text:
\chapter{Introduction}
\REF\rTak{L. Takhtajan, L.D. Faddeev - {\it Russian Math. Surveys} {\bf
34}, 11 (1979)}
\REF\rSkyanin{E.K. Sklyanin and P.P. Kulish - {\it Lecture Notes in
Physics} {\bf 151}, 61 (1982)}
\REF\rFad{L.D. Faddeev - {\it Les Houches Lectures 1982}, ed. Elsevier,
Amsterdam (1984)}\REF\rFAD{L.D. Faddeev, N.Yu. Reshetikin and L.A. Takhtajan -
 {\it Algebra and Analysis} {\bf 1}, 178 (1989)}\REF\rW{S.L. Woronowicz -
{\it Publ. RIMS, Kyoto Univ.} {\bf 23}, 117 (1987)}\REF\rWor{S.L.
Woronowicz - {\it Comm. Math. Phys}
 {\bf 111}, 613 (1987)}
 The appearance of noncommutative entries in some matrices describing
 quantum inverse scattering methods for spin systems (see \eg\
 [\rTak-\rFad]) has led to the introduction of the concept of a quantum
 matrix group.\refmark\rFAD$\sp{-}$\refmark\rWor\   From the
algebraic point of view the description of a quantum group as a
quasitriangular Hopf algebra was first given by Drinfeld\Ref\rD{V.G.
Drinfeld - {\it Proc of XX International Math. Congress, Berkeley} Vol.I
, 798 (1986)}\rlap, with the basic object being the noncommutative algebra of
functions on a quantum group. The quantum extensions of all classical
matrix groups (Cartan $A_n$, $B_n$, $C_n$ and $D_n$ series), describing the
generators of Drinfeld's quantum algebra have also been
given\refmark\rFAD\rlap.
\REF\rPod{P. Podles - {\it Lett. Math. Phys.} {\bf 14},
193 (1987); ibid, {\bf 18},
107 (1980)}\REF\rNou{M. Noumi and K. Mimachi - {\it Comm. Math. Phys.} {\bf
128}, 521 (1990)}\REF\rYam{M. Noumi, H. Yamada and K. Mimachi - {\it Proc.
Jap. Acad. Sci.} {\bf 65A}, 169 (1989)}

Moreover, the quantum counterparts of the homogeneous coset spaces
(\eg\ spheres - $S_q\sp{n}$, projective spaces - $CP_q(n)$ \etc) have also been
found\refmark\rFAD\refmark\rPod$\sp{-}$\refmark\rYam.
 \REF\rz{W.J. Zakrzewski - {\it Low dimensional $\sigma$ models} - Adam
Hilger (1989)}

\par In this paper we will consider fields taking values in quantum groups
and we will discuss the corresponding $\sigma$ models. Let us recall that
the usual $\sigma$-field $\phi(x)$ describes the mappings from the
coordinate manifold $S$ into the target space $M$ (see \eg\ [\rz]). In
principle we can ``$q$-deform" the target space $M$ ($M\rightarrow M_q$)
as well as the coordinate manifold $S$ ($S\rightarrow S_q$), \ie\ we can
introduce three kinds of $\sigma$ models: \break
\vskip-8mm\item{a)} with quantum deformation of the target manifold
$$\phi\sp{q}_a(x):\qquad x\in S\;\rightarrow\;\phi_a\sp{q}\in M_q,
\eqn\eone$$ where the index $a$  ennumerates the local coordinates
on $M_q$.
\item{b)} with quantum deformation of the base manifold
$$\phi_a(x_q):\qquad x_q\in S_q\;\rightarrow\;\phi_a\in M,\eqn\oneb$$
\item{c)} with both manifolds deformed
$$ \phi\sp{q}_a(x_q):\qquad x_q\in S_q\;\rightarrow\;\phi_a\sp{q}\in M_q.
\eqn\eonec$$

Let us add that, analogously, there exist three types of supersymmetric
models, corresponding to the three types of mappings, and that we believe
that in the near future, the quantum group $\sigma$ models of all three
types will be studied. In the present paper we will discuss mainly the case
\eone.
\par The description of quantum group $\sigma$ models can be presented in
two different ways: \hfil\break
1) We can consider quantum group $\sigma$ fields as fields satisfying
at a point $x\in S$ the quantum algebra, and so we can study the general
properties which follow from the basic formulae of noncommutative
differential geometry on quantum groups. Such an approach, which we shall
call algebraic, was recently used by Arefeva and Volovich\Ref\rA{I.Y.
Arefeva and I.V. Volovich - {\it Mod. Phys. Lett.} {\bf 6A}, 893
(1991)}\Ref\rV{I.Y. Arefeva and I.V. Volovich - {\it Phys. Lett.} {\bf
264B}, 62 (1991)}.\REF\rWW{S.L. Woronowicz - {\it Comm. Math. Phys.}
 {\bf 122}, 125 (1989)}
 In the algebraic formulation of a quantum group $\sigma$
model one can repeat the major part of the geometric formulation of the
standard approach to the $\sigma$ models (Cartan forms, Cartan structure
relations, algebra of the covariant derivatives \etc), provided that the
exchange relations between quantum group valued $\sigma$ fields and their
derivatives are properly introduced. For the $SU_q(2)$ case the formulae of
noncommutative geometry are well known (see \eg\ ref [\rW,\rWW]) and so in
the next section we shall present the algebraic formulation of the $SU_q(2)$
$\sigma$ model. \hfil\break
2) We can assume that the quantum group $G_q$ $\sigma$ fields
$\phi_A\sp{q}(x)$ are products of ``ordinary" functions $f_A(x)\in H$ and
of the $x$ independent algebra $A$ related to the quantum group algebra.
This approach gives us expressions which belong to the tensor product
$H_G\otimes T_A$, where for an $n\times n$ matrix quantum group $\sigma$
model the first part $H_G$ is parametrised by an $n\times n$ matrix of
``classical" fields (suitably constrained standard $GL(n)$ $\sigma$ fields),
and  $T_A$ carries the realisation of the algebra $A$. If $A=f(G_q)$, the
natural realisation on the polynomial basis of the functions on the quantum
group is infinite dimensional, and for $q$ real it can not be reduced to a
finite dimensional case. In the second part of section 2 we will present
our discussion of the $SU_q(2)$ quantum group $\sigma$ model for the
solutions satisfying the separability condition described above.
\REF\rc{C. de Concun and V.G. Kac - ''Representations of quantum groups at
root of 1", {\it Progress in Math.} Vol {\bf 92}, 471 (1990); ed.
Birkhauser (Basel)}
\REF\rDa{E. Date, M. Jimbo, K. Miki and T. Miwa - {\it Com. Math. Phys.}
{\bf 137}, 133 (1991)}
\REF\rAa{D. Arnandon and A. Chakrabarti - {\it Com. Math. Phys.} {\bf 139},
605 (1991)}
\par The realisation $T_A$ can be described by finite matrices in one case:
when $q$ is complex and $q\sp{N}=1$ ($q=e\sp{i{2\pi\over N}}$). It is worth
mentioning that some realisations of the quantum groups for $q$ being the
$N$-th root of unity have recently been found to have physically relevant
applications (see \eg\ ref [\rc,\rDa,\rAa]). It appears, however, that the
quantum deformations $O_q(n)$ and $SU_q(n)$ of the semisimple groups $O(n)$ and
$SU(n)$  which are the natural candidates for quantum group $\sigma$ fields,
 do not
permit complex $q$\refmark\rFAD.  \par The simplest example of a compact
quantum group with $q\sp{N}=1$, the  quantum group $U_q(2)$, will be considered
in section 3. There, we will first show that the $U_q(2)$ for complex
values of $q$ can be obtained as a special case of a two-parameter
deformation $GL_{q,p}(2,C)$ of the $2\times 2$ general linear group. Then,
using the results on the matrix realisations of $GL_q(n)$ for
$q\sp{N}=1$\Ref\rF{E.G. Floratos - {\it Phys. Lett.} {\bf B233}, 395
(1989)} \Ref\rWey{J. Weyers - {\it Phys. Lett.} {\bf B240}, 396( 1990)}
we will embed the $U_q(2)$ $\sigma$ model with $q\sp{n}=1$ (``anyonic
$\sigma$ model") into the conventional $U(2n)$ $\sigma$ model. It appears
that when we use such a representation we describe solutions that,
 for $D=1$ (see eq. (3.2)), satisfy
the assumptions made by Arefeva and Volovich in
their discussion of quantum group sigma models\refmark\rA. Finally, some
open problems are discussed in section 4.
\REF\rVaks{L.L. Vaksman and Y.S. Sorbelman - {\it Funct. Anal. and its
Appl.} {\bf 22}, 170 (1989)}
\REF\rWess{A. Schirrmecher, J. Wess and B. Zumino - {\it Zeit. f. Phys.}
{\bf C49}, 317 (1991)}
\REF\rEguchi{T. Eguchi and H Kawai, {\it Phys. Lett.} {\bf B110}, 143 (1982)}
\REF\rKoorn{T.H. Koornwinder - {\it Neder. Acad. Wetensch. Proc. Ser.} {\bf
A92}, 97 (1989)}
\REF\rBaulieu{L. Baulieu and E.G. Floratos - {\it Phys. Lett.} {\bf B258},
 171-178 (1991)}
\REF\rTze{F. Gursey and H.C. Tze - {\it Ann. of Phys.} {\bf 128}, 29
(1980)}
\REF\rreview{see \eg\ C.M. Hull, {\it Lectures on nonlinear $\sigma$ models},
in {\it Super Field Theories}, New York, Plenum (1987)}

\chapter{$SU_q(2)$\ $\sigma$\ model}
\section{Algebraic formulation}
Let us first introduce the quantum group $SL_q(2,C)$ as the following Hopf
bialgebra ($q$ complex):\refmark\rFAD \hfil\break
a) multiplication:
$$\eqalign{U\,&=\,\pmatrix{a&b\cr c&d\cr};\cr
ab\,&=\,qba\qquad ac\,=\,qca\qquad cd\,=\,qdc\cr
bc\,&=\,cb \qquad bd\,=\,qdb\cr
ad\,&-\,qbc\,=\,da-q\sp{-1}cb\,=\,1.\cr}\eqn\etwoone$$
b) coproduct
$$\Delta\pmatrix{a&b\cr c&d\cr}\,=\,\pmatrix{a&b\cr c&d\cr}\times \pmatrix
{a&b\cr c&d\cr},\eqn\etwotwo$$
c) inverse (antipode) and co-unit
$$S\pmatrix{a&b\cr c&d\cr}\,=\,\pmatrix{d&-q\sp{-1}b\cr -qc&a\cr}\qquad
\epsilon\pmatrix{a&b\cr c&d\cr}\,=\,\pmatrix{1&0\cr 0&1\cr}.\eqn\ethree$$
For $q$ real we can introduce the following unitarity condition. For
$$U\,=\,\pmatrix{u_{11}&u_{12}\cr u_{21}&u_{22}\cr}=\pmatrix{a&b\cr
c&d\cr}\eqn\eaaaa$$ we put
$$U\sp{\dagger}\;=\;\pmatrix{a\sp{\star}&b\sp{\star}\cr
c\sp{\star}&d\sp{\star}\cr}\sp{T}\;=\;\pmatrix{d&-q\sp{-1}b\cr
-qc&a\cr}\;=\;S(U),\eqn\etwofour$$
which defines the $SU_q(2)$ quantum group, the matrix elements of which
have the form $$U=\pmatrix{a&-qc\sp{\star}\cr c&a\sp{\star}\cr}.\eqn\eee$$
In this case the relations \etwoone\ take the form
$$\eqalign{ac\,=&\,qca\qquad ac\sp{\star}\,=\,qc\sp{\star}a\qquad
cc\sp{\star}\,=\,c\sp{\star}c\cr aa\sp{\star}+&q\sp2 cc\sp{\star}=1\qquad
a\sp{\star}a+c\sp{\star}c=1.\cr}\eqn\etwofive$$
\par In order to define a quantum group $\sigma$\  model we introduce the Cartan
one-forms on $SU_q(2)$
$$\Omega\,=\,U\sp{\dagger}dU\,\leftrightarrow\,\Omega_{ik}\,=\,U_{ij}
\,dU_{jk}.\eqn\esix$$
The formula \esix\ describes the left-invariant one-forms
($\Omega=\Omega_l$). The right-invariant forms are given in terms
of the left-invariant forms in the same way as in the $q=1$
case, and so are given by
$$\Omega_R\;=\;-U\Omega_lU\sp{\dagger}.\eqn\eright$$
We  also have
$$\eqalign{dU\,&=\,U\Omega_l\,=\,-\Omega_RU,\cr
dU\sp{\dagger}\,&=\,U\sp{\dagger}\Omega_R\,=\,-\Omega_lU\sp{\dagger}.\cr}
\eqn\esixc$$
For the Cartan one-form \esix\ we can introduce the linear basis.
Following the so-called $4D_+$ bi-covariant calculus of
 Woronowicz\refmark\rPod\refmark\rWW\  we can choose
$\Omega=\omega_A\tau_A$ ($A=0,1,2,3)$, where
$$\tau_1\,=\,\pmatrix{0&1\cr 0&0\cr}\qquad \tau_2\,=\,\pmatrix{0&0\cr
{-1\over q}&0\cr}\quad \tau_3\,=\,{1\over 1+q\sp2}\pmatrix{-{1\over q}&0\cr
0&{1\over q}\cr}\eqn\etwoseven$$
and $\tau_0={q\sp3-1\over (1+q)(1+q\sp2)}I,$ where $I$ denotes the unit matrix.
Observe that we have chosen a nonsingular normalisation (compare with ref
 [\rPod], second paper, p.108).

\par The $SU_q(2)$ $\sigma$\ fields are then introduced by the mapping (1.1)
\ie\ $U_{ij}\rightarrow U_{ij}(x)$. If we now write
$$ \eqalign{\Omega_{ik}&=U_{ij}\sp{\dagger}{\partial U_{jk}\over \partial
 x_{\mu}
} dx_{\mu}\,=\,\Omega_{ik}\sp{\mu}dx_{\mu},\cr \Omega_{\mu}\,=&\,\omega_{A\mu}
\tau_{A},\cr}
\eqn\etwoeight$$
the action of the $SU_q(2)$ model can be written as:
$$\tilde S\,=\,-\int d\sp{d}x\,\Tr{(\Omega_{\mu}\Omega\sp{\mu})}\,=\,-
\int\,d\sp{d}x\,G_q\sp{AB}\omega_{A\mu}\omega_B\sp{\mu},\eqn\etwonine$$
where $G_q\sp{AB}=\Tr{(\tau_A\tau_B)}$ is given by
$$G_q\sp{AB}\;=\;\pmatrix{0&-{1\over q}&0&0\cr
-{1\over q}&0&0&0\cr 0&0&{2\over q\sp2(1+q\sp2)\sp2}&0\cr
0&0&0&{2(q\sp3-1)\sp2\over (1+q)\sp2(1+q\sp2)\sp2}\cr}.\eqn\etwoten$$
The Cartan forms $\omega_A$ describe the $\sigma$\ model currents.
We see that the contribution of the scalar current $\omega_0$ vanishes in
the
limit $q\rightarrow 1$. $G_q\sp{00}$ vanishes
at $q=1$ by the choice of $\tau_0$. This choice is consistent as
 in this limit we get the $SU(2)$ model and no
 $U(1)$ current.
\par We can now consider $\sigma$\ fields and take currents $\omega_A$ as our
basic variables.
Due to the unitarity ($\Omega=U\sp{\dagger}dU=-dU\sp{\dagger}U$) we can
rewrite the action \etwonine\ as
$$\tilde S=\int d\sp{d}x{\partial U_{ji}\sp{\star}\over \partial
x\sp{\mu}}{\partial
U_{ji}\over \partial x_{\mu}}\,=\,\int d\sp{d}x{\partial(SU)_{ij}\over\partial
x\sp{\mu}}{\partial U_{ji}\over \partial x_{\mu}}.\eqn\etwoeleven$$
Denoting $$U(x)=\pmatrix{A(x)&B(x)\cr C(x)&D(x)\cr}\eqn\ebbb$$ we obtain
$$\tilde S\,=\,\int
d\sp{d}x(A_{,\mu}\sp{\star}A\sp{,\mu}+q\sp2C_{,\mu}C\sp{\star,\mu}
+C_{,\mu}\sp{\star}C\sp{,\mu}+A_{,\mu}A\sp{\star,\mu},\eqn\etwotwelve$$
where the field operators $A(x),A\sp{\star}(x),C(x),C\sp{\star}(x)$
 satisfy the algebra
\etwofive\ at every coordinate point $x$.  But as we have the operators
and their derivatives, we  need to know the algebra
at points $x$ and $x+\epsilon$,  with $\epsilon$ infinitesimal.
To do this we have
to
determine
the complete algebra for our basic fields, \ie\ for $U_{ij}$ and $\omega_A$.
This algebra, in the case of $SU_q(2)$, is known in an explicit
form\refmark\rW\refmark\rWW.
\hfil\break

\section{Separable realisations}
Let us assume that
$$U_{ij}(x)\,=\,f_{ij}(x)\otimes \hat U_{ij},\quad
i,j=1,2\eqn\etwotheirteen$$
where the functions $f_{ij}(x)$ are classical and $\hat U_{ij}$ describe the
coordinate independent operators. Further, let us assume that the quantum
$\sigma$\ field \etwotheirteen\ satisfies the unitarity condition $UU\sp{\dagger}=
U\sp{\dagger}U=1.$
\par We shall consider here only two cases:\hfil\break
$\alpha$) The operators $\hat U_{ij}$ describe the generators of the
$SU_q(2)$ quantum algebra \etwofive\ separately (obviously the total $U_{ij}$
should).
\par In this case the unitarity condition, with  $f_{11}=f$, $f_{12}=g$,
$f_{21}=h$, $f_{22}=k$ and $ \hat c=\hat u_{21}$ is
$$ \eqalign{h\,=\,g\sp{\star},\quad& k\,=\,f\sp{\star}\cr
\vert f\vert \sp2(1-q\sp2\hat c\sp{\dagger}\hat c)\,&+\,q\sp2\vert g\vert\sp2
\hat c\sp{\dagger} \hat c\,=\,1\cr
\vert f\vert \sp2(1-\hat c\sp{\dagger}\hat c)\,&+\,\vert g\vert\sp2
\hat c\sp{\dagger}\hat c\,=\,1.}\eqn\etwofourteen$$
We see that for $q\ne1$ we have
$$\vert f\vert \sp2\,=\,\vert g\vert \sp2\,=\,1, \eqn\etwofifteen$$
\ie\ we obtain the $U(1)\times U(1)$ classical $\sigma$\ model.\hfil\break
$\beta$) The functions $f_{ij}(x)$ describe the $SU(2)$ $\sigma$\ fields and
so besides the conditions in the first line of \etwofourteen\
they satisfy also $$ \vert f\vert \sp2\,+\,\vert g\vert
\sp2\,=\,1.\eqn\etwosixteen$$
\par Then, we can write \etwotheirteen\ as an $SU_q(2)$ matrix
$$U\;=\;\pmatrix{f(x)\cdot A&-qg(x)\cdot C\sp{\dagger}\cr
g\sp{\star}\cdot C&f\sp{\star}\cdot A\sp{\dagger}\cr}, \eqn\twoseventeen$$
where $A$ and $C$ do not depend on $x$. Thus
we obtain for $A,$ $A\sp{\dagger}$, $C$ and $C\sp{\dagger}$ the relations
of the first line of \etwofive\  and
$$\eqalign{\vert f\vert \sp2 AA\sp{\dagger}\,+&\,q\sp2\vert g\vert \sp2
CC\sp{\dagger}\,=\,1\cr
\vert f\vert \sp2 A\sp{\dagger}A\,+&\,\vert g\vert\sp2
C\sp{\dagger}C\,=\,1,\cr}\eqn\twoeightteen$$
 whose solutions exist only for $q=1$, and only when
$AA\sp{\dagger}=A\sp{\dagger}A=CC\sp{\dagger}=1$.
\par Thus, in conclusion, we see that the assumption \etwotheirteen\ for
$q$ real and with either case $\alpha)$ or $\beta)$,
 is too restrictive and should be extended to, say
$$U_{ij}(x)\;=\;\sum_n\,f_{ij}\sp{(n)}(x)\otimes \hat
U_{ij}\sp{(n)},\eqn\etwonineteen$$
where $\hat U_{ij}\sp{(n)}$ describe a polynomial basis of the ring
of noncommutative functions $f(G_q)$ (see \eg\ ref [\rVaks]).
Such an assumption corresponds to the considering of the mapping
(1.2) with $S_q=S\otimes G_q$, and $M=GL(2)$. \hfil\break
\section{Embeddings in the $U(\infty)$ $\sigma$\ model}
Another way of representing  the operators $A$,
$A\sp{\dagger}$, $C$ and $C\sp{\dagger}$ of \etwotwelve\ corresponds to the
use of the parameter dependent irreducible representations of the
functions $f(SU_q(2))$ in a separable Hilbert space $H$. Promoting the
parameters to the functions generates an $\infty$-dimensional $\sigma$\ model.
\par The irreducible representations of the $SU_q(2)$ algebra in a Hilbert
space are known\refmark\rW\refmark\rVaks. There are only 2 series of
irreducible representations of $f(SU_q(2))$, each one parametrised
by the parameter of the unit circle $t=e\sp{i\phi}\in S\sp2$. One series
is degenerate, as for it  only the element $a$ of \etwofive\ is
represented in a nontrivial way. The other irreducible representation
is nontrivial. It is described by the operators $\rho_{\phi}$
which act as
$$\eqalign{
\rho_{\phi}(a)e_0=0\quad&\rho_{\phi}(a)e_k=(1-q\sp{-2k})\sp{1\over2}
e_{k-1}\cr
\rho_{\phi}(c)e_k\,=&\,(e\sp{i\phi}\cdot q\sp{-k})e_k,\cr}\eqn\etwotwenty$$
where $e_k$, for  $k=0,1,\cdots\infty$ describes an orthonormal basis in $H$.
\par If we now replace $\phi$ by a function $\phi(x)$ we obtain from
\etwotwenty\ the embedding of the $U(1)$ $\sigma$\ field into $U(\infty)$, in
analogy with the separable realisations \etwotheirteen.

\chapter{$U_q(2)$ $\sigma$\ model for $q\sp{N}=1$}
In order to obtain a solution of a quantum group $\sigma$\ model in the separable
form (see \etwotheirteen), it will
turn out that  we should consider the deformation parameter
$q$ as being complex and satisfying $\vert q\vert=1$. Then, if $q$ equals
the $N$-th root of unity ($q\sp{N}=1$) one can represent the operators
$\hat U_{ij}$ by $N\times N$ dimensional matrices (see refs [\rF], [\rWey]).
As we want to consider a quantum group $\sigma$\ model defined on a quantum
compact group, we shall discuss here the simplest such compact quantum group
permitting complex values of $q$, namely, the quantum group $U_q(2)$.
\section{Quantum group $U_q(2)$}
We shall define the quantum group $U_q(2)$, for complex $q$,
 as a real form of a two parameter
deformation of $GL(2,C)$, denoted by $GL_{p,q}(2)$\refmark\rWess.
Then we will know that the real Hopf algebra is valid for our $U_q(2)$.
\par The formulae \etwoone\ have to be extended and they become
$$\eqalign{ab\,&=\,pba\quad ac\,=\,qca\quad cd\,=\,pdc\cr
pbc\,&=\,qbc\qquad bd\,=\,qdb\cr
ad\,-&\,da\,=\,(p-q\sp{-1})bc\cr}\eqn\ethreeone$$
with a coproduct still defined by \etwotwo. If we now introduce
the determinant
$$\eqalign{D\,=&\,ad-pbc\,=\,ad-qcb\cr
=&\,da-p\sp{-1}cb\,=\,da-q\sp{-1}bc\cr}\eqn\ethreetwo$$
then one can check from \ethreeone\ that the following relations hold:
$$\eqalign{[D,\,a]=0,&\qquad [D,\,d]=0\cr
qDb=pbD,&\qquad pDc=qcD.\cr}\eqn\threethree$$
We see that only if $q=p$ we can put $D=1$, \ie\ we have the quantum
group $SL_q(2)$ defined by \etwoone.
\par The quantum group $GL_{p,q}(2)$ is a genuine Hopf algebra for any
complex $q$ and $p$. In particular, the antipode of $$U=\pmatrix{a&b\cr
c&d\cr}\eqn\ezzz$$ is given by the formulae
$$S\pmatrix{a&b\cr c&d\cr}=D\sp{-1}\pmatrix{d&-q\sp{-1}b\cr
-qc&a\cr}=\pmatrix{d&-p\sp{-1}b\cr -pc&a\cr}D\sp{-1},\eqn\ethreefour$$
where we have used $DD\sp{-1}=D\sp{-1}D=1$.
If we now impose the unitarity condition $U\sp{\dagger}=S(U)$, \ie\
$$\eqalign{a\sp{\star}\,&=\,D\sp{-1}d\,=\,dD\sp{-1}\cr
b\sp{\star}\,&=\,-qD\sp{-1}c\,=\,-pcD\sp{-1}\cr
c\sp{\star}\,&=-q\sp{-1}D\sp{-1}b\,=\,-p\sp{-1}bD\sp{-1}\cr
d\sp{\star}\,&=\,D\sp{-1}a\,=\,aD\sp{-1},\cr}\eqn\ethreefive$$
we find as the consistency conditions, that
$$D\sp{\star}\,=\,D\sp{-1},\qquad p=q\sp{\star}\eqn\ethreesix$$
and so  obtain the following $U_q(2)$ algebra
$$\eqalign{ac=qca,&\qquad ac\sp{\star}=q\sp{\star}c\sp{\star}a,\cr
&c\sp{\star}c=cc\sp{\star}.\cr}\eqn\ethreeseven$$
Moreover, from \ethreetwo\ and \ethreefive\ we find that
$$a\sp{\star}a+c\sp{\star}c=1,\qquad aa\sp{\star}+\vert
q\vert\sp2cc\sp{\star}=1.\eqn\ethreeeight$$
\par
\noindent  The quantum matrix $U_q(2)$  given by
$$U\;=\;\pmatrix{a&-q\sp{\star}c\sp{\star}D\cr c&a\sp{\star}D\cr}.
\eqn\ethreenine$$
 It describes the generators of the Hopf algebra with standard comultiplication
rule
$$\Delta(U_{ik})\,=\,\sum_{j=1,2}U_{ij}\otimes U{jk}\eqn\ethreeten$$
and the following antipode condition (using $qD\sp{-1}c=q\sp{\star}cD\sp{-1}$)
$$U\sp{\dagger}\,=\,S(U)\,=\,\pmatrix{a\sp{\star}&c\sp{\star}\cr
-qD\sp{-1}c&D\sp{-1}a\cr}.\eqn\ethreeeleven$$
It should be stressed that even for $\vert q\vert=1$ but with $q\ne1$, we
cannot
put $D=1$ as in this case \ethreeeight\ would give
$$a\sp{\star}a+c\sp{\star}c=1 \qquad
a\sp{\star}a=aa\sp{\star}.\eqn\ethreetvelve$$

\section{$2N\times 2N$ matrix realization of $U_q(2)$ for $
q\sp{N}=1$}
If  $ q\sp{N}=1$ the elements $a$, $c$, $a\sp{\star}$ and
$c\sp{\star}$ can be represented by $N\times N$ matrices.
Following ref. [\rF,\rWey] we introduce the following matrices\foot{The
matrices (3.13) were introduced earlier by Eguchi and Kawai in their
construction of the ''twisted Eguchi-Kawai" models\refmark\rEguchi.}
$$Q\,=\,\pmatrix{q&..&0\cr 0&\ddots&0\cr 0&..&q\sp{N}\cr}\qquad
P=\pmatrix{0&1&..&0\cr
0&0&\ddots&0\cr 0&0&..&1\cr 1&0&..&0\cr}.\eqn\ethreethirteen$$
These matrices satisfy $PQ=qQP$, and with $q=exp({2\pi i\over N})$
they generate the algebra of all $N\times N$ matrices.
Moreover, as $Q\sp{N}=P\sp{N}=1$ we find that $Q\sp{\dagger}=Q\sp{N-1}$
and $P\sp{\dagger}=P\sp{N-1}$ and, as is easy to see,
$$q\sp{ij}Q\sp{i}P\sp{j}\,=\,P\sp{j}Q\sp{i}.\eqn\ethreefourteen$$
This observation suggests that if we restrict ourselves to the $q$'s being
the $N$'th root of identity, we can seek a solution of our conditions
\ethreeseven-\ethreeeight\ with the elements of $U$ given in terms of
various linear combinations of the products of $Q$'s and $P$'s. The
simplest of such
solutions corresponds to the case when $a\sim Q$ and $c\sim P$ and so is
given by
$$\eqalign{a=sin\alpha e\sp{i\psi}P,&\quad a\sp{\star}=sin\alpha
e\sp{-i\psi} P\sp{N-1}\cr
c=cos\alpha e\sp{i\phi}Q,&\quad c\sp{\star}=cos\alpha
e\sp{-i\phi}Q\sp{N-1},\cr}\eqn\ethreefifteen$$
where $\alpha$ and $\psi$ are real numbers. Then it is easy to check that
all conditions (3.7) and (3.8) (with $\vert q\vert=1$) are
satisfied if we choose $$D\,=\,e\sp{i\xi}P\sp2.\eqn\ethreefifteenb$$
Hence a $2N\times 2N$ matrix \ethreenine\ with its entries given by
\ethreefifteen\ and \ethreefifteenb\ is a representation of $U_q(2)$
for $q=exp({2\pi i\over N})$.
\par In the following we shall express the matrix \ethreenine\ as a product
of two matrices $U=T\tilde D$, $U\sp{\dagger}=\tilde
D\sp{\dagger}T\sp{\dagger}$, where
$$T\,=\,\pmatrix{sin\alpha e\sp{i\psi}P& -q\sp{\star}cos\alpha e\sp{-i\phi}
Q\sp{\dagger}\cr cos\alpha e\sp{i\phi}Q& sin\alpha
e\sp{-i\psi}P\sp{\dagger}\cr}\eqn\ethreesizteena$$
and $$\tilde D\,=\,\pmatrix{1&0\cr
0&e\sp{i\xi}P\sp2\cr}.\eqn\ethreesixteenb$$
 The basic Lagrangian then becomes
$$L=-{1\over
2}\Tr{(U\sp{\dagger}\partial_{\mu}U)(U\sp{\dagger}\partial\sp{\mu}U)}\eqn
\ethreeseventeen$$ with the trace taken with respect to the $U_q(2)$
matrix indices as well as the ones describing the realisations
(3.18) and (3.19). One can write
$$\Omega\sp{\mu}\,=\,U\sp{\dagger}\partial\sp{\mu}U\,=\,\tilde D\sp{\dagger}L
\sp{\mu}\tilde D\,+\,\tilde D\sp{\dagger}\partial\sp{\mu}\tilde D,
\eqn\ethreeeighteen$$
where
$$L\sp{\mu}\,=\,T\sp{\dagger}\partial\sp{\mu}T\,=\,\pmatrix{V\sp{\mu}&
Z\sp{\mu}\cr K\sp{\mu}& -V\sp{\mu}\cr}.\eqn\ethreenineteen$$
We obtain
$$V\sp{\mu}\,=\,i(\psi,\sp{\mu}sin\sp2\alpha\,+\,\phi,\sp{\mu}cos\sp2
\alpha)\eqn\ethreetwenta$$
and
$$\eqalign{Z\sp{\mu}\,=&\,Q\sp{\dagger}P\sp{\dagger}y\sp{\dagger \mu}
=Q\sp{\dagger}P\sp{\dagger}e\sp{-i\psi-i\phi}[\alpha,\sp{\mu}
-i(sin\alpha cos\alpha)(\psi,\sp{\mu}-\phi,\sp{\mu})]\cr
K\sp{\mu}\,=&\,-PQy\sp{\mu}=-PQe\sp{i\psi+i\phi}[\alpha,\sp{\mu}
+ i(sin\alpha
cos\alpha)(\psi,\sp{\mu}-\phi,\sp{\mu})]=-Z\sp{\mu\dagger}
\cr}\eqn\ethreetwentyb$$
where
$$\rho,\sp{\mu}={\partial \rho\over \partial
x\sp{\mu}}.\eqn\ethreetwentyone$$
Hence $L\sp{\mu}$ can be resolved into
$$L\sp{\mu}=V\sp{\mu}\pmatrix{1&0\cr0&-1\cr}+y\sp{\dagger\mu}
\pmatrix{0&Q\sp{\dagger}P\sp{\dagger}\cr0&0\cr}-y\sp{\mu}\pmatrix{0&0\cr
PQ&0\cr}\eqn\ethreetwentyonea$$
and so we observe the explicit appearance of the $SU_q(2)$ algebra
generators (which in this case are also the $SU(2)$ generators).
Notice that as $D=e\sp{i\xi}P\sp2$
$$\Omega\sp{\mu}\,=\,\pmatrix{V\sp{\mu}&Z\sp{\mu}D\cr D\sp{-1}K\sp{\mu}&
-\tilde V\sp{\mu}\cr},\eqn\ethreetwentytwo$$
where $$\tilde
V\sp{\mu}=V\sp{\mu}-D\sp{-1}\partial\sp{\mu}D
=V\sp{\mu}-i\partial\sp{\mu}\xi\eqn\ethreetwentythree$$
and the Lagrangian (3.20) is given by the formula
$$L=-{1\over 2}V\sp{\mu}V_{\mu}+y\sp{\mu}y\sp{\dagger}_{\mu}
+{1\over 2}\xi_{,\mu}\xi,\sp{\mu}-iV\sp{\mu}\xi_{,\mu}\eqn\threetwentufour$$
where $\xi(x)$ describes the $U(1)$ field extending $SU(2)$ to $U(2)$.
Indeed, one can show that if we put $\xi=0$ we recover the classical
action for the $SU(2)$ $\sigma$\ model. On the other hand, we can generalise
the Lagrangian \ethreeseventeen\ to
$$L\,=\,-{1\over 2}\Tr{(U\sp{\dagger}}\bigtriangledown_{\mu}U)(U\sp{\dagger}
\bigtriangledown\sp{\mu}U)\,=\,-\Tr{}{1\over
2}\tilde\Omega_{\mu}\tilde\Omega\sp{\mu},\eqn\ethreetwentyfive$$
where we have introduced the $U(1)$ covariant derivative
$$\bigtriangledown_{\mu}\,=\,\pmatrix{\partial_{\mu}\cdot 1_N&0\cr
0&(\partial_{\mu}-A_{\mu})\cdot 1_N\cr},\eqn\ethreetwentysix$$
and so
$$\tilde\Omega\sp{\mu}\,=
\pmatrix{{V\sp\mu}&{Z\sp\mu
 D}\cr{D\sp{-1}K\sp\mu}&
 {-(\tilde{V}\sp\mu+A\sp\mu)}\cr}.\eqn\ethreetwentyseven$$

Then, if in particular, we choose the pure gauge mode for the $A_{\mu}$
field
$$A_{\mu}\,=\,D\sp{-1}\partial_{\mu}D\,=\,i\xi_{,\mu}\cdot
1_N,\eqn\ethreetwentyeight$$
we find that \ethreetwentyfive\ reduces to the conventional $SU(2)$
$\sigma$\ model. We see that the $U(1)$ gauge field $A_{\mu}$ leads to the
appearance of the gauge invariance which allows us to set $D=1$ in
\ethreetwentytwo.

\chapter{Outlook}
In this paper we have considered some particular solutions of $\sigma$ models
taking values in quantum groups. Our main example corresponded to
the $U_q(2)$ $\sigma$\ model with $q\sp{N}=1$. The ``classical" fields of this model
were described by $2N\times 2N$ matrices, \ie\ we considered the embedding
 $U_q(2)\subset\,
   U(2N)$. In the general case, however, the embeddings must involve infinite-
dimensional $\sigma$\ models \ie\ $U(\infty)$ or $O(\infty)$. Indeed, the quantum
 group generators can be represented in terms of the Heisenberg algebra, which
 can be
realised using infinite-dimensional matrices. In  fact, it is only when
we  impose the relation
$$[a,\,a\sp{\star}]=0\eqn\eCOM$$
that we can realise the generators of $SU_q(2)$ or $U_q(2)$ in terms of finite
dimensional matrices. For $SU_q(2)$, eqs. (2.7) and (4.1)
imply $q\sp2=1$. For $U_q(2)$, eqs. (3.9) and (4.1) imply $\vert q\vert=1$.
\par We hope that our paper will be treated as a preliminary study of some
 aspects of sigma models taking values in a quantum group.
Let us mention some of these aspects:\hfil\break
a) For a given quantum group one can consider any finite dimensional
 representation ($u_{ij}\rightarrow T_{AB}(u_{ij})$).  For example, for the
 quantum group $SU_q(2)$  one can consider any $(2j+1)$ - dimensional
 representation (see
\eg\ ref [23]). If $j=1$,  this would give us the $O_q(3)$ quantum group $\sigma$\
 model.
\hfil\break
b) In the algebraic formulation of the models one has to make precise the
 commutation relations between the quantum group $\sigma$\ fields and their
 variations.
 Only when
these are known can we  derive the field equations from the action.\hfil\break
c) The action of the algebraic quantum group $\sigma$\ model can be considered as the
argument (after exponentiation) of the generalised Feynman path integral
provided that the suitable  formulae  for the integration over  the quantum
 group
functions $f(G_q)$ are found. This problem bears some analogy with the Berezin
integration rules for Grassmann algebras, and in the case of an arbitrary
 quantum
group, is still to be determined . (See, however, ref [24] for a discussion of
a
 simple
two-dimensional noncommutative case). \hfil\break
d) The concept of a quantum group $\sigma$\ field should be useful when
one wants to introduce the notion of generalised gauge fields, with {\bf local}
gauge transformations described by quantum group parameters. In section 3 we
have introduced the $U(1)$ gauge field but because of its Abelian nature
the field's noncommutative aspects were absent. It would be interesting to
consider \eg\ the $U_q(3)$ $\sigma$\  model coupled to non-Abelian $U_q(2)$ gauge
fields. This should  allow us (by gauge fixing) to formulate the quantum
group $\sigma$\ model on the coset ${U_q(3)\over U_q(2)}$.
\par Finally, we would like to add that although in this paper we have
 considered
$\sigma$\ fields taking values in noncommutative algebras the two best known choices,
namely, the quaternionic algebra (see \eg\ ref [25]) and the Grassmann algebra
(see \eg\ ref [26]) are finite dimensional. In the case of quantum groups,
for a generic $q$, the algebra of noncommuting functions $f(G_q)$ is infinite
dimensional and so in order that we can
 extend \eg\ the superfield formalism of
 supersymmetric
theories to the case of quantum groups, the new formal tools still have to be
developed.

\ack
Most of the work reported in this paper was performed when one of us (WJZ)
visited the Weizmann Institute of Science and the other (JL) the University
of Bordeaux I. WJZ wishes to thank the Weizmann Institute of Science
for its hospitality and the Einstein Center
for Theoretical Physics for the support of his stay in Israel.
  JL wishes to thank Prof. Minnaert for the hospitality at Bordeaux I.

 In addition JL would like to thank S.L. Woronowicz for
numerous discussions.

\endpage
\refout
\endpage
\bye


text:
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Welcome to the San Marcos Golf Resort
1912 was a big year for Arizona. Still a territory at the time, Arizona officially became the 48th state on February 14, 1912, and that same year Dr. Alexander J. Chandler founded a small town just southeast of Phoenix. Naming it after himself, Dr. Chandler had big plans for his little town hoping to create a "jewel in the desert." Dr. Chandler believed his town could one day become a totally planned community and an oasis in the desert where the rich and famous would flock. Hiring architect Arthur Burnett Benton, who was an authority on California mission design, the construction on the resort began and when it opened in 1913 the original San Marcos Resort had 35 guest rooms, a lobby, dining room, ballroom, and a full range of resort amenities like golf, horseback riding, and polo. One of the design accents that became a trademark feature of the resort was the construction of pergolas, which framed almost the entire structure. To this day, the resort remains one of the best examples of integrated Mission Revival design in Arizona.
The 100-acre golf course was built in 1913 and was seeded with Bermuda grass, making it the first green grass golf course in Arizona. Fifteen years later in 1928 the course was redesigned and was built to include a sprinkler system. Fresh off a multi-million dollar renovation, today the 18-hole San Marcos Golf Course sits on 90 well-manicured acres and includes a state-of-the-art irrigation system that produces premium playing conditions year-round, expanded lakes, new tee boxes, reconstructed bunkers with new sand, and green enlargements on a handful of holes. The result is a strategically-diverse and visually stunning layout that makes for an enjoyable challenge for golfers of all playing abilities.
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text:
প্রখম পাতা
প্রসঙ্গ
আমাদের বিষয়ে
বিষয়বস্তু
জ্বালানী/শক্তি
আঞ্চলিক সহযোগিতা
জল
নদী অববাহিকা সমূহ
ব্রক্ষ্মপুত্র অববাহিকা
গঙ্গা অববাহিকা
ইন্দাস অববাহিকা
মেকং অববাহিকা
ইয়াংসি অববাহিকা
ইয়েলো অববাহিকা
দেশসমূহ
মিয়ানমার
আরো তথ্য জানতে ব্লিক করুন
A GLOF threatens Bhutan right now
(English) Bhutan sounds the alarm as its biggest glacier lake swells dangerously due to sustained high temperatures
Workers from the United Nations Development Programme in Bhutan are trying to reduce the risk of floods around Thorthormi lake (Image: United Nations Development Programme)
(English) Nidup Gyeltshen; Dawa Gyelmo, জুলাই 10, 2019
বিভাগসমূহ:
ট্যাগসমূহ:
দূর্যোগসমূহ
দুঃখিত , এই গল্পের মধ্যে শুধুমাত্র উপলব্ধ%LANG:
Sustained heat waves, a delayed monsoon and a spike in temperature has recently melted huge volumes of ice and snow in Bhutan's largest glacial lake, the Thorthormi Tsho (lake), causing the water level to rise by almost two meters. This has prompted the government to issue a nationwide flood advisory, warning of the potential risk of a GLOF. A team of senior glaciologists, hydrologists and technicians has visited the site in Lunana to monitor the development.
Any major disturbance on Thorthormi Lake could result in a Glacial Lake Outburst Flood (GLOF), potentially with huge cost to life, property and infrastructure.
Satellite images showed much of the melting on Thorthormi took place between June 6 and 15, as temperatures spiked. On June 20 excess water drained into two smaller lakes located downstream, causing one of them to burst due to its weak moraine entrapment.
Fresh scouring/landslides can be seen on the moraine dam that separates lakes Thorthormi with Raphstreng [image courtesy: National Centre for Hydrology and Meteorology]
The incident led to a minor flood that washed away a small bridge in the nearest village in Thanza. Three other villages located close by were immediately evacuated. It also exposed an Automatic Water Level Sensor (AWLS) on one of the lakes. The sensor has been relocated.
Senior glaciologist Karma Toeb, from Bhutan's National Centre for Hydrology and Meteorology (NCHM) said the excess water was due to the melting of ice in the glacier lake.
In recent years, these glacier lakes have been forming and expanding all over the Hindu Kush Himalayas due to climate change. As the temperature rises and glaciers melt faster, water starts accumulating at their snouts, because the rate of melt becomes faster than the ability of the loose scree and moraine surrounding the snout to drain the water away. The accumulated water forms a lake that keeps getting bigger each year as the problem keeps getting worse. And since these lakes are bounded only by loose pebbles and moraine, the walls are prone to collapse as the weight of water, ice and snow keeps increasing. A GLOF of this kind had caused havoc in India's Uttarakhand state in 2013.
See Climate change, poor policies, multiply Himalayan flood effects
NCHM officials said the rise in temperature this June was due to a sustained heat wave lasting more than 30 days. The heat wave caused average temperatures in Bhutan to rise by around two degrees Celsius. Temperature sensors that have been installed in Lunana recorded the highest average temperature in the last nine years at nearly 15 degree Celsius for the months of April, May and June.
Toeb explained that when ice melts, water fills up the space between ice and the lake bedrock. This results in less friction. So the ice can move and slide, eventually causing a glacial surge.
A visual representation of what really happened at Thorthormi on 20 June. Excess water on Thorthormi spills over into Subsidiary lake I and then onto Subsidiary lake II. The moraine entrapment of Sub II is weak and it eventually bursts and empties itself, causing a minor flood [image courtesy: National Centre for Hydrology and Meteorology]
As water levels rose by almost two metres on Thorthormi Lake, it eroded several portions of the moraine dam that separates it from another fully formed glacial lake: Raphstreng Lake located 70 metres downstream.
A major breach on the moraine dam will combine the two lakes that will release more than 53 million cubic metres of water gushing down the Pho chhu valley. This would be three times bigger than Bhutan's last major GLOF in 1994 that killed 20 people.
A 2009 satellite photograph of Bhutan's dangerous lakes [image by: Robert Simmon / NASA Earth Observatory]
Sangay Tenzin, a hydrologist, said that currently the moraine dam has sufficient mass to withstand the hydrostatic (water) pressure from Thorthormi Lake. But if the melting continues due to a rise in temperature the moraine will not be able to hold.
If the two lakes combine and cause a GLOF, it will travel much faster this time because the river channels have already been cleared of debris and other hindrances by the 1994 flood, Tenzin said.
One of the worst scenarios for Bhutan is a combination of heavy rainstorm and GLOF happening at the same time. This could also have serious implication across the border in downstream India.
Tenzin explained that, unlike Bhutan, the bordering areas of India were plains. There the floodwater would inundate surrounding land for weeks. This would mean much greater damage.
The director of NCHM, Karma Dupchu, said Bhutan had good communication mechanisms with the Central Water Commission of India and any water and flood related information is shared with India on a sub-hourly basis during the monsoon. This is often not enough, and a system of WhatsApp information sharing between Indian and Bhutanese NGOs and civilians has helped supply timely flood warnings.
See: WhatsApp messages from Bhutan save lives in Assam
In the wake of the current incident, the government is now looking to step up monitoring of temperature in the Thorthormi area by installing more sensors. Over the last few years most of the ice blocks and snow inside Thorthormi have melted and disintegrated.
Threat to development and the economy
The International Centre for Integrated Mountain Development has warned of the potential risk of costly damages from GLOFs on key development sectors like agriculture, hydropower and forests in the Hindu Kush Himalayas.
The GLOF poses a direct threat to the economy, since Bhutan's core revenue rests on hydropower. Two of the biggest hydropower projects in the country, the 1,200 MW Punatsangchu-I and 1,020 MW Punatsangchu-II, are being built downstream of the Thorthormi Lake. What is more, the new Bhutanese government, has prioritised the construction of the 2,560 MW Sankosh hydropower project on the same river system.
As climate change brings sustained high temperatures to the Hindu Kush Himalayas, infrastructure across the region is increasingly under threat. A GLOF in Pakistan's Khyber Pakhtunkhwa province has led to a great deal of damage over the last few days. The sister of the Pakistani prime minister was among the dozens of people who had to be rescued from the sudden flood.
Source of water channel for 108mw Golen Gol powerhouse before and after GLOF. pic.twitter.com/ptKm9hHMBW
— Chitral Today (@ChitralToday) July 8, 2019
As heatwaves and rising temperatures become the new normal across the Hindu Kush Himalayas, the countries in the region will have to rethink both disaster preparedness as well as infrastructure development.
With additional inputs from Omair Ahmad
(English) The future of Bhutan's hydropower
Almost all Everest glaciers may disappear by end of century
Prof Pillai | Wednesday October 9th, 2019 at 15:48
very good and appreciable information One of my students is working on climate change and am telling him to follow up the updates, but should have a scientific basis


text:
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CRADLE OF FILTH: Total Fucking Darkness Demo to Get Vinyl Treatment
Posted on February 28, 2014 by dr.park
After more than two decades, Total Fucking Darkness, the infamous third demo from extreme metal pioneers, CRADLE OF FILTH, will be unleashed on gatefold double LP and digipak CD via Mordgrimm May 5, 2014.
The undertaking was helmed by vocalist Dani Filth, former guitarist Paul Ryan (currently of THE KING IS BLIND) and Cacophonous Records founder Frater Nihil, who first signed CRADLE OF FILTH, in an effort to bring fans old and new a remastered version of the demo along with material that had been sitting in the vaults for twenty years.
Given new life at Turan Studio in Oxford with engineer Tim Turan (who has worked on everything from EMPEROR to THIN LIZZY), Total Fucking Darkness commences with "Spattered In Faeces," the only surviving track from the now mythical Goetia sessions, followed on Side A and the rest of Side B with the recordings from the Total Fucking Darkness demo. Side C offers up another previously unreleased hymn with "Devil Mayfair (Advocatus Diaboli)," recorded in a rehearsal studio on Samhain 1992, followed by three more tracks from the same session, a further hint of the material that would have featured on the aborted Goetia album. Two short instrumental movements from former CRADLE OF FILTH keyboardist Ben Ryan are also included.
Housed in a heavy gatefold sleeve with paintings featuring the perverse visions of Radio 1 Presenter and occult artist, Daniel Carter, the legendary recording will be released in an initial mail-order edition of 666 copies, unleashed in an unholy trinity of three two-color, 180-gram vinyl variants limited to 222 copies each; orange/black, blue/grey and grey/black. Each copy includes a limited edition booklet with a foreword by Dani Filth and formerly unseen photos from the band archives. A retail version in blue vinyl with insert, and a digipak CD, will be available from Amazon and other outlets on May 5, 2014, with the limited edition version slated for release two weeks earlier.
This entry was posted in News and tagged cradle-of-filth. Bookmark the permalink.


Due to growth, our client a leader in the manufacturer of specialist machinery, currently has a new on-going temporary opportunity for a Panel Wirer to work from their Stamford location.
- Basic mechanical fitting i.e. drilling, hole punching, tapping, etc.


Callin' 'dibs' on her lyrics!
Kelsea Ballerini has shot to country star fandom at only 24 years-old. She's had numerous hits, award-winning records, and an upcoming tour. She knows how to sing a song about love, relationships, and break-ups. So what better person to quote than Ms. Ballerini. If you're looking for a good instagram caption, your girl has got you covered!
• "And I ain't one to be nobody's second best, no"
• "Stop holding onto me when the truth is you can't let go"
• "You're just a lost boy with your head up in the clouds"
• "And you're always gonna fly away, just because you know you can"
• "If you're gonna hold me, hold me like I'm leaving"
• "If you're gonna kiss me, kiss me like you need it"
• "So tell me, baby, are you just crazy, or crazy over me?"
• "If you're gonna talk the talk, you better walk it"
• "I'm calling dibs on your lips on your kiss on your time"
• "If you got a heart that ain't afraid of love ain't afraid to break"
• "If you're gonna love me, love me like you mean it"
• "I'm just trynna make you mine, boy"
• "I'm unapologetically in love"
• "The night is young and so are we, so you what you say we go and see it all for ourselves?"
• "We can rename constellations while we're making wishes"
• "I want a one way ticket where the creek and crickets are the only sound"
• "I talk the talk like I'm a tough girl cause there ain't room for weakness in a rough world"
• "Yeah, I wear my pain like stilettos"
• "You and me all summer, bright and living color"
• "14th of February, gotta feeling my heart still carries"
• "I hate catching bouquets, the honeymoon phase and letterman jackets don't fit"
• "Your eyes can't hold stars and you'd die if your heart really skipped"
• "And he's stuck in high school but this ain't high school"
• "We were golden, we were fire, we were magic"
• "No, I ain't gonna choke on that secondhand smoke"
• "Nighttime, bonfire, grab a guitar and sing along"
• "Barefeet on the boardwalk"
• "The sun may be making me crazy well, this could be a memory in the making"
• "Can't help but sit back and smile, summertime, yeah"
• "Is a backroad, breaking curfew, I was smiling and climbing out the sun roof"
• "We were crazy, we were stupid, never thought about what we were doing"
• "I see the moonlight on your skin, taking my hand, running and jumping in"
• "What happens in the dark always gets brought to life"
• "'Cause it's all fun and games it's all poker face"
• "Maybe one day, we'll take this big top down"


Here's a sweet little vid sent in by Laura Crane from Croyde.
Not only does she do the whole modelling thing, but she really can surf as well.
The Two times British Champion likes to get out and about with extra activities, which include motocross, snowboarding, fishing and football etc etc.
I think it would be more than fair to say, she's the real deal on all counts.


text:
Q: Error while Packaging Project for Windows The error with my textures, that are used for my HUD. I tried to change textures, but the result is same. They were imported from ordinary png images. What could go wrong?

Texture Crosshair has source art in an invalid format.

Packaging log image:

A: When I make HUD from C++ I've created UPROPERTY from UTexture2D, when I change them to Utexture without 2D all works.

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