text:
\section{Introduction}

Yang-Mills theory in two dimensions has been an object of
intensive studies during almost two decades. It is known that
the classical theory is trivial and the quantum theory has no
propagating degrees of freedom. However, this theory possesses
very many interesting properties when formulated on
topologically nontrivial spaces \cite{Witten} and also in the
large $N$ limit ($N$ is related to the rank of the gauge group
$G=SU(N)$) \cite{DK}. A 2D Yang-Mills theory is almost
topological in the sense that, for example, for compact
spacetime manifolds of area (two-dimensional volume) $V$
its partition function
depends only on $e^{2} V$, where $e$ is the gauge coupling
constant.
Also, as we will see, the vacuum expectation value of
a Wilson loop depends only on the area of the
region surrounded by the loop and is independent of the points
where the loop is located.
This is a manifestation of the invariance under the
area preserving diffeomorphisms. It is believed that 2D
Yang-Mills theories share some of the important qualitative
features of 4D ones, in particular the area law behaviour.
Recently there has been a revival of interest in
studying 2D Yang-Mills theories because it was shown that in the
strong coupling limit they can be related to
lower-dimensional strings \cite{Gross} and because of a
special role such theories play in
M-theory \cite{YM-M}. Rich mathematical structures appearing in
two-dimensional gauge theories were studied in a number of
papers \cite{Witten}, \cite{AB} - \cite{BT1} (see lecture \cite{Moore}
for a review and references). The partition function and the
vacuum expectation values of Wilson loop functionals were
calculated by many authors with various techniques \cite{W-calc} -
\cite{ALMMT}. Physical aspects of these theories, in particular
the analysis of Polyakov loops and the $\theta$-vacua were
discussed in Refs. \cite{Grig}.

In the present contribution we analyze vacuum expectation values
of the Wilson loop functionals. As a (Euclidean) space-time
a smooth two-dimensional orientable Riemannian manifold $M$
will be considered. The gauge group
$G$ is supposed to be a compact Lie
group. The gauge potential $A_{\mu}$ can be characterized by the
1-form $A=A_{\mu} dx^{\mu}$ on $M$ associated with the connection
form on the principal fibre bundle $P(M,G)$ \cite{KN}
in a standard way via local sections (see Refs. \cite{gauge-geom}
for geometrical desciption of gauge theories).
We denote by ${\cal A}$ the space of smooth connections.

Let us fix a point $x_{0}$ in $M$ and consider based loops
defined in a standard way as continuous mappings of the unit
interval $I=[0,1]$ into $M$:
\[
\gamma: I \rightarrow M, \; \; \; s  \rightarrow \gamma(s) \in M,
\; \; \; s \in I
\]
with $\gamma(0)=\gamma(1)=x_{0}$. For a given $A \in {\cal A}$
we associate with each loop $\gamma$ the element of $G$ called
holonomy
\begin{equation}
H_{\gamma}(A) = {\cal P} \exp \left( ie\oint_{\gamma} A \right),
          \label{H-def}
\end{equation}
where ${\cal P}$ means the path ordering. We will be
interested in traced holonomies
\begin{equation}
  T_{\gamma}(A) \equiv \frac{1}{d_{R}} Tr R \left(H_{\gamma}(A) \right) =
  \frac{1}{d_{R}} \chi_R (H_{\gamma}(A)),
   \label{trh}
\end{equation}
called also Wilson loop variables \cite{Wi}, where $R$ is an
irreducible representation of $G$, $d_{R}$ is its dimension and
$\chi_R$ is its character. Let us denote by ${\cal T}$ the
group of local gauge transformations, i.e. the group of smooth
vertical automorphisms of $P$. It is known that $T_{\gamma}$
suffice to separate points in ${\cal A}/{\cal T}$ representing
classes of gauge equivalent configurations and, therefore, enable
to reconstruct all smooth gauge connections up to gauge
equivalence \cite{Giles}. Due to this property the Wilson loop
functionals (\ref{trh}) form a natural set of gauge invariant
functions of connections in the classical theory.

We will study the vacuum expectation
value $<T_{\gamma}>$ of the Wilson loop functional given by
\begin{equation}
< T_{\gamma} > = \frac{1}{Z(0)} Z(\gamma), \; \; \;
  Z(\gamma) =  \int {\cal D}A e^{-{\cal S}} T_{\gamma}(A),
 \label{Zg-def}
\end{equation}
where ${\cal S}$ is the Yang-Mills action.
In Sect. 2 we will discuss as an example calculation of
(\ref{Zg-def})
in the abelian case for arbitrary $M$. In Sect. 3 $<T_{\gamma}>$
will be calculated on the lattice for a general gauge group.
In Sect. 4 we will study a special class of connections,
namely invariant connections, and their contribution to $<T_{\gamma}>$
in the case $M=S^{2}$ and $G=SU(2)$. In Sect. 5
this contribution will be compared with the complete vacuum expectation
value (\ref{Zg-def}).

\section{Continuum case, $G=U(1)$}

The action ${\cal S}$ in (\ref{Zg-def}) is given by the standard
expression
\[
{\cal S} = \frac{1}{4}\int_{M} Tr \left( F \wedge \ *F \right)=
 \frac{1}{8}\int_{M} Tr \left( F_{\mu \nu} F_{\mu \nu} \right)
 \sqrt{ \det g_{\mu \nu}} d^{2}x.
\]
In general the space of connections ${\cal A}$ consists of a number of
components, or sectors ${\cal A}_{\alpha}$, labelled by elements
$\alpha$ of an index set ${\cal B}$,
$\alpha \in {\cal B}$.
The functional integral is represented as a sum
over the elements of ${\cal B}$, each term of the sum
being the functional integral over the connections in ${\cal
A}_{\alpha}$. This feature has an analog in quantum mechanics:
in the case of multiply connected space $M$ evaluation of the functional
integral includes summation over the
connected components of the space of paths in $M$ and
integration over the paths within each component \cite{QM-analog}.

The set ${\cal B}$ of connected
components of ${\cal A}$ is in 1-1 correspondence with the space of
non-equivalent principal $G$-bundles $P(M,G)$ over the manifold $M$.
Let us denote this space as ${\cal B}_{G}(M)$, ${\cal B} \cong
{\cal B}_{G}(M)$. The problem of classification of fibre bundles
was considered in a number of books and articles.
Following closely the lectures by Avis and Isham \cite{AI} we
obtained that in the case when $M$ is a two-dimensional
manifold (actually, even a CW-complex) ${\cal B}_{G}(M)$ is equal to
\[
   {\cal B}_{G}(M) \cong H^{2} (M,\pi_{1}(G)),
\]
the second cohomology group of $M$ with coefficients given by
elements of the first homotopy group of the gauge group $G$
(see details in \cite{AK}).
An equivalent classification of principal fibre bundles over a
two-dimensional surface in terms of elements of the group
$\Gamma$, specifying the global structure of $G$ through the
relation $G = \tilde{G}/\Gamma$, where $\tilde{G}$ is the
universal covering group, was given in the second article of
Ref. \cite{Witten}.
For completeness, we present here a list of the first homotopy groups
$\pi_{1}(G)$ for  some Lie groups which are of interest in gauge theories:
$\pi_{1}(SU(n)) = \pi_{1}(Sp(n)) = 0$, $\pi_{1}(SO(n)) = Z_{2}$
($n = 3$ and $n \geq 5$), $\pi_{1}(U(n)) = Z$.

Let us consider the abelian case.  $U(1)$-bundles are classified
by elements of the second cohomology group $H^{2}(M,Z)$. The
class $c_{1}(P) \in H^{2}(M,Z)$, corresponding to the bundle $P$,
is known as the first (integer) Chern class
\cite{KN}, \cite{Span}. For a closed orientable
2-dimensional manifold $M$ $H^{2}(M,Z) \cong Z$.
For the bundle $P_{n}$, characterized by the label $n \in {\cal B} \cong
Z$, the integer cohomology class can be represented by
$e F^{(n)} / (2 \pi)$ with $F^{(n)}$ being the curvature 2-form
defined locally through $F^{(n)} = dA^{(n)}$, where $A^{(n)}$ is the
gauge 1-form given by a connection on $P_{n}$. The integral
\[
    \frac{e}{2 \pi} \int_{M}  F^{(n)} = n,
\]
does not depend on the choice of the connection and gives the
1st Chern number, also called the topological charge.

From the discussion above we conclude that
\begin{equation}
Z(\gamma) = \sum_{n=-\infty}^{\infty} \int {\cal D} A^{(n)}
   e^{-{\cal S}(A^{(n)})} T_{\gamma}(A^{(n)}) = \sum_{n=-\infty}^{\infty}
    Z^{(n)}(\gamma),          \label{Z-sum}
\end{equation}
where in each term $Z^{(n)}$ the integration goes over the gauge
potentials with the Chern number equal to $n$. We
represent $A^{(n)}$ as $A^{(n)} = \tilde{A}^{(n)} + a$, where
$\tilde{A}^{(n)}$ is a potential with
\begin{equation}
     \int_{M} \tilde{F}^{(n)} = \frac{2\pi n}{e},  \label{F-int}
\end{equation}
$\tilde{F}^{(n)} = d \tilde{A}^{(n)}$, and $a$ is a 1-form with
zero Chern number.  As $\tilde{A}^{(n)}$ we take instanton
configurations, i.e. solutions of the equation of motion. In the
literature they are often referred to as monopoles, and we will
follow this
terminology in the present paper.
Then $Z^{(n)}(\gamma)$ in (\ref{Z-sum}) becomes
\begin{eqnarray}
Z^{(n)}(\gamma) & = & e^{-{\cal S}(\tilde{A}^{(n)})} T_{\gamma}
(\tilde{A}^{(n)}) Z_{0}(\gamma),
     \label{Zn}  \\
Z_{0}(\gamma) & = & \int_{{\cal A}^{(0)}} {\cal D}a e^{-{\cal S}(a)}
   T_{\gamma}(a),    \label{Z0}
\end{eqnarray}
and the functional integral in $Z_{0}(\gamma)$ goes over the
connections in the trivial bundle $P_{0}$ with zero topological
charge.

It turns out that for the evaluation of (\ref{Zn}) we
do not need explicit expressions for the monopole solutions
$\tilde{A}^{(n)}$. Indeed, it is easy to show using Eq.
(\ref{F-int}) that
\[
{\cal S}(\tilde{A}^{(n)}) = \frac{2\pi^{2} n^{2}}{Ve^{2}}.
\]
If $\gamma = \partial \sigma$, where $\sigma$ is the interior of
$\gamma$ and $\tilde{A}^{(n)}$ is regular in $\sigma$,
then by using Stoke's theorem we obtain that
\[
\oint_{\gamma} \tilde{A}^{(n)}  =
\oint_{\partial \sigma} \tilde{A}^{(n)} =
\int_{\sigma} d\tilde{A}^{(n)} =
 \int_{\sigma} \tilde{F}^{(n)} =
\frac{2\pi n}{Ve} S,
\]
where we denoted the area of the surface $\sigma$ by $S$.
Then the contribution of the monopoles to the full function
(\ref{Z-sum}) is equal to
\begin{eqnarray}
 & & Z_{mon}(\gamma) := \sum_{n=-\infty}^{\infty} e^{-{\cal S}
(\tilde{A}^{(n)})} T_{\gamma} (\tilde{A}^{(n)})   \nonumber \\
  & &  =  \sum_{n=-\infty}^{\infty} \exp
  \left( -\frac{2\pi^{2}n^{2}}{Ve^{2}} +
  i \frac{2\pi n}{V}\nu S \right) =
 e \sqrt{\frac{V}{2\pi}} \sum_{l=-\infty}^{\infty}
  \exp \left[ - \frac{Ve^{2}}{2} \left( \frac{S}{V}\nu + l
 \right)^{2} \right].   \label{Zmon}
\end{eqnarray}
The last equality was obtained using the Poisson summation
formula. We have taken the Wilson loop variable in an
arbitrary irreducible representation so that
$T_{\gamma}(A) = \exp \{ i\nu e
\oint_{\gamma} A\}$, $\nu$ being an integer.

Now let us turn to the calculation of the contribution
$Z_{0}(\gamma)$ given by the functional integral Eq. (\ref{Z0}).
The power of the exponent in $T_{\gamma}(A)$ can be understood
as a functional $J_{\gamma}$ acting on $A$ and written
as
\[
J_{\gamma}[A] \equiv e \oint_{\gamma} A =
\int_{M} dv A_{\mu}(x) J_{\gamma}^{\mu}(x).
\]
Then the integral (\ref{Z0}) becomes gaussian and can be easily
calculated. However, the components $J_{\gamma}^{\mu}$ are not
smooth functions,
\[
  J_{\gamma}^{\mu}(y) = e\oint_{\gamma} dx^{\mu} \frac{1}
{\sqrt{\det g_{\mu \nu}}} \delta (y-x),
\]
so that $J_{\gamma}[A]$ should be treated as a singular form, or
weak form, \cite{Fels} for the accurate calculation of the
contribution of fluctuations around the monopoles. The result is
\[
Z_{0}(\gamma) = {\cal N} e^{-\frac{e^{2}}{2}\nu^2
\frac{S(V-S)}{V}}.
\]

Combining this with the contribution (\ref{Zmon}) of monopoles
we obtain the final expression for the
expectation value of the Wilson loop variable for a
homologically trivial loop $\gamma$ in the abelian case:
\begin{equation}
<T_{\gamma}> = \frac{
  \sum_{l=-\infty}^{\infty} \exp \left[ - \frac{e^{2}}{2} V
  \left( \frac{S}{V}\nu + l \right)^{2} - \frac{e^{2}}{2}\nu^2
  \frac{S(V-S)}{V} \right]}{\sum_{l=-\infty}^{\infty}
  \exp \left[ - \frac{e^{2}}{2} V l^{2} \right]}.
  \label{Tg}
\end{equation}
Expression (\ref{Tg})
depends only on the total area $V$ of the compact
two-dimensional manifold $M$ and the area $S$ of $\sigma$
surrounded by $\gamma$. It is invariant under the transformation
$S \rightarrow (V - S)$, as it must be, because any of the two regions
of $M$, in which the loop $\gamma$ divides it, can be considered
as the interior of $\gamma$.

\section{Two-dimensional gauge theories on the lattice}

Here we consider the case of arbitrary gauge group $G$ and arbitrary
compact orientable two-dimensional manifold $M$.
We assume that $\gamma$ divides $M$ into two regions
$\sigma_1, \sigma_2$ of genera $r_1, r_2$ and areas $S_1, S_2$
respectively. The genus of $M$ is $r=r_1+r_2$ and its area
$V=S_1+S_2$.

To perform the calculation we introduce a connected graph, or
lattice, $\Lambda$ on
the manifold $M$ which is a union of finite sets whose elements
are 0-cells $s$ (sites), 1-cells $l$ (links) and 2-cells
$p$ (plaquettes). We consider only such lattices that the closed path
$\gamma \subset \Lambda$, that is $\gamma$ is a sequence of
links of $\Lambda$. We denote the area
of the plaquette $p$ as $|p|$. The ``lattice spacing" $a$
is defined as the minimum distance such that every plaquette is
contained in a circle of the radius $a$. As a
system of loops let us choose  $\gamma_{p} = \partial p$, i.e.
loops which are boundaries of the plaquettes, and a set of
generators $a_i, b_i$ of the fundamental group of $M$.
The index $i=1,\ldots ,g$ labels the handles of $M$.
Any loop on the lattice can be obtained from the system
$\{\gamma_{p},a_i,b_i\}$.
The loops $a_{i}$, $b_{i}$ allow us to be in any homotopy class while
the loops $\gamma_{p}$ generate continuum deformations. This system
is not independent but subjected to one relation:
\begin{equation}
      \prod_{p \in \Lambda} \gamma_{p} =
     \prod_{i=1}^g a_ib_ia_i^{-1}b_i^{-1}    \label{loop-rel}
\end{equation}
from which one can construct an independent system of loops
$\beta_{j}$ $(j = 1, 2, \ldots, n = \dim \pi_{1}(\Lambda))$.
The product in (\ref{loop-rel}) has to be taken in a certain order
which can be shown to exist.
Also we would like to note that for the rigorous treatment
independent hoops, that are classes of holonomically equivalent
loops, should be considered. We will skip this distinction here.

Using results of \cite{AL} (see also \cite{ALMMT}) one can give a
well-defined meaning to the heuristic measure ${\cal D}A $ in Eq.
(\ref{Zg-def}) as a measure constructed out of copies of the
Haar measure on the
group $G$. This is possible if the action can be written as a
cylindrical function on the space ${\cal A} / {\cal T}$.
The standard Yang-Mills action
${\cal S}(A)$ is not well defined on such space, hence
one has to use some regularized action ${\cal S}_{\Lambda}(A)$
on the lattice $\Lambda$, calculate $Z_{\Lambda}(\gamma)$, which
is a lattice analog of (\ref{Zg-def}), and then take the limit
$Z(\gamma) = \lim_{a \rightarrow 0} Z_{\Lambda}(\gamma)$.

In this article we consider a class of actions such that
\[
   e^{-{\cal S}_{\Lambda}(A)} = \prod _{p \in \Lambda}
  e^{-{\cal S}_{1}(H_{\gamma_{p}}(A))},
\]
where the holonomy $H_{\gamma}(A)$ is defined by Eq.
(\ref{H-def}).
The function ${\cal S}_{1}$ is a real function over $G$
which satisfies the following conditions:
1) ${\cal S}_{1}(g) = {\cal S}_{1}(g^{-1})$;
2) it has the absolute minimum on the identity element;
3) $\lim_{a \rightarrow 0}{\cal S}_{1}(H_{p}(A))|p|^{-1} =
Tr( F_{\mu \nu}(x))^{2}/2$.
An important example is given by the Wilson action:
${\cal S}_{1}(g) = 1 - Re \chi_F (g)/d_{F}$, where $F$ stands for the
fundamental representation.

With such actions the integrand in (\ref{Zg-def}) is
a cylindrical function. Let us write the action in terms of the Fourier
coefficients:
\begin{equation}
\mu_{p}(g) \equiv e^{-{\cal S}_1(g)} = \sum_R d_R \chi_{R}(g)
B_{R}(p),    \label{mu-p}
\end{equation}
where the sum goes over all irreducible representations of $G$.
Any action can be specified through the coefficients $B_{R}$.
A relevant case is the heat-kernel action given by
$B_{R} (p) = (\tilde{B}_{R})^{|p|}$,
$\tilde{B}_{R} = \exp (- e^{2}c_2(R)/2)$,
where $c_{2}(R)$ is the value of the second Casimir operator.
When $G=U(1)$ the heat-kernel action is called Villain action.

Let us denote by $g_{a_{i}}$ and $g_{b_{i}}$ the holonomies
corresponding to the generators $a_{i}$ and $b_{i}$ of the
homotopy group and by $g_{p} \equiv g_{\gamma_{p}}$ the holonomies
corresponding to the loops $\gamma_{p}$.
The functional integral $Z_{\Lambda} (\gamma)$ is equal to
\begin{equation}
 Z_{\Lambda}(\gamma) =
	\int_{G^{n+1}} (\prod _{i} dg_{a_i}dg_{b_i})
	\left( \prod _{p \in \Lambda}
	dg_{p} \mu_{p}(g_{p})\right)\Delta [\{g_q\}]
	\frac{1}{d_R} \chi_R \left(\prod_{q
      \in \gamma} g_{q} \right).   \label{Zp-def}
\end{equation}
Here $dg$ is the Haar measure on $G$ and
$\Delta [\{g_q\}]$ is a factor which imposes the relation
between the group variables implied by (\ref{loop-rel}):
\[
\Delta [\{g_q\}]=
\delta_G (\prod_{p\in \Lambda} g_{p} ,
\prod_i g_{a_i}g_{b_i}g_{a_i}^{-1}g_{b_i}^{-1} ),
\]
where $\delta_G(g,h)$ is the Dirac distribution on the group $G$.
A specific decomposition of $\gamma$, which
appeares in the argument of the character $\chi_{F}$ in
(\ref{Zp-def}), is given in terms of
the loops located in one of the regions, say, $\sigma_1$,
namely
\[
\gamma =\prod_p^{(1)}\gamma_p
[\prod_i^{(1)}a_ib_ia_i^{-1}b_i^{-1}]^{-1}
\]
where the superscript $(1)$ means the restriction to $\sigma_1$.

Using Eq. (\ref{mu-p}) and interchanging
the summation in (\ref{mu-p}) with the product over the
plaquettes in (\ref{Zp-def}) we obtain that
\begin{eqnarray}
Z_{\Lambda}(\gamma )  & = &
 \sum_{\{R_p\}}(\prod_{p \in \Lambda} B_{R_p}(p))
\int_{G^{n+1}} (\prod _{i} dg_{a_i}dg_{b_i})
\left( \prod _{p \in \Lambda}
	dg_{p}\right) (\prod _{p \in \Lambda} d_{R_p}
	\chi_{R_p}(g_p))      \nonumber \\
   & \times &   \Delta [\{g_q\}] \frac{1}{d_R} \chi_R
     \left(\prod_{q \in \gamma} g_{q} \right).   \nonumber
\end{eqnarray}
Here the first sum goes over all possible ``colorations'' of the
surface, i.e. over all possible configurations of the
irreducible representations, where each configuration $\{ R_{p}
\}$ is a set of irreducuble representations assigned to
each plaquette.

To perform the $g_p$ integration we use a procedure of
``lattice reduction". If plaquettes $p_1$ and $p_2$ share
a link that does not belong to $\gamma$ (then, they are
in the same $\sigma_i$ component) we change the group
variable $g_{p_1}\rightarrow g_{p_1}g_{p_2}^{-1}$.
The measure of integration is invariant under such changes,
$g_{p_2}$ disappears in $\Delta$ and $\chi_F$, and we can
integrate over $g_{2}$ using the properties of invariant
group integration.
The result is non-zero if both plaquettes are in the same
representation, in which case the common link is ``erased" and
the two plaquettes merge into one plaquette while the form of the
partition function remains the same. Such reductions can be done
until we arrive to a lattice consisting only of two plaquettes,
$p_1$ and $p_2$ with representations $R_1$ and $R_2$ respectively.
Integration over the variables of the homotopically non-trivial
loops gives rise to the factors $d_{R_{1}}^{1-2r_{1}}$ and
$d_{R_{2}}^{1-2r_{2}}$. Finally we obtain that
\[
Z_\Lambda(\gamma )  =
 \sum_{R_1, R_2}(\prod_{p \in \sigma_1} B_{R_1}(p))
                 (\prod_{p \in \sigma_2} B_{R_2}(p))
	d_{R_1}^{1-2r_1}d_{R_2}^{1-2r_2}\{R_1,R,R_2\},
\]
where
\[
\{R_1,R,R_2\} = \int_{G} dg \chi_{R_1}(g)
   \chi_{R_2}(g^{-1}) \chi_{R}(g)
\]
is the number of times the representation
$R_2$ is contained in $R_1\otimes R$.

To obtain the continuum limit one should, in general,
compute the coefficients $B_{R}(p)$ and take the
limit $|p| \rightarrow 0$. This is the case if we work
with the Wilson action. If instead we use the heat-kernel action,
$Z_{\Lambda} (\gamma)$ can be written in the form
\[
Z_{\Lambda} (\gamma)  =
 \sum_{R_1, R_2} \tilde{B}_{R_1}^{S_1}
                 \tilde{B}_{R_2}^{S_2}
	d_{R_1}^{1-2r_1}d_{R_2}^{1-2r_2}\{R_1,R,R_2\}
\]
which is lattice independent, so it gives already the continuum
limit. The final expression for the continuum limit of the
vacuum expectation value of
the Wilson loop functional is
\begin{equation}
<T_\gamma >  =
\frac{
 \sum_{R_1, R_2} \tilde{B}_{R_1}^{S_1}
                 \tilde{B}_{R_2}^{S_2}
	d_{R_1}^{1-2r_1}d_{R_2}^{1-2r_2}\{R_1,R,R_2\} }
{ d_{R}\sum_{R_3}\tilde{B}_{R_3}^{V}d_{R_3}^{2-2r}}.    \label{Tfin}
\end{equation}

For abelian groups $d_{R}=1$. In this case the topology of
the surface is irrelevant.
If $G=U(1)$ the irreducible
representations are labelled by integers $n\in Z$. Since
$B_n =e^{-e^2n^2/2}$, taking $R=\nu$ we get
\[
<T_{\gamma} >=\frac{
\sum_{n=-\infty}^\infty B_n^{S_1}B_{n+\nu}^{S_2}}
{\sum_{n=-\infty}^\infty B_n^V}
  =\frac{\sum_{n=-\infty}^\infty e^{-S_1e^2n^2/2}
  e^{-S_2e^2(n+\nu)^2/2}} {\sum_{n=-\infty}^\infty
  e^{-Ve^2n^2/2}}.
\]
This coincides with (\ref{Tg}) after the identifications
$S_{1}=S$ and $S_{2}=V-S$.

We would like to make a few comments. The result (\ref{Tfin})
is invariant under the transformation $S \leftrightarrow V-S$.
Although we have considered only compact surfaces,
the same techniques can be applied in the non-compact case.
The important difference is that the infinite component at one
side of $\gamma$ appears to be in the trivial representation.
This case can also be analized as the limit $V\rightarrow \infty$
of a compact case. Performing it in (\ref{Tfin}) we see
that the Wilson loop expectation value for $R^{2}$
or any non-compact
surface has the typical area law behaviour
$<T_{\gamma}> \propto \exp \left(-e^{2}c_{2}(R) S/2 \right)$,
where $S$ is the area of the compact part enclosed by $\gamma$.
For finite $V$ similar behaviour is obtained in
the strong coupling limit $e^2 V \gg 1$
in which case only the region with the smaller area
contributes. In the weak coupling limit $e^2 V \ll 1$ for
the abelian case
\[
<T_{\gamma }>\sim e^{-e^2(S-S^2/V)\nu^2/2}.
\]

\section{Wilson loop variables for invariant connections.}

In Sect. 2, while calculating the vacuum expectation value of
the Wilson loop functional
in the abelian case, we analyzed the contribution of the monopoles.
In the present section we
will study this quantity for a special class of
connections, called invariant connections, for $M=S^{2}$ and $G=SU(2)$.
The result of the calculation will be compared with the exact
formula derived above. This will allow
us to understand how much of the information is captured by the
invariant connections. Our interest
in this class of connections is motivated by the fact that the
analogous calculation can be
carried out in Yang-Mills theory in any dimension provided,
of course, that non-trivial invariant connections exist.

Let us first discuss the definition of invariant connection
and explain how they can be constructed. After that a
concrete example with $M=S^{2}$ and $G=SU(2)$ will be considered.
Assume that there is a group $K$ which acts on the
space-time $M$ and its action can be lifted to the fibre bundle
$P(M,G)$. We will call $K$
symmetry group. A connection in $P$ is said to be invariant with
respect to transformations $L_{k}$ of $K$ if its connection
1-form $w$ satisfies $L_{k}^{*} w = w$
for all $k \in K$.
This class of connections includes many known monopole and instanton
configurations and they were intensively
used for the coset space dimensional reduction of gauge theories
(see \cite{Manton}).
Gauge potentials corresponding to invariant
connections are called symmetric potentials and were introduced
in \cite{Schwarz}. The condition of invariance implies that for any
$k \in K$ there exists a gauge tranformation $g_{k}(x) \in G$
such that the gauge potential $A_{\mu}$, corresponding to the
invariant connection form $w$, has the property
\begin{equation}
    (O_{k} A)_{\mu} = g_{k}(x)^{-1} A_{\mu}(x) g_{k}(x) +
\frac{1}{ie}g_{k}(x)^{-1} \partial_{\mu} g_{k}(x),   \label{sympot}
\end{equation}
where the l.h.s. is the standard
change of the field under the space-time transformation $O_{k}$
on $M$ (which is the projection of $L_{k}$ to $M$). This
formula tells that a symmetric potential is invariant under
transformations of $K$ up to a gauge transformation.

Here we consider the case when $K$ acts transitively on $M$
(see Ref.
\cite{KN} for general theory and Ref. \cite{KMRV} for review).
Then $M$ is a coset space $K/H$, where $H$ is a subgroup
of $K$, called the isotropy group, and $K$ acts on $K/H$ in the
canonical way. The 1-form $A = A_{\mu} dx^{\mu}$, which
describes a symmetric potential satisfying (\ref{sympot})
and is a pull - back of an
invariant form with respect to a (local) section
of $P(M,G)$, can be constructed in the following way.
Let ${\cal G}$, ${\cal K}$ and ${\cal H}$ be the Lie algebras
of the groups $G$, $K$ and $H$ respectively.
If $H$ is a closed compact subgroup of $K$, the case we
have in mind, then there exists the decomposition
\begin{equation}
    {\cal K} = {\cal H} + {\cal M}     \label{K-decomp}
\end{equation}
with $ad ({\cal H}) {\cal M} = {\cal M}$, where $ad$ denotes
the adjoint action of the algebra on itself, i.e. $ad (h) (u) =
[h,u]$, $h,u \in {\cal K}$. Let $\theta$ be the canonical
left-invariant 1-form on the Lie group $K$ with values in ${\cal
K}$, and $\bar{\theta}$ is its pull-back to $M=K/H$. It can be
calculated as $\bar{\theta} = k(x)^{-1} dk(x)$, where $k(x) \in
K$ is a local representative of the class $x \in K/H$.  We
decompose the 1-form $\bar{\theta}$ into the ${\cal H}$- and
${\cal M}$-components in accordance with (\ref{K-decomp}):
$\bar{\theta} = \bar{\theta}_{\cal H} +
\bar{\theta}_{\cal M}$. It can be shown that
the invariant connections are given by
\begin{equation}
        A = \frac{1}{ie} \left( \tau (\bar{\theta}_{\cal H}) +
        \phi (\bar{\theta}_{\cal M}) \right),   \label{A-inv}
\end{equation}
where $\tau$ is the homomorphism of algebras induced by a group
homomorphism $H \rightarrow G$ and $\phi$ is a mapping $\phi :
{\cal M} \rightarrow {\cal G}$ satisfying the equivariant
condition
\begin{equation}
      \phi \left( ad (h) u \right) = ad (\tau (h)) \phi(u),
    \; \; h \in {\cal H}, \; \; u \in {\cal M}.    \label{equiv1}
\end{equation}
This condition can be viewed as the intertwining condition
between representations of ${\cal H}$ in ${\cal M}$ and ${\cal
G}$.
An effective technique for solving constraint (\ref{equiv1})
was developed in Refs. \cite{VK} (see also \cite{KMRV}).

As a concrete example we will consider the Yang-Mills theory on
the two-dimensional sphere $S^{2}$ with the gauge group
$G=SU(2)$. The sphere is realized as a
coset space $S^{2} = SU(2)/U(1)$. Let us construct first the
1-forms $\bar{\theta}_{\cal H}$ and $\bar{\theta}_{\cal M}$
which appear in Eq. (\ref{A-inv}). As usual, we cover the
manifold $M=S^{2}$ with two charts $U_{1}$ and $U_{2}$, the
northern and southern hemispheres, so that $U_{1} \cup U_{2} \cong
S^{2}$ and $U_{1} \cap U_{2} = U_{12} \cong S^{1}$ is the equator.
More precisely, if $\vartheta$ and $\varphi$ are two angles
parametrizing points of the sphere, then $U_{1} =
\{ 0 \leq \vartheta \leq \pi/2, 0 \leq \varphi < 2 \pi \}$,
$U_{2} = \{ \pi/2 \leq \vartheta \leq \pi, 0 \leq \varphi < 2
\pi\}$. We take the generators of $K=SU(2)$ to be $Q_{i} =
\tau_{i}/2$ ($i=1,2,3$), where $\tau_{i}$ are the Pauli
matrices. The subgroup $H=U(1)$ is generated by $Q_{3}$
and the vector space ${\cal M}$ is spanned by
$Q_{1}$ and $Q_{2}$. Let us consider the decomposition of the
algebra ${\cal K}$ in the case under consideration. Denote by
$e_{\pm \alpha}$ the root vectors and by $h_{\alpha}$ the
corresponding Cartan element of this algebra and take
\[
e_{\pm \alpha} = \tau_{\pm} = \frac{1}{2} \left(\tau_{1}\pm
i\tau_{2}\right), \; \; \;
h_{\alpha}= \tau_{3}.
\]

We choose local representatives $k^{(j)}(\vartheta, \varphi) \in
K=SU(2)$, $j=1,2$ of the
points of the coset space $S^{2} = SU(2)/U(1)$ as follows:
\[
  k^{(1)}(\vartheta, \varphi) = e^{i\varphi \frac{\tau_{3}}{2}}
  e^{i\vartheta \frac{\tau_{2}}{2}}
e^{-i\varphi \frac{\tau_{3}}{2}} \; \; \mbox{and} \; \;
  k^{(2)}(\vartheta, \varphi) = e^{-i\varphi \frac{\tau_{3}}{2}}
  e^{i(\vartheta-\pi) \frac{\tau_{2}}{2}}
  e^{i\varphi \frac{\tau_{3}}{2}}.
\]
By straightforward computation one obtains the forms
$\bar{\theta}_{\cal H}$ and $\bar{\theta}_{\cal M}$ as
\[
    \bar{\theta}^{(i)}  =  \left(k^{(i)}\right)^{-1} d k^{(i)} =
    \bar{\theta}_{\cal H}^{(i)} + \bar{\theta}_{\cal M}^{(i)}.
\]

According to general formula (\ref{A-inv}) the invariant
gauge connection depends on the gauge group and the embedding
$\tau(H) \subset G$.
The decomposition of the vector space ${\cal
M}$ into irreducible invariant subspaces of ${\cal H}$
corresponds to the following decomposition of representations:
\begin{equation}
    \underline{2} \rightarrow (2) + (-2),  \label{M-decomp}
\end{equation}
where in the r.h.s. we indicated the eigenvalues of $ad
(h_{\alpha})$, and the reducible representation carried by the
space ${\cal M}$ is indicated by its dimension in the l.h.s.

Now let $E_{\alpha}$, $E_{-\alpha}$ and $H_{\alpha}$ be the root
vectors and the Cartan element of the algebra $A_{1}$, which
appears as complexification of ${\cal G} = su(2)$. We assume
that they are given by the same combinations of the Pauli
matrices as the corresponding elements of complexified ${\cal
K}$ described above. Define the group homomorphism $\tau:
H=U(1) \rightarrow G=SU(2)$ by the expression
\[
\tau \left( e^{i\alpha_{3}\frac{\tau_{3}}{2}} \right) =
e^{i \alpha_{3} \kappa \frac{\tau_{3}}{2} } =
\cos (\kappa \frac{\alpha_{3}}{2}) +
i \tau_{3} \sin (\kappa \frac{\alpha_{3}}{2}).
\]
It is easy to check that this definition is consistent if
$\kappa$ is integer. Thus the corresponding homomorphism of
algebras is labelled by $n \in Z$ and is given by
$\tau_{n} (h_{\alpha}) = n H_{\alpha}$.
The three-dimensional space ${\cal G}$ of the adjoint
representation of $A_{1}$ decomposes into three 1-dimensional
irreducible invariant subspaces of $\tau_{n}({\cal H})$:
\begin{equation}
    \underline{3} \rightarrow (0) + (2n) + (-2n)  \label{G-decomp}
\end{equation}
(in brackets we indicate the eigenvalues of
$ad \left(\tau ( h_{\alpha})\right)$).
Compare decompositions (\ref{M-decomp}) and
(\ref{G-decomp}). For $n \neq \pm 1$ there are no equivalent
representations in the decompositions of ${\cal M}$ and ${\cal
G}$, and the intertwining operator $\phi: {\cal M} \rightarrow
{\cal G}$ is zero.  It also turns out to be zero for $n=0$.
In accordance with (\ref{A-inv}) for $n \neq \pm 1$
\begin{equation}
A^{(1)}_{n} = \frac{n}{2e} \tau_{3}
       (1 - \cos \vartheta) d \varphi,  \; \;
A^{(2)}_{n} =  -\frac{n}{2e} \tau_{3}
       (1 + \cos \vartheta) d \varphi.
                           \label{A-SU2}
\end{equation}
If $n=1$ or $n=-1$ the results are more interesting.
Let us consider the case $n=1$ in detail.
Comparing again (\ref{M-decomp}) and (\ref{G-decomp}) we see
that there are pairs of representations with the same
eigenvalues and, therefore, the operator $\phi$ is non-trivial.
It is determined by its action on basis elements of ${\cal M}$:
$\phi (e_{\alpha}) = f_{1} E_{\alpha}$, $\phi (e_{-\alpha}) =
f_{2} E_{-\alpha}$,
where $f_{1}$, $f_{2}$ are complex numbers. The fact that the
initial groups and algebras are compact implies a reality
condition \cite{KMRV} which tells that $f_{1} = f_{2}^{*}$. Thus,
the operator $\phi$ and the invariant connection are
parametrized by one complex parameter $f_{1}$ (we will suppress
its index from now on). Using the formulas above we obtain that
\begin{equation}
A^{(1)} = \frac{1}{2e} \left( \begin{array}{cc}
 (1 - \cos \vartheta) d\varphi &
         f e^{-i\varphi} (-id\vartheta - \sin \vartheta d \varphi) \\
         f^{*}e^{i\varphi}(id\vartheta -
       \sin \vartheta d \varphi) & -(1 - \cos \vartheta) d \varphi
     \end{array}  \right),    \label{A-SU2-1}
\end{equation}
\begin{equation}
A^{(2)} = \frac{1}{2e} \left( \begin{array}{cc}
-(1+\cos \vartheta) d\varphi &
        f e^{i\varphi} (-id\vartheta - \sin \vartheta d \varphi) \\
f^{*}e^{-i\varphi}(id\vartheta -
       \sin \vartheta d \varphi) & (1+\cos \vartheta) d \varphi
     \end{array}  \right).  \label{A-SU2-2}
\end{equation}
The curvature form $F$ is given by
\begin{equation}
    F =  dA + \frac{ie}{2} [A,A] =
     - \frac{1}{2e} \tau_{3} \left( |f|^{2} - 1 \right) \sin
    \vartheta  d\vartheta \wedge d\varphi,        \label{F-SU2}
\end{equation}
and the action for such configuration is equal to
\begin{equation}
   {\cal S}_{inv}(f) = \frac{\pi}{2e^{2}} \frac{1}{R^{2}}
    \left(|f|^{2} - 1 \right)^{2},     \label{S-SU2}
\end{equation}
where $R$ is the radius of the sphere. Due to the invariance
property any extremum of the action found withing the subspace of
invariant connections is also an extremum in the space of all
connections \cite{KMRV}. From Eq. (\ref{S-SU2}) we see that
there are two extrema: the trivial one with $f=0$ and the
non-trivial one with $|f|=1$. The trivial
extremum was found in Ref. \cite{RP} as a spontaneous
compactification solution in six-dimensional Kaluza-Klein
theory.

It turns out that potentials (\ref{A-SU2}), (\ref{A-SU2-1})
and (\ref{A-SU2-2}) are related to known non-abelian monopole
solutions (see, for example, \cite{CHM}). For $n
\neq \pm 1$ and for $n = \pm 1$ with $f=0$ these expressions
coincide with the monopole solutions with the even monopole
charge $m = 2n$.
In fact the solution with charge $m > 0$ can be transformed
to the solution with charge $(-m)$ by the gauge transformation
$A \rightarrow S^{-1} A S$ with the constant matrix $S=-i \tau_{1}$.
All these monopoles live in the trivial
principal fibre bundle $P(S^{2},SU(2))$, which is the only bundle
with such structure group. The latter result also
follows from our discussion in Sect. 2. Indeed, in the case under
consideration ${\cal B}_{SU(2)}(S^{2}) = H^{2}(S^{2},
\pi_{1}(SU(2))) = 0$ since $SU(2)$ is simply connected.
Thus, all the monopoles can be described by a unique function on the
whole sphere. This is indeed the case. Namely, there exist gauge
transformations, different for the northern and southern
patches, so that the tranformed potentials coincide. Let us
show this for the case $n=1$. In fact we will construct such
transformations  for the whole family of
the invariant connections (\ref{A-SU2-1}), (\ref{A-SU2-2}). The
matrices giving these gauge transformations are
\[
V_{1} = i\left( \begin{array}{cc}
 \cos \frac{\vartheta}{2} & e^{-i\varphi} \sin \frac{\vartheta}{2}  \\
 e^{i\varphi} \sin \frac{\vartheta}{2}  & -\cos \frac{\vartheta}{2}
     \end{array}  \right), \; \;
V_{2} = i\left( \begin{array}{cc}
 e^{i\varphi} \cos \frac{\vartheta}{2}  & \sin \frac{\vartheta}{2}  \\
 \sin \frac{\vartheta}{2}  & e^{-i\varphi \cos \frac{\vartheta}{2} }
     \end{array}  \right).
\]
By calculating $A^{(i)'} = V_{i}^{-1} A^{(i)} V_{i} +
(ie)^{-1}V_{i}^{-1} d V_{i}$ for $i=1$ and $i=2$ one can easily
check that the transformed fields are equal to
\begin{equation}
A^{(1)'} = A^{(2)'} = \frac{1}{2e} \left(\tau_{+} c_{+} +
\tau_{-} c_{-} + \tau_{3} c_{3} \right)
\label{A-SU2-gen}
\end{equation}
with
\begin{eqnarray}
 c_{+} & = & c_{-}^{*} = e^{-i\varphi}
 \left\{ \left[ -\sin \vartheta \cos \vartheta +
 \left( f \cos ^{2} \frac{\vartheta}{2}
 - f^{*} \sin^{2} \frac{\vartheta}{2} \right)
  \sin \vartheta \right] d\varphi
                                         \right. \nonumber \\
 & + & \left. i(-1 + f \cos ^{2} \frac{\vartheta}{2} +
     f^{*}\sin^{2} \frac{\vartheta}{2} ) d\vartheta \right\}
                                  \nonumber \\
 c_{3} & = &
\left( 1 - \frac{f + f^{*}}{2} \right) \sin^{2}\vartheta d\varphi -
i\frac{f-f^{*}}{2} \sin \vartheta d\vartheta.   \nonumber
\end{eqnarray}
Note that in general expressions (\ref{A-SU2-1}),
(\ref{A-SU2-2}) and (\ref{A-SU2-gen}) the phase of the complex
parameter $f$ can be rotated by residual gauge transformations
which form the group $U(1)$.

For $f=0$ this formula gives the known expression for the $m=2$
$SU(2)$-monopole \cite{CHM}:
\begin{equation}
A^{(1)} = A^{(2)} = \frac{1}{4e} \left( \begin{array}{cc}
(1-\cos 2\vartheta) d\varphi &
  e^{-i\varphi} (-2id\vartheta - \sin 2\vartheta
 d \varphi) \\
 e^{i\varphi}(2id\vartheta -
   \sin 2\vartheta d \varphi) & -(1-\cos 2\vartheta) d\varphi
     \end{array}  \right).  \label{A-SU2-0}
\end{equation}
One can easily see that the forms (\ref{A-SU2-gen}) and (\ref{A-SU2-0})
do not have singularities on the sphere.

For $f=f^{*}=1$ the form (\ref{A-SU2-gen}) vanishes. This
shows that this configuration, which is also an extremum of the
action, describes the trivial $SU(2)$-monopole with
$m=0$.  Note that in the original form the potentials
(\ref{A-SU2-1}) and (\ref{A-SU2-2}) do not seem to be trivial.
Of course, one can check that they are pure gauge configurations and
correspond to the flat connection.  Vanishing of the gauge field
(\ref{F-SU2}) for this value of $f$ confirms this.

A similar situation occurs for $n=-1$. Again, there exists a
family of in\-va\-ri\-ant con\-nec\-tions pa\-ra\-met\-riz\-ed by a
complex parameter, say $h$, analogous to $f$. The action
possesses two extrema. One, with $h=0$, describes the
$SU(2)$-monopole solution with $m=-2$ and the other extremum, with
$|h|=1$, describes the trivial monopole with $m=0$. However,
this family of invariant connections is gauge equivalent to the family
for $n = +1$ and can be obtained from (\ref{A-SU2-1}), (\ref{A-SU2-2})
by the gauge transformation with the constant matrix $S=-i\tau_{1}$
and the identification $h=f$.

The picture we obtained is the following. Various homomorphisms
$\tau_{n}: H \rightarrow G$ give rise to various invariant
connections describing $SU(2)$-monopole solutions. All monoples
are reproduced in this way. Moreover there is a continuous
family of invariant gauge connections, parametrized by one
complex parameter $f$ which passes
through the points corresponding to $SU(2)$-monopoles with
$m=0$ ($n=0$), $m=2$ ($n=1$) and $m=-2$ ($n=-1$) in the space of
all connections of the theory.
Classes of gauge equivalent invariant connections are labelled by
values of $|f|$. Thus, $|f|=0$ corresponds to the class of the $m=2$
monopole (the $m=-2$ monopole is in the same class, as we already
explained above), $|f|=1$ corresponds to the class of the
$m=0$ monopole.

Let us calculate now the contribution of the invariant
connections to the vacuum
expectation value of the Wilson loop functional. For this we
choose a loop $\gamma (\vartheta_{0})$ on $S^{2}$ given by
\begin{equation}
 \gamma (\vartheta_{0}) = \{(\vartheta_{0}, \varphi'),
  \vartheta_{0} =
     \mbox{const}, 0 \leq \varphi' < 2\pi \}.  \label{loop}
\end{equation}
This loop is parallel to the equator, labelled by the angle
$\vartheta_{0}$ and parametrized by the polar angle $\varphi'$.
Here for definiteness we consider the case when the loop lies in
the northern hemisphere, i.e. $0 \leq \vartheta_{0} \leq \pi/2$.

We notice that the group element
\[
U(\varphi) = {\cal P} \exp \left[ ie\int_{\eta(\varphi;\vartheta_{0})}
A^{(1)}_{\phi} d\varphi' \right],
\]
where $\eta (\varphi; \vartheta_{0})$ is the same path as (\ref{loop})
but with $\varphi'$ changing from $0$ to $\varphi$, satisfies the
matrix equation
\begin{equation}
\frac{dU(\varphi)}{d\varphi} = ie U(\varphi) A^{(1)}_{\varphi}
(\vartheta_{0},\varphi).    \label{U-eq}
\end{equation}
The holonomy $H_{\gamma (\vartheta_{0})} (A^{(1)}) = U(2\pi)$.
It turns out that Eq. (\ref{U-eq}) can be solved and
the traced holonomy is equal to
\begin{equation}
T_{\gamma (\vartheta_{0})} (A^{(1)})
  =  - \cos \left( \pi \sqrt{ 1 + \frac{4S}{V} \left(
  1 - \frac{S}{V} \right) \left( |f|^{2} -1 \right) } \right),
  \label{T-SU2}
\end{equation}
where $S = 2 \pi R^{2} (1 - \cos \vartheta_{0})$ is the area of the
surface surrounded by the loop $\gamma (\vartheta_{0})$ and $V = 4
\pi R^{2}$ is the total area of the two-dimensional sphere of the
radius $R$.
Note that for $|f|=1$ $T_{\gamma (\vartheta_{0})}=1$ as it should
be for a flat connection. The expression (\ref{T-SU2}) is
invariant under the transformation $S \rightarrow (V - S)$.

Finally we evaluate the quantity
$  < T_{\gamma (\vartheta_{0})}>_{inv} = Z_{inv}
( \gamma (\vartheta_{0}) )/Z_{inv}(0), $
characterizing the contribution of the invariant connections to
the vacuum expectation value of the Wilson loop functional,
where
\begin{equation}
  Z_{inv}( \gamma (\vartheta_{0}) ) =
  - \int df \  df^{*} e^{-\frac{2 \pi^{2}}{e^{2}V}
  \left( |f|^{2} - 1\right)^{2} }
  \cos \left( \pi \sqrt{ 1 + \frac{4S}{V} \left(
1 - \frac{S}{V} \right) \left( |f|^{2} -1 \right) } \right).
                               \label{Z-inv}
\end{equation}
This integral is an analog of the functional integral
(\ref{Z-sum}). It mimics a ``path-integral quantization" of the
gauge model where the configuration space (the space of
connections) is finite-dimensional and consists of $SU(2)$ - invariant
connections. The integral takes into
account the contribution of the monopoles with $m=0,\pm 2$ and
fluctuations around them along the {\it invariant} direction.
The action on the invariant connection in the formula
above is given by Eq. (\ref{S-SU2}).

Of course, the complete contribution of the invariant connections,
given by Eqs. (\ref{A-SU2-1}), (\ref{A-SU2-2}) or Eq.
(\ref{A-SU2-gen}) to the true functional integral (\ref{Z-sum})
is different because it includes also contributions of all
fluctuations around them. In the next section we
are going to discuss which part of the true vacuum expectation
value $<T_{\gamma (\vartheta_{0})}>$ is captured by
$<T_{\gamma (\vartheta_{0})}>_{inv}$.

We would like to make a few remarks. In Ref. \cite{Mottola} it was
argued that the Faddeev-Popov determinant on invariant
connections turns out to be zero, hence the contribution of such
connections to the functional integral vanishes. This issue was
analyzed in \cite{AGV} in a different context, namely the
authors considered similar ``quantization"
for $SU(2)$-invariant connections in $(3+1)$ - dimensional
Ashtekar's gravity. They found that zeros of the
Faddeev-Popov determinant, which are responsible for the
vanishing of the determinant, are cancelled by the contribution
of the delta-functions of constraints and the path
integral measure is regular on invariant connections.

\section{Discussion of the result}

In Sect. 2 and Sect. 3 we calculated the vacuum expectation
value of the Wilson loop functional.
Let us analyze the quantity $E(g^{2}V,S/V) \equiv
-\log < T_{\gamma (\vartheta_{0})}> $ which in a theory with fermions
characterizes the potential of the interaction. Recall that the
closed path $\gamma (\vartheta_{0})$ on $S^{2}$ was defined by Eq.
(\ref{loop}), $S$ is the area of the region surrounded by this loop and
its ratio to the total area of the surface is given by
$S/V = (1 - \cos \vartheta_{0})/2$. $E(g^{2}V,S/V)$ as a function
of $S/V$ for two values of $e^{2}V$ in
the {\it abelian} case is given by Eq. (\ref{Tg}) and is presented
in Fig. 1.

\vskip 0.1cm
\begin{figure}[ht]
\epsfxsize=0.9\hsize
\epsfbox{f1.ps}
\caption{ $E(e^{2}V,S/V)$ (dashed line) and $E_{mon}(e^{2}V,S/V)$
(solid line) as functions of $S/V$ for $e^{2} V=10$ (lower lines)
and $e^{2} V=20$ (upper lines) in the abelian gauge theory}
\end{figure}

In the same plot we also show the contribution
$E_{mon}(g^{2}V,S/V)$ of the {\it abelian} monopoles calculated from
(\ref{Zmon}). We see from Fig. 1 (also from the exact formula
(\ref{Zmon})) that the dependence of $E_{mon}(g^{2}V,S/V)$ on
$S/V$ at small $S/V$ is quadratic almost
till $S/V = 0.5$, where the curve reaches its maximum.
This, being combined with the contribution of the fluctuations,
gives the linear dependence (the area law) almost for
all $S/V$ in the interval $0 < S/V < 0.5$.
The area law for the Wilson loop in the pure
Yang-Mills theory is considered as an indication of
the regime of confinement in the corresponding gauge
theory with quarks. The behaviour of $E(g^{2}V,S/V)$
for the {\it non-abelian} theory with $G=SU(2)$
is qualitatively the same. In this case contributions of
fluctuations around a monopole depend on the monopole solution,
so that the complete $Z(\gamma)$ cannot be written in the factorized
form as in the abelian case. However, the sum of the
terms $\exp(-{\cal S}_{mon})T_{\gamma}(A_{mon})$ over all
monopoles gives rise to qualitatively the same behaviour as
$E_{mon}(g^{2}V,S/V)$ in the abelian case.

As we have pointed out
the linear behaviour of the total $E(g^{2}V,S/V)$ is a manifestation
of the area law for ``not very large" $S/V$, i.e. when the area
surrounded by the loop $\gamma$ is far below the half of the
total area of the compact space-time $M$.
In this respect the quadratic behaviour of the monopole
contributions $E_{mon}(g^{2}V,S/V)$ seems to be an important feature
which gives rise to the linear dependence of the complete function.

\vskip 0.1cm
\begin{figure}[ht]
\epsfxsize=0.9\hsize
\epsfbox{f2.ps}
\caption{Contribution of invariant connections $E_{inv}(e^{2}V,S/V)$
as a function of $S/V$ for $e^{2} V=10$ (lower line)
and $e^{2} V=20$ (upper line) in the $SU(2)$ Yang-Mills theory.}
\end{figure}

The plot of the energy $E_{inv}(g^{2}V,S/V) \equiv -\log < T_{\gamma
(\vartheta_{0})}>_{inv}$ as a function of $S/V$ is given by Eq.
(\ref{Z-inv}) and is shown in Fig. 2.
Comparing it with Fig. 1 we see qualitative similarity in the
contribution of invariant connections and monopoles, and
we want to attract attention to this feature.
It can be argued that the quadratic behaviour of the
contribution of invariant connections, similar to that of
the monopoles, also leads to the linear dependence of the complete
$E(g^{2}V,S/V)$ in the $SU(2)$ case and, thus, serves as an idicator
of the area law behaviour of Wilson loop variables in the theory.
If this is also true in higher dimensions, one can study just the
contribution of invariant connections and from the
form of the behaviour of the function $E_{inv}$
get some hints about the type of the behaviour of the complete
function $E$. We are going to study this possibility in a future
publication.

\bigskip

\noindent{\large \bf Acknowledgments}

We would like to thank Jose Mour\~ao and Sebasti\`a Xamb\'o
for useful discussions and valuable remarks.
J.M.A. acknowledges financial support from CICYT (project PB94-1196).
Y.K. acknowledges financial support from
CIRIT (Generalitat de Catalunya), RFFR (grant 96-0216413-a),
INTAS (grant INTAS-93-1630.EXT) and JNICT
(grant JNICT/CERN/S/FAE/1111/96).

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NYC SHOOTINGS: Man shot in Times Square fracas, Harrison blames gangs for half of gun violence incidents
By Todd Maisel
Detectives look for clues at mass shooting at 15 Albany Avenue in Bedford Stuyvesant, Brooklyn, part of gang violence in the city. Chief Rodney Harrison discuss the case this morning. (Photo by Lloyd Mitchell)
A 20-year-old man was shot during what may have been a dispute on West 42nd Street in Midtown, a block from nearly empty Times Square, on Tuesday night, police officials said.
This follows several days of relative quiet as cooler temperatures keep much of the violence tamped down. In addition, aggressive moves against gangs, especially in Brooklyn, have kept pressure on those crews to keep violence in check.
In this latest shooting, at 8:14 p.m. on Dec. 1, a man pulled a gun on the victim in front of 360 West 42nd Street and shot him in the buttocks as he tried to flee the attacker. The victim was rushed to Bellevue Hospital in stable condition.
The assailant was described as a Black man with a dark complexion and dark dread locks, who was last seen wearing a black jacket, blue jeans and black sneakers and fleeing on foot southbound from the corner of West 33rd Street and 7 Avenue.
Surveillance photos, taken from the incident were provided by the NYPD.
Anyone with information in regard to this shooting is asked to call the NYPD's Crime Stoppers Hotline at 1-800-577-TIPS (8477) or for Spanish, 1-888-57-PISTA (74782). The public can also submit their tips by logging onto the Crime Stoppers website at WWW.NYPDCRIMESTOPPERS.COM, or on Twitter @NYPDTips. All calls are strictly confidential.
Half of shootings gang-related
Chief of Detectives Rodney Harrison said in an interview on WPIX-11 Wednesday morning that about 50% of the more than 1,400 shootings this year are gang related. The shootings also contributed to a 37.2% increase in homicides this year.
Harrison pointed to the long-term closure of courts during the pandemic as a factor complicating the NYPD's efforts to combat gang shootings.
"The courts being closed because of COVID-19 made things difficult for us and the pandemic has taken a toll on the NYPD as a whole," Harrison said this morning. "Combating the violence has been a struggle with much of it in Brooklyn. Of those 1,400 shootings, 50 percent is gang [related], and it will be precision policing that will get them off the streets."
Harrison called the court and prosecution backlogs "problematic" and he said it has been difficult to get some criminals to trial. However, he said his detectives have been working to identify those committing the most heinous crimes to get them off the streets.
Harrison cited the Nov. 22 shooting of seven young people at a sweet sixteen party on Albany Avenue in Bedford-Stuyvesant, in which a 20-year-old woman visiting from Virginia was killed.
"Several individuals walked into the place with firearms and began shooting, then to the lobby and then the third floor where they shot several more people," Harrison said. "I'm optimistic that the 79th Precinct detectives and Brooklyn North Homicide [Squad] will identify those responsible."
He added that he believe the investigation was in the "right direction" and "we are looking at a couple of people."
Shots fired in Brooklyn
The NYPD is seeking a gunman wanted for firing shots at a store window on Nov. 12. in Bedford Stuyvesant, Brooklyn.
Police say that at 6 p.m., a man fired multiple rounds through the window of a store at 494 Marcus Garvey Blvd. No injuries were reported in that shooting.
The individual is described as a man with a dark complexion in his 20s with a thin build and facial hair. He was last seen wearing a gray hoodie sweater, black jacket, light blue ripped jeans and white sneakers.
Surveillance photo of the attacker was provided by the NYPD after the incident.
Staten Island murder
In an unrelated case, police are seeking the assailants in two vehicles wanted for the shooting murder of a 52-year-old woman innocent bystander on Nov. 16 in Staten Island.
Investigators from the 120th Precinct say that at 6:58 p.m., an unknown assailant fired shots at at group standing in front of 225 Park Hill Ave. in Mariners Harbor, Staten Island. One of those shots struck Sue Doe, 52, of Park Hill Avenue, in the head.
She was rushed to Staten Island University Hospital and died on Nov. 25.
Police described the vehicle as a white 2019 Kia Sorento. Only an hour earlier, two vehicles engaged in a gun battle on Forest Avenue, only blocks from this incident. It was unclear if the two incidents were linked. Nobody was hit in the earlier incident.
Surveillance photo of Kia was provided by detectives.
Police also provided photos of the two vehicles from the earlier incident.
Anyone with information in regard to any of the aforementioned shootings is asked to call the NYPD's Crime Stoppers Hotline at 800-577-TIPS (8477) or for Spanish, 888-57-PISTA (74782). You can also submit tips online at nypdcrimestoppers.com, or on Twitter @NYPDTips. All calls and messages are kept confidential.
Eat it. Drink it. Do it. Tackle the city, with our help.
'Don't go back to work until you get paid what you're worth!' AOC, Williams rally around striking Bronx Teamsters
Three women sought for hitting victim with a bottle and taking off with her phone: cops
Still can't breathe: How NYPD officers continue to use chokeholds on civilians
Biden will order masks on planes and trains, increase disaster funds to fight coronavirus
NFL preview: A viewer's guide to the 2021 NFC, AFC Championship Game
Offensive flare there for Irving, Harden, Durant, but Nets ball security, defense needs work
Are the Jets gaining momentum in potential Deshaun Watson sweepstakes?
Jets to introduce new head coach Saleh on Thursday
Powerful exhibit 'Not Another Second' explores the stories and 'years lost' of LGBTQ+ Elders
Here are some rainy day activities that kids of all ages can enjoy in New York City
Here's why kids need camp after a year of remote learning
NYC SHOOTINGS: Three people wounded citywide overnight in ongoing gun violence
NYC SHOOTINGS: 40 people shot in 5 days as cops battle gang violence surge
NYC SHOOTINGS: Brooklyn man slain, six others injured as gun violence spikes again
NYC SHOOTINGS: Two dead, six wounded in citywide gunfire, cops arrest three after Bronx battle
Opinion: Making the right choices so we can 'have a nice life'
QNS
Bayside Times: January 22, 2021
City Planner Turned Developer Seeks Variance to Put Up 15-Story 'Model Block' in Red Hook
© amNY 2021 Schneps Media

redpajama_set_name: RedPajamaCommonCrawl


text:
Hamadryas epinome est une espèce de lépidoptères (papillons) de la famille des Nymphalidae, de la sous-famille des Biblidinae et du genre Hamadryas.

Dénomination
Hamadryas epinome a été décrit par Cajetan Freiherr von Felder et Rudolf Felder en 1867 sous le nom initial d' Ageronia epinome.

Synonymes : Ageronia fall Staudinger, 1886 ; Ageronia epinome florentia Fruhstorfer, 1916 ; Ageronia epinome aldrina Martin, [1923] ; Peridromia epinome argentina Bryk, 1953.

 Nom vernaculaire Hamadryas epinome se nomme Epinome Cracker en anglais.

 Description Hamadryas epinome est un papillon aux ailes antérieures à bord costal bossu et bord externe concave, au dessus marbré de beige, de vieux rose, de bleu-gris et de blanc avec une ligne submarginale d'ocelles cernés de gris et pupillés, complète aux ailes postérieures, uniquement de trois ocelles vers l'apex aux ailes antérieures. Une marque rouge en croissant est visible au 1/3 du bord costal des ailes antérieures.

Le revers des ailes antérieures présente une partie basale blanche et le reste de couleur marron suffusé de vert très largement taché de blanc. Les ailes postérieures sont blanches avec uniquement la marge marron taché de blanc et une ligne submarginale de cercles foncés.

 Biologie

 Plantes hôtes
Les plantes hôtes des chenilles des Hamadryas sont des Euphorbiaceae.

 Écologie et distribution Hamadryas epinome'' est présent en Bolivie, au Paraguay, en Argentine et au Brésil.

Biotope

Protection
Pas de statut de protection particulier.

Notes et références

Annexes

Articles connexes
 Biblidinae
 Nymphalidae

Liens externes

Lépidoptère (nom scientifique)
Biblidinae
Lépidoptère d'Amérique

redpajama_set_name: RedPajamaWikipedia


Its such as you read my thoughts! You seem to know a lot about this, like you wrote the e book in it or something. I feel that you just can do with some percent to force the message home a bit, but instead of that, this is wonderful blog. A great read. I will definitely be back.


What is Zoho Campaigns ?
Zoho Campaigns helps you to send content-rich emails and makes your email marketing campaign more effective. Its pre-designed templates and drag-and-drop editor lets you create and send attractive newsletters within minutes. Its marketing automation tools let you send follow-up emails easily and keep your audience engaged. With Zoho Campaigns, get insightful and detailed reports about your campaign performance that will help you take informed decisions. It also helps you make your email look more personalized and dynamic, so that every subscriber feels the message is only for them. It also conducts A/B Testing between two versions of your campaign to find out which one is more effective.
How much does Zoho Campaigns cost ?
The pricing for Zoho Campaigns starts at $12.0 per month. Zoho Campaigns has 2 different plans such as - Subscription at $12.0 per month and Pay-as-you-go at $40.0. Zoho Campaigns offers a Free Plan with limited features. They also offer an Enterprise Plan for their product.
Send 12,000 emails per month to up to 2,000 Subscribers.
Fill in your requirement here to know about our High Volume Plans.
"Simple interface and best features"
We tried many other email marketing softwares / tools before, but we liked Zoho because of its easy to use interface and no compromise on complex features like workflows.
We tried many other email marketing softwares / tools before, and we don't like anything in Zoho apart from it's UI.
We're now able to smoothly connect with our prospects and clients.
It has very friendly user interface. Using is easy and results are stunning.
Keeping in touch with customers. Providing new products.
It's pretty easy to set up. Very smooth and flexible.
The 24 hour support, they are very fast.
A bit of trouble with RSS in the beginning but was sorted out quite fast.
"Easy to use Zoho Campaigns"
Zoho allows us to manage our campaigns easily. Making lead forms and it also have robust marketing tools. It integrates with may software. Pricing is cheap and gives 2k subscribers for free every month.
In zoho campaigns the least I like about is the email templates and the flow of designing the templates.
It is the most efficient, simple and effective tool in market that helps us manage our campaigns. It provides metadata which is one of its unique features. They provide customer service round the clock.
It has allowed us to create some exciting campaigns and marketing tools to get visibility to our customer base. Overall it has increased our income by a lot in the last year since we start to use the Campaigns feature.
Email campaigns have never been easier, thanks Zoho!
I have nothing to bad to say. This software works as it should.
An award given to products that are appreciated by our users.


Got an idea to grow your business?
click This year, NEIdeas will award 26 existing businesses in Detroit, Hamtramck, and Highland Park $10,000 each for their ideas for growth. Applications will be accepted May 1-June 1, online at NEIdeasDetroit.org or at one of 28 ambassador locations throughout Detroit, Hamtramck, and Highland Park.
http://ridbedre.tv/severo-lopez-2er-changementer/ More information about rules and eligibility is available here.
cialis patent expiration Want to learn more? enter Attend the NEIdeas kickoff info session on Tuesday, May 1, from 5-8 p.m. at the Bel Air Cinema in northeast Detroit (10100 East 8 Mile Road, Detroit, MI 48234). RSVP here.
levitra for women NEIdeas staff will be on hand to give you tips on submitting a successful application.
NEIdeas is a program of the New Economy Initiative (NEI) and the Community Foundation for Southeast Michigan (CFSEM).
Since 2014, NEIdeas has awarded a total of $1.9 million to 118 businesses in Detroit, Hamtramck, and Highland Park to help them realize their ideas for growth. To learn more about last year's winners, click here.
http://mail.dementiabangladesh.org/index.php?option=com_content cialis wiki Since 2014, NEIdeas has granted a total of $1.9 million in cash awards to 118 existing businesses in Detroit, Hamtramck, and Highland Park for their ideas for growth.
Congratulations to the 22 winners of the 2017 NEIdeas challenge! Click here to learn about their ideas for growth.


This pc is alot harder to hook up than my other one and all the manuals it came with dont help alot. Sofar I have the mobo, the ram, and the cpu and the heatsink all in and the cd and hdd hooked up but thats about it. I got mainly just the fans 3 pin things hooked up and the main PSU cable and the 4 pin one beside it. Also I have all but 2 front panel wires hooked up but theres 2 left, one is micro atx and the other is a stray wire to hook down there that comes from the back by the door case. And I have no idea how to hook up the stuff coming from the top of the case. >.< This is way beyond my own capabilities. Plugging wires in is not my strongpoint. And figuring out the right way to get all the fans plugged in together.
I am totally lost, if someone has the DFI mobo ad the Tsunami case can you help me? I need it alot. Or even if you dont im sure all of you guys would know.
that doesnt really help me. I already have that stuff done that it says. Theres alot of stuff for my psu and case thats what i need help on.
I am no fan of TT cases. That said on their webpage is a guide that I have used before. just got a screenie of TT mainpage being hacked, wow. Anyway I googled "thermaltake wiring guide"
See if these links dont answer your questions.
middle of the page also has some hardcano stuff.
[b]EDIT ah a tsunami and the frontpanel stuff. O h, I just leave it unhooked, but you can probably find some documentation on the TT site or this one.
try tsunami frontpanel wiring (or similar combo) Personally, I just bundled and shoved them out of the way in my tt case.
gah why cant they have the tsunami case. This is sad, i guess ill have to take it in somewhere just to have them hook it up because its totally different from my other case and mobo and theres no manuals at all. I still dont understand how to get all the fans hooked up, not the small 3 pin ones. And theres like 50 different wires to hook up from the case and the psu I dont know what they do.
ok i looked off my other pc and tried to do the same thing, i think i got most of it but im not sure. Everything is hooked up except the Power SW and Micro SW wires from the case for the led for the frontpanel. It doesnt say anything about those two and where to put them in at. A question though, is, for the mobo, should the PSU fan be plugged into Fan 2? Because I have the psu fan plugged into fan 2, cpu fan to the heatsink, back fan on fan 5, front fan on fan 4 and the mobo fan is on fan 3. Is this right? Also, there are two plugin wires that say Fan only that have 4 holes but only 2 have the gold peice in it and I dont know what to use them for. I plugged in all the fan wires together and pluged them into the wire that had alot of 4 pin plugins and used that wire for the cddrive and the fan.
Aside from that i just have to plug in the usb stuff. Does all this sound right?
Also, i never had a sata hdd before, so do i have to use that floppy disk it came with? or can i just use it immediatly like non sata hdd.
The wires coming from the back of the case are probably for an intrusion alarm. Disregard it, your board doesn't support it. The "fan only" leads from the psu are for, guess what, fans. Fans with 4-pin connectors. PM me and tell me where in Kansas you live. I may be able to help.
eh the fan only things theres just 2 of them and they say fan only and haver 4 pins, but only 2 wires and there yellow and black. Someone else told me not to touch them.
I live near Lawrence and Topeka.
As I said, these are to be used to power fans ONLY. The yellow wire is +12V and the black wire is ground. Some fans have a third wire that can report fan speed, if the motherboard header supports it. I live in Lawrence and can help you if needed. PM me.
I would recommend skipping adding all of your fringe components until you get things running stable. Use one stick of ram, one harddrive, one cd/DVD drive, and minimum fans. I would not connect any of the case wires but would use the on/off button on the motherboard.
No drivers are needed for Sata hard drives as the OS supports them fine.
Using this method you will be able to get into bios if needed and be able to get past the post tests and into the OS installation.
Many people have problems before getting to the OS point because they try to hook everything up at once. Some folks do not have any problems if they do the one shot method so use judgement since you know your experience level.


text:
<?php

/**
 * The "albums" collection of methods.
 * Typical usage is:
 *  <code>
 *   $photoslibraryService = new Google_Service_PhotosLibrary(...);
 *   $albums = $photoslibraryService->albums;
 *  </code>
 */
class Google_Service_PhotosLibrary_Resource_Albums extends Google_Service_Resource
{
  /**
   * Adds an enrichment to a specified position in a defined album.
   * (albums.addEnrichment)
   *
   * @param string $albumId Identifier of the album where the enrichment will be
   * added.
   * @param Google_Service_PhotosLibrary_AddEnrichmentToAlbumRequest $postBody
   * @param array $optParams Optional parameters.
   * @return Google_Service_PhotosLibrary_AddEnrichmentToAlbumResponse
   */
  public function addEnrichment($albumId, Google_Service_PhotosLibrary_AddEnrichmentToAlbumRequest $postBody, $optParams = array())
  {
    $params = array('albumId' => $albumId, 'postBody' => $postBody);
    $params = array_merge($params, $optParams);
    return $this->call('addEnrichment', array($params), "Google_Service_PhotosLibrary_AddEnrichmentToAlbumResponse");
  }
  /**
   * Creates an album in a user's Google Photos library. (albums.create)
   *
   * @param Google_Service_PhotosLibrary_CreateAlbumRequest $postBody
   * @param array $optParams Optional parameters.
   * @return Google_Service_PhotosLibrary_Album
   */
  public function create(Google_Service_PhotosLibrary_CreateAlbumRequest $postBody, $optParams = array())
  {
    $params = array('postBody' => $postBody);
    $params = array_merge($params, $optParams);
    return $this->call('create', array($params), "Google_Service_PhotosLibrary_Album");
  }
  /**
   * Returns the album specified by the given album id. (albums.get)
   *
   * @param string $albumId Identifier of the album to be requested.
   * @param array $optParams Optional parameters.
   * @return Google_Service_PhotosLibrary_Album
   */
  public function get($albumId, $optParams = array())
  {
    $params = array('albumId' => $albumId);
    $params = array_merge($params, $optParams);
    return $this->call('get', array($params), "Google_Service_PhotosLibrary_Album");
  }
  /**
   * Lists all albums shown to a user in the 'Albums' tab of the Google Photos
   * app. (albums.listAlbums)
   *
   * @param array $optParams Optional parameters.
   *
   * @opt_param string pageToken A continuation token to get the next page of the
   * results. Adding this to the request will return the rows after the pageToken.
   * The pageToken should be the value returned in the nextPageToken parameter in
   * the response to the listAlbums request.
   * @opt_param int pageSize Maximum number of albums to return in the response.
   * The default number of albums to return at a time is 20. The maximum page size
   * is 50.
   * @return Google_Service_PhotosLibrary_ListAlbumsResponse
   */
  public function listAlbums($optParams = array())
  {
    $params = array();
    $params = array_merge($params, $optParams);
    return $this->call('list', array($params), "Google_Service_PhotosLibrary_ListAlbumsResponse");
  }
  /**
   * Marks an album as 'shared' and accessible to other users. This action can
   * only be performed on albums which were created by the developer via the API.
   * (albums.share)
   *
   * @param string $albumId Identifier of the album to be shared. This album id
   * must belong to an album created by the developer. .
   * @param Google_Service_PhotosLibrary_ShareAlbumRequest $postBody
   * @param array $optParams Optional parameters.
   * @return Google_Service_PhotosLibrary_ShareAlbumResponse
   */
  public function share($albumId, Google_Service_PhotosLibrary_ShareAlbumRequest $postBody, $optParams = array())
  {
    $params = array('albumId' => $albumId, 'postBody' => $postBody);
    $params = array_merge($params, $optParams);
    return $this->call('share', array($params), "Google_Service_PhotosLibrary_ShareAlbumResponse");
  }
}

redpajama_set_name: RedPajamaGithub


Snorting Chocolate Is Now Apparently A Thing, But Why Would You Do It?
•Snorting Chocolate Is Now Apparently A Thing, But Why Would You Do It?
•By Bruce Y. Lee, Forbes, July 5, 2017.
•I do like chocolate and I do like sushi but, in both cases, in my mouth. •Up my nose? •Not so sure about that.
•While they don't seem to have found a way to get a California roll easily through your nostrils, apparently now you can get chocolate up your nose.
•And this isn't just about laughing while drinking chocolate milk through a straw. •For $24.99 you can get a 10- serving tin of Coco Loko, a cacao-based powder (along with gingko biloba, taurine and guarana, which are common in energy drinks) that you can snort.
•Why in the world would you snort chocolate (along with energy drink ingredients) when you can just eat it?
•What's next? •Sticking Hershey's kisses in your ear? •Well, as Abha Bhattarai writes for the Washington Post, Nick Anderson, the 29-year-old founder of Legal Lean, the company that makes Coco Loko, heard about a "chocolate-snorting trend" in Europe, tried it and eventually worked to create his company's own blend.
•According to him, for 30 minutes to an hour you can get an "almost like an energy-drink feeling, like you're euphoric but also motivated to get things done." The company website includes the following claims: Endorphin rush: When endorphins are released it triggers a positive feeling of wellbeing in your body similar to morphine. •For example, the feeling after a long workout known as "runner's high" where you feel euphoric, positive and exhilarated.
•Serotonin rush: A sudden rush of serotonin will produce an elevated mood and a state of euphoria similar to the feeling of ecstasy. •This is the feeling that will make the music sound better and [increase] overall happiness.
•Euphoric energy: Raw cacao will give you a steady rush of euphoric energy and motivation that is great for partygoers to dance the night away without a crash. •It is even used by athletes to give them the natural competitive edge.
•Calm focus: Raw cacao will give you a sense of calm focus and reduce the chatter and worry in your brain so you are able to accomplish many tasks at once and maximize your time and focus. •It is also known to help with anxiety and to reduce stress.
•All of these supposedly from putting chocolate and then some up your nose? •And here you thought better speakers help music sound better, relationships and purpose in life lead to happiness, and training and mental toughness give athletes their competitive edge. •Where exactly are the scientific studies backing any of these claims? •Who nose.
•Während sie scheinbar keinen Weg gefunden haben, dass man eine kalifornische Rolle durch die Nasenlöcher ziehen kann, kann man anscheinend jetzt Schokolade schnupfen.
•Und, das ist nicht zum Lachen, während du Schokomilch durch einen Strohhalm trinkst. •For $24.99 you can get a 10- serving tin of Coco Loko, a cacao-based powder (along with gingko biloba, taurine and guarana, which are common in energy drinks) that you can snort.
•Warum in aller Welt solltest du Schokolade (zusammen mit Energydrink-Zutaten) schnupfen, wenn du sie einfach essen kannst?
•Was kommt als Nächstes? •Hersheys Küsse in dein Ohr stecken? •Well, as Abha Bhattarai writes for the Washington Post, Nick Anderson, the 29-year-old founder of Legal Lean, the company that makes Coco Loko, heard about a "chocolate-snorting trend" in Europe, tried it and eventually worked to create his company's own blend.
By Bruce Y. Lee, Forbes, July 5, 2017.
I do like chocolate and I do like sushi but, in both cases, in my mouth.
While they don't seem to have found a way to get a California roll easily through your nostrils, apparently now you can get chocolate up your nose.
And this isn't just about laughing while drinking chocolate milk through a straw.
For $24.99 you can get a 10- serving tin of Coco Loko, a cacao-based powder (along with gingko biloba, taurine and guarana, which are common in energy drinks) that you can snort.
Why in the world would you snort chocolate (along with energy drink ingredients) when you can just eat it?
Sticking Hershey's kisses in your ear?
Well, as Abha Bhattarai writes for the Washington Post, Nick Anderson, the 29-year-old founder of Legal Lean, the company that makes Coco Loko, heard about a "chocolate-snorting trend" in Europe, tried it and eventually worked to create his company's own blend.
According to him, for 30 minutes to an hour you can get an "almost like an energy-drink feeling, like you're euphoric but also motivated to get things done." The company website includes the following claims: Endorphin rush: When endorphins are released it triggers a positive feeling of wellbeing in your body similar to morphine.
For example, the feeling after a long workout known as "runner's high" where you feel euphoric, positive and exhilarated.
Serotonin rush: A sudden rush of serotonin will produce an elevated mood and a state of euphoria similar to the feeling of ecstasy.
This is the feeling that will make the music sound better and [increase] overall happiness.
Euphoric energy: Raw cacao will give you a steady rush of euphoric energy and motivation that is great for partygoers to dance the night away without a crash.
It is even used by athletes to give them the natural competitive edge.
Calm focus: Raw cacao will give you a sense of calm focus and reduce the chatter and worry in your brain so you are able to accomplish many tasks at once and maximize your time and focus.
It is also known to help with anxiety and to reduce stress.
All of these supposedly from putting chocolate and then some up your nose?
And here you thought better speakers help music sound better, relationships and purpose in life lead to happiness, and training and mental toughness give athletes their competitive edge.
Where exactly are the scientific studies backing any of these claims?
"Marlene: bitte Pünktchen entfernen, ein 't' zuviel"
"Schokolade zu schnupfen ist offensichtlich etwa..."
"Warum in aller Welt solltest du Schokolade (zus..."
"Und, das ist nicht zum Lachen, während du Schok..."
"siehe Reverso Passt auch, denke ich. :-)"
"Duch die Nase ziehen und schnupfen sind Bezeich..."
"Während sie scheinbar keinen Weg gefunden habe..."
"Hier würde ich im Deutschen hinter beides kein ..."
"Ich mag Schokolade und ich mag Sushi, aber beid..."
"Hersheys Küsse in dein Ohr stecken?"
"Warum in aller Welt würdest du Schokolade schnu..."
"War nur ein kleiner Schreibfehler drin :-)"
"Während sie anscheinend keinen Weg gefunden hab..."
"Da bin ich mir nicht so sicher"
"Von Bruce Y. Lee, Forbes, 5. Juli 2017"
"Schokolade zu schnupfen ist offentsichtlich etw..."
I do like chocolate and I do like sushi but, in both cases, in my mouth. Up my nose? Not so sure about that.
serving tin of Coco Loko, a cacao-based powder (along with gingko biloba, taurine and guarana, which are common in energy drinks) that you can snort.
Legal Lean, the company that makes Coco Loko, heard about a "chocolate-snorting trend" in Europe, tried it and eventually worked to create his company's own blend.
Endorphin rush: When endorphins are released it triggers a positive feeling of wellbeing in your body similar to morphine. For example, the feeling after a long workout known as "runner's high" where you feel euphoric, positive and exhilarated.
sound better and [increase] overall happiness.
Euphoric energy: Raw cacao will give you a steady rush of euphoric energy and motivation that is great for partygoers to dance the night away without a crash. It is even used by athletes to give them the natural competitive edge.
Calm focus: Raw cacao will give you a sense of calm focus and reduce the chatter and worry in your brain so you are able to accomplish many tasks at once and maximize your time and focus. It is also known to help with anxiety and to reduce stress.
lead to happiness, and training and mental toughness give athletes their competitive edge. Where exactly are the scientific studies backing any of these claims? Who nose.


KUALA LUMPUR, Dec 6 (Bernama) -- The ringgit closed lower against the US dollar for the second consecutive day today on the lack of a positive boost, said a dealer.
At 6 pm, the ringgit traded at 4.1620/1670 versus the greenback fom Wednesday's close of 4.1525/1555.
A dealer said market sentiment was weak in most regional currencies amid an inversion of US bond yield, which signalled a possible economic downturn.
"Apart from that, market participants also are looking towards the outcome of the Organisation of the Petroleum Exporting Countries (OPEC) meeting in Austria later today," he said.
The ringgit traded mostly lower against other major currencies.
It was almost flat against the Singapore dollar at 3.0362/0409 against 3.0366/0392 on Wednesday but fell against the yen to 3.6953/6007 from 3.6754/6784.
The domestic unit depreciated against the euro to 4.7197/7262 from 4.7077/7115 and weakened versus the pound to 5.3016/3096 from 5.2832/2891 yesterday.


n. 1. (Zool.) In the most general sense, any one of the Quadrumana, including apes, baboons, and lemurs.
This is the monkey's own giving out; she is persuaded I will marry her.
b - A half-decked boat used on the River Thames.
(Naut.) a small single block strapped with a swivel.
(Bot.) a plant of the genus Mimulus; - so called from the appearance of its gaping corolla.
(Naut.) a light gaff attached to the topmast for the better display of signals at sea.
a short closely fitting jacket, worn by sailors.
(Naut.) a second and lighter rail raised about six inches above the quarter rail of a ship.
v. t. & i. 1. To act or treat as a monkey does; to ape; to act in a grotesque or meddlesome manner.
to handle in a meddlesome manner.
Verb 1. monkey - play around with or alter or falsify, usually secretively or dishonestly; "Someone tampered with the documents on my desk"; "The reporter fiddle with the facts"
2. monkey - do random, unplanned work or activities or spend time idly; "The old lady is usually mucking about in her little house"


text:
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Move over, Oscars….here Come The Pawscars™!
LOS ANGELES, CALIFORNIA, February 25, 2016 — Today, American Humane is revealing the REAL winners of this year's top acting awards with the PAWSCARS™, honoring the top animal stars in…
Congressional briefing outlines advances and pioneering research on cancer in children and animals
February 24, 2016 — For more than 1.5 million Americans diagnosed with cancer each year and 40,000 U.S. children now in treatment for the deadly disease, some of the best…
Two new Chicken Soup for the Soul books showcase the zany antics and enduring love of dogs and cats
WASHINGTON, DISTRICT OF COLUMBIA, February 9, 2016 — Dogs and cats do so much to save and even improve our lives, and that's exactly what the newest entries in the…
Movie alert for animal lovers: American Humane gives high marks to Universal Pictures' "Hail, Caesar!"
WASHINGTON, DISTRICT OF COLUMBIA, February 4, 2016 — Few things excite moviegoers more than a new film written and directed by the Coen brothers, and the latest comedy from the…
Major new national resource launched for America's active-duty military, veterans, their families, and military animals
PALM BEACH, FLORIDA, February 3, 2016 — American Humane, the country's first national humane organization and one that is celebrating a century of working with the U.S. military in 2016,…
Packing party prepares provisions for K-9 Battle Buddy teams
WASHINGTON, DISTRICT OF COLUMBIA, January 29, 2016 — There are hundreds of military dog teams serving on U.S. military bases around the world, but some of these teams only have…
Honor the Hero Veterinarian or Hero Veterinary Technician in your animal's life!
WASHINGTON, DISTRICT OF COLUMBIA, January 28, 2016 — For millions of Americans, animals are not just their best friends, but their heroes. Whether they are lending eyes to the unsighted,…
Calling all dogs! Nominations open today for the 2016 American Humane Hero Dog Awards®
WASHINGTON, DISTRICT OF COLUMBIA, January 27, 2016 — American Humane, the country's first national humane organization, today announced the official start of the 2016 American Humane Hero Dog Awards®, sponsored…
American Humane and philanthropist Lois Pope proudly present full schedule of Palm Beach season events
PALM BEACH, FLORIDA, January 19, 2016 — Heroes of every kind will be celebrated all season long in Palm Beach by American Humane and local philanthropist Lois Pope, one of…
American Humane's Red Star® Rescue team involved in dramatic rescue effort underway in animal cruelty case involving 46 cats seized from Memphis-area home
BARTLETT, TENNESSEE, January 12, 2016 — American Humane's renowned Red Star® Rescue team, accompanied by one of their 50-foot Lois Pope Red Star Rescue Vehicles and two F-350 trucks, has…
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Down's syndrome (also called Down syndrome) is a genetic condition known as trisomy 21, where a person inherits an extra copy of one chromosome. People with the syndrome have three copies of chromosome 21 rather than the usual two.
This additional genetic material changes the finely tuned balance of the body and results in characteristic physical and intellectual features.
It's not known why this abnormality occurs, but it affects around one in 1,000 babies born in the UK - about 750 babies a year - and is the most common inherited cause of learning disability.
Regular trisomy 21 - all the cells have an extra chromosome 21. Around 94 per cent of people with Down's syndrome have this type.
Translocation - the extra chromosome 21 material is attached to another chromosome and one of the parents may carry the translocated chromosome without any signs of the condition themselves. This accounts for around 4 per cent of cases.
Mosaic - only some of the cells have an extra chromosome 21. Around two per cent of people with Down's have this type, which tends to result in milder features.
People with Down's syndrome tend to look different – they typically have a flat facial profile and eyes that slant upwards. Other facial features include smaller ears, a flat back of the head and protruding tongue.
People with the syndrome also tend to be shorter than average with poor muscle tone and have short, broad hands with a single crease across the palm.
Almost half of people affected have heart defects, some of which can be treated, some of which can't. Many also have gut problems, which can make eating difficult and increases the risk of problems such as constipation, and disorders of the thyroid gland.
Other physical problems include cataracts, hearing and sight problems, and a susceptibility to infections. Later in life there's also an increased risk of leukaemia and Alzheimer's dementia.
People with Down's syndrome have varying degrees of learning disability, which may range from moderate to severe. Autistic spectrum disorders are also more common.
There's no cure, but treatment of any accompanying health problems and support for learning difficulties allows many people with the syndrome to lead relatively normal and semi-independent lives. Others, however, need full-time care. Many people with the condition live well into adulthood, with an average life expectancy of around 60 years.
Physiotherapy, speech therapy and special educational programmes have an important role to play, while specific medical conditions associated with the syndrome are treated as appropriate.
20 years - 1 in 1,500 25 years - 1 in 1,300 30 years - 1 in 900 35 years - 1 in 350 40 years - 1 in 100 45 years - 1 in 30Please note these are approximate figures and that maternal age is taken by convention to refer to age at estimated or actual delivery of the baby. Also note that the risk of conceiving a baby with Down's syndrome is higher than the above figures but many are lost early in the pregnancy through spontaneous miscarriage.
The chance of having a baby with Down's syndrome increases with the age of the mother, especially over the age of 35. But as the majority of women have their children when they're younger, most Down's babies are born to younger parents.
Having a child with Down's syndrome is more likely if a woman already has a child with the syndrome, or if a close family member has had an affected child. But most cases are sporadic (the parents aren't affected) and the risk of recurrence in further pregnancies is very small.
Down's syndrome affects people from all ethnic groups and cultures.
It's not possible to prevent the genetic abnormality that causes Down's syndrome, but in recent years it has become possible to identify more accurately during pregnancy the likely risk that the baby is affected.
The NHS Fetal Anomaly Screening Programme (FASP) uses a combination of tests to screen for Down's syndrome. These tests cannot give a definite answer as to whether the baby does or does not have the syndrome but provide a reasonable estimate of the level of risk (such as a 1 in 100 chance or a 1 in 24,000 chance).
The tests used depend on the stage of pregnancy at which a woman requests testing, but they all combine blood tests with a special type of ultrasound, known as a nuchal translucency scan. This type of ultrasound is similar to a normal ultrasound scan, but specifically measures the space between the spine and the nape of the baby's neck. All babies tend to collect fluid behind the neck but those with Down's syndrome tend to have more fluid in the neck than most. Measuring the thickness of fluid or "nuchal translucency" helps to estimate the risk of Down's syndrome.
Ideally, combined testing is done during the first trimester or third of pregnancy. Blood is taken from the mother at between 10-14 weeks to check for levels of certain proteins and hormones. At this time the hormones that are measured are usually free beta-hCG and placenta associated plasma protein A (PAPP-A). The levels of these chemicals, combined with the Nuchal Translucency measurement, are used to calculate the risk of the baby having Down's Syndrome.
If combined testing is done a bit later, during the second trimester of pregnancy, the hormones that are measured are usually alpha-fetoprotein (AFP), free beta-human chorionic gonadotrophin (beta-hCG) and uE3.
Those women who are found to be at a higher risk of having a baby with Down's syndrome are offered the opportunity of further tests, known as prenatal diagnostic tests, to establish more clearly whether or not their child is affected. The tests offered are amniocentesis or chorionic villus sampling (CVS), which involve taking samples of either the fluid around the unborn baby or some of the cells from the placenta. These provide a definitive answer, but are invasive and carry a 1 per cent risk of triggering a miscarriage.
Screening for Down's syndrome isn't compulsory. Women are offered screening if they wish to know the likelihood of having a baby with Down's syndrome. If they are at high risk, they can then decide whether to have a prenatal diagnostic test to get a definite answer.
If this test confirms Down's syndrome, women can then make an informed decision about continuing with their pregnancy or not.
Some companies have started offering tests called Non Invasive Prenatal Diagnosis (NIPD) tests. These look a pregnant woman's blood to check for genetic material from the baby, which passes across from the placenta into the mother's blood during pregnancy. This genetic material or DNA is called cell free fetal DNA (cffDNA). The tests would have the advantage of not being invasive like amniocentesis or CVS. But research into this is in the very early stages and scientists aren't sure how reliable these tests are, so they are not yet available on the NHS.


We offer a few differnt children's aprons. The School aprons are an ideal choice to use in lessons such as Science,Art, Craft, Technology and cookery.The school aprons are available in two size - Junior, and Senior.
Al the children's aprons can be personalised using either printing or embroidery.
We offer a wide range of Children's Aprons, all of which can be personalised with your company or school logo, using embroidery or printing.
Our most popular school apron in the Children's Craft and Technology Apron. This apron is available in white, natural and navy blue. It is mainly used for art, science, technology or cookery in schools.
The Craft and Technology apron is also popular for Children's cookery parties. Some customers have had the childrens names embroidered onto each apron, and some have bought special pens which can be used by the children to decorate the aprons.


Our best hiking in late summer! Join one of our experienced guides on a hike from Skansbukta to the Pyramiden.
Our best hiking in late summer!
Join one of our experienced guides on a hike from Skansbukta to the Pyramiden.
Participants meet at 4 pm the day before departure to get to know each other and the guide goes through the chosen route, informs about private and shared equipment, clothing and safety. You will be able to go shopping if you need anything before departure.
All the shared equipment participants must help carrying during the hike. Backpack can be up to 20-25 kilos.
Early morning on the day of departure, the guide will pick up all the participants for transport to our starting point in Skansbukta. This is a peaceful and beautiful place with a lot of interesting history that the guide will share with you before the hike begins.
Included: Transport to/from your accommodation, return boat trip to/from the expedition. Food, tent, sleeping mat, experienced guide with necessary safety equipment.
The price does not include sleeping bag which you can rent for NOK 500.


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Understanding the True Meaning of Civility
This article was based on the TED@BCG Talk, "Is Civility a Sham?" by Teresa Bejan:
Teresa Bejan - Is civility a sham?
Teresa Bejan begins her TED talk by describing her experience writing a book about civility. She wrote the book because she was convinced that civility did not exist. However, in the course of writing the book and studying the long history of civility and religious tolerance, she came to discover that civility is an essential part of a tolerant society. Civility is what allows a society to promise, not only to protect diversity, but also the heated and sometimes even hateful disagreements that that diversity inspires.
Disagreeing Is Offensive
The English philosopher, Thomas Hobbes, pointed out in 1642, that the mere act of disagreement is offensive. When two people disagree, each person thinks that they are right and cannot make sense of the fact that the other person is wrong. It couldn't possibly be that they have just come to different conclusions in good faith. The mere fact of that they are in disagreement is taken implicitly as an insult not only to each person's views but also their intelligence. The disagreements at stake are the ones that we somehow consider to be fundamental, whether to our worldviews or to our identities.
These fundamental disagreements are precisely the ones that tolerant societies like the United States propose to tolerate. Perhaps this explains why, historically, at least, tolerant societies haven't been the happy communities of difference that you sometimes hear about. Instead, they tend to be places where people have to pretend they are indifferent despite their mutual contempt.
Modern Civility
When most people talk about civility today, they seem to have something else in mind. Civility is the virtue that makes it possible to tolerate disagreement so that we can actually engage with our opponents. Talking about civility seems to be a strategy of disengagement. It's a little bit like threatening to take your ball and go home when the game isn't going your way.
Today's big civility talkers tend to be vague and fuzzy when it comes to what they think civility actually entails. We're told that civility is simply a synonym for respect, for good manners, for politeness. At the same time, it's clear that accusing someone of incivility is much worse than calling them impolite. To call someone uncivil is a way of communicating that they are not worth engaging with at all.
Despite its confusing nature, civility is precious because it's the virtue that makes fundamental disagreement not only possible but even sometimes productive. As important as it is, civility is also extremely difficult.
A Crisis of Civility
Politicians and intellectuals have been warning us for decades that the United States is facing a crisis of civility, and they tend to blame it on technological developments like cable TV, talk radio, and social media. Yet, any historian will tell you that there was never a golden age of disagreement. In her book, Bejan argues that the first modern crisis of civility began about 500 years ago, when theology professor, Martin Luther, called the Pope the Antichrist, and thus inadvertently launched the Protestant Reformation.
Even back then, as the Catholics and Protestants traded insults, you could call out your opponent for going low, and then take advantage of the moral high ground to go as low or lower. This is because calling for civility sets up the speaker as a model of decorum while implicitly, subtly stigmatizing anyone with the temerity to disagree as uncivil.
Civility talk became an effective way for members of the religious establishment to silence, suppress, and exclude dissenters outside of the established church. Anglican ministers could lecture atheists on the offensiveness of their discourse. Everyone could complain about the Quakers for refusing to doff and don their hats or their "uncouth" practice of shaking hands. These accusations of incivility soon became pretexts for persecution.
Civility Talk Versus the Virtue of Civility
These trends have continued, and modern civility talk is making the problem worse. It's saving us the trouble of speaking to each other, allowing us to speak past each other or at each other while signaling our superior virtue and letting the audience know which side we're on.
It's presumptuous to assume that just because civility talk is fruitless, civility doesn't exist. The same early crisis of civility that launched the Reformation also gave birth to tolerant societies, places like Rhode Island, Pennsylvania, and eventually the United States. These places aspired to protect disagreement as well as diversity, and what made that possible was the virtue of civility. What made disagreement tolerable, what made it possible for us to share a life, even when we didn't share a faith, was the virtue of civility.
Civility is not the same as being respectful or polite because we need civility precisely when we're dealing with those people that we find the most difficult, or maybe even impossible, to respect. Similarly, being civil is not the same as being nice, because being nice means not telling people what you really think about them or their wrong views. Being civil means speaking your mind, but to your opponent's face, not behind their back. Civility allows us to fundamentally disagree without denying or destroying the possibility of a common life tomorrow with the people that we think are standing in our way today.
It's Not Normal! - The Coaching Room