text:
\section{Introduction.}
The Schramm-Loewner evolutions (SLEs) are random growth processes that
generate conformally invariant curves in two dimensions (2D). SLEs yield
probability measures on the continuum limit of non-crossing interfaces in 2D statistical lattice models at criticality with conformal
invariance. SLEs has been proved successful to a more complete, and in some
case mathematically rigorous, description of fractal curves in critical
percolation \cite{Smirnov}, loop erased walk \cite{Lawler}, level lines of
height models \cite{Schramm} and domain boundaries at phase transitions (see e.g.
\cite{Bernard_review} and references therein).  The SLEs consider directly the
geometrical characterization of non-local objects in 2D critical systems and
complement the powerful tools provided by the conformal field theories (CFTs)
techniques.

On the other hand, CFTs focus on correlation functions of local operators
which are the scaling limit of lattice variables.  A first family of CFTs,
the minimal CFTs, calculates these correlation functions by studying the
infinite constraints imposed by the conformal invariance in 2D systems.
The Hilbert space of these theories is constructed from the highest weight
representations of the Virasoro algebra formed by  the generators of the
conformal symmetry.

The relation between SLEs and CFTs has been worked out in
\cite{Bernard_connection1,Bernard_connection2,Bernard_connection3}.  In this respect, an
important role is played by the boundary conditions changing operators (b.c.c.)
which generate the boundary conditions from which the curve grows \cite{Cardy_bcft1,Cardy_bcft2}.
The key property of these operators is to satisfy particular relations under the action of the
conformal symmetry generators which lead to second order
differential equations for their correlation functions.  The link
between the SLEs and CFTs is derived by comparing these second order equations
to the Fokker-Plank equations coming from the Brownian motion driving the SLE.

The SLEs/CFTs connection is well established in the case of minimal CFTs,
which have a central charge $c\leq 1 $. However, many critical models in
condensed matter and statistical physics posses, in addition to the
conformal invariance, symmetries in some internal degree of freedom, such as
the $SU(2)$ spin-rotational symmetry in critical quantum spin chains \cite{Affleck} or
the $Z_N$ symmetry in spin lattice models \cite{Zamo1,Zama_lat2}. In the continuum limit, these
additional symmetries are enhanced by the presence of chiral currents which
form, together with the energy-momentum tensor, more structured algebras which
present the Virasoro one as a sub-algebra.  The space of local fields occurring
in a CFT with additional symmetry, the non-minimal CFT,  corresponds to the representations of the
associated chiral algebra. These theories have in general a central charge
$c\geq 1$.

The  connection between SLEs and non-minimal CFTs  has been first addressed in \cite{Rasmussen1,Rasmussen2}, where the relation between stochastic evolutions  and superconformal field theory was investigated. More recently,  the connection between SLE and Wess-Zumino-Witten models, i.e. CFTs with additional Lie-group symmetries,  has been studied  by very different approaches    \cite{Rasmussen3, Ludwig}. In particular,  it has been shown   in \cite{Ludwig}  that a
consistent SLE approach is possible provided that an additional stochastic
motion in the internal symmetry group space in introduced. For an $SU(2)$ symmetry, for
instance, the SLE describes a process which carries a fluctuating additional spin
$1/2$ degree of freedom.

Inspired by these results, we discuss in this paper a possible SLE approach to
CFTs with additional $Z_N$ symmetry. The conserved currents associated to the
extra symmetry, called parafermions, have fractional spin.  We consider the
first of such parafermionic theories with central charge $c=2(N-1)/(N+2)$
\cite{Zamo1}.  The $Z(2)$ and $Z(3)$ models coincide with some particular
minimal CFT where the domain wall boundary conditions and the associated
b.c.c.  operator are known. We exploit these pieces of information to
identify the possible b.c.c. operators for general $Z_N$ symmetry.
We show that, under the action of the parafermionic conserved currents, these
operators satisfy algebraic relations similar to the ones found in the case of
minimal models. From these CFT predictions and in analogy with the approach
proposed in \cite{Ludwig}, we introduce an additional stochastic motion driven
by the action of the parafermionic currents.

The strong motivation to study these theories lies on the fact that
they describe the continuum limit of critical lattice models, the $N$-states
generalization of the Ising model which reduce to the Ising and three
state Potts model for $N=2$ and $N=3$.
   This allows for  the identification of the  SLE interfaces on the lattice and opens the possibility of  a direct numerical verification.

The paper is organized as follows. We review the connection between SLE and
minimal CFTs in Section \ref{sle_review}. Section \ref{slelie} provides the
extension to CFTs with extra Lie symmetries. In Section \ref{sle/para}, after
an introduction to the operator content of the parafermionic theories, we
define a stochastic motion in the $Z_N$ internal space. Further, by studying
the parafermionic algebra representations, we derive some algebraic relations
related to the b.c.c. operators. In Section \ref{lattice
  model} we shall discuss the definition of interfaces on the lattice.
We conclude the paper in Section \ref{conclusion}.

\section{Stochastic Loewner evolutions and conformal field theories.}
\label{sle_review}
\subsection{The SLE$_{\kappa}$/Minimal models connection.}

\subsubsection{Chordal SLE: definition.}
Throughout this paper, we consider chordal SLE which describes random curves
joining two boundary points of a connected planar domain.  For a comprehensive
and detailed introduction to SLE, see e.g.
\cite{Walter,Cardy_review,Bernard_review}. The definition of SLE is most
conveniently given in the upper half complex plane $\mathbb{H}$: it describes
a fluctuating self-avoiding curve $\gamma_{t}$ which emanates from the origin
($z=0$) and progresses in a properly chosen time t. If $\gamma_{t}$ is a
simple curve, this evolution is defined via the conformal map $g_t(z)$ from
the domain $\mathbb{H}_t=\mathbb{H}/\gamma_{]0,t]}$, i.e. the upper half plane
from which the curve is removed, to $\mathbb{H}$. In the more general case of
non-simple curves, the function $g_t(z)$ produce conformal maps from
$\mathbb{H}_t=\mathbb{H}/K_t$ to $\mathbb{H}$ where $K_t$ is the SLE hull at
time $t$.  The SLE map $g_t(z)$ is uniquely determined by fixing three real
parameters.  A conventional normalization (the so called hydrodynamic
normalization) is such that $g_t(z)=z+2t/z+\cdots$ near $z=\infty$. With this
choice the time $t$ corresponds to the upper half-plane capacity $c_{K_t}$ of
the hull $K_t$, $c_{K_t}=2t$. The function $c_{K_t}$ is a positive quantity
and satisfies the additive law $c_{K_{t}\cup
  g_{t}^{-1}(K_{t'})}=c_{K_{t}}+c_{K_{t'}}$. The SLE map $g_t(z)$ is a
solution of the Loewner equation:
\begin{equation}
  \frac{d}{d t} g_t (z)=\frac{2}{g_t(z)-\xi_t} \quad g_{t=0}(z)=z,
\label{Sle_definition}
\end{equation}
where $\xi_t$ is a real valued process, $\xi_t \in \mathbb{R}$, which drives the evolution
of the curve. For a system which satisfies the Markovian and conformal
invariance properties, together with the left-right symmetry, the process
$\xi_t$ is shown \cite{Schramm_2} to be proportional to a Brownian motion: {\bf E}$[\xi_t]=0$
and {\bf E}$[\xi_t \xi_s]= \kappa\,\mbox{min}(s,t)$. In the following, we use the
symbol {\bf E}[...] for the stochastic average over the Brownian
motion.

\subsubsection{Martingales in SLE processes.}
In
\cite{Bernard_connection1,Bernard_connection2,Bernard_connection3,Bernard_connection4}
it was understood how to associate CFT states with the growing curves of the
Loewner process.  To reveal the SLE/CFT connection, the first step is to
define stochastic process such that conditioned correlation function are
martingales for this process.  The main idea is to arrange the statistical sum
by first summing over all the configurations $\mathcal{C}_{\gamma_{t}}$
presenting a trace $\gamma_{t}$ of the interface with fixed shape and capacity
$t/2$ and then summing aver all the possible shapes of the trace. The first sum
defines a conditioned correlation function while the second sum can be seen as
the stochastic mean of this correlation function. To make it more concrete,
consider an observable $\mathcal{O}$ of a lattice model defined on the domain
$\mathbb{H}$. The statistical sum $\prec O\succ_{\mathbb{H}}$ can be written
as:
\begin{equation}
  \prec \mathcal{O} \succ_{\mathbb{H}}=
\mbox{{\bf E}}[\prec \mathcal{O} \succ _{\gamma_{t}}]
=\sum_{\gamma_{t}}P[\mathcal{C}_{\gamma_{t}}]\prec
  \mathcal{O} \succ _{\gamma_{t}},
\label{martingaletrick}
\end{equation}
where $\prec 0 \succ_{\gamma_{t}}$ is the statistical average conditioned to
the presence of the trace $\gamma_{t}$ and $P[\mathcal{C}_{\gamma_{t}}]$ is
the probability of its occurrence.  The correlator $\prec
\mathcal{O}\succ_{\mathbb{H}}$ does not depend on the choice of $t$: the
stochastic mean of the correlator $\prec 0 \succ_{\gamma_{t}}$ is thus time
independent and it is a martingale.

\subsubsection{Martingales and conformal correlators.}
The relation (\ref{martingaletrick}) becomes extremely useful at the critical
point where the model is expected to be described in the continuum limit by a
conformal field theory. In this case, the
operator $\mathcal{O}$ corresponds generally to a product of, say, $l$ primary
fields $\phi_i(z_i)$, $\mathcal{O}(\{z_i\})=\prod_{i=1}^{l} \phi_i(z_i)$ at
positions $z_i$. The image $^{f}\phi$ of $\phi$ under a conformal transformation
$f(z)$ is
\begin{equation}
^{f}\phi_i=[\partial_z f(z)]^{\Delta_{i}}\phi_i(f(z_i)),
\label{primary_cond}
\end{equation}
where $\Delta_{i}$ is the conformal dimension of $\phi_{i}$.  The statistical
expectation values can then be expressed in terms of CFT correlation functions
$\mathcal{F}(\{z_i \})_{\mathbb{H}_{t}}$:
\begin{equation}
  \prec \mathcal{O} \succ_{\gamma_{t}}
  \rightarrow \mathcal{F}(\{z_i\})_{\mathbb{H}_{t}}=\frac{<\mathcal{O}(\{z_i\}) \psi
    (\infty)\psi(z_t)>_{\mathbb{H}_{t}}}{< \psi
    (\infty)\psi(z_t) >_{\mathbb{H}_{t}}}
\label{martingale_cft}
\end{equation}
where $<\dots>_{\mathbb{H}_{t}}$ indicates the conformal correlation function
computed in the domain $\mathbb{H}_{t}=\mathbb{H}/\gamma_t$, i.e. the upper
half plane with the trace $\gamma_t$ removed.  The fields $\psi(z_t)$ and
$\psi(\infty)$, inserted respectively at the tip $z_t$ of $\gamma_{t}$ and at
the infinity, are the b.c.c. operators implementing the boundary
conditions at which the interface anchor.

Using the conformal invariance, the correlation functions
$\mathcal{F}(\{z_i\})_{\mathbb{H}_{t}}$ in the domain wall ${\mathbb{H}_{t}}$
can be expressed as the correlation functions in the upper-half plane
${\mathbb{H}}$:
\begin{equation}
 \mathcal{F}(\{z_i\})_{\mathbb{H}_{t}}=\frac{<^{g_t}\mathcal{O} (\{z_i \})\, \psi
    (\infty)\,\psi(\xi_t)>_{\mathbb{H}}}{< \psi
    (\infty)\,\psi(\xi_t) >_{\mathbb{H}}}.
\label{martingale_cft_up}
\end{equation}
Note that the Jacobians coming from the conformal transformation on the $\psi$
fields cancel between the numerator and the denominator in the above expression.

We are now in the position to understand the SLE/CFT connection.  Under the
SLE, the trace $\gamma_t$ evolves and the iterated sequence of infinitesimal
conformal mappings $g_t(z)$ satisfying Eq.(\ref{Sle_definition}) leads to a
Langevin dynamics for the conformal correlator
$\mathcal{F}(\{z_i\})_{\mathbb{H}_{t}}$.  Imagine evolving the trace for a
time $t$ and then, for an infinitesimal time $dt$. Using
Eq.(\ref{Sle_definition}), the variation $d (^{g_t}\phi_i(z_i))=^{g_{t+d
    t}}\phi_i(z_i)\, -\, ^{g_{t}}\phi(z_i)$ is given by:
\begin{equation}
  d (^{g_t}\phi_i(z_i))=\frac{1}{2\pi i}\oint_{z_i} d g_t(w) T(w)\, ^{g_t}\phi_i(z_i)=
2 dt \left(-\frac{\Delta_i}{(g_t(z_i)-\xi_t)^2}+\frac{\partial_{g_t(z_i)}}{g_t(z_i)-\xi_t}
\right)\, ^{g_t}\phi_i(z_i)
\label{spect_dynamic}
\end{equation}
where $T(z)$ is the energy-momentum tensor. The variation for the $\psi$'s are
given by the Ito differential:
\begin{equation}
d (\psi(\xi_t)) =  \partial_{\xi_t}\psi(\xi_t) d \xi_t +\frac{\kappa}{2}
\partial^2_{\xi_t} \psi (\xi_t) dt.
\label{psi_dynamic}
\end{equation}
Using the Eqs.(\ref{psi_dynamic})-(\ref{spect_dynamic}) in
Eq.(\ref{martingale_cft_up}) and averaging over all the realization of
$\gamma_t$, one obtains the diffusion equation:
\begin{equation}
  \partial_{t} \mbox{{\bf E}}[\mathcal{F}(\{z_i\})_{\mathbb{H}_{t}}]=\left(2\sum_i\left[-\frac{\Delta_i}{(g_t(z_i)-\xi_t)^2}+\frac{\partial_{g_t(z_i)}}{g_t(z_i)-\xi_t}\right]+\frac{\kappa}{2}\partial^2_{\xi_t}
 \right)\mbox{{\bf E}}[\mathcal{F}(\{z_i\})_{\mathbb{H}_{t}}].
\label{diff_equation}
\end{equation}
As previously said, the correlation function
$\mathcal{F}(\{z_i\})_{\mathbb{H}_{t}}$ is a martingale of the SLE process and
satisfies therefore the following differential equation
\begin{equation}
\left(2\sum_i\left[-\frac{\Delta_i}{(g_t(z_i)-\xi_t)^2}+\frac{\partial_{g_t(z_i)}}{g_t(z_i)-\xi_t}\right]+\frac{\kappa}{2}\partial^2_{\xi_t}
 \right)\mathcal{F}(\{z_i\})_{\mathbb{H}_{t}}=0.
\label{diff_eq_F}
\end{equation}

\subsubsection{Operator formalism and minimal models.}
In order to show the consequences of the above relation, it is convenient to
introduce the operators $L_n$ which are the modes of the energy-momentum
tensor $T(z)$:
\begin{equation}
L_n \phi(z)=\frac{1}{2\pi i} \oint_{z} d w\, w^{n+1} T(w) \phi(z).
\end{equation}
The $L_n$ are the generators of the conformal transformation in the CFT
Hilbert space. A primary operator $\phi$, satisfying Eq. (\ref{primary_cond}),
is annihilated by the positive modes of
$L_n$,  $L_n \phi=0$ for $n>0$,  and the eigenvalue of the zero mode is its
conformal weight, $L_0 \phi =\Delta \phi$.
The $L_{n}$ satisfy the Virasoro
algebra
\begin{equation}
[L_n,L_m]=(n-m)L_{n+m} + \frac{c}{12}n^2(n-1)\delta_{n+m}.
\label{Virasoro}
\end{equation}
All local fields of the theory can be obtained by applying the $L_{-n}$,
$n\geq 1$, to a primary fields $\phi$. Each set of states of the form
$L_{k_1}\cdots L_{k_m} \phi$ forms a conformal family $[\phi]$ and corresponds
to the representation of the highest weight of the Virasoro algebra, also
called Verma module, the primary field $\phi$ corresponding to the highest
vector.  The descendant state $L_{-k_1}\cdots L_{-k_m} \phi$, $\sum k_m= n$, is
said to be the $n-$th level of the module of $\phi$ and its conformal weight is
$\Delta+n$.

For $c\leq 1$, the unitary representations of the Virasoro algebra has central
charge \cite{Friedan}:
\begin{equation}
c=1-\frac{6}{m(m+1)}\quad m=2,3,\cdots
\label{c_minimal}
\end{equation}
and correspond to the simplest family of conformal theories, called minimal
models $M_m$. The minimal model $M_m$ is characterized by a finite number of
Verma modules $[\phi_{r,s}]$ with conformal weight:
\begin{equation}
\Delta_{r,s}=\frac{[(m+1)r-m s]^2 -1}{4 m(m+1)}, \quad r=1,2,\cdots m-1, \quad
s=1,2,\cdots r.
\label{kac_minimal}
\end{equation}

\subsubsection{Null vectors in CFT and SLE. }
In CFT, the differential equations satisfied by the correlation function can
be derived from the structure of the representation modules and, in
particular, from  the relations between the descendant states.
One can show that $\mathcal{F}(\{z_i\})_{\mathbb{H}_{t}}$ is a solution of
(\ref{diff_eq_F} )
if the b.c.c. operator $\psi$ obeys the
condition:
\begin{equation}
(L_{-2} -\frac{\kappa}{4}L_{-1}^2) \psi=0,
\label{sle_nul_vect}
\end{equation}
The equation Eq.(\ref{sle_nul_vect}) means that the descendant states
$L_{-1}^2 \psi$ and $L_{-2} \psi$ are linearly dependent. The operator $\psi$
transforms then as a degenerate representation of the Virasoro algebra which
has a null state at level 2.  The operator $\psi$ can thus be identified with
the operators $\phi_{1,2}$ or $\phi_{2,1}$ of the minimal models $M_m$ which
are shown to satisfy:
\begin{eqnarray}
(L_{-2} - \frac{3}{2(2 \Delta_{1,2}+1)} L_{-1}^2) \phi_{1,2}&=&0,
\label{vir_nul_vect_1} \\
(L_{-2} - \frac{3}{2(2 \Delta_{2,1}+1)} L_{-1}^2)\phi_{2,1}&=&0.
\label{vir_nul_vect_2}
\end{eqnarray}
Comparing Eq.(\ref{vir_nul_vect_1}) and Eq.(\ref{vir_nul_vect_2}) to Eq.(\ref{sle_nul_vect}) and using
Eq.(\ref{c_minimal}) and Eq.(\ref{kac_minimal}), one can finally make explicit
that the SLE/CFT connection: an SLE process with parameter $\kappa$ describes
interfaces in CFT with central charge:
\begin{equation}
c_{\kappa}=\frac{(6-\kappa)(3\kappa-8)}{2 \kappa}
\label{sle_cft_minimal},
\end{equation}
and  the boundary conformal operator $\psi$ has scaling dimension:
\begin{equation}
\Delta_{\kappa}=\frac{(6-\kappa)}{2 \kappa}
\label{sle_cft_bcc},
\end{equation}
This derivation shows that martingales for SLE
processes are closely related with the existence of null vectors in the the
appropriate Verma module.

\section{SLEs for CFTs with extra Lie group symmetries.}
\label{slelie}

As mentioned above, the operator content of the minimal models $M_m$, which
have central charge $c\leq 1$, corresponds to the space of representations of
the Virasoro algebra (\ref{Virasoro}). In this case, the conformal symmetry
determines completely the spectrum of the theory and the differential
equations satisfied by the correlation functions.

The applications of the SLEs defined in the previous section are limited to
CFTs with central charge $c\leq 1$. This can be directly understood from the
fact that, for a CFT with extended symmetry, not all the local fields can be
obtained by the application of the $L_n$. In particular, one expects at each
level additional states and the  linear relations
 of the type (\ref{vir_nul_vect_1})-(\ref{vir_nul_vect_2}) will include additional terms.

\subsection{SLEs and CFTs with Lie group symmetries.}
The $SU(2)$ WZW models are among the most important CFTs with additional
symmetries.  These CFTs posses an internal $SU(2)$ continuous symmetry which,
together with the conformal one, is realized by the set of chiral currents
$J^{a}(z)$, $a=+,-,0$, with scaling dimension $\Delta_{J}=1$.  The current
algebra, derived from the current-current operator product expansion (OPE),
takes the form:
\begin{equation}
[J^{a}_{n},J^{b}_{m}]=f^{a,b}_{c} J^{c}_{n+m}+N\, n \,\delta_{n+m},
\label{kac_moody}
\end{equation}
where $J^{a}_n$ are the current modes, $J^{a}_{n}=1/(2\pi i)\oint_{z} d w
\,w^n \,J^{a}(w)$, the $f^{a,b}_{c}$ are the structure constants of the
$su(2)$ Lie algebra and $N$ is the level of the algebra. The representations
of the Kac-Moody algebra , leads to a family of CFTs, denoted
$SU(2)_{N}$, with central charge $c_N=3N/(N+2)$.
The local fields $\phi_{j}(z)$ of
the theory transform under the action of the chiral currents in the
representation space of dimension $j(j+1)$ of $SU(2)$. Analogously to the
Virasoro case, a WZW primary field is defined by:
\begin{equation}
J^{a}_{n}\phi_{j}(z)=0 \quad \mbox{for}\,\, n>0; \quad J^{a}_{0}\phi_{j}(z)=t^{a}_{j}\phi_{j}(z),
\end{equation}
where the $t^{a}_{j}$ are the $SU(2)$ generator matrices in the $j$
representation.
The conformal weight $\Delta_{j}$ of a primary field  $\phi_{j}(z)$ is
\begin{equation}
L_0 \phi_{j}(z)=\Delta_{j}\phi_{j}(z)= \frac{j(j+1)}{N+2}\phi_{j}(z) .
\label{dim_wzw}
\end{equation}
Under an infinitesimal conformal $d g(z)$ transformation together with an
infinitesimal $SU(2)$ gauge transformation $d \theta^{a}(z)$, the primary
fields $\phi_{j}(z)$ changes as:
\begin{equation}
d (\phi_{j})(z)=\frac{1}{2\pi i}\oint_{z} d\,w\,d g (w) T(w)\, \phi_{j}(z)+ \frac{1}{2\pi i}\oint_{z} d\,w\,d \theta^{a} (w) J^{a}(w)\, \phi_{j}(z)
\label{wzw_varia}
\end{equation}

In \cite{Ludwig}, the SLE approach has been extended to these theories. It
proposes  to describe this
model by a composition of two independent Brownian motions,
one in the physical space, defined as in Eq.(\ref{Sle_definition}), and one in the
internal $SU(2)$ spin degrees of freedom:
\begin{equation}
d \theta^{a}_t(z)=\frac{d \theta^{a}_t}{g_t(z)-\xi_t}, \quad \mbox{{\bf
    E}}[\theta^{a}_t\theta^{b}_s]=\tau\,\delta^{a,b}\,\mbox{min}(s,t),
\label{tau_sle}
\end{equation}
where the variance $\tau$ is an independent parameter of the SLE evolution.
The idea is then to attach to trace $\gamma_t$ a spin $1/2$ degrees of
freedom. The interface then undergoes both a standard SLE evolution in the
physical space and a stochastic $SU(2)$ rotation.

The derivation of the SLE/CFT connection is strictly analogous to the one
shown before. The difference here is that the b.c.c. operator
$\psi(\xi_t)$ in Eq.(\ref{martingale_cft_up}) carries a spin $1/2$,
$\psi\equiv\psi_{1/2}(\xi_t)$ which transforms under the representation
$\phi_{1/2}$. Using Eq.(\ref{Sle_definition}) and Eq. (\ref{tau_sle}) in
Eq.(\ref{wzw_varia}), the Ito formula for $\psi_{1/2}(\xi_t)$, written in terms of
the modes $L_n$ and $J^a_n$, is:
\begin{equation}
d( \psi_{1/2}(\xi_t))= d \xi_t L_{-1}\psi_{1/2}(\xi_t)  + d \theta^{a} J^{a}_{-1} \psi_{1/2}(\xi_t)
+ dt\left( \frac{\kappa}{2}
L_{-1}^2 +\frac{\tau}{2} J^{a}_{-1}J^{a}_{-1}\right)\psi_{1/2}(\xi_t).
\label{wzw_psi_dinamic}
\end{equation}
The operator $\mathcal{O}$ is now a product of WZW primaries and remain fixed
while the trace evolves. This means one has only to consider their variation
(\ref{spect_dynamic}) under the conformal transformation $g_t$ which maps the
domain $\mathbb{H}_t$ in the domain $\mathbb{H}_t$.  The martingale condition
(\ref{sle_nul_vect}) takes the form:
\begin{equation}
\left(L_{-2} -\frac{\kappa}{4}L_{-1}^2-\frac{\tau}{4}J^{a}_{-1}J^{a}_{-1}\right) \psi_{1/2}=0.
\label{sle_nu_vect_wzw}
\end{equation}
The correlation functions in WZW theories satisfy first order differential
equations, called Knizhnik-Zamolodchikov equations, which are derived from a
relation between descendant states at the first level:
\begin{equation}
\left(L_{-1}+\frac{1}{N+2}J^{a}_{-1}t^{a}\right)\phi^{m}_{j}=0.
\label{KZ}
\end{equation}
Applying the operator $L_{+1}$ to Eq.(\ref{sle_nu_vect_wzw}) and using the
commutation relation (\ref{kac_moody}) together with $[L_{n},J^{a}_m]=-m
J^{a}_{n+m}$, one fixes a first relation between $\kappa$, $\tau$ and $N$,
$\tau/(6-\kappa(2\Delta_{1/2}+1))=2/(N+2)$. It was shown \cite{Ludwig} that a second
condition, $\kappa+\tau=4$, can be obtained by demanding the
one-point function of the current to exhibit a simple pole at the tip of the
trace $z_t$. The final result is:
\begin{equation}
\kappa=4\frac{N+2}{N+3}\quad \tau=\frac{4}{N+3}.
\end{equation}

\section{SLEs in $Z(N)$ parafermionic theories.}
\label{sle/para}
We consider the CFTs with extended $Z_N$ symmetries, the so calles $Z(N)$
parafermionic theories. In particular, we focus on the $Z(N)$ parafermionic
theories with central charge $c=2(N-1)/(N+2)$ ($c>1$ for $N>4$). These
theories describe the continuum limit of the $N$-states spin models
interacting via a $Z_N$
invariant nearest-neighbor coupling, at their
self-dual critical points (see section \ref{lattice model}).  For $N=2$ and
$N=3$ one finds the well known Ising and three-state Potts model. The
corresponding CFTs has central charge $c=1/2$ and $c=4/5$ respectively and
coincide with the minimal models $M_3$ and $M_5$: the operator content of
these theories can also be determined by studying the representations of the
Virasoro algebra.  In this sense, the $Z_2$ and $Z_3$ symmetry of these models
are trivially realized.  The CFTs describing the critical point of the Ising
and the three-state Potts are then expected to be described by the SLE in
Eq.(\ref{Sle_definition}). The b.c.c.  conformal operators
implementing the SLE interface boundary conditions have been identified.  We
will show that these results admit a natural extension to general $N$.

\subsection{Parafermionic current algebra.}
We briefly review the $Z(N)$ parafermionic theories with central
charge $c=2(N-1)/(N+2)$.  These theories were introduced and constructed in
\cite{Zamo1,Zamo2}. Here we shall enounce the main results in a slightly
different manner.
The general arguments and the notations used here are strictly analogous to the
ones discussed  in a series of papers \cite{Raoul1,Raoul2,Raoul3,Raoul4}  where a second series of
parafermionic theories was studied.

Extra $Z_N$ group symmetries in two-dimensional conformal field theories are
generally realized by a set of holomorphic currents $\Psi^{k}(z)$
($k=1,\cdots,N-1$), satisfying the following operator product expansion:
\begin{eqnarray}
\Psi^{k}(z)\Psi^{k'}(z') &=&
\frac{\lambda^{k,k'}_{k+k'}}{(z-z')^{\Delta_{k}^{\Psi}
 +\Delta_{k'}^{\Psi}-\Delta_{k+k'}^{\Psi}}} \label{opepara1} \\
 &\times& \left \{\Psi^{k+k'}(z')+0(z-z')\right \},\quad k+k'\neq 0 \nonumber \\
 \Psi^{k}(z)\Psi^{-k}(z') &=& \frac{1}{(z-z')^{2\Delta_{k}^{\Psi}}}
 \left \{1+(z-z')^{2}
 \frac{2\Delta_{k}^{\Psi}}{c}T(z')+\ldots \right \}
 \label{opepara2}
\end{eqnarray}
where $\Delta_{k}^{\Psi}$ is the conformal dimension of the parafermions
$\Psi_{k}(z)$ and $\lambda_{k+k'}^{k,k'}$ are the structure constants of the
algebra. We shall be interested in parafermionic theories in which the
dimensions $\Delta_{k}^{\Psi}$ of the parafermions $\{\Psi^{k}\}$ take the minimal possible values
admitted by the associativity constraint:
\begin{equation}
 \Delta_{k}^{\Psi}=\Delta_{N-k}^{\Psi}=\frac{k(N-k)}{N} \quad k=0,1\cdots N-1.
 \label{chidim}
\end{equation}
Note that the above formula is not symmetric under the exchange $k\to -k$ :
the field $\Psi^{-k}$ in (\ref{opepara2}) is assumed to have
dimension $\Delta_{N-k}^{\Psi}$, in the sense that the indices $k$ referring
to the $Z_N$ charge are always defined modulo $N$. Thus,
\begin{equation}
 \Psi^{N-k}\equiv\Psi^{-k}\equiv(\Psi^{k})^{+},\quad \Delta_{N-k}^{\Psi}
 \equiv\Delta_{-k}^{\Psi}.
\end{equation}
The structure constants $\lambda_{k+k'}^{k,k'}$ and the central
charge
$c$ (of the Virasoro algebra) are given by the expressions:
\begin{eqnarray}
  (\lambda^{k,k'}_{k+k'})^{2} &=&
  \frac{(k+k')!(N-k)!(N-k')!}{k!k'!(N-k-k')!N!} \label{struct} \\
  c &=& \frac{2(N-1)}{N+2}. \label{c}
\end{eqnarray}

\subsection{Representation space: the $Z_N$ sector.}

The CFTs we are considering describes the self-dual (Kramers-Wannier
invariance) critical point of $Z_N$-invariant lattice model \cite{Zamo1}. In addition to
the $Z_N$ symmetry, the theory possesses the dual $\tilde{Z}_N$ invariance,
which is enanched by the conservation of  the antiholomorphic fields $\overline{\Psi}(\overline{z})$. The
Hilbert space of a $Z_N\times \tilde{Z}_N$ invariant theory, splits into
subspaces characterized by the $Z_N \times \tilde{Z}_N$ charges $\{p,q\}$. A
field $\phi_{\{p,q\}}$ in one of these subspaces transform as
$\phi_{\{p,q\}}(z)\to \exp[2i\pi (p m+ q n)/N]\phi_{p,q}(z)$ under global
rotations of angles $2\pi m/N$ and $2\pi n/N $.  The lattice spin operator
$\sigma_k$ (see \ref{lattice model}) and its dual $\mu_k$ are described in the continuum limit
by fields of charge $\{k,0\}$ and $\{0,k\}$ respectively. The
(anti-)holomorphic currents $\Psi^{k}(z)$ ( $\overline{\Psi}(\overline{z})$),
which appear in the OPE of $\sigma_k \mu_k$ ($\sigma_k \mu_k^{+}$), have
charges $\{k,k\}$ ($\{k,-k\}$). Recently, the lattice holomorphic realization
of the parafermionic currents has been discussed in \cite{Riva}.

It is in general
convenient to express the charge ${k,k'}$ as $[q^{*},p^{*}]=\{k+k',k-k'\}$, where now
$q$ and $p$ are defined mod $2N$ and $q^{*}+p^{*}$ even. In this notation, the
currents $\Psi^{k}$ and $\overline{\Psi}(\overline{z})$ have charge $[2k,0]$
and $[0,2k]$ respectively.

In the following we concentrate on the action of the holomorphic field
$\Psi^{k}(z)$, all the results being valid also for
$\overline{\Psi}(\overline{z})$. Thus, without losing any generality, one can
study the structure of the $Z_N$ representations which we denote with
$\Phi^{\pm q^{*}}$. Moreover, we find convenient to use a different convention
$q$ for the $Z_N$ charges, defined by $q^{*}= 2 q \,\mbox{mod} 2 N$. With this
choice, the product $\Psi^{k}(z)\Phi^{\pm q}$ is a field with
charge $k \pm q$.

For $N$ odd, we consider then the representation fields
\begin{equation}
\Phi^{\pm q}(z,\bar{z})\quad q=0,\pm 1,\cdots,\pm (N-1)/2 \quad(N \,\mbox{odd}).
\end{equation}

For $N$ even, it turns out that the modules of the representation
corresponding to $\Phi^{\pm q}$ with $\lfloor N/4 \rfloor <q \leq \lfloor N/2
\rfloor$ are identical to those of $0 \leq q \leq \lfloor N/4 \rfloor$. In
order to recover the right number of $Z_N$ representations one has to consider
half-integer value of $q$:
\begin{equation}
\Phi^{\pm q}(z,\bar{z})\quad q=0,\pm \frac{1}{2},\pm 1,\cdots,\pm \lfloor N/4 \rfloor \quad(N \,\mbox{even}).
\end{equation}
Naturally, the physical meaning of the the half-integer charge is recovered in
the usual notation $q^{*}$.

\subsubsection{Currents modes in $Z_N$ sector.}

The currents $\{\Psi^k\}$  can be decomposed into mode operators
$A^{k}_{\cdots+n}$, whose action is to change the $Z_N$ charge of the representation fields $\Phi^{q}$:
\begin{eqnarray}
\Psi^{k}(z)\Phi^{q}(0)&=& \sum_{n}\frac{1}{(z)^{\Delta_{k}^{\Psi}-\delta_{k}^{k+q}+n}}
  A^{k}_{-\delta^{q}_{k+q}+n}\Phi^{q}(0)
\label{eq3} \\
\delta^{q}_{k}&=&\frac{q^{2}-k^{2}}{N} \mbox{ mod }\, 1.
\label{level_structure}
\end{eqnarray}
The value $\delta^{q}_{k}$ is the first level in the module of
$\Phi^{\pm q}$ corresponding to the $Z_N$ charge $k$: it determines thus the
level structure of the modules induced by the $Z_N$ representation fields.
The values of $\delta^{q}_{k}$ can be easily obtained by considering the
module of the identity whose descendant states are the chiral currents
$\Psi^{k}(z)$. The levels of these operators correspond to their conformal
dimensions $\Delta_{\pm k}^{\Psi}$. Taking into account that, owing to the Abelian
monodromy of the fields $\Psi^{k}(z)$ in the $Z_N$ sector, the level spacing
is equal to 1, one can readily obtain from Eq.(\ref{chidim})
$\delta^{0}_{k}=-k^2/N \,\mbox{mod}\,1$. The level structure of
a generic field $\Phi^{\pm q}(z)$, Eq.(\ref{level_structure}), is then
extracted from the module of the identity by inspecting its corresponding
submodule. Note also that $\delta^{q}_{k+q}=0$ for $k=-2 q$.  This results in the
presence of zero modes $A^{\pm q}_{0}$:
\begin{equation}
 A^{\mp 2q}_{0}\Phi^{\pm q}(0)=h_{q}\Phi^{\mp q}(0). \label{zeroeigen}
\end{equation}
which associate at each field $\Phi^{q}$, with conformal dimension
$\Delta_{q}^{\Phi}$,  the field $\Phi^{-q}$ with
opposite charge and with the same conformal dimension, $\Delta_{q}^{\Phi}=\Delta_{-q}^{\Phi}$.
The eigenvalues $h_q$ defined in (\ref{zeroeigen}) characterize the representations of the
parafermionic algebra together with the conformal dimension the fields $\Phi^{q}$.
As usual, primary fields are defined by
$A^{k}_{-\delta^{q}_{k+q}+n}\Phi^{q} = 0$ for $n>0$.
Each representation module is then characterized by the two primary fields $\Phi^{\pm q}$ which can be obtained
one from the other as in (\ref{zeroeigen}).
An example of the module structure associated to the field $\Phi^{\pm 1}$ and of the action of the currents is given in
Fig.\ref{phi1modulo}  for $N=5$. Using the formula (\ref{level_structure}), we
have $\delta^{\pm 1}_{0}=1/5$ and $\delta^{\pm 1}_{\pm2}=4/5$.

Using the expansion (\ref{eq3}), the action of the modes in each sector can be given in terms of a
contour integral:
\begin{equation}
 A^{k}_{-\delta^{q}_{k+q}+n}\Phi^{q}(0) = \frac{1}{2\pi i}\oint_{C_{0}}
 {\rm d}z \,(z)^{\Delta_{k}^{\Psi}-\delta^{q}_{k+q}+n-1}\Psi^{k}(z)\Phi^{q}(0).
 \label{modes}
\end{equation}
In the $Z_N$ sector, one can simply consider the action of the currents
$\{\Psi^{\pm 1} \}$ because, via the Eq. (\ref{opepara1}), they completely
determine the full algebra of the other currents $\{\Psi^{\pm k}\}$,
$k=2,\cdots, \lfloor N/2 \rfloor$.

The commutation relations of the mode operators can be
deduced from Eq.(\ref{modes}) by using standard techniques in the complex
plane. These relations are given in Appendix \ref{commutation_relation_zn}.

As shown in the Appendix \ref{commutation_relation_zn}, the representation
space contains $\lfloor N/2 \rfloor$+1 primary operators $\Phi^{q}$ with
$q=0,1,\cdots, \frac{N}{2}$ for $N$ odd and with $q=0,\pm \frac{1}{2},\pm
1,\cdots,\pm \lfloor N/4 \rfloor$ for $N$ even.  The conformal dimension of
these operators turn out to be:
\begin{equation}
\Delta_{q}^{\Phi}=\frac{q(N-2 q)}{N(N+2)} \quad \, q=0,
1,\cdots, \frac{N-1}{2} (N\,\mbox{odd}),\quad q=0, \frac{1}{2}, 1,\cdots,  \frac{N}{4}  \quad(N \,\mbox{even}) .
\label{kac_zn}
\end{equation}
As said above, to each primary $\Phi^{q}$ corresponds another primary field
$\Phi^{-q}$ of the same dimension.

In the case $N=2$ and $N=3$ (i.e. Ising and three state Potts model), one has
only two primaries of the parafermionic algebra. For $N=2$ one finds the
identity operator with $\Delta_{0}=0$ and the operator $\Phi^{\pm 1/2}$, with
dimension $\Delta_{\pm 1/2}=1/16$, corresponding to the Ising model spin
operator. Analogously, for $N=3$ one has, together with the identity operator,
the operator $\Phi^{\pm 1}$ with $\Delta_{\pm 1}=1/15$.  As previously said,
the case $N=2$ and $N=3$ are special because they can be identified with the
minimal model $M_3$ and $M_5$. The operators $\Phi^{\pm 1/2}$ and $\Phi^{\pm
  1}$ are identified respectively to the operators $\phi_{1,2}$ and
$\phi_{2,3}$ of the corresponding $M_3$ and $M_5$ Kac tables, see
Eq.(\ref{kac_minimal}).  As we will show more in detail below, the
representation module corresponding to $\Phi^{\pm 1}$ contains one descendant
state, $\varepsilon^{(N=3)}=A^{-1}_{-1/3}\Phi^{+1}=A^{1}_{-1/3}\Phi^{-1}$
singlet under $Z_3$ transformation (i.e. $q=0$). This singlet (energy)
operator has dimension $\Delta_{\varepsilon}=1/15+1/3=2/5$, it is a primary of
the Virasoro algebra and can be identified with the operator
$\varepsilon^{(N=3)}=\phi_{2,1}$ of the $M_5$ Kac table.

\subsection{Representation space: the disorder sector.}
\label{Rsector}
A key observation is that the theory we are considering is actually invariant
under the dihedral group $D_N$ which includes $Z_N$ as a subgroup.  This can
directly be seen from the symmetry of Eq.(\ref{opepara1})-(\ref{struct}) under
the conjugation of the $Z_N$ charge, $q\to N-q$. At the level of the lattice
model (see \ref{lattice model}), this comes from the invariance of the Hamiltonian under the
transformation $\sigma_k(x)\to \sigma_k^{+}(x)$.

The space of representation thus includes the $N-$plet of $Z_2$
operators which we denote as:
\begin{eqnarray}
 \{R_a(z,\bar{z}),\quad a=1,\ldots,N\}.
\end{eqnarray}
In the following, we refer to these operators as disorder operators.

The theory of disorder operators has been fully developed in \cite{Zamo2}.
A detailed discussion about the general properties (products, analytic
continuations) of these operators is given in \cite{Raoul1}.

The most important feature of the disorder operators is their non-abelian monodromy
with respect to the chiral fields $\{\Psi^{\pm k}\}$. This amounts to the
decomposition of the local products $\Psi^{k}(z) R_{a}(0)$ into half-integer
powers of $z$:
\begin{equation}
 \Psi^{k}(z)R_{a}(0) =
\sum_{n}\frac{1}{(z)^{\Delta_{k}^{\Psi}+\frac{n}{2}}}\, A^{k}_{\frac{n}{2}}R_{a}(0),
\qquad k=1,2,\ldots,\lfloor N/2 \rfloor. \label{mode1R}
\end{equation}
The expansion of the
product $\Psi^{-k}(z)R_{a}(0)$ (with $k=1,2,\ldots,\lfloor N/2 \rfloor$) can be
obtained by an analytic continuation of $z$ around 0 on both sides of
Eq.~(\ref{mode1R}). The result is:
\begin{equation}
 \Psi^{-k}(z)R_{a}(0) =
\sum_{n}\frac{(-1)^{n}}{(z)^{\Delta_{k}^{\Psi}+\frac{n}{2}}} A^{k}_{\frac{n}{2}} \,
R_{a-k}(0), \qquad k=1,2,\ldots,\lfloor N/2 \rfloor.
\label{rexpansion}
\end{equation}

In accordance with these expansions, the mode operators
$A^{k}_{\frac{n}{2}}$ can be defined by the contour integrals
\begin{equation}
 A^{k}_{\frac{n}{2}}R_{a}(0)=\frac{1}{4\pi i}\oint_{C_{0}}
 {\rm d}z \, (z)^{\Delta_{k}^{\Psi}+\frac{n}{2}-1}\Psi^{k}(z)R_{a}(0),
\label{rmodes}
\end{equation}
where the integrations are defined by letting $z$ turn twice around the
operator $R_{a}(0)$ at the origin, exactly as described in \cite{Zamo2,Raoul1}.  The
commutation relations between the current modes in this sector can be computed
and the dimension of the primary disorder operators is (see Appendix
\ref{rsector_kac}):
\begin{equation}
\Delta^{R}_{s}=\frac{N-2+(N-2 s)^2}{16(N+2)} \quad s=0,1,\cdots, \lfloor N/2
\rfloor.
\label{disorder_kac}
\end{equation}
Note that for $N=2$ one has the operators $R_{a}^{(0)}$ and $R_{a}^{(1)}$ with
dimension $\Delta^{R}_{0}=1/16$ and $\Delta^{R}_{1}=0$. This is of course expected as
the dihedral group $D_2=Z_2$, i.e. the cyclic $Z_N$ elements and the
reflection $Z_2$ elements of the group $D_N$ coincide for $N=2$. In particular
one has $R_{a}^{0}=\Phi^{\pm 1/2}$ while $R_{a}^{1}$ coincides with the
identity.  For $N=3$, the disorder sector contains the operator $R_{a}^{0}$
($\Delta^{R}_{0}=1/8$) and $R_{a}^{1}$ ($\Delta^{R}_{1}=1/40$). In terms of primaries
of the Virasoro algebra they coincide respectively with the $\phi_{1,2}$ and
$\phi_{2,2}$ operators of the $M_5$ Kac table, see again
Eq.(\ref{kac_minimal}).

\subsection{Stochastic motion in the internal space.}
\label{sle_zn}

We discuss now a possible SLE approach to describe
parafermionic theories.  In the previous section we have seen that the action
(\ref{modes}) and (\ref{rmodes}) of the chiral currents modes on the
representation fields amounts to a twist either of the $Z_N$ charge $q$ for
fields $\Phi^{q}(z)$ belonging to the $Z_N$ sector, or of the $Z_2$ index
$a$ for fields $R_{a}(z)$ in the disorder sector.  The b.c.c.
operator is expected to transform as an operator in the $Z_N$ or in the $Z_2$
sector. By analogy to the case of the $SU(2)_N$ WZW \cite{Ludwig}, we
consider an independent stochastic motion in the additional degrees of freedom.
We provide a parafermionic field the following evolution:
\begin{equation}
X(z)\to X(z)+\frac{1}{2\pi i}\oint_{\mathcal{C}_z} d\, g_t(w) T(w)
X(z)+\frac{1}{2\pi i}\oint_{\mathcal{C}_z}d\, w (d \theta)^{k} (w)\Psi^{k}(w)
X(z),
\label{evolution}
\end{equation}
where $X(z)=\Phi^{q}$ or $X(z)=R_a(z)$ and the contour $\mathcal{C}_z$ is a
general closed contour around $z$ (remember that in the $R$ sector the
Riemann surface is two-fold). This evolution contains, in addition to the variation under an
infinitesimal conformal transformation $d g_t(w)$, a component with shifted
$Z_N$ charge $q\to q+k$ or $Z_2$ charge, $a \to a+k$. The (\ref{evolution})
can be seen as a motion in the parafermionic representation module.
In the sequel, we shall concentrate on considering only
the action of the most fundamental $\Psi^{\pm 1}$ fields.

The definition of the internal stochastic motion is given in such a way that
the corresponding Ito derivative of  $\psi(z)$ takes the
form (\ref{psi_dynamic}) or (\ref{wzw_psi_dinamic}). More precisely, we want
to derive a martingale condition as a linear relation between descendants at
the level two of some representation module. This allows for a direct
comparison with the CFT results.

In the case the boundary operator $\psi(z)$
coincides with a primary $\Phi^{q}$ in the $Z_N$ sector, we define:
\begin{equation}
(d \theta)_t^{k} (z)=\frac{(d \theta)^{k}_t}{(g_t(z)-\xi_t)^{-\Delta_{k}^{\Psi}+\delta^{q}_{k+q}+1}}, \quad \mbox{{\bf
    E}}[\theta^{k}_t\theta^{k'}_s]=\tau\,\delta^{k,-k'}\,\mbox{min}(s,t).
\label{tau_sle_para_zn}
\end{equation}
The exponent in the denominator has been chosen according to
Eq.(\ref{eq3}) which encodes the OPE between the currents $\Psi^{k}$
and the fields $\Phi^{q}$.  Note that this evolution is very similar to
(\ref{tau_sle}) in the $su(2)$ Lie algebra. This is somehow expected since the
$Z(N)$ parafermionic theory corresponds to the coset $SU(2)_N/U(1)$ (see
appendix \ref{wzw_para}).

In the case  when $\psi(z)$ transforms as a disorder $R_{a}$ field, we
define, according to Eq.(\ref{mode1R}):
\begin{equation}
(d \theta)_t^{k} (z)=\frac{(d \theta)^{k}_t}{(g_t(z)-\xi_t)^{-\Delta_{k}^{\Psi}+2}}, \quad \mbox{{\bf
    E}}[\theta^{k}_t\theta^{k'}_s]=\tau\,\delta^{k,-k'}\,\mbox{min}(s,t).
\label{tau_sle_para_r}
\end{equation}

We point out that the definition of the stochastic motion given above is by
now purely algebraic and based on a formal analogy with the case $SU(2)$.
However, the CFT predictions and the identification of the possible interface
on the lattice model motivate the interest of such construction.

\subsection{Martingale conditions and parafermionic results.}
\label{mart_cond}

As we will discuss in some detail later, the Ising model ($N=2$ and $c=1/2$)
shows in its FK representation (see \ref{lattice model}) a domain wall described by the
SLE$_{16/3}$. The b.c.c operator creating such interface
is identified with the operator $\phi_{1,2}$ of the $M_3$ Kac table,
what is consistent with the Eq.(\ref{sle_cft_minimal}) and Eq.(\ref{sle_cft_bcc}). In
the parafermionic representations, we have seen that this operator corresponds
to the disorder $R_{a}^{(0)}$ operator or, equivantely, to the $\Phi^{\pm
  1/2}$ operator (remember that $N=2$ is special as $D_2$=$Z_2$). In the three
state Potts model ($N=3$), the operator $\phi_{1,2}$ of the $M_5$ Kac table is
again identified again with the disorder $R_{a}^{(0)}$ and it is expected to
create an SLE$_{24/5}$, in agreement again with Eq.(\ref{sle_cft_minimal}) and
Eq.(\ref{sle_cft_bcc}). The operator $\phi_{2,1}$, on the other
hand, can be associated to an interface described by the SLE$_{10/3}$ and it
corresponds to the
$\varepsilon^{(N=3)}=A^{1}_{-1/3}\Phi^{-1}=A^{-1}_{-1/3}\Phi^{1}$ operator
mentioned before.

On the basis of these correspondences, it is then natural to identify for
general $N$ the b.c.c.  operator $\psi (z)$ with the disorder
primary operator $R_{a}^{(0)}$ with dimension $\Delta_{0}=(N^2+N-2)/(16(N+2))$
and with the $Z_N$ singlet (i.e.  charge $q=0$) operator
$\varepsilon^{(N)}=A^{-1}_{-1/N}\Phi^{1}=A^{1}_{-1/N}\Phi^{-1}$ with dimension
$\Delta_{\varepsilon}=2/(N+2)$ . Note that this singlet operator is not a primary of the
parafermionic algebra but it appears as  descendant in the module of
$\Phi^{\pm 1}$ (see appendix \ref{nullvector}).

Using Eq.(\ref{chidim}), Eq.(\ref{level_structure}) and Eq.(\ref{evolution}),
the stochastic motions (\ref{tau_sle_para_zn}) and (\ref{tau_sle_para_r})
amounts respectively to the following dynamics for $\psi(z)$ in the two cases
when $\psi(z) \equiv \psi_{\varepsilon}=\varepsilon(z)$ and $\psi(z) \equiv \psi_{R}=R_{a}^{(0)}$:
\begin{eqnarray}
d( \psi_{\varepsilon} (\xi_t))= d \xi_t L_{-1}\psi_{\varepsilon}(\xi_t)  + (d \theta)^{1}
A^{1}_{1/N-1} \psi_{\varepsilon}(\xi_t)+ (d \theta)^{-1}
A^{-1}_{1/N-1} \psi(\xi_t)_{\varepsilon}\nonumber \\
+ dt\left( \frac{\kappa}{2}
L_{-1}^2 +\frac{\tau}{2} \left[A^{1}_{-1/N-1}A^{-1}_{1/N-1}+A^{-1}_{-1/N-1}A^{1}_{1/N-1}\right]\right)\psi_{\varepsilon}(\xi_t).
\label{para_psi_dinamic_zn}
\end{eqnarray}
and $\psi(z) \equiv \psi_{R}=R_{a}^{(0)}$:
\begin{eqnarray}
  d( \psi_{R} (\xi_t))= d \xi_t L_{-1}\psi_{R}(\xi_t)  + (d \theta)^{1}
  A^{1}_{-1} \psi_{R}(\xi_t)+ (d \theta)^{-1}
  A^{-1}_{-1} \psi_{R}(\xi_t)\nonumber \\
  + dt\left( \frac{\kappa}{2}
    L_{-1}^2 +\frac{\tau}{2} \left[A^{1}_{-1}A^{-1}_{-1}+A^{-1}_{-1}A^{1}_{-1}\right]\right)\psi_{R}(\xi_t).
\label{para_psi_dinamic_R}
\end{eqnarray}
The martingale condition, to be compared with Eq.(\ref{sle_nul_vect}) and
Eq.(\ref{sle_nu_vect_wzw}), now reads:
\begin{eqnarray}
  \left(L_{-2}-\frac{\kappa}{4}
  L_{-1}^2 -\frac{\tau}{4}
  \left[A^{1}_{-1/N-1}A^{-1}_{1/N-1}+A^{-1}_{-1/N-1}A^{1}_{1/N-1}\right]\right)\psi_{\varepsilon}(\xi_t)&=&0
\label{sle_nul_vect_para_1}
\\
\left(L_{-2}- \frac{\kappa}{4}
    L_{-1}^2 -\frac{\tau}{4} \left[A^{1}_{-1}A^{-1}_{-1}+A^{-1}_{-1}A^{1}_{-1}\right]\right)\psi_{R}(\xi_t)&=&0.
\label{sle_nul_vect_para_2}
\end{eqnarray}

We can now enounce the main results of this paper. We have studied the
representation modules of the $\varepsilon$ and $R_{a}^{(0)}$ operator. For  $N\geq 4$ one finds that there are at least two independent states at the second level of these operators.  We
have found the following relations at the second level of these modules:
\begin{eqnarray}
  \left(L_{-2}-\frac{N+1}{N+2}
  L_{-1}^2 -\frac{N^3}{(N+2)^3}
  \left[A^{1}_{-1/N-1}A^{-1}_{1/N-1}+A^{-1}_{-1/N-1}A^{1}_{1/N-1}\right]\right)\psi_{\varepsilon}(\xi_t)&=&0 \label{para_nul_vect_1}\\
\left(L_{-2}- \frac{N+2}{N+1}
    L_{-1}^2 -\frac{ 2^{-4/N} N^3}{4(N+2)(N+1)}\left[A^{1}_{-1}A^{-1}_{-1}+A^{-1}_{-1}A^{1}_{-1}\right]\right)\psi_{R}(\xi_t)&=&0,
\label{para_nul_vect_2}
\end{eqnarray}
The derivation of these relations is shown in
appendix \ref{nullvector}. We have verified that, for $N=2$, the relation
(\ref{para_nul_vect_2}) reduces to (\ref{vir_nul_vect_1}) and for $N=3$, the
Eq.(\ref{para_nul_vect_1}) and Eq.(\ref{para_nul_vect_2}) are respectively
equivalent to Eq.(\ref{vir_nul_vect_2}) and Eq.(\ref{vir_nul_vect_1}). In
these cases, which as already said coincide with minimal models $M_3$ and
$M_5$, all the descendants of the theory can be obtained by applying negative
Virasoro modes. In our case, this amounts to the fact that the states at the
second level constructed from the application of parafermionic modes can be
written as a linear combination of $L_{-1}^2$ and $L_{-2}$.

Comparing Eq.(\ref{sle_nul_vect_para_1}) and Eq.(\ref{sle_nul_vect_para_2}) to
Eq.(\ref{para_nul_vect_1}) and Eq.(\ref{para_nul_vect_1}), we obtain, for
$N\geq 4$ and  for the
two SLEs evolutions:
\begin{eqnarray}
\kappa_1&=&4 \frac{N+1}{N+2}\quad \tau_1= 4\frac{N^3}{(N+2)^3} \label{Sle1} \\
\kappa_2&=&4 \frac{N+2}{N+1}\quad \tau_2= 2^{-4/N}\frac{N^3}{(N+2)(N+1)}\label{Sle2}
\end{eqnarray}

\section{$Z(N)$  self-dual critical  spin models.}
\label{lattice model}

Consider a square lattice with the spin variables $\sigma_{j}$ at each
sites $j$ taking $N$ possible values:
\begin{equation}
\sigma_{j}=\exp\left[\frac{i2\pi}{N}n(j)\right] \quad n(j)=0,1,\cdots,N-1.
\label{spin_variables}
\end{equation}
The most general $Z_N$-invariant spin model  with nearest-neighbor
interactions is defined by the reduced Hamiltonian \cite{Zama_lat,Dotsi_lat,Fradkin_lat}:
\begin{equation}
H[n]=-\sum_{m=1}^{\lfloor N/2 \rfloor} J_{m}\left[\cos \left(\frac{2\pi m n}{N}\right)-1\right].
\label{reducedH}
\end{equation}
and the associated partition function reads:
\begin{equation}
Z=\sum_{\{\sigma\}}\exp\left[-\beta \sum_{<ij>} H[n(i)-n(j)]\right]
\label{partition1}
\end{equation}
In fact, the Potts model is recovered in the case
$J_m=J$ for all $m$. The model presents, in this case, a permutational $S_N$
symmetry. Another meaningful model is the clock model which is defined by the
partition function
(\ref{partition1}) with $J_m=J\delta_{m,1}$.
Defining the Boltzmann weights:
\begin{equation}
x_n=\exp\left[-\beta H(n)\right], \quad n=0,1,\cdots,N-1,
\end{equation}
the general $Z(N)$ spin system is described by $\lfloor N/2\rfloor$ independent
parameters $x$. The Kramers and Wannier (order-disorder) duality has proven to
be a powerful tool for examining the behavior of these models. The phase
diagram of the model (\ref{partition1}) for $N=5$ states of spin is shown in
Fig.(\ref{diaZ5}). It has been show that the $Z(5)$ model is self-dual on the
line $x_1+x_2=1/2(\sqrt{5}-1)$. Here we are interested in the
existence of two critical points $(x_1^{*}, x_2^{*})$ and $(x_2^{*},
x_1^{*})$, where $x_1^{*}=\sin(\pi/20)/\sin(3\pi/20)\approx 0.34$ and
$x_2^{*}=\sin(\pi/4)/\sin(7\pi/20)x_1^{*}\approx 0.27$, where the model is
completely integrable.  The $Z(5)$ parafermionic theory defined in the previous
sections describes the continuum limit of the lattice model at these points.
In general, in the self-dual subspace of (\ref{reducedH})-(\ref{partition1}),
which contains also the Potts model the clock model, the $Z(N)$ spin model is
completely integrable at the points \cite{Zama_lat2} :
\begin{eqnarray}
x^{*}_0&=& 1 \nonumber \\
x^{*}_n&=& \prod_{k=0}^{n-1} \sin \left(\frac{\pi k}{N}+\frac{\pi}{4
    N}\right)\left[\sin \left(\frac{\pi (k+1)}{N}-\frac{\pi}{4 N}\right)\right]^{-1}.
\label{integrablecond}
\end{eqnarray}
These critical points are described by the $Z(N)$ parafermionic field theories.

\subsection{ Boundary conditions and SLE interfaces.}
Consider the three-states Potts model.  From the Eq.(\ref{sle_cft_minimal})
and the Eq.(\ref{sle_cft_bcc}), as previously mentioned, the b.c.c.
operator $\psi_{\varepsilon}^{(N=3)}$ is expected to create an
SLE$_{10/3}$ trace. In this paragraph we suggest a definition of the
interfaces within the lattice models whose scaling limit is described in these
SLE processes.  In the following we
indicate the possible values of the spins with the letters $A, B\cdots$. The
b.c.c.  operator $\psi_{\varepsilon}^{(N=3)}$ has been shown to generate the
boundary condition where the spins are fixed to (say) the value $A$ on the
left side of the origin while can take the value $B$ or $C$ with equal
probability on the right side \cite{Cardy_bcft2}. With this boundary condition, to which we refer
with the short-hand notation $A|B+C$, there exists a single domain wall
between the spin $A$ and the spins $B$ and $C$, as shown in
Fig.\ref{slekl4}. The statistical properties of this interface has recently
been analyzed by numerical means in \cite{Cardy_potts} and are shown to be well
described in the continuum limit by an SLE$_{10/3}$.

\begin{figure}
\begin{center}
 \includegraphics[scale=0.5]{slekl4}
 \end{center}
 \protect\caption[3]{\label{slekl4}a)Three state Potts model: interface (green
   curve) between the $A$ spins (black circles) and the $B$ (full blue circles) and
   $C$ (empty blue circles). The boundary conditions $A|B+C$ as in the
   text. b) The interface defined in the $Z(5)$ spin model with boundary conditions
   $A|B+C+D+E$.}
\end{figure}

The generalization to the $N$-states spin model is straightforward. Indeed,
one can show (see appendix \ref{boundarystates}) that the b.c.c. operator $\psi_{\varepsilon}^{(N)}$
produces the boundary conditions $A|B+C+D+\cdots$, where the spins on the left
of the origin take the value $A$ while the ones on the right take with equal
probability the $N-1$ values of spin different from $A$.  As in the case of
$N=3$, there exists a single boundary domain between the $A$ spin and the
$B,C,\cdots..$. Note that the boundary conditions do not satisfy the
reflection symmetry and the measure on the curve should reflect this
asymmetry.  However, in the continuum limit the reflection symmetry is
expected to be restored as explicitely shown in \cite{Cardy_potts}. This can
be argued from the fact that, in this limit, the statistical expectation values
can be expressed in terms of CFT correlation functions which contain the
b.c.c.  operators, see Eq.(\ref{martingale_cft}). These correlation functions
satisfy indeed the reflection symmetry.

It is interesting to note that one can assign a ``color'' to the interface
depending on the values of the spins ($B,C,\cdots$) at the right of the domain wall.  The
point is that, for $N<4$, the Boltzmann weights $x_n^{*}$,
$n=1,2,\cdots,N-1$, are equal: the energy of one bond does not depend on the
difference between the values of the nearest-neighbor spins. In this sense,
the $Z_N$ symmetry is trivially realized and this is essentially the reason
why, for $N=2,3$, the correspondent CFT can be constructed from the conformal
invariance alone. However this is not true for $N\geq 4$ where the Boltzmann weights
$x_n^{*}$, $n=1,2,\cdots \lfloor N/2 \rfloor$ are different.  In the continuum
limit, this amounts to a more structured chiral algebra with a central charge
$c\geq1$.  From the point of view of the lattice model, one can observe that for $N=2,3$ the
energy cost to have a spin $B$ or $C$ at one side of the interface is the
same. Thus, as $N\geq 4$, one expects the internal spin degree of freedom
to play an important role in the spatial evolution of the interface.  This is
somehow in agreement with the need to introduce an additional stochastic
motion in the $Z_N$ group. However a clear the geometrical interpretation of the
stochastic evolution in the representation modules (see section \ref{sle_zn})
is missing.

From the results (\ref{para_nul_vect_1}) and (\ref{Sle1}), we hypothesize that
the geometric properties of this interface for $N\geq 4$ are described by a
$SLE$ with $\kappa= 4(N+1)/(N+2)$. In particular, since $\kappa\leq 4$,
we expect the domain under consideration to be a simple curve for each $N$.

We discuss now the identification of the interface on the lattice associated
to the b.c.c operator $\psi_{R}$, see Eq.(\ref{para_nul_vect_2}).  In the
Ising and three states Potts model, this operator is associated to an
SLE$_{16/3}$ and SLE$_{24/5}$ respectively. The correspondent interfaces on
the lattice can be identified with the help of the Fortuin-Kasteleyn (FK)
representations.

The partition function (\ref{partition1}) for the $N=2$ and $N=3$ critical
models take the form:
\begin{equation}
Z=\sum_{\{ \sigma\}} \exp \left[ -\beta \sum_{<ij>} H(n(i)-n(j))\right]=\sum_{\{
  \sigma\}} \prod_{<ij>}\left[x_1^{*}+(1-x_1^{*})\delta_{n(i),n(j)}\right],
\label{fk1}
\end{equation}
where $x_1^{*}=\sqrt{2} -1$ or $x_1^{*}=x_{-1}^{*}=(\sqrt{3} -1)/2$ is the
critical point of the Ising and of the three states Potts model respectively.
By expanding the product (\ref{fk1}) and summing over all the spin
configurations, one obtains the FK random cluster representation:
\begin{equation}
Z=\sum_{\mathcal{C}} (x_1^{*})^{M-b}(1-x_1^{*})^{b} N^{c}
\label{fk2}
\end{equation}
where the sum is over the subgraphs $\mathcal{C}$ of $\mathcal{G}$,
$\mathcal{C}\subseteq \mathcal{G}$, the graph $\mathcal{G}=(V,E)$ being
composed by the $V$ sites and the $E$ edges of the square lattice. Each graph
$\mathcal{C}$ is specified by the $b$ bonds on the edges and the $c$ clusters
(=connected component which include the single sites) in a given bond
configuration.

Consider the following boundary conditions: all the edges on the negative axis
carry bonds while all the edges on the positive axis carry no bonds.  For the
spin variables this means that  all the spins located at
the left of the origin take the same value while the spins on right side are
unconstrained, i.e. they are free (F) to take all the $N$ possible values.

In such an arrangement, each graph $\mathcal{C}$ presents a cluster growing from the
negative axis into the upper half-plane. The
boundary of this cluster, starting from the origin, is conjectured (see for
istance \cite{Walter}, chapter 5) to be
statistically equivalent in the continuum limit to the SLE$_{16/3}$ for $N=2$
(Ising model) and to the SLE$_{24/5}$ for $N=3$ (the three states Potts
model).  This is in
agreement with the properties of the b.c.c. $\psi_{R}^{(N)}$ operators,
$N=2,3,4$, which have been shown \cite{Bauer_R} to be associated to the $A|F$ boundary
conditions.

The generalization to the case $N\geq 4$ is more cumbersome. To our knowledge,
the modular properties of the $R-$sector characters, and thus the boundary
conditions associated to these operators, have not been yet studied for these
cases. This motivates a more detailed analysis of the boundary conformal
properties of this sector. However, we point out that at the level of the
parafermionic algebra representations the main features of the $\psi_{R}$
fields, such as the structure and the degeneracies of the representation
module or the current zero modes eigenvalues (see Appendix \ref{rsector_kac}), directly
generalize to each $N\geq 2$.  The formula (\ref{disorder_kac}) is an example.
It is thus quite natural to suppose that the operators $\psi_{R}^{(N)}$
produce the $A|F$ boundary conditions for each $N$. Another observation is
that it is quite direct to find an alternative formulation of
(\ref{partition1}) in terms of colored clusters on a graph. For sake of
clarity, we focus our attention on the case $N=5$.
\begin{figure}
\begin{center}
 \leavevmode
\includegraphics[scale=0.5]{diaZ5}
 \end{center}
 \protect\caption[3]{\label{diaZ5}Phase diagram of th $Z(5)$ spin model}
\end{figure}

The partition function (\ref{partition1}) for $N=5$ at the critical
point $x_n^{*}$, see Eq.(\ref{integrablecond}), can be rewritten as:
\begin{equation}
Z^{(N=5)}=\sum_{\{\sigma\}}\prod_{<ij>} \left[x_1^{*}+(1-x_1^{*})\delta_{n(i),n(j)}+(x_2^{*}-x_1^{*})\delta_{n(i),n(j) \pm 2}\right].
\label{partition2}
\end{equation}
If $x_1^{*}=x_2^{*}$ one recovers the five states Potts model
in the usual FK representation.

Expanding the factors in (\ref{partition2}), one has to take into account two
type of bonds: the bonds (type 1) connecting spins which take equal values
and the bonds (type 2) connecting spins which differ by an angle $4\pi/5$.
Then, summing over the spin configurations, the partition function can be cast
in terms of a sum over ``colored'' graphs $\mathcal{C'}$:
\begin{equation}
  Z^{(N=5)}=\sum_{\mathcal{G'}} (x_1^{*})^{E-b_1-b_2}(1-x_1^{*})^{b_1}(x_2^{*}-x_1^{*})^{b_2}5^{c}.
\label{z5_fk}
\end{equation}
Each graph $\mathcal{C'}$ is specified by the number $c$ of connected
component and by the numbers $b_1$ and $b_2$ of the bonds of type 1 and type 2.
Naturally, the sum over the differences of
the spins along a loop must be zero (modulo $2\pi$).  For instance, a loop
formed by three bonds of type 2 and one bond of type 1 is not permitted. The
prime in $\mathcal{C'}$ indicates that the sum in (\ref{z5_fk}) is taken over
the allowed graphs. A typical configuration is shown in
Fig.(\ref{fk_colorato}).

\begin{figure}
\begin{center}
 \leavevmode
 \includegraphics[scale=0.5]{fk_colorato}

 \end{center}
 \protect\caption[3]{\label{fk_colorato}Cluster expansion of the $Z(5)$ spin
   model. The spins at the vertices of a black bond have the same value while
   the ones at the vertices of a red bond  differ by an angle $4\pi/5$}
\end{figure}

By analogy to the Ising and three states Potts model, we consider the boundary
conditions in which the spins have a fixed value on the negative axis while
they are free on the positive axis. With such boundary conditions,
there is a cluster of bonds of type $1$ growing from the negative axis, see
Fig.(\ref{slekp4}) .
\begin{figure}
\begin{center}
 \leavevmode
 \includegraphics[scale=0.5]{slekp4}

 \end{center}
 \protect\caption[3]{\label{slekp4} a) A typical cluster configuration of
   the $Z(5)$ spin model with type 1 (black) and type 2 (red) bonds. The
   boundary conditions on the real axes are as in the text. The green curve is the boundary of
   the cluster growing from the negative axes. b) A piece of the interface
   which crosses red or empty bond.}
\end{figure}

From (\ref{Sle2}), we conjecture the geometric properties of the boundary of this cluster are
described in the continuum limit by the SLE$_{14/3}$.

The above discussion can be directly generalized to each value of $N$. In
particular, the $Z(N)$ spin model admits a random cluster representations
with $\lfloor (N-1)/2 \rfloor$ types of bonds. Imposing the conditions on the
real axis discussed above, we conjecture that the measure of the boundary of the
cluster growing from the negative real axis is described by an SLE with $\kappa=4(N+2)/(N+1)$.

Finally we remark that, during its evolution, the interface crosses edges
without  bonds or with bonds of the type $2,3,\cdots \lfloor (N-1)/2 \rfloor$ which
have different energetic costs.  Analogously to the case of the interface
generated by the $A|B+C+\cdots$ boundary conditions, one can color the interface
depending on the type of bond it crosses.  Again, the interface is naturally
provided of an $Z_N$ additional internal degree of freedom, in agreement with
the approach proposed in Section (\ref{sle_zn}).

\section{Conclusions.}
\label{conclusion}
In this paper we have considered the possibility to extend an SLE approach to
CFTs with additional $Z_N$ discrete symmetries.  These theories describe the
continuum limit of the self-dual critical $Z(N)$ spin models. The case $N=2$
and $N=3$ correspond to the Ising and three states Potts model where the
identification of the interface on the lattice and of the correspondent b.c.c.
operators are known. Using these results, we have identified the possible
b.c.c. operators for general $N$. We show that these operators satisfy a two
level null vector condition. For $N\geq 4$, an additional term enter these
relations. This term is obtained from the action of the parafermionic currents
and, for $N\geq 4$ ($c\geq 1$), it cannot be derived by the Virasoro modes
action. These results led us to propose an additional stochastic motion in the
different $Z_N$ charge sectors and in the $Z_2$ sectors which describe the
representation modules of the parafermionic algebra. We have finally discussed
the possible interfaces in the lattice to which the CFT/SLE results should
apply. The fact that SLE candidates have been identified in lattice spin
models described by non-conformal minimal theories opens the possibility to
study the SLE/non-minimal CFT connection by a numerical approach.

It will be interesting to verify our predictions by numerical studies. Further
we believe these studies can also better clarify the geometrical
interpretation of the additional stochastic motion in the internal space.

{\bf Acknowledgments:} We would like to thank M.~ Bauer, J.~Cardy, I.~Gruzberg, C.~Hagendorf,
A.~Ludwig, V.~Riva for very helpful discussions.  We are particularly grateful
to D.~Bernard and P.~Le Doussal for very useful advises and support all along
this work.

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Originally from West Sussex, I am a 29 year old environment artist who now resides in the North West with a little over three years worth of experience and one shipped title with DLC.
I am extremely passionate about the development of computer games and creating 3D environments / props. From Sept. 2009 to May 2012 I attended the University of Derby and studied 'Computer Games Modelling & Animation' in which I successfully achieved an upper second class honours (2:1).
I am currently working on Squadron 42 for StarCitizen at Foundry 42 / Cloud Imperium Games.
Can comfortably work as part of a team.
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text:
ParksSAFeedback@sa.gov.au
Southern Flinders Precinct Partners
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Southern Flinders Precinct
Project and Partner News
Designs underway for an 'EPIC' mountain bike trail in Melrose
Concept designs for an international-standard mountain bike trail around Mount Remarkable are now underway.
This is one of the key projects for the Southern Flinders Precinct with trail design experts World Trail coming on board to undertake concept work.
This mountain bike loop will be built to a standard which will be sensitive to the environment and deliver exceptional views, a great experience and a sense of adventure to riders. The trail will be assessed by the International Mountain Biking Association with the view to gain 'EPIC' accreditation, which will attract mountain bike enthusiasts from all over Australia and the world to the region.
World Trail are one of the largest and most experienced mountain biking trail companies in the world and have designed, planned and constructed hundreds of projects in 20 countries – including Australia's well-known Blue Derby trail network.
In addition to designing the loop in Mount Remarkable, World Trail will also assess the potential for other cycling opportunities in the Southern Flinders Ranges, including any downhill trails.
Developing new facilities in the precinct will bring together a unique partnership of environmental organisations, economic agencies, Councils and State and Australian Government.
The South Australian Government has committed $5million to establish the precinct as part of the Parks 2025 strategy.
This funding will be matched through a community partnership.
The "Remarkable Southern Flinders" proposal, is a concept put forward by community partners including the Foundation for National Parks & Wildlife, Northern Areas Council, District Council of Mount Remarkable, Port Pirie Regional Council and Regional Development Australia Yorke and Mid-North to further advance and promote the Southern Flinders region as a tourism destination.
Together these partners put forward the "Remarkable Southern Flinders" proposal to the Australian Government and were successful in securing $5million from the Building Better Regions Fund.
The addition of the funding for the Remarkable Southern Flinders proposal will expand the infrastructure able to be built across the new national park precinct.
The establishment of the park will closely involve the traditional owners of the land, the Nukunu Nation. The Nukunu will also be co-managers of the Telowie Gorge Conservation Park, inside the precinct.
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text:
Q: jQuery Datatables: Custom copy-to-clipboard function The built-in copy-to-clipboard function in Datatables can copy the table head with the selected rows, so it pastes like this (Title, Number and Comment being columns):
Title Number Comment
Test   102    "nice"
Test2  103   "ok"

I need it like this:
Title: Test Number: 102 Comment: "nice"
Title: Test2 Number: 103 Comment: "ok"

My Datatables setup for the copy-button is currently like this:
dom: 'Bfrtip',
buttons: {
       buttons: [
         {
            extend: 'copyHtml5',
            text: 'Copy Selected Rows',
            header: false,
            exportOptions: {
                 modifier: {
                        selected: true
                 }
            }
         }
       ]
}

Is there a function to archive this? Or how can I modify the copying process?

A:
SOLUTION

You can use orthogonal option to specify data type copy requested for copy operations and columns.render to render appropriate content when data type copy is requested.
$('#example').DataTable({
    dom: 'Bfrtip',
    columnDefs: [{
        targets: "_all",
        render: function (data, type, full, meta) {
            if (type === 'copy') {
                var api = new $.fn.dataTable.Api(meta.settings);

                data = $(api.column(meta.col).header()).text() + ": " + data;
            }

            return data;
        }
    }],
    buttons: [{
        extend: 'copyHtml5',
        text: 'Copy Selected Rows',
        header: false,
        exportOptions: {
            modifier: {
                selected: true
            },
            orthogonal: 'copy'
        }
    }]
});

DEMO

See this jsFiddle for code and demonstration.

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text:
USA EB-5 VISA
Home » Citizenship » USA
HOW TO OBTAIN EB-5 VISA (GREEN CARD) BY INVESTMENT
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text:
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PM announces judicial appointments
Prime Minister Stephen Harper today announced three judicial appointments.
The Honourable Paul S. Crampton, a judge of the Federal Court, is appointed Chief Justice to replace the Honourable A.F. Lutfy who became a supernumerary judge on October 1, 2011. Mr. Justice Crampton was appointed to the Federal Court in 2009.
The Honourable Austin F. Cullen, a judge of the Supreme Court of British Columbia, is appointed Associate Chief Justice to replace Madam Justice A.W. MacKenzie who was appointed to the British Columbia Court of Appeal effective December 31, 2011. Mr. Justice Cullen was appointed to the Supreme Court of British Columbia in 2001.
The Honourable Martel D. Popescul, a judge of Her Majesty's Court of Queen's Bench for Saskatchewan, is appointed Chief Justice to replace Mr. Justice R.D. Laing who elected supernumerary status effective January 1, 2012. Mr. Justice Popescul was appointed to the Court of Queen's Bench in 2006.
"These are all remarkable individuals and outstanding legal jurists," said Prime Minister Harper. "They will all assume key leadership roles in our Canadian superior courts."
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text:
The slate-colored solitaire (Myadestes unicolor) is a species of bird in the family Turdidae. It is found in Belize, El Salvador, Guatemala, Honduras, Mexico, and Nicaragua. Its natural habitat is subtropical or tropical moist montane forests.

Gallery

References

External links
Stamps (for El Salvador)
Slate-colored solitaire photo gallery VIREO

slate-colored solitaire
Birds of Mexico
Birds of Guatemala
Birds of Honduras
Birds of Nicaragua
slate-colored solitaire
slate-colored solitaire
Taxonomy articles created by Polbot

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text:
Home/In the news (page 145)
Joe Biden signs executive order ending Trump's ban on transgender Americans serving in the military
25 January 2021 Diasporas, In the news, Politics, United States 0
Joe Biden the new American president revoked the Trump administration's ban on openly transgender individuals serving in the U.S. military through executive order Monday morning, calling it "the right thing to do." "President Biden believes that gender identity should not be a bar to military service, and that America's strength …
Uganda: the authorities threaten those who use VPN
25 January 2021 Africa, East Africa, In the news 0
Few days after the announcement of the presidential elections results, social networks are still blocked in the country. The connection had been restored but Facebook, Whatsapp, Twitter, and the other social networks remain inaccessible. The population however found a solution to that problem. Despite the blockage of social networks for …
Nick Mangwana apologises for "medical assassins" slur
25 January 2021 Africa, Coronavirus, In the news, Southern Africa 0
Zimbabwe's government spokesman Nick Mangwana has apologised after the country's medical association reacted furiously to his Tweet that doctors were behind the deaths of top government officials who succumbed to Covid-19. Nick Mangwana tweeted "medical assassins" to insinuate that the politicians were being deliberately eliminated. Four cabinet ministers have now …
Ghana pays last tribute to former President John Jerry Rawlings
25 January 2021 Africa, In the news, People, West Africa 0
Former Ghanaian President Jerry Rawlings will lie in state from Monday in the capital, Accra, although strict Covid-19 protocols will be in place for people who want to pay their respects. John Jerry Rawlings, who was Ghana's longest-serving leader, died in November, aged 73, after a short illness. Funeral events …
The former Malawian president no longer has access to his bank accounts
25 January 2021 Africa, East Africa, In the news, Politics 0
The High Court in Malawi has refused to allow former President Peter Mutharika access to his bank accounts which were frozen by the country's anti-corruption body pending trial. Mr Mutharika's bank accounts, as well as those of his wife, Gertrude, were frozen last August by the Anti-Corruption Bureau (ACB) which …
Seen without a mask, a senior South-african official under investigation
Bheki Cele, the South African Police Minister, has ordered investigation into the alleged conduct of the head of Mpumalanga province who was seen not wearing a mask while attending a funeral on Sunday. Refilwe Mtsweni-Tsipane said she was not aware that her mask fell off when she stepped out of …
Death of eleven people near the Somalia-Kenya border
In a Somali town, near the Kenyan border, took place a heavy fighting, in which at least 11 people died and 14 other injured according to eyewitnesses. The fighting started late Sunday night and continues until Monday morning in Baled-Hawo town in the South-western region of Gedo. Somalia's information ministry …
Burundi: new leader at the head of the ruling party
Burundi's ruling party, CNDD-FDD, has elected Révérien Ndikuriyo as its new leader. He will take over the secretary general role from incumbent President Evariste Ndayishimiye, who held the post until May 2020 when he won the election. Mr Ndikuriyo on Twitter, thank "companion" President Ndayishimiye and said he "will not …
Zimbabwe: doctors dissatisfied after a reaction from Nick Mangwana
25 January 2021 Africa, In the news, Southern Africa 0
Doctors in Zimbabwe have protested against a Twitter slur by the government spokesperson, who referred to them as "medical assassins" in the wake of the deaths of senior government officials from Covid-19. Medical Assassins— Nick Mangwana (@nickmangwana) January 24, 2021 The doctors' association said it was unfortunate that a government …
Central African Republic: Abdel Kani handed to ICC
25 January 2021 Africa, central Africa, In the news, Politics 0
The International Criminal Court says the Central African Republic has handed over a former rebel commander accused of war crimes and crimes against humanity. Mahamat Said Abdel Kani is charged with offences dating back to 2013. He was a prominent figure in the "Seleka" faction, a mainly Muslim coalition which …
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More 'extremely significant' archaeological discoveries made at Newgrange
George Morahan
Archaeologists have have made further 'extremely significant' discoveries around the great neolithic passage tomb at Newgrange following the discovery of a large prehistoric enclosure at the Co Meath site through drone photography this week.
Anthony Murphy and Ken Williams discovered a large crop marked enclosure at the heritage site that had become visible due to the recent dry conditions before reporting it to the government.
The persistently hot weather in recent weeks has led to many previously unknown and archaeological sites appearing as crop marks in fields across Europe.
Archaeologists have discovered a host of historic sites at Newgrange using drone photography. Pic: DCHG
Archaeologists from the Department of Culture, Heritage and the Gaeltacht conducted a further aerial survey on Friday, which resulted in 'the discovery of an array of very significant additional archaeological features in the vicinity across the River Boyne floodplain south of the Newgrange passage tomb,' according to Minister Josepha Madigan.
It is understood the features likely date back to the Neolithic period 5,000 years ago, and the scale of these discoveries is believed to be unparalleled in Ireland.
Further study of them will have the potential to transform experts' understanding of the Bru na Boinne archaeological landscape and Neolithic society. They will now be mapped to give historians a better understanding of their spatial relationship to each other and other monuments in the area.
The sites have become visible due to the hot, dry conditions. Pic: DCHG
Minister Madigan said: 'After my department being informed of a significant archaeological discovery last week near Newgrange, which we are very grateful to Mr Murphy and Mr Williams for reporting, my department 's National Monuments Service carried out an aerial survey on Friday to scan the Newgrange landscape for any additional features, in the knowledge that these rare ground conditions and crop mark phenomena affords a rare opportunity to uncover further secrets held in our landscape.
'The reconnaissance carried out Friday by archaeologists from our National Monuments Service has resulted in the discovery of an array of very significant additional archaeological features in the vicinity across the River Boyne floodplain south of the Newgrange passage tomb.
A range of similar discoveries have been made across Europe. Pic: DCHG
'With further research we know these fascinating finds will add greatly to our knowledge of the wonderful Bru na Boinne World Heritage landscape'.
Man, 20s, dies after horror crash involving truck and two cars in Meath
Body of woman in her 60s pulled from River Boyne
Woman drives car into back of another vehicle before screaming at driver
'Absolute heroes!' -- Gardai in Trim hailed after saving three-month-old baby's life


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BBC apologises over Jimmy Savile parody on CBeebies
Children's channel airs repeat of 2001 episode of The Tweenies featuring a character dressed as disgraced presenter
Screen grab from the Cbeebies show "Tweenies" which shows Max dressed like Jimmy Savile presenting a Top of the Pops-style show. Photograph: SWNS.com/ BBC
Sun 20 Jan 2013 08.51 EST
The BBC has apologised after screening a repeat of a children's television programme featuring a character dressed as Jimmy Savile.
An episode of The Tweenies filmed in 2001 was shown on CBeebies before 9am on Sunday.
In the episode, a character called Max presents a Top of the Pops-style show. Wearing one of Savile's trademark tracksuits topped off with a blond wig, he uses the disgraced presenter's accent and utters the catchphrase: "Now then, guys and gals."
A BBC spokesman said: "This morning CBeebies broadcast a repeat of an episode of The Tweenies, originally made in 2001, featuring a character dressed as a DJ impersonating Jimmy Savile. This programme will not be repeated and we are very sorry for any offence caused."
Viewers tweeted about the gaffe. Kenny Senior wrote: "Are BBC trying to self destruct? Max from Tweenies dressed as Jimmy Savile just now nearly chokes on my cornflakes."
A recent Metropolitan police report on allegations of sexual abuse against Savile, who died in 2011 aged 84, concluded he was a "prolific, predatory sex offender" who abused more than 200 people over a 60-year period.
Jimmy Savile


The Otaki Electorate Labour nominee Jade Teki says there has been a gaping hole for Labour since the electorate turned Blue.
"Our neighbours are enjoying the grassroots style representation that Labour is famous for in Iain Lees-Galloway, MP for Palmerston North and Kris Faafoi, MP for Mana. It has been wonderful have had Annette King, our "Buddy MP" go to such great efforts to not leave us out in the cold to freeze and turn completely blue. It's time to gather together, get warm and melt some cold blue hearts."
Ms Teki says the successful candidate will have some big shoes to fill with Peter Foster deciding not to stand again.
"There are so many considerations to be made when selecting a candidate. They must tick all the boxes, uphold Labour values, be an on the ground community advocate, be skilled and able to contribute at a high level and almost more importantly, to return the area to Labour they must be recognisable. Meaning, when they enter a room, you should know they have done so. If the candidate is bland they really have little chance of winning against Nathan Guy in 2014 or 2017."
"A strong and visible advocate for our region will see us not disappear into the abyss, without on the ground local representation for another three years. Grassroots democracy is about engagement, empowerment and participation, this is what is missing. It is time to claim it back," says Ms Teki.
"I believe the Otaki electorate is very winnable for Labour. We have a strong organisation and a dedicated team and the public want national out."
Ms Teki says if elected to be the candidate for Labour on December the 14th she will undertake to contact each Labour member registered in the Otaki Electorate regularly to discuss the issues that are important to them. She says an advocate for the people has to reach out past their own inner circle to ensure they have an accurate overview of the entire region, the people they represent and the issues that are important to them.
"With a vast area such as the Otaki Electorate it is important to understand that our issues differ for each town. Coming from a provincial town, Wanganui, and choosing to uproot my family in 2012 to join the region I have a firm grasp on community issues and I feel it would be an attribute to know that I have no bias and my time and energy would be distributed throughout the entire electorate," says Ms Teki.
Selection day is Saturday the 14th of December, 3pm at Otaki College Hall. There is expected to be a good turn out with Labour energised and well prepared for the 2014 election.


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Stéphane Lucas peut désigner :
 Stéphane Lucas (1978-), footballeur français.
 Stéphane Lucas (19?-), boxeur, mannequin et chanteur français.


Both of these are issued by Broadland District Council.
This is needed if you run a scrap metal business from a fixed business premises. A site manager must be named for each site.
A site licence also allows you to transport scrap metal to and from sites in any local authority area.
A collector's licence allows you to operate as a collector in the area of the issuing local authority. It does not allow you to collect scrap metal in any other local authority area. For example, if you get a licence in the Broadland District Council area, you can't then operate in Norwich City Council's area unless you get a separate licence.


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Kickoff to Disneyland's Diamond Celebration: 60 Facts and Secrets about the Disneyland Resort
As a Disney fanatic, I find myself wandering the internet quite often looking for various facts about the wonderful world that is Disney. Searching for these random facts, is something I do to occupy my time when I am bored. In honor of the kickoff of Disneyland's 60th Diamond Celebration, I decided to put together a list of 60 interesting facts and secrets about the park that started it all.
1. Walt Disney decided to build a theme park after taking his daughters to a merry-go-round at L.A.'s Griffith park, what a sweet inspiration!
2. There is a basketball court inside of the Matterhorn. I've always wanted to secretly sneak in there and shoot some hoops!
3. It cost $3.50 to get into Disneyland when it first opened. Wouldn't we love to be able to pay that now?! Although, park guests still had to buy tickets for the rides, like you would do today at a fair or carnival. I'm glad the ticket system for the rides are a thing of the past.
4. All of the plants in Tomorrowland are edible. Maybe the imagineers envisioned this as a futuristic detail that we will have one day, this is definitely an odd one!
5. On opening day at Disneyland, the asphalt did not set in time! Apparently, the high heel shoes of the women that were visiting that day were sinking into the ground!
6. Doritos were invented at Disneyland. Cast members came up with this idea when they were trying to figure out what to do with leftover stale tortilla chips.
7. Walt's Main Street firehouse apartment: Walt loved being at Disneyland and being able to watch everyone enjoy the parks. He had an apartment built for him and his family overlooking Main Street located in the fire station. When he was in the park, he would light a lantern to let everyone know he was there. However, now it is lit at all times to signify his constant presence.
8. Whenever the water rides are drained and cleaned, the coins thrown in the water are collected and donated to charity.
9. Keep Disneyland clean! Trash cans are strategically placed within 30 feet of you at all times. Walt had researchers do a study to see on average how many steps it took before a person would drop trash on the ground, apparently 30 ft. was the magic number.
10. Harold the Snowman: The abominable snowman inside the Matterhorn is named Harold!
11. Working drawbridge: Sleeping Beauty's castle has a working drawbridge that has been lowered twice in its history. Once at opening day and again at the unveiling of the Fantasyland remodel in 1983.
12. There is a special railroad car on the Disneyland Railroad named after Walt's wife, Lillian. It is the caboose car and it's called Lily Belle Presidential Train Car. However, access to riding this car is limited to special events and mostly VIP guests.
13. The Hollywood Tower Hotel is the tallest building in Anaheim. It stands a whopping 183 feet tall.
14. Steve Martin used to work at the Magic Shop on Main Street before he made it big in Hollywood, wouldn't that have been a site to see!
15. Main Street USA is mirrored after Walt's boyhood hometown: Marceline, Missouri.
16. Disneyland's address is 1313 Harbor Blvd., which is code for Mickey Mouse's initials. Since "M" is the 13th letter of the alphabet, 1313 stands for "MM".
17. Mainstreet USA and Tomorrowland are set in 1910 and 1986, both of the years in which Halley's comet was visible to Earth.
18. Mickey Mouse's birthday is November 18th, 1928.
19. Disneyland was built in a year.
20. The fastest teacups at the Mad Tea Party ride are the purple teacup or the orange with diamonds on it. For a slower route, choose the two teacups with hearts on them.
21. Originally, Walt envisioned Disneyland being built across from Walt Disney studios in Burbank, Ca.
22. Every year, 2.8 million churros are sold at Disneyland!
23. Feral cats live at Disneyland to keep rodents such as rats and mice under control. Ironic isn't it?
24. Real human skeletons were used as props when Pirates of the Caribbean first opened. Imagineers felt the skeletal props were not realistic at the time, so they purchased real ones from UCLA Medical School. There is still a real skeleton head located on the headboard in one of the scenes.
25. Three babies have been born at Disneyland!
26. King Arthur's Carousel is older than Disneyland. It was built in 1922 for Sunnyside Beach Park in Toronto.
27. The Matterhorn was the first tubular steel roller coaster in the world.
28. Pirates of the Caribbean was the last ride Walt supervised the construction of.
29. Splash Mountain was originally going to be called Zip-a-Dee River Run.
30. Walt originally wanted live animals in the Jungle Cruise ride! Looks like his vision came true in a big way at Animal Kingdom.
31. Disneyland has the 4th largest Navy in the world based on its ships.
32. The fastest you'll ever go on Space Mountain is 35 mph. Fans that are installed throughout the ride make you feel like you are going faster than you really are.
33. "Smellitizers" are placed throughout the parks to enhance your experience. The machines spray smells into the air according to each land or ride you're in. For example, on Main Street during Christmas, sweet smells of peppermint fill the air.
34. Cast members display their first name only on their name tags, due to the fact that Walt despised being called Mr. Disney.
35. You can drive the Mark Twain river boat if you are lucky enough! Ask a cast member before you board the vessel if you can visit the wheel house. From there if luck is on your side, they'll lead you to the captain and you'll be able to drive the boat!
36. 11-year-old George Lucas visited Disneyland with his family on opening day. I'm sure he never fathomed the ongoing relationship he'd have with Disney one day!
37. All costume characters cannot answer or acknowledge anything about the outside/real world.
38. There used to be "live mermaids" (actresses in mermaid fins) that swam in the area where Finding Nemo's Submarine Voyage is located now. I would've loved that job!
39. Disneyland cost $16 million to build.
40. "It's A Small World" represents 100 different regions from around the world.
41. King Arthur's Carousel has 68 horses on board, each have their own name and no two are alike.
42. Disneyland's monorail stretches 2.5 miles and takes 9 minutes to ride round trip.
43. An estimated 400 items are collected at Disneyland's Lost and Found each day.
44. There is a "no sad kids rule" at the parks that all cast members must abide by. Whether it's a dropped ice cream cone, popped balloon, lost toy, etc. Cast members are told to replace the items at no cost. After all, it is the happiest place on earth!
45. California Adventure was originally going to be called Westcot, the west coast version of Epcot. For budget reasons, this plan never went through.
46. Cast members are not allowed to point with one finger, it is considered rude in some cultures. Therefore, they either use the "two finger point" or guide with their whole hand.
47. The Matterhorn attraction is 1/100th of the size of the real Matterhorn in Switzerland.
48. California Screamin' reaches a top speed of 55 mph, the fastest at Disneyland Resort.
49. Radio Spring Racers cost $200 million dollars to build, which is the most expensive ride out of any theme park! Even with inflation adjustments, that is a considerable amount more than the $16 million it took to build ALL of Disneyland in 1955.
50. There were 18 attractions on opening day at Disneyland. Today the resort has 92 attractions.
51. There is a fake pet cemetery at the Haunted Mansion.
52. You used to be able to fish on Tom Sawyer's Island.
53. Disneyland spends $42,000 a night on fireworks.
54. Disneyland has a strict no fly zone, except for Tink and Dumbo of course!
55. It costs $25,000 to join the exclusive Club 33 located in New Orleans Square, and it's around a 10 year wait to become a member!
56. Water attractions have brown and green water to hide the vehicle track system and filtration. They are always trying to preserve the magic!
57. Disneyland is older than 110 different countries.
58. The reason you get a stamp on your left hand when you park hop, is because Walt was left-handed.
59. Disneyland could fit into Magic Kingdom's parking lot in Florida.
60. Finally, my favorite "fact" from Walt. "Disneyland will never be completed. It will continue to grow as long as there is imagination left in the world."
– Kelsee
Tagged as: Disney facts, Disneyland, Disneyland 60th anniversary, Disneyland facts
Enjoy Disneyland Magic All-Year-Long: Picking an Annual Passport That is Right for You
Question of the Week (5/24/15)
Christopher Lampton says:
Hope everybody caught the live video feed of the beginning of the Celebration. We Chromecasted it onto our widescreen TV. The Paint the Night parade and the new fireworks with projections are spectacular. We have annual passes and plan to go next Thursday and at least once the following week before us SoCal Select people get blocked out for the summer. (We're also moving down to Long Beach this weekend, only ten miles from the park, but that's just coincidence.)
Great trivia!
kelseekrueg says:
Thank you, Christopher! How exciting that you will get to see all the new additions for the celebration and be so close! Enjoy it for us here at Wandering in Disney until we get to see it for ourselves!


Well-maintained house with 4 bedrooms. The house looks like a brand new one and very clean. Spacious living room and family room. Closed to hw6 Freeway. Hurry and call us to make an appointment! Thank you for showing. READY TO MOVE IN NOW.