaligned in phase with
the bare mass $M$. In that case, quarks at finite radius have
double-valued wave functions and Aharonov-Bohm scatter off the flux
tube, which we identify as an Alice string. \footnote{We consider here
  quarks whose trajectories remain well-separated from degeneracies at
  the origin; that is, for which $r^2 >> m_i^{-2},$ where $m_i$ are
  the quark mass eigenvalues. Thus we rely on an asymptotic regime
  inside the flux tube, in which $m_i^{-1} << m_\pi^{-1}.$ The flux
  tube background affects general quark wave functions more subtly.
  Since the mass eigenstates depend explicitly on $\theta$,
  off-diagonal $\hat{\theta} \cdot \hat{\tau}$ terms appear in the
  quark equations of motion, mixing mass eigenstates at small radius.
  This causes corrections to maximal Aharonov-Bohm scattering, much as
  in \cite{wilczek}.}

Motivated by its gauge theoretic counterpart, we calculate the
skyrmion number of a twisted flux tube:
\begin{equation}
\Sigma(z, r, \theta) = A(z) \ \Sigma(r,\theta)\ A^{-1} (z) \ ,
\end{equation}
where $A(z) = \exp{(i z\, l\, n_h T_h)} \in H.$ Imposing
$2\pi$-periodicity in $z$ constrains $l$: for planar $n_h=n_\Box$, $l$
must be integral; for nonplanar $n_h$, it must be even. Thus any
twisted flux loop can deform to the planar flux loop $ \Sigma(z, r,
\theta) = h(\theta + lz) \ b(F(r)) \ h^{-1} (\theta + lz)$, which
takes values only in the two-flavor subspace $SU(2)/SO(2) \sim S^2.$
This simplification allows us to identify the loop's $\pi_3$ index
with its Hopf number, {\em i.e.} the linking number between any
two fibers of constant $\Sigma$ in physical space. As discussed in
\cite{wuetc}, this linking number is precisely $l$ --- the number of
times a nontrivial fiber $\Sigma_0$ twists around the loop's core,
which has $\Sigma= -{ {\mbox{1\hskip-0.22em\relax l}}\,}_\Box$. Thus
flux loops with an $l=1$ planar twist form fundamental skyrmions; flux
loops with nonplanar twist have $l>1$, which is trivial for $N_f > 3$.

A nicer parametrization of the skyrmion stems from the exact sequence
\[
\pi_3\left( SO(N_f)\right)  \ \  \rightarrow \ \
\pi_3\left( SU(N_f)\right)  \ \  \rightarrow \ \
\pi_3\left( SU(N_f)/SO(N_f)\right)\ \  \rightarrow \ \
\pi_2\left( SO(N_f)\right) = 0 \  .
\]
This implies that fundamental skyrmions in the theory can be
constructed from the fundamental skyrmions
\begin{equation}
g(r, \hat{u} ) = \exp{(iF_s (r) \ \hat{u}_i\,\, T_{i \Box} )}
\end{equation}
in $SU({N_f})$. Here $r$ and $\hat{u}$ are the radius and unit direction
vector in 3-space, and $F_s (r)$ approaches
$2\pi$ at $r=0$ and zero at $r=\infty$. This determines an
axisymmetric skyrmion
\begin{eqnarray}
\label{skyrmion}
\lefteqn{\Sigma_s = {{\mbox{1\hskip-0.22em\relax l}}\,} - 2 \sin^2 (F_s/2)\,\,
(1 - u_z^2) \,\,{{\mbox{1\hskip-0.22em\relax l}}\,}_\Box
           + 2i\, \sin (F_s/2) \,\, \cdot } \nonumber\\
&& \quad\left(
\cos(F_s/2) \left(u_y \ \tau_{x\Box} + u_x \ \tau_{z\Box} \right)
+ \sin(F_s/2) \,\,u_z
\left(-u_x \ \tau_{x\Box} + u_y \ \tau_{z\Box} \right)
 \right)\ ,
\end{eqnarray}
after a global spatial rotation fixing the
$z$-axis as the axis of spatial axisymmetry. In the $xy$-plane, this
gives
\begin{equation}
\Sigma_s(z=0) = {{\mbox{1\hskip-0.22em\relax l}}\,} + (\cos F_s - 1) \
{{\mbox{1\hskip-0.22em\relax l}}\,}_\Box
           + i \,\sin F_s\  \left(
 u_y \ \tau_{x\Box} +
 u_x \ \tau_{z\Box} \right).
\end{equation}
Comparing with equation \ref{Sigrexp} for the flux tube allows us
to identify the angular winding $n_b (\theta)\,\, T_{b \Box}$ of a flux
tube with that of a skyrmion in the $xy$-plane. Thus, if their
radial boundary conditions coincided, we could deform the lower
hemisphere of the skyrmion into a flux tube.  However, because
both $F$ and $F_s$ vary from $0$ to $2\pi$ over this plane, the
skyrmion cannot end in a single flux tube. The boundary
conditions instead allow the skyrmion to join only to flux tube
configurations where $F(r)$ ranges from $0$ to $4\pi$ --- that is,
configurations with two flux tubes, deformable to the trivial
configuration. Thus skyrmions cannot be confined in this theory.

However, objects which combine to form skyrmions can
interact with the flux tubes. Such ``half-skyrmions'' could arise as
external spinor sources in the underlying theory. They should
be confined, as fundamentals cannot screen them. As mappings on $G/H$,
they appear in our theory precisely as half-skyrmions, that is, as
objects of the form \ref{skyrmion} with $F_s(r)$ ranging from $0$ to
$\pi$. Such objects are not defects in the conventional sense, since
they have linearly divergent energy --- just like an unscreened point
source.  Their boundary conditions allow them to join to their
opposite winding counterparts via single flux tubes, confining their
linearly divergent energy to a length scale set by the tube
length.  We thus see that confinement of sources in an
$SO(N_c)$ gauge theory can induce a relic phenomenology, which persists
in the low energy Skyrme model.

\acknowledgements
 We thank Aneesh Manohar, David Kaplan, Tom Imbo, Sidney
Coleman and Glenn Boyd for helpful conversations.  This work was
supported by the Department of Energy under Grant
No.~DOE-FG03-90ER40546 and by the University of California
President's Postdoctoral Fellowship program.

redpajama_set_name: RedPajamaArXiv


text:
Q: convert XML to PostgreSQL using MATLAB In a weeks time I should be able to transfer the GIS data provided in the XML format to PostgreSQL database. I have prepared the PostgreSQL database (empty) and looking at Matlab scripts. (I am on ubuntu platform).
Matlab is our existing platform.The work is passed on to me by another person who used Mapping tools on MATLAB to export the data to viewers like ArcGIS/Google Earth.This was created for a prototype. (However, in the final product I will be exporting PostgreSQL to QGIS to view the data).
I am pretty new to the GIS data/table.Could you suggest me the steps in-detail to have the data in PostgreSQL?

A: It depends what kind of XML format you have. Do you have GML, KML, GPX  or some homegrown XML format?  If you are using a standard spatial XML, I would suggest looking at OGR2OGR which is part of GDAL tool kit.  It's a commandline tool that can import several formats into PostGIS and other spatial databases and spatial formats.  http://www.gdal.org/ogr2ogr.html - you can download binaries from http://trac.osgeo.org/gdal/wiki/DownloadingGdalBinaries
If you have some homegrown format or are picky about how the conversion is done, you could use the built in functions in PostgreSQL to import the XML and parse it into the fields you want e.g. I use pg_read_file (requires super user rights)  to load from filesystem and then the various xpath functions to parse the xml.  Here is an example of that approach I wrote about a while ago:
http://www.postgresonline.com/journal/archives/116-Loading-and-Processing-GPX-XML-files-using-PostgreSQL.html


Representative play has made a modest comeback, with teams from the Rugby South, Capital, Northern, and Southern California unions in action in December.
Home to some of America's more populous leagues, the latter three teams were perennial contenders before USARFU abandonned its 30-year-old representative system last late decade.
The resurgence springs from players' fundamental urge to compete. Athletes who aren't on an international track nonetheless want to test themselves, to see how good they can be. Coaches, referees, and union officials see intrinsic merit in the pursuit, as well as opportunity for their own improvement.
Arguments against all-star play are numerous, most incorporating the view that geographic separation and lacking of funding fatally cripple efforts to help players systematically improve. Training is nothing more than haphazard preparation for sporadic matches. Better to concentrate energy and monies in the school or club environment.
Boulder never declared an end to representative competition. But in shying from organizational expense, hoping professionalized franchises were just around the corner, and failing to consider how the pell-mell transition from territorial to geographic unions would encompass rep-side matches, its lack of action stripped the all-star championships of their national dynamic.
In the current decade, the most interesting new competitions have originated outside USARFU: the 7s Collegiate Rugby Championship, the Varsity Cup, the hoped-for PRO Rugby league. But such expansion means the schedule has grown more crowded and demand for players multiplied.
While the representative game's future is far from clear, there may be a model in the South Panthers. Save for Atlanta's Life University, Southern teams haven't produced many Eagle hopefuls. But the South has forged a regular schedule against Caribbean opponents, and even won the North American-Carribean Rugby Association (NACRA) title in 2013.
any perspective on NM hiring JM, Kurt?


FOR INFO ON HOW TO JOIN CARNEGIE PERFORMING ARTS CENTER FORTE COMPETITION TEAM CALL 412-279-8887.
Eligibility begins at competition birth-date of age 8.
You MUST be a current student of the Carnegie Performing Arts Center Dance Program to Participate.


Brigus Gold Corp. is a growing gold producer committed to maximizing shareholder value through a strategy of efficient production, targeted exploration and select acquisitions. The company operates the wholly owned Black Fox mine and mill in the Timmins gold district of Ontario. The Black Fox complex encompasses the Black Fox mine and adjoining properties in the township of Black River-Matheson, Ontario. Brigus is also advancing the Goldfields project located near Uranium City, Saskatchewan.


Glen Affric, Plodda Falls, Dog Falls, Loch Affric, Inverness, Cullodon, Black Isle.
Loch Ness is the largest of the three lochs along the Great Glen, approximately 23 miles in length and 1 mile width with an average depth of 600ft !
Fort Augustus village is located at the southern end of Loch Ness in the heart of The Great Glen, where the Caledonian canal drops and flows into Loch Ness through an impressive series of five locks, built by Thomas Telford in the early 1800's.
Cruise along historic Loch Ness on The Royal Scot enjoying the breathtaking surroundings and scenery of Urquhart Castle or enjoy a few hours searching for a glimpse of Nessie, the infamous Loch Ness Monster!
Plockton, Applecross, Torridon, Gairloch, Ullapool, The West Highlands.
Heading towards Skye through the mountainous scenery of Glenshiel and Kintail you reach Eilean Donan Castle which overlooks loch Duich and Loch Alsh. Just a short distance further is Kyle of Lochalsh and the Isle of Skye Bridge. An alternative route onto Skye is over the Mam Ratigan pass to Glenelg and on to Skye via the small "Glenahulish" ferry crossing to Kylerhea.
The main high mountains on Skye are "The Cuiliin Mountain Range" and two great positions to view them from lower down are from Ord on the southern end of Skye (garden of Skye) or Elgol, accessed via a single track road winding around Loch Slapin.
There are fantastic views from Elgol across Loch Scavaig and The Cuillin Sound to the small Isles of Soay, Canna and Rum. Scenic boat trips on "The MIsty Isle" also leave from Elgol and a lovely cafe sits on the hilltop overlooking the bay.
Also on the southern part of Skye is Armadale Castle with gardens, a visitor centre and cafe restaurant. Armadale also has a larger Caledonian MacBrayne car & passenger ferry going back across onto the mainland to the fishing port of Mallaig.
Further attractions north on Skye are the Whisky Distillery at Carbost, the Aros Centre just prior to the main Skye town of Portree which offers shops, cafes, restaurants and galleries. On the far north west is Dunvegan Castle and Neist Point Lighthouse can be found overlooking a remote and rugged coastline and to the far north east is Kilt Rock waterfall, Staffin Bay and Duntulm Castle Ruins.
Back on the mainland from The Skye Bridge, a short trip around the coastal road is Plockton, a beautiful fishing village overlooking Loch Carron, there are numerous cafes, restaurants and pubs.
central for touring Loch Ness, The Isle of Skye and The Highlands.
"Tailor made tours just for you"
that make the highlands such a wonderful place to visit"
own personal driver/guide and an itinerary designed around you.


The online gaming industry is enjoying massive success; something which has necessitated it to rank as one of the world's fastest growing industries across a number of countries such as Poland for example. It is for this reason that we decided to look into what the online casino industry looks like in Poland to find out why it's so popular.
Before we start exploring everything there is to this niche, let's briefly learn about Poland online casinos which are also referred to as online kasyno in the Polish native language. Poland has the highest number of online casinos, most of which are dual licensed. Virtually, all of them hold such an operating license from the Malta Gambling Authority, Curacao, or the UK Gambling Commission which enables players from Poland to access and play casino games. These licenses also allow players from all over the world to access online casinos in Poland. It is for this reason as to why Poland online casinos are popular.
Now, let's see what players at online casinos in Poland can expect in terms of services offered.
There are hundreds of online casinos in Poland that offer a solid collection of casino games. For many of these online operators, you can expect to find a vast number of slot games in the casino lobby, along with a list of other games. Slots are loved by many (both beginners and professionals) due to their simplicity, and as such, in order to cater to the needs of most players, online casinos in Poland feature only the best that the industry has to offer. Apart from slots games, online casinos in Poland also offer other types of casino games such as table games which include blackjack, roulette, and baccarat, amongst others. These sorts of games are also highly popular in this market, raking in millions in revenue per annum. In recent times though, online casinos in Poland were improved upon due to advancements in technology and as it stands, we can now see a number of live dealer casino games available for play. What this means is that there's a whole new world to explore in 'live' casino games that can be played in real time and in HD quality.
Poland is in Europe and as we all know, the continent is huge on sports. European football is a perfect example of how popular sports betting is and not just on the continent, but across the whole world as well. The same applies to other sports such as rugby, tennis and cricket, among others. In order to cater to the needs of players who love placing bets on sports games, most online casinos in Poland offer sports betting services. Sports bettors who place their bets at online casinos in Poland can expect to bet on virtually any sport and on any match.
In more recent times, most online casinos in Poland have added novelty betting products to their offerings to cater to the needs of those who want to partake in online betting, but who aren't fond of neither playing casino games nor placing sports bets. Novelty betting is a form of betting that allows punters to bet on almost anything from politics, economics to entertainment, awards, and royal proceedings in terms of the outcomes and possibilities. Currently, punters who are into novelty betting can place bets on markets such as 'The Day of the week that Harry and Meghan's 1st baby is to be born?' and 'The Gender and/or the Name of Harry and Meghan's first baby' among many other choices.


This past week during the Democrat presidential debate, Hillary Clinton told America that nothing would come from the FBI's investigation into her private email account. The former Secretary of State told NBC's Chuck Todd that she was "100 percent confident. This is a security review that was requested. It is being carried out." However, someone with intimate knowledge of the FBI's inner workings and about how they run their criminal investigations said that Mrs. Clinton was lying about what was going on. Steve Pomerantz worked for the FBI for 28 years, climbing as high as Assistant Director, the 3rd highest-ranking position with the FBI, and on Sunday he told Fox News that Clinton's comments during the debate were inaccurate.
Not only does the FBI not do "security reviews," other sources explained that the criminal investigation is (and will remain) connected to national security concerns.
The point to all this is that Clinton's comments in this past week's debate were, at best, misleading, but at worst… an outright lie. That the FBI investigation continues should be proof enough that Hillary Clinton should be very concerned about what her future might look like and that the American people should most definitely be taking Clinton's possible future stay in a federal prison into account when they enter the voting booth during the primaries and again in November.
Obama's Last-Ever Budget Proposal is $4.1 TRILLION Worth of Tax Increases and Wasteful Spending!


You're no doubt working hard to complete your due diligence analysis and finalize your Form SD (and Conflict Minerals Report) as required by the US Conflict Minerals Rule. But, around the corner is another conflict minerals regime that could impact your business — the EU Conflict Minerals Regulation. The EU's due diligence and disclosure obligations do not take effect until 2021, but if you are a Union importer, there are things you will want to do before that date to minimize your compliance costs and avoid possible disruptions to your supply chains.
This LexisNexis article gives an introductory look into the EU Conflict Minerals Regulation and warns businesses to start working to manage their supply chains in order to avoid some additional burdens that are built into the regulation.
The article, "The EU conflict minerals regime," can be accessed on LexisNexis here.
Those without a LexisNexis subscription may reach out via email to obtain access to the article.