text:
\section{Introduction}
In a few recent studies, we have considered Kaluza-Klein reductions of 11-dimensional (11D) supergravity (SUGRA) over $AdS_4 \times S^7/Z_k$ when the internal space $S^7/Z_k$ is an $U(1)$ bundle on $CP^3$. As a result, we found localized or partially localized objects in the bulk of AdS$_4$, and explored their boundary CFT$_3$ holographic duals according to the well-known AdS/CFT correspondence rules (see \cite{Aharony99} as an original review). Following the special truncation considered in \cite{Me7} in probe approximation, here we include also the backreaction so that the truncation would be consistent.

In fact, after considering a general 4-form ansatz of the 11D SUGRA and facing our setups with those in \cite{Gauntlett03}, where a more general 4-form ansatz and truncation are analyzed, we obtain the equations of motion (EOM's) for the involved (pseudo)scalars in Euclideanized Anti-de Sitter space ($EAdS_4$) and get some solutions including the backreaction. Next, we take a specific 4-form ansatz and obtain a second-order nonlinear partial deferential equation for an included (pseudo)scalar that could be massive, massless or tachyonic and is self-interacting. To address the backreaction, we compute the energy-momentum (EM or stress) tensors of the Einstein equations because of the bulk flux turned on, and get a new set of scalar equations, in external and internal spaces, which must in turn be satisfied to insure that the resulting solution or object does not backreact on the background geometry. By solving them together with the main bulk equation, we see that to have instantons we have to take an exactly marginal or a marginally relevant deformation on the boundary \footnote{It is noteworthy that the exactly marginal solution, which is obtained by setting the bulk stress tensors to zero, might be attributed to a bulk (pseudo)scalar profile with $SO(4,1)$ symmetry--we have already analysed such massless modes in \cite{Me4}.}.

The resulting (pseudo)scalars are singlets of $SU(4) \times U(1) \equiv H$ (as the isometry group of the whole internal space) and, at least because of the associated (anti)M-branes wrapping around the mixed directions, break all supersymmetries (SUSY's) ($\mathcal{N}=6 \rightarrow 0$) according to the intersection rules of M-branes, for instance, in \cite{Bergshoeff96}. Meanwhile, we notice that a reasonable way to justify the SUSY breaking and get the needed singlet operators on the boundary theory, is to swap the fundamental representations (reps) of $SO(8)\equiv G$ (as the isometry group of $S^7$) for supercharges, fermions and scalars; and then discuss how we can get the wished $H$-singlets under the branching $G \rightarrow H$. In addition, the mass term in the bulk action breaks the scale invariance (SI) and that, although the bulk solution including the backreaction is scale-invariant in leading order, a relevant or mass term beside a marginal term in the boundary Lagrangian breaks the conformal $SO(4,1)$ symmetry.

Indeed, as a dual description for the bulk solution including the backreaction, we first consider a scalar Lagrangian with a marginal deformation term (or a triple-trace deformation of a relevant dimension-one operator) plus a so-called mass term that meet our needs well; see \cite{Elitzur05}. Next, we make instanton solutions for the massless case and argue that they should be $SO(4)$-invariant on $S^3$ (or $SO(3,1)$ on 3D de Sitter space-time $dS_3$ after the Lorenz analytical continuation), cause vacuum instability and are dual to big crunch singularities in the bulk. The solution's moduli $a$ and $\vec{u}_0$ mark size and location of the boundary instanton and of a thin-wall bubble that may nucleate everywhere in the bulk. The boundary duals might be realized in 3D Chern-Simon (CS) $U(N)$ and $O(N)$ vector models, although we continue to use the former model \footnote{The ABJM model \cite{ABJM} in large $k$ reduces to a 3D $O(N)$ vector model and on that basis, a marginal triple-trace deformation with a well-known Ultra-Violet (UV) fixed-point at $g_6^*=192$ is studied in \cite{CrapsHertog}; we return to this issue in subsection \ref{subsec.5.2}.}. Especially, we focus on an $U(1)$ part of the ABJM quiver gauge group with matters in fundamental reps of $SU(4)$, because of symmetry arguments and correspondence rules.

The organization of this paper is as follows: In section 2, we introduce the background and a general 4-form ansatz of 11D SUGRA. Then, we obtain its corresponding equations and solutions in $EAdS_4$ space, without and with including the backreaction, in Appendix A from facing our ansatz with that in \cite{Gauntlett03}. In section 3, we use a special version of the general ansatz, which results in an interesting bulk equation; and to get the solutions including the backreaction, we first compute the associated EM tensors of the Einstein equations in Appendix B (with some useful formulas used in computations in subappendix B.1) and then write the EOM's by zeroing both the external and internal components of the stress tensors. As a result, we obtain the main solution and conditions arisen from solving the latter equations and the main one in the bulk simultaneously in subsection 3.1. In subsection 3.2, we evaluate the correction to the background 11D action based on the solution including the backreaction. In section 4, we discuss the symmetries of the bulk setup and solutions and argue how they can help to fulfill the state/operator correspondence and find the correct boundary counterparts. Section 5 deals with the field theory duals of the main bulk solution including the backreaction. There, we present a suitable dual Lagrangian, find a plain solution and interpret implications of it. In particular, we discuss the relation of the latter setup to a massive deformation of the ABJM model and other issues such as SI breaking in subsection 5.2. In section 6 is a summary and more comments especially on the instability and false vacuum decay because of the instanton.

\section{The background and Genaral Ansatz} \label{sec02A}
The background we consider is
\begin{equation}\label{eq01A}
 \begin{split}
 & ds^2_{11D}= {R_{AdS}^2}\, ds^2_{AdS_4} + R_7^2\, ds^2_{S^7/Z_k}, \\
 & G_4^{(0)}= d\mathcal{A}_3^{(0)} = 3\, R_{AdS}^3\, \mathcal{E}_4 = N \mathcal{E}_4,
 \end{split}
\end{equation}
where the geometry is $AdS_4 \times S^7/Z_k$ of 11D SUGRA with $S^7$ as a $S^1$ fiber-bundle on $CP^3$, and the 4-form ansatz is that in ABJM \cite{ABJM}.
The general 4-form ansatz we are considering here is
\begin{equation}\label{eq01}
 \begin{split}
G_4 = & \left( 3 f_1 + R_{AdS} \ast_4(df_2 \wedge \mathcal{A}_3^{(0)}) \right) \wedge J^2 - R_{AdS}^{-1} \left( df_3 - f_4\, \ast_4 \mathcal{A}_3^{(0)} \right) \wedge J \wedge e_7  \\
& + \frac{1}{4 R_{AdS}^3} \ast_4(df_5 \wedge \ast_4 \mathcal{A}_3^{(0)}) \wedge J + \frac{3}{16 R_{AdS}^5} \ast_4 df_6 \wedge e_7 + \frac{3}{64 R_{AdS}^3}  f_7\, \mathcal{E}_4,
 \end{split}
\end{equation}
where $f_1, f_2, \ldots$ are scalar functions in the external space, $R=R_7= 2 R_{AdS}$ is the $AdS$'s radius of curvature, $\mathcal{E}_4$ is the unit-volume 4-form on $AdS_4$, the 2-form $J=d \omega$ is the K\"{a}hler form on $CP^3$, and $e_7 = (d\acute{\varphi}+\omega)$ with $\acute{\varphi}$ as the fiber coordinate.

For the general 4-form anstaz (\ref{eq01}), we have derived the equations and conditions arising from satisfying the Bianchi identity ($dG_4=0$) and the Euclidean EOM
\begin{equation}\label{eq01B}
d \ast_{11} G_4 - \frac{i}{2}\, G_4 \wedge G_4=0
\end{equation}
in \cite{Me7}. However, it is also interesting to discuss the backreaction, that is considering the Einstein's equations as well; and of course we have done it in Appendix \ref{Appendix.AA}, where equations and solutions without and with including the backreaction are analysed in accordance with computations in \cite{Gauntlett03} where a more general ansatz is employed.

\section{Special 4-Form Ansatz and Solutions} \label{sec03}
We employ a special 4-form ansatz from (\ref{eq01}), made of the plain forms $e_7, J, \mathcal{E}_4$, whose clear form reads
\begin{equation}\label{eq11}
\frac{\tilde{G}_4}{(2 R_{AdS})^4} = 8\, \bar{f}_1 J^2 - 2\, df_3 \wedge J \wedge e_7 + \frac{3}{8} f_7\, \mathcal{E}_4,
\end{equation}
where $f_1 N =\bar{f_1}$; and the resulting EOM becomes
\begin{equation}\label{eq12}
     \ast_4 d \left(\ast_4 d\bar{f}_1 \right) - \frac{4}{R^2} \left(1 \pm 3\, \bar{C}_7 \right) \bar{f}_1 - 2 \times 192\, \bar{f}_1^3=0,
\end{equation}
where for different values of $\bar{C}_7$, towers of tachyonic, massless (with $\bar{C}_7 =\frac{1}{3}$) and massive (pseudo)scalars in the bulk of $EAdS_4$ are possible.

It is interesting to compare the ansatz (\ref{eq11}) with (2.5) in \cite{Gauntlett03}. As a result, we see that with $U = V = \chi = A_1 = B_1 = B_2 = 0$ and
\begin{equation}\label{eq13a}
     f = 6\, f_7, \quad h= 4\, R^4 \bar{f}_1, \quad dh= - R^4\, df_3,
\end{equation}
the formalisms match, and the counterpart of (2.23) in \cite{Me7} is
\begin{equation}\label{eq13b}
     f = \frac{6}{R^7} \left(\epsilon + h^2 \right), \quad \epsilon =\pm\, \bar{C}_7\, R^6,
\end{equation}
which comes from the equation (B.11) of \cite{Gauntlett03}. In particular, we read from (B.13)
\begin{equation}\label{eq13c}
     \ast_4 d (\ast_4 dh)- (16 + 24 \epsilon)\, h - 2 \times 12\, h^3 =0,
\end{equation}
which is the same as  (\ref{eq12}) up to some scaling and notice that $R=1$ is set. Then, from the latter equation, we read $m^2 R_{AdS}^2=-2$ with $\epsilon = -1$ (skew-whiffed) and $m^2 R_{AdS}^2=10$ with $\epsilon = 1$ (Wick-rotated) \footnote{Remember that with $\eta = 2 e_7$, the modes in (\ref{eq13c}) match with those in (\ref{eq12}).}, which were already discussed in \cite{Me6} and \cite{Me7} respectively. \\
To continue, we note that considering only the EOM (\ref{eq12}) means working in probe approximation, that is excluding the backreaction, for which we presented some solutions in \cite{Me7} with dual descriptions. Here we include the backreaction as well.

\subsection{Solutions Including the Backreaction} \label{subsec03.01}
To get solutions including the backreaction, we should first compute the stress-energy tensors of the replying Einstein equations. The details of such computations are given in Appendix \ref{Appendix.A0}, where (\ref{eq20a}), (\ref{eq20b}) and (\ref{eq20c}) come from zeroing the external, internal and seventh components of the EM tensors, respectively. Next, by combining the latter equations with the main one (\ref{eq12}), we arrive at
\begin{equation}\label{eq23a}
     \Box_4 \bar{f}_1 =0,
\end{equation}
\begin{equation}\label{eq23b}
     \Box_4 \bar{f}_1 + \left(8 \pm 12\, \bar{C}_7 \right) \frac{\bar{f}_1}{R^2}=0,
\end{equation}
\begin{equation}\label{eq23c}
     \Box_4 \bar{f}_1 + \frac{8}{3 R^2} \bar{f}_1=0,
\end{equation}
respectively, which are conditions imposed on the (pseudo)scalar from including the backreaction on the background geometry of the external and internal spaces. In other words, if we find solutions to these equations, the corresponding objects (e.g. instantons as topological objects) do not backreact on the background geometry.

However, satisfying (\ref{eq23a}), (\ref{eq23b}) and (\ref{eq23c}) simultaneously results in $\bar{f}_1=0$; Still, we may discuss each of them separately. In particular, the solution
\begin{equation}\label{eq24a}
      \bar{f}_1(u,\vec{u}) = c_{8} + \frac{c_{9}\, u^3}{\left[u^2 + (\vec{u}-\vec{u}_0)^2 \right]^3}
\end{equation}
of (\ref{eq23a}), which corresponds to a \emph{marginal} deformation with the boundary operator $\Delta_+ =3$, does not backreact on the external space geometry.

On the other hand, satisfying (\ref{eq23b}) and (\ref{eq23c}) simultaneously, which means taking the backreaction in the whole internal space into account, again results in the trivial solution $\bar{f}_1=0$; but if we take the massless solution (\ref{eq24a}), which is in turn realized with $\bar{C}_7 = \frac{1}{3}$ (given that $(1 \pm 3\, \bar{C}_7)={m^2} {R_{AdS}^2}$) in the skew-whiffed version of (\ref{eq12}), in the internal $CP^3$ space equation (\ref{eq23b}), we will have
\begin{equation}\label{eq24b}
 \Delta_{\pm} = \frac{3}{2} \pm \frac{\sqrt{5}}{2},
\end{equation}
which corresponds to a \emph{marginally relevant} operator with $\Delta_+ < 3$ (and the same behavior for \ref{eq23c}); That means for this solution does not have any backreaction on the background metric, one must take a marginally relevant deformation in the corresponding boundary theory; We return to this interesting case when discussing dual solutions in section \ref{sec.5}. \footnote{It is also interesting to consider solutions for each of the equations (\ref{eq23a}), (\ref{eq23b}) and (\ref{eq23c}) separately, in the same way done in subsection (2.4) of \cite{Me7} for the equation (\ref{eq12}).}

\subsection{Correction to the Action} \label{subsec03.02}
In is also interesting to compute the 11D action correction based on the solution including the backreaction. To this end, we use the 11D SUGRA action in Euclidean space as
\begin{equation}\label{eq25}
  S_{11D}^E = -\frac{1}{2 \kappa_{11}^2} \left[ \int d^{11}x \, \sqrt{g} \, \mathcal{R}_{11} + \frac{1}{2} \int \left(\tilde{G}_4 \wedge \ast_{11} \tilde{G}_4 - \frac{i}{3} \tilde{\mathcal{A}}_3 \wedge \tilde{G}_4 \wedge \tilde{G}_4 \right) \right],
\end{equation}
where $\mathcal{R}_{11}$ is the Ricci scalar and $\kappa_{11}^2 = 8\pi \mathcal{G}_{11}$ with $\kappa_{11}$ as the gravitational constant. To evaluate the correction, we should employ (\ref{eq11}) for $\tilde{G}_4$ and its 11D dual as
\begin{equation}\label{eq26a}
  \frac{\tilde{G}_7}{(2 R_{AdS})^7} = \bar{f}_1\, \mathcal{E}_4 \wedge J \wedge e_7 + \ast_4 df_3 \wedge
  J^2 +  f_7\, J^3 \wedge e_7,
\end{equation}
and that
\begin{equation}\label{eq26b}
\tilde{G}_4 = d\tilde{\mathcal{A}}_3, \quad \tilde{\mathcal{A}}_3 = \tilde{\mathcal{A}}_3^{(0)} + (8\, R^8) (\bar{f}_1\, J \wedge e_7), \quad \tilde{G}_4^{(0)} = d\tilde{\mathcal{A}}_3^{(0)} = \frac{3}{8} R^4 f_7\, \mathcal{E}_4.
\end{equation}
Next, plug (\ref{eq26a}) and (\ref{eq26b}) into (\ref{eq25}) together with
\begin{equation}\label{eq27}
 df_3 = -4\, d\bar{f}_1, \quad  f_7 = +i\, 32\, R\, \bar{f}_1^2 \pm i\, \frac{\bar{C}_7}{R},
\end{equation}
which in turn come from (\ref{eq13a}) and (\ref{eq13b}), the right part of the action for us (the second and third terms) becomes $\bar{S}_{11D}^E = S_0 + S_{11}^{modi.}$, where
\begin{equation}\label{eq28a}
  S_0 = \frac{9}{R^2\, \kappa_{11}^2} \texttt{vol}_4 \wedge \texttt{vol}_7, \quad \texttt{vol}_7 = \frac{R^7}{3!} \int  J^3 \wedge e_7 = \frac{\pi^4\, R^7}{3\, k},
\end{equation}
is from the ABJM background realized with $\bar{C}_7=1$, and
\begin{equation}\label{eq28b}
  S_{11}^{modi.} = \frac{3\, R^4}{2 \kappa_{11}^2}\, \texttt{vol}_7 \int \bigg(8\, R^2\, \bar{f}_1^2\, \mathcal{E}_4 + 32\, d\bar{f}_1 \wedge \ast_4 d\bar{f}_1 + 384\, R^2\, \bar{f}_1^4\,  \mathcal{E}_4 \bigg).
\end{equation}
Then, by putting the solution (\ref{eq24a}) with $c_8=0$ in the latter action and taking (see \cite{Me6})
\begin{equation}\label{eq29}
  \mathcal{E}_4 = \frac{1}{u^4}\, dx \wedge dy \wedge dz \wedge du, \quad \kappa_{11}^2 = \frac{16\, \pi^5}{3} \sqrt{\frac{R^9}{3\, k^3}},
\end{equation}
we arrive at, the correction in the unit 7D internal volume,
\begin{equation}\label{eq28c}
  S_{corr.} = \frac{9\, c_9^2}{32\, \pi^3} \sqrt{3\, k^3\, R^3} \left[\frac{35}{48} \frac{1}{\epsilon^6} + \frac{4199}{8192} \frac{c_9^2}{\epsilon^{12}}\right],
\end{equation}
as the finite part of the action, where $\epsilon>0$ is a cutoff parameter used instead of $u=0$ to evade the infinity of integrals with respect to (wrt) $u$.

\section{Dual Symmetries and Correspondence} \label{sec.04}
We first remind that the ansatz (\ref{eq11}) and the (pseudo)scalars therein are singlets of $SU(4) \times U(1)$ and so, we look if we can find the wished singlet (pseudo)scalars in the spectrum of the involved 11D SUGRA over $AdS_4 \times CP^3 \ltimes S^1/Z_k$. This task was already done in \cite{Me7}, where we considered three massive (pseudo)scalars. But, for the solution including the backreaction, the (pseudo)scalars should be almost massless and so, we should look whether we can find any singlet massless (pseudo)scalar in the spectrum or not.

To this end, we first note that the massless multiplet ($n=0$) of $G$ (as the isometry group of $S^7$) includes a graviton ($\textbf{1}$), a gravitino ($\textbf{8}_s$), 28 spin-1 fields ($\textbf{28}$), 56 spin-$\frac{1}{2}$ fields ($\textbf{56}_{s}$), 35 scalars ($\textbf{35}_{v}$) of $0_1^+$ emerging from the external ingredients ($\mathcal{A}_{\mu \nu \rho}$), and 35-pseudoscalars ($\textbf{35}_{c}$) of $0_1^-$ emerging from the internal ingredients ($\mathcal{A}_{m n p}$). In the higher KK multiplets ($n>0$), the massless pseudoscalar sets in $\acute{\textbf{840}}_{s}$ of $0_1^-$ with $n=2$ while the massless scalar sets in $\textbf{1386}_{v}$ of $0_1^+$ with $n=4$ of $G$ (see, for instance, \cite{FreedmanNicolai}, \cite{NilssonPope} and  \cite{Biran}), and there is not any $H$-singlet under the branching $G\rightarrow H$.

On the other hand, we read from the ansatz structure that it breaks all SUSY's because of the mixed internal directions around which the associated (anti)M-branes wrap-- see also \cite{DuffNilssonPope84} that states the solutions with 4-form components all in the internal space break SUSY's and parity; and as a result, the boundary duals might be realized in CS-matter $U(N)$ and $O(N)$ vector models. Thus, a consistent way to meet this need is to swap the fundamental reps $\textbf{8}_s$, $\textbf{8}_c$ and $\textbf{8}_v$ of $SO(8)$. On it, after swapping $\textbf{s} \leftrightarrow \textbf{v}$ with $\textbf{c}$ rep fixed (that means exchanging supercharges(spinors) with scalars while keeping the fermions unchanged), we have the rep $\textbf{1386}_{s}$ whose $U(1)$-neutral reps under the branching read
\begin{equation}\label{eq30}
      \textbf{1386}_{s} \rightarrow \textbf{1}_{0}\oplus \acute{\textbf{20}}_{0} \oplus \textbf{105}_{0} \oplus \textbf{336}_{0} \oplus ...\ ,
\end{equation}
which include an $H$-singlet mode. On the other hand, for the massless pseudoscalar of the original model ($\acute{\textbf{840}}_{s}$), there is not any singlet under $H$ even after both swappings.

As another point, we note that the ansatz breaks the inversion symmetry and thus conformal transformation of $SO(4,1)$ (as isometry of $EAdS_4$) besides the fact that the mass term in the bulk equation (\ref{eq12}) breaks the SI; and as a result, we argue that our solution must be $SO(4)$ invariant; see \cite{Me7} for more details. It is also notable that although the resulting equation (\ref{eq23a}) and main solution (\ref{eq24a}) preserve full conformal symmetry, the marginally relevant deformation breaks the SI as we will discuss in the next section.

\section{Boundary Field Theory Duals} \label{sec.5}
For the general ansatz (\ref{eq01}), the scalar profiles of the forms (\ref{eq03a}) and (\ref{eq03d2}) are already discussed in \cite{Me4} and \cite{N} respectively and so, we do not pay more attention to them. Instead, we focus on the duals for the solutions including the backreaction. \\
Indeed, from the bulk description with backreaction in subsection \ref{subsec03.01}, we see that the dual boundary operator have to be for an exactly marginal or a marginally relevant deformation. Although in \cite{Me4}, \cite{Me5} and \cite{Me3} we studied samples of marginal operators and deformations, here we concentrate on a special sample of the (exactly) marginal deformation $\Delta_+=3$, valid as an approximation for another case too, and study aspects of it. In our formalism, we make this operator from the singlet (pseudo)scalar after the swapping $\textbf{8}_s \leftrightarrow \textbf{8}_v$ of the original supersymmetric theory. On the other hand, besides breaking SUSY's, the $H$-singlet states break the even-parity symmetry of the ABJM model, which might in turn be understood through fractional M2-brane \cite{ABJ} associated with the probe (anti)M5-brane wrapping around $R^3 \times S^3/Z_k$. As a result and after gauging, we remain with just the diagonal $U(1)$ part of the quiver gauge group $SU(N)_k \times SU(M)_{-k}$, for which we set $A_i^-=0$, as it is for the boundary baryonic symmetry under which our modes are singlet \footnote{It should be noted that the singlet sections of $U(N)$ and $O(N)$ CS vector models, with complex and real (pseudo)scalars in fundamental reps, are dual to the non-minimal and minimal Vasiliev higher-spin (HS) bulk theories (see \cite{Giombi01} for a review), respectively, with parity breaking scheme \cite{Aharony1110}, \cite{Minwalla1207}. As a result, our setups may be recast in that framework; For instance, our bulk massive modes may be considered as arisen from some loop corrections or interactions in the minimal HS model or as fluctuations about the minimal solution of $m^2 R_{AdS}^2=-2$.}. In other words, the fundamental fields are neutral to the diagonal $U(1)$ that couples to $A_i^+ \equiv (A_i  + \hat{A}_i)$ while $A_i^-$ acts as the baryonic symmetry and, since our (pseudo)scalars are neutral, we assign zero to it.

By the way, we could consider the marginal deformation as a triple-trace deformation \footnote{It is notable that with mixed boundary conditions, corresponding to multi-trace deformations, the effective action reads ${\Gamma}_{eff.} [\alpha] = S_{on}[\alpha] + \int d^3 \vec{u}\ \tilde{f}(\alpha)$, where $S_{on}$ is the bulk on-shell action and $\tilde{f}(\alpha)$ is a function of the local operator $\mathcal{{O}}_{\Delta_-}$ with which the action is deformed. In general, the mixed boundary conditions lead to conformal field theories only if $\tilde{f}(\alpha) \sim \alpha^{3/\Delta_-}$ or $\beta = f_0\, \alpha^{({3}/{\Delta_-})-1}$, where $\alpha$ and $\beta$ act as vacuum expectation value and source for the operator $\Delta_-$ and conversely for $\Delta_+$, and different values of $f_0$ correspond to various points along the lines of marginal deformations. \label{ftn.10.}} of the operator $\mathcal{O}_1 \sim \texttt{tr}(y \bar{y})$ used in \cite{Me5}. The most recognized case with the latter operator is the $O(N)$ vector models made of it and its multi-trace deformations. The interesting case is the tri-critical model, which includes just the kinetic term and a term proportional to a triple-trace deformation of it; see for instance \cite{Elitzur05}. With respect to the bulk studies, as a reasonable proposal consistent with our discussions, we match the bulk to boundary field as $\bar{f}_1 \rightarrow \varphi^2$ that in turn means the bulk instantons are square of the boundary ones \cite{deharo06}, in leading order of course. As a result, from the EOM (\ref{eq12}), we can take a dual Lagrangian for the boundary theory. Fortunately, a form for such an effective Lagrangian is already known (see \cite{Elitzur05} and \cite{deHaro2}) and follows our wishes. In fact, next to the CS term (here we continue to use the $U(N)$ model), we can consider \footnote{Note that we use the metric signature $(-, +, +,\ldots +)$ for both gravity and field theories; and after Wick-rotation, we reach to the fully positive signature metric along with $t_M \rightarrow i t_E$ and $e^{-i S_M} \rightarrow e^{-S_E}$, where $S_E$ is the positive Euclidean action.}
\begin{equation}\label{eq31}
     \mathcal{L}_{3}^{eff.} = \frac{1}{2} (\partial_i \varphi)^2+ \frac{1}{16} \mathcal{R}_3\, \varphi^2 - \frac{\lambda_6}{6 N^2}\, \varphi^6,
\end{equation}
where $\mathcal{R}_3$ is the boundary 3D scalar curvature that is $\frac{6}{R_0^2}$ for the three-sphere ($S^3$) of the radius $R_0$ \footnote{In fact, wrt the footnote \ref{ftn.10.} and fact that the solution (\ref{eq24a}) could be interpreted as a marginal triple-trace deformation, we can read the holographic effective action $\tilde{\Gamma}_{eff.} [\alpha]$, from the bulk analysis as shown in \cite{deHaro2}, which agrees with (\ref{eq31}).}. Such a Lagrangian is actually used (with $\lambda_6 > 0$) as a dual to describe the dynamics of Coleman-de Luccia large-expanding vacuum bubbles of $AdS_4$ in thin-wall approximation; see \cite{Barbon1003} and \cite{Maldacena010} for related discussions. In our setup, these shells might emerge from the special (anti)M5-brane wrapping that results in domain-walls interpolating among the bulk vacua; see also \cite{Bena}. In other words, note that with $\lambda_6 > 0$, we have an unbounded potential from below signaling instability near the potential extrema, which in turn describes the bulk big crunch singularities. Indeed, the $O(3,1)$ invariant bubble solution includes an open and infinite Friedmann-Lema\^{\i}tre-Robertson-Walker universe inside $AdS_4$ space-time that collapses to a big crunch singularity. A field theory dual for the latter is obtained from a marginal triple-trace deformation of the ABJM model in \cite{CrapsHertog}; see also \cite{BzowskiHertog} for a recent related study.

\subsection{An Explicit SO(4)-Invariant Solution} \label{subsec.5.1}
Because the classical solutions for the EOM from the Lagrangian with different $\mathcal{R}_3$'s could be related by conformal transformations, to have a simple analytical solution, we consider a massless instanton on $S^3_\infty$(the three-sphere at infinity, $R_0\rightarrow \infty$) that is in turn equivalent to the solution on $R^3$ and so, we set the second term of (\ref{eq31}) to zero for now and provide complementary discussions in the next subsection \footnote{An analysis with a Lagrangian like (\ref{eq31}), when the boundary is defined on $S^3$, is presented in \cite{SmolkinTurok}.}. Equally and to comply with our formalism of using the ABJM boundary action, we set the fermions to zero with keeping only the $U(1)_{diag}$ part of the gauge group beside the scalar part; That is
\begin{equation}\label{eq32}
  \mathcal{\bar{L}}_{3} = \mathcal{L}_{CS}^+  - \texttt{tr} (\partial_i Y_A^{\dagger}\, \partial^i Y^A ) - V_{bos},
\end{equation}
where $\mathcal{L}_{CS}^+$ is the CS Lagrangian associated with $A_i^+$ and $V_{bos}$ is the bosonic potential of ABJM; see, for instance, \cite{Me4}. Then, to make a proper solution, we take the following ansatz
 \begin{equation}\label{eq33}
     Y^A = i\, \varphi(r)\, S^A, \quad  Y_A^\dagger = \varphi(r)\, S_A^\dagger, \quad S^A= S^B S_B^\dagger S^A - S^A S_B^\dagger S^B,
\end{equation}
where the presence of $i$ factor and using different $Y^A$ and $Y_A^\dagger$ are because of being in Euclidean space; and further set $Y^3=Y^4=0$. A clear solution for $S^1$ and $S^2$ reads \cite{Gomiz}, \cite{Terashima}
\begin{equation}\label{eq33a}
    (S_1^\dagger)_{m,n} = \sqrt{m-1}\, \delta_{m,n}, \quad (S_2^\dagger)_{m,n} = \sqrt{N-m}\, \delta_{m+1,n}.
\end{equation}
Thereupon, the scalar EOM of $Y_A^\dagger$ becomes
\begin{equation}\label{eq34}
    \partial_i \partial^i \varphi(r) + \frac{12 \pi^2}{k^2}\, \varphi(r)^5 =0,
\end{equation}
from which a solution reads
\begin{equation}\label{eq34a}
   \varphi(r)= \sqrt{\frac{k}{2 \pi}}\, \left( \frac{a}{a^2 + (\vec{u}-\vec{u}_0)^2} \right)^{1/2},
\end{equation}
which is Fubini-like \cite{Fubini1} and the $O(4)$-invariant one we have been looking for. Meanwhile, we notice that by the ansatz (\ref{eq33}), the gauge field equations are satisfied trivially. As a commentary, note that because of the SI in its UV fixed-point, the model admits an infinite family of instantons responsible to form the bulk vacuum bubbles, which in turn break the $AdS_4$ isometry to $SO(3,1)$. The broken generators, with four free parameters $a$ and $\vec{u}_0$ marking the size and position of the boundary instanton respectively, act to translate the bubbles around in the bulk 4D volume. Further, the finite contribution for the action on $S^3_\infty$ based on the solution (\ref{eq34a}), after linearization of $S^A$ matrices, reads
\begin{equation}\label{eq34b}
   \int_0^\infty \frac{r^2}{(a^2 + r^2)^2}\, dr = \frac{\pi}{4\, a} \ \ \Rightarrow \ \ S_{b} \cong -\frac{\pi}{2 k\, \lambda^2},
\end{equation}
where in the last term on the RHS we have used the large $N$ normalization coefficient $1/N^4$ for the marginal deformation term and $\lambda= N/k$.

Still, a subtle point is that as our bulk solutions are $H$-singlets, how we can use the ansatz (\ref{eq33}). To resolve this, we first note that $Y^A$ and $Y_A^\dagger$ may be considered as independent degrees of freedom in Euclidean space, which signals the parity breaking as well. Next, we notice that with $Y^3=Y^4=0$, one of the $SU(2)$'s is considered as a spectator and then, from one of the two complex scalar fields of the remaining $SU(2)$, we may write
 \begin{equation}\label{eq35}
     Y \equiv Y_1^\dagger + Y^1, \quad  Y^\dagger \equiv Y_2^\dagger + Y^2;
\end{equation}
Last, wrt the swapping $\textbf{s} \leftrightarrow \textbf{v}$ and with a linear combination of these, we can build the singlet scalars as
 \begin{equation}\label{eq35a}
     y = \varphi(r) (S_1^\dagger S^2 - S^1 S_2^\dagger), \quad \bar{y} = i \varphi(r) (S^1 S_2^\dagger - S_1^\dagger S^2).
\end{equation}

\subsection{Relation to Mass Deformation and SI Breaking} \label{subsec.5.2}
An ansatz like (\ref{eq33}) is already used as a tool to make the fuzzy $S^3$ solutions of the mass-deformed ABJM model; see \cite{Gomiz}, \cite{Terashima} and \cite{HanakiLin}. In fact, the mass deformation of the original ABJM model, associated with turning the bulk flux on, is done through a relevant operator, $\mathcal{L}_{mass}=\mu^2 \texttt{tr}(Y^A Y_A^\dagger)$, which breaks the global R-symmetry $SU(4)$ down into $SU(2) \times SU(2) \times U(1)$ while breaks the conformal symmetry $SO(3,2)$ entirely. The vacua of the mass-deformed model have interpretations as fuzzy three-spheres. In other words, the M5-branes are understood as M2-branes puffing into a fuzzy sphere near the M5-brane core. The fuzzy-funnels solutions describing M2-M5 brane intersections have interpretations in the field theory on M2-branes as domain-walls. It is interesting that the second term of the Lagrangian (\ref{eq31}) has a similar structure to this mass deformation term.

There also are other interesting points and discussions with the Lagrangian and our setup. First note that the potentials of $O(N)$ vector models in three dimensions are renormalizable in the large $N$ limit; and by including relevant and marginal terms up to the so-called $\varphi^6$, the potentials are stable for $0\leq g_6 \leq g_6^c =(4 \pi)^2$ and unstable for $g_6 > g_6^c$ and $g_6<0$ because of the potential unbounded from below--note that $g_6 \equiv -\lambda_6$ and that the sign of $g_6$ controls the behavior of the system in the classical case. There is the tri-critical model just with $g_6 (\varphi^2)^3$ for which the beta function is zero at $N\rightarrow \infty$, while $1/N$ corrections break the SI and a massless Goldstone boson (dilaton) is appeared as a dynamical bound-state of $(\vec{\varphi}.\vec{\varphi})$--we remind that for any finite and positive $g_6$, the operator becomes irrelevant quantum mechanically. In the next-to-leading order of the $1/N$ expansion, the dilaton gets a tachyonic mass and the spontaneously broken phase becomes unstable. In other words, in infinite $N$ limit, there is the UV fixed-point $g_6^c \cong 158$ and by increasing $g_6$, one reaches to the UV fixed-point $g_6^* = 192$, which in turn is an instability region where non-perturbative effects are dominated \cite{Bardeen1984}; see also \cite{Elitzur05}. As a result, the nonzero vacuum expectation value $\langle \varphi^2 \rangle \neq 0$ is equivalent to the massive deformation discussed above or a relevant deformation by the $\Delta_{-}=1$ operator. In addition, this marginal plus relevant deformations are consistent with the marginally relevant operator we have got in (\ref{eq24b}) by including the backreaction.

Further, according to the discussions in \cite{RabinoviciSmolkin011}, in the presence of CS term, only the dilaton causes a bounded potential and for $g_6 > g_6^c$ just the neutral states under $U(1)$ are formed; and this is the unstable phase we have met as well. Still, a more interesting discussion is in \cite{BardeenMoshe014}, where spontaneous breaking of SI in 3D $U(N)_k$ CS theories coupled to a scalar in fundamental rep is studied in the large $N$ limit. By adding a self-interacting term like $\lambda_6 (y^\dagger y)^3$, they have shown that there is a massive phase for a critical combination of $\lambda=N/k$ (the t'Hooft coupling) and $\lambda_6$; Indeed, for the tri-critical model, there is a spontaneous SI breaking for $\lambda^2 + \frac{\lambda_6}{8 \pi^2}=4$. Near the critical phase, the $U(N)$-singlet massive bound-state plays role as a pseudo-dilaton. Meanwhile, this phase of the SI breaking is dual to the parity broken version of the Vasiliev's HS theory in $AdS_4$ bulk \cite{Vasiliev01}.

\section{Final Comments}
In this study, we first considered a general 4-form ansatz of 11D supergravity over $AdS_4 \times CP^3 \ltimes S^1/Z_k$ and after comparing its structure with that in \cite{Gauntlett03}, analyzed the resulting equations and solutions, especially by including the backreaction. Next, making use of a special 4-form ansatz with the same settings, we obtained a particular nonlinear EOM, from the identity and equation of the 4-form, that included all massive, massless and tachyonic modes for an involved (pseudo)scalar in $EAdS_4$. Then, we referred to the Einstein equations and got the equations (\ref{eq20a}), (\ref{eq20b}), (\ref{eq20c}) from setting the bulk energy-momentum tensors to zero. Then, by solving them simultaneously with the bulk equation (\ref{eq12}), we obtained an instanton solution in subsection \ref{subsec03.01}, from the consistent truncation, corresponding to an exactly marginal or a marginally relevant deformation of the boundary theory. After that, we evaluated a correction to the main background supergravity action in subsection \ref{subsec03.02}. After an analysis of dual symmetries, we finally discussed the corresponding field theory counterparts, solutions and some related points.

In fact, for the boundary dual of the bulk solution with considering the backreaction, we employed a scalar Lagrangian including a marginal deformation term, which could be considered as a triple-trace deformation of a dimension-one operator, plus a relevant mass term of the latter operator, like those used in the standard $O(N)$ and $U(N)$ vector models. Then, considering the correspondence rules, we built a $SO(4)$-invariant Fubini-like instanton solution. The condition to break scale invariance, except by a boundary mass term, bulk interpretations of the solutions and other related issues were also addressed. In particular, we noticed that the marginal deformation in the CFT$_+$ fixed-point triggers an instability of CFT$_3$ on $S^3$ (or $dS_3$ in Lorentzian signature) \footnote{Note also that multi-trace deformations in general destabilize $AdS_4$ vacua, see \cite{HertogMaeda01}, although the triple-trace deformation here preserves the conformal invariance in leading order.}.

As the last comment, we notice that because of the special (anti)M-brane wrapping, there are domain-wall flows corresponding to thin-wall bubbles of $AdS_-$ (for the true vacuum) that expand exponentially within $AdS_+$ (for the false vacuum) and show Bose-Einstein condensations on $dS_3$ space-time. In the case that the condenses lead to the bulk crunches, the instability dynamics is described by finite $N$ corrections including formation and collisions of multi-bubbles; see \cite{Elitzur05} and \cite{Barbon1003}. In other words, a $AdS_-$ thin-wall bubble expanding within $AdS_+$ is equivalent to a flow between the UV (CFT$_+$) and IR (CFT$_-$) fixed points. Arriving the bubbles in a finite time to the boundary has a CFT interpretation as rolling in the potential of the unstable marginal boundary operator. Meantime, it is argued in \cite{BarbonRabinovici011} that the crunches are associated with \emph{negative energy falls} at least for marginally relevant operators. One should also note that although the constant-field arrangements $\bar{\varphi}=0$ and $\bar{\varphi} = \pm \left[{3}/{(4 R_0^2\, \lambda)} \right]^{1/4}$ are the local minimum and maximums of the potential $V(\varphi)=\frac{3}{8\, R_0^2}\, \varphi^2 - \frac{\lambda_6}{6 N^2}\, \varphi^6$  in (\ref{eq31}) respectively, and have \emph{bounce} nature, the bubble dynamic is however described by a field similar to $\varphi(r)$ in (\ref{eq34a}). For discussions on \emph{generalized Fubini instantons} and \emph{oscillating Fubini instantons} of double-hump potentials of this type, see \cite{BumHoonLee014A} and \cite{BumHoonLee014}. \\

\begin{appendices}

\section{Solutions from Matching with \cite{Gauntlett03}} \label{Appendix.AA}
In this Appendix, we discuss the equations, solutions and backreaction issue for the general ansatz (\ref{eq01}) briefly. To this end, we first face the formalism and reduction here for the metric and 4-form with $ds^2$ and $G_4$ given, respectively, in the equations (2.4) and (2.5) of \cite{Gauntlett03} \footnote{Note that our ansatz (\ref{eq01}) is a general one that could be constructed from the scalar functions in the external space next to the given forms $\omega, e_7$ and $\mathcal{A}_3^{(0)}$ of the background solution, while $G_4$ in \cite{Gauntlett03} includes more ingredients. Except for this overall likeness, we have our own objectives and to succeed them, we study the bulk modes, equations, solutions and other related topics as well as their field theory duals in details, none of which has been covered in \cite{Gauntlett03}.} that read
\begin{equation}\label{eq02a}
ds_{11}^2 = ds_4^2 + e^{2 U} ds^2 ({K E_6}) + e^{2 V} \left( \eta + A_1 \right)^2,
\end{equation}
 where $ds_4^2$ and $ds^2 ({K E_6})$ (a K\"{a}hler-Einstein metric) are equivalent to our full $AdS_4$ and $CP^3$ metric respectively, $U, V$ are scalar fields and $A_1$ is an 1-form on the external 4D space, and
\begin{equation}\label{eq02b}
 \begin{split}
 G_4^{(1)} = & 2 h\, J \wedge J + H_1 \wedge J \wedge  \left( \eta + A_1 \right) + H_2 \wedge J + H_3 \wedge \left( \eta + A_1 \right) + f\, \texttt{vol}_4 \\
   & + \sqrt{3}\, \left(\chi_1 \wedge \Omega + \chi\, \left( \eta + A_1 \right) \wedge \Omega + c.c. \right),
 \end{split}
\end{equation}
where $h, f$ are real scalars, $H_r\, (r=1,2,3,)$ are $r$-forms, $\chi_1$ is a complex 1-form and $\chi$ is a complex scalar all on the external 4D space, and $\Omega$ is the complex $(3,0)$ form on $CP^3$.

To adjust the formalisms, we should first take $e_7 \rightarrow 2 e_7 \equiv \eta$  and as a result $J \rightarrow 2J$, and set $U = V = \chi_1 = \chi = A_1 = B_1=0$. Next, from comparing the ansatzs and satisfying the Bianchi identity $d G_4=0$, we have
\begin{equation}\label{eq03a}
 H_1=dh=0, \quad f_1= f_2= c_1 + c_2\, u^3, \quad h=\frac{3}{2} c_1, \quad df_3 = f_4\, \ast_4 \mathcal{A}_3^{(0)},
\end{equation}
where $c_1, c_2, \ldots$  are constants of integration, $u$ is the horizon coordinate in the Poincar$\acute{e}$ metric
\begin{equation}\label{eq03aa}
 ds^2_{EAdS_4} = \frac{1}{u^2} \left(du^2 + dx_i ^2 \right),
\end{equation}
with $i=(1,2,3)$ for $(x,y,z)=\vec{u}$ respectively, and
\begin{equation}\label{eq03b}
 f= \pm \frac{6}{R^7} \epsilon, \quad  \texttt{vol}_4 =\frac{R^4}{16}  \mathcal{E}_4,
\end{equation}
where $\epsilon = f_7 = - \frac{i}{3} R^3$, with $i$ for being in Euclidean space \footnote{One should note that with $G_4 \rightarrow i G_4$, the Euclidean equation (\ref{eq01B}) goes to that in \cite{Gauntlett03}; and that with $\epsilon =1$ in \cite{Gauntlett0912}, the matching term is just the ABJM background solution $G_4^{(0)}$.} and the lower sign for the skew-whiffed solutions in general, and
\begin{equation}\label{eq03c}
H_2 = \frac{2}{R^3} \ast_4(df_5 \wedge \ast_4 \mathcal{A}_3^{(0)}), \quad H_3 = \frac{3}{R^5} \ast_4 df_6,
\end{equation}
where $H_2 = 2 B_2$, $H_3 = dB_2$ \footnote{It is remarkable that, according to the discussion after the equation (2.18) in \cite{Gauntlett03}, the massive ($m^2 R_{AdS}^2=12$) 2-form $B_2$ could be dualised to a pseudoscalar ($a$) with $\Delta_+ = 5$, which might in turn be identified with $f_5$ we have considered in \cite{Me7}.}. Note also that to satisfy the Bianchi identity $dH_3 =0$ and $dH_2 =2 H_3$, the conditions read \footnote{Note that if we set $f_5$=0 in the main ansatz (\ref{eq01}), we have a massless (pseudo)scalar in $AdS_4$ that is already studied in \cite{Me4}.}
 \begin{equation}\label{eq03d}
 f_5=-\frac{3}{2 R} f_6, \quad \partial_i \partial^i f_5 = 0
\end{equation}
with $r=\sqrt{x_i x^i}$ and the solution
 \begin{equation}\label{eq03d2}
  f_5(r)=c_3 + \frac{c_4}{r}.
\end{equation}
It is also notable that to satisfy the EOM of $G_4$, an extra condition is setting $c_1=0$.

Now, we try to present solutions including the backreaction. To do this, from the action (2.10) of \cite{Gauntlett03} we read
\begin{equation}\label{eq04}
     S_{4E} = \int d^4x\, \sqrt{g_4}\, R^7 \left(- \left(\mathcal{R}_4-\frac{10}{3} \Lambda \right) + \frac{3}{4 R^4} H_{\mu \nu}\, H^{\mu \nu} + \frac{1}{12 R^2} H_{\mu \nu \rho}\, H^{\mu \nu \rho} \right),
\end{equation}
where $\mathcal{R}_4$ is the scalar curvature of $EAdS_4$ and $\Lambda = - \frac{12}{R^2}$ is the cosmological constant, and that with (\ref{eq03c}), we have
\begin{equation}\label{eq05}
     H_{\mu \nu}\, H^{\mu \nu} = \frac{32}{R^8}\, u^2 (\partial_i f_5)(\partial^i f_5), \quad H_{\mu \nu \rho}\, H^{\mu \nu \rho} = \frac{384}{R^{10}}\, u^2 (\partial_i f_5)(\partial^i f_5).
\end{equation}
As a result, the (pseudo)scalar equation from the action is (\ref{eq03d}) that in turn satisfies
\begin{equation}\label{eq06}
     d( R^2 \ast_4 H_3) + 6 \ast_4 H_2 =0,  \quad  d( R^3 \ast_4 H_2)=0,
\end{equation}
which are the (pseudo)scalar equations from (B.9)-(B.13) of \cite{Gauntlett03}.

On the other hand, from the 11D Einstein equation
\begin{equation}\label{eq07}
 \mathcal{R}_{MN}= \frac{1}{6} \left(\frac{3}{3!} {G}_{MPQR}\, {G}_N^{PQR} - \frac{1}{4!} g_{MN}\, {G}_{PQRS}\, {G}^{PQRS} \right),
\end{equation}
we read the equation
\begin{equation}\label{eq08}
(\partial_i f_5)(\partial^i f_5) + {C}_0\, \frac{R^{10}}{u^2} =0,
\end{equation}
with ${C}_0=\frac{2}{7}, \frac{1}{2}$ and $-\frac{2}{5}$ corresponding to $\mathcal{R}_{\mu \nu}$, $\mathcal{R}_{mn}$ and $\mathcal{R}_{77}$ from (B.19), (B.21) and (B.22) of \cite{Gauntlett03}, respectively. Then, a solution from (\ref{eq08}) is also realized as (\ref{eq03d}) with
\begin{equation}\label{eq09}
 c_4 = c_5\, \frac{r^2}{u},
\end{equation}
where $c_5$ has different values for different $C_0$'s. It is noteworthy that the solution is structurally similar to the (pseudo)scalar $m^2 =+4$ asymptotic solution near the boundary ($u = 0$), associated with the non-normalizable mode $\Delta_- = -1$ of course. It should also be noted that it is not possible to have a unique solution including the backreaction on the whole background geometry; In fact, to have just one solution (one $c_5$), we must consider the backreaction in one part of 11D space (e.g. the external space) and neglect it on the remaining parts.

\section{Details of Computing the Stress Tensors} \label{Appendix.A0}
Although we could discuss on the backreaction according to relations (B.19-22) of \cite{Gauntlett03} directly, we do it in our own way in this Appendix. For the Einstein equation
\begin{equation}\label{eq14a}
    \mathcal{R}_{MN} - \frac{1}{2} g_{MN} \mathcal{R} = 8 \pi \mathcal{G}_{11} T_{MN}^{\tilde{G}_4},
\end{equation}
we consider
\begin{equation}\label{eq14b}
 T_{MN}^{\tilde{G}_4} = \frac{1}{4!} \left[4 \tilde{G}_{MPQR} \tilde{G}_N^{PQR} - \frac{1}{2} g_{MN} \tilde{G}_{PQRS} \tilde{G}^{PQRS} \right],
\end{equation}
where we use the capital indices $M, N,...$ for the 11D space-time directions and small indices $m, n,...$ for the 6D internal $CP^3$ space, and the Greek indices $\mu, \nu,....$ for the 4D external $AdS_4$ space.

We compute the external and internal components of the above EM tensor for the ansatz (\ref{eq11}) wrt its dual 7-from, which in components read
\begin{equation}\label{eq15a}
 \tilde{G}_{PQRS} \equiv \hat{c}_1\ \tilde{G}_{m n p q} + \hat{c}_2\ \tilde{G}_{\mu m n 7} + \hat{c}_3\ \tilde{G}_{\mu \nu \rho \sigma},
\end{equation}
\begin{equation}\label{eq15b}
 \tilde{G}_{PQRSTUV} \equiv  \bar{f}_1\ \tilde{G}_{\mu \nu \rho \sigma m n 7} + \tilde{G}_{\mu \nu \rho m n p q 7} +  f_7\ \tilde{G}_{m n p q r s 7},
\end{equation}
respectively, where we use $e_7$ as the seventh vielbein (i.e. as a coordinate base) and that
\begin{equation}\label{eq16a}
 \begin{split}
 & \tilde{G}_{m n p q} = 6\, F_{m n p q},\quad \mathcal{F}_{m n p q} = J_{[{m n}} J_{{p q}]} = \frac{1}{3} \left(J_{m n}J_{p q} - J_{p n}J_{m q} - J_{q n}J_{p m}\right),  \\
 & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \tilde{G}_{\mu m n 7} = (\partial_\mu f_3)\, J_{m n} = - \tilde{G}_{7 m n \mu}, \quad \tilde{G}_{\mu \nu \rho \sigma} = \varepsilon_{\mu \nu \rho \sigma}, \\
 & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \hat{c}_1 = 8\, R^4\, \bar{f}_1, \quad  \hat{c}_2 = -2\, R^4 \quad  \hat{c}_3 = \frac{3}{8} R^4\, f_7,
 \end{split}
\end{equation}
\begin{equation}\label{eq16b}
 \begin{split}
\tilde{G}_{\mu \nu \rho \sigma m n 7} & = (105)\, \varepsilon_{\mu \nu \rho \sigma}\, J_{m n}, \quad \tilde{G}_{\mu \nu \rho m n p q 7} = (210)\, A_{\mu \nu \rho}\, \mathcal{F}_{m n p q}, \quad \tilde{G}_{m n p q r s 7} = \frac{7!}{8}\, \mathcal{F}_{m n p q r s}, \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A_{\mu \nu \rho}=\sqrt{g_4}\ g^{\sigma \sigma} \varepsilon_{\sigma \mu \nu \rho}\, \partial^{\sigma} f_3, \\
\mathcal{F}_{m n p q r s} & = J_{[{m n}} J_{p q} J_{{r s}]} = \frac{1}{15} (J_{m q} J_{n s} J_{p r} + J_{m s} J_{n p} J_{q r} - J_{m s} J_{n q} J_{p r} - J_{m r} J_{n p} J_{q s} + J_{m r} J_{n q} J_{p s} \\
  & + J_{m p} J_{n r} J_{q s}- J_{m p} J_{n s} J_{q r} - J_{m q} J_{n r} J_{p s} + J_{m n} J_{q r} J_{p s} + J_{n p} J_{m q} J_{r s} + J_{n r} J_{p q} J_{m s} \\
   & + J_{m n} J_{p q} J_{r s} - J_{m n} J_{q s} J_{p r} - J_{n q} J_{m p} J_{r s} - J_{n s} J_{p q} J_{m r} ).
 \end{split}
\end{equation}

To continue, we write
\begin{equation}\label{eq17a}
 \tilde{G}_{PQRS} \tilde{G}^{PQRS} = \hat{c}_1\, \tilde{G}_{m n p q} \tilde{G}^{m n p q} + 12\, \hat{c}_2\, \tilde{G}_{\mu m n 7} \tilde{G}^{\mu m n 7} + \hat{c}_3\, \tilde{G}_{\mu \nu \rho \sigma} \tilde{G}^{\mu \nu \rho \sigma},
\end{equation}
\begin{equation}\label{eq17b}
 \tilde{G}_{\mu PQR} \tilde{G}_\nu^{PQR} = 3\, \hat{c}_2\, \tilde{G}_{\mu m n 7} \tilde{G}_\nu^{m n 7} + \hat{c}_3\, \tilde{G}_{\mu \rho \sigma \delta} \tilde{G}_\nu^{\rho \sigma \delta},
\end{equation}
\begin{equation}\label{eq17c}
 \tilde{G}_{m PQR} \tilde{G}_n^{PQR} = \hat{c}_1\, \tilde{G}_{m p q r} \tilde{G}_n^{p q r} + 6\, \hat{c}_2\, \tilde{G}_{m p 7 \mu} \tilde{G}_n^{p 7 \mu},
\end{equation}
\begin{equation}\label{eq17d}
 \tilde{G}_{7 PQR} \tilde{G}_7^{PQR} = 3\, \hat{c}_2\ \tilde{G}_{7 m n \mu} \tilde{G}_7^{m n \mu},
\end{equation}
in which the numerical coefficients are for permutations of indices. As a result, we obtain
\begin{equation}\label{eq18a}
 \tilde{G}_{PQRS} \tilde{G}^{PQRS} = 96 \left[\frac{8}{R^7} \bar{f}_1^2 + \frac{1}{2 R^5} (\partial_\mu f_3)(\partial^\mu f_3) + \frac{3}{8 R^7} f_{7}^2\right],
\end{equation}
\begin{equation}\label{eq18b}
 \tilde{G}_{\mu PQR} \tilde{G}_\nu^{PQR} = \frac{9}{R^7} f_{7}^2\ g_{\mu \nu} + \frac{12}{R^5} (\partial_\mu f_3)(\partial_\nu f_3),
\end{equation}
\begin{equation}\label{eq18c}
 \tilde{G}_{m PQR} \tilde{G}_n^{PQR} = \frac{128}{R^7} \bar{f}_1^2\ g_{m n} + \frac{4}{R^5} (\partial_\mu f_3)(\partial^\mu f_3)\ g_{m n} ,
\end{equation}
\begin{equation}\label{eq18d}
 \tilde{G}_{7 PQR} \tilde{G}_7^{PQR} = \frac{12}{R^5} (\partial_\mu f_3)(\partial^\mu f_3)\ g_{7 7},
\end{equation}
with a $4!$ coefficient for all terms. In obtaining the latter results, we have used the differential-geometry formulas in Appendix \ref{Appendix.A1} together with permutations of the indices depended on their locations on the forms besides the following relations for the K\"{a}hler form on $CP^3$:
\begin{equation}\label{eq19}
 \begin{split}
  & \ \ \ \ J_{m n} = - J_{n m}, \quad J_{m n} J^{m n} = 6, \quad g_{m p} g_n^{\ p} = g_{m n}, \\
  & J_{m n} J_{p q} J_{r s}\ \varepsilon^{\acute{m} n p q r s} = 8 g_m^{\ \acute{m}}, \quad J_{m n} J_{p q} J_{r s}\ \varepsilon^{m n p q r s} =48.
 \end{split}
\end{equation}

Then, plugging (\ref{eq18a}) with (\ref{eq18b}), (\ref{eq18c}) and (\ref{eq18d}) back into (\ref{eq14b}), making use of (\ref{eq13b}), taking traces and using the Euler-Lagrange equation, we finally arrive at
\begin{equation}\label{eq20a}
     \Box_4 \bar{f}_1 - \left(2 \pm 6\, \bar{C}_7 \right) \frac{\bar{f}_1}{R^2} - 192\, \bar{f}_1^3=0,
\end{equation}
\begin{equation}\label{eq20b}
     \Box_4 \bar{f}_1 - \left(1 \pm 9\, \bar{C}_7 \right) \frac{\bar{f}_1}{R^2} - 288\, \bar{f}_1^3=0,
\end{equation}
\begin{equation}\label{eq20c}
     \Box_4 \bar{f}_1 - \left(-1 \pm 3\, \bar{C}_7 \right) \frac{\bar{f}_1}{R^2} - 96\, \bar{f}_1^3=0,
\end{equation}
respectively.

\subsection{Some Useful Formulas}\label{Appendix.A1}
The main differential-geometry relations, in order to do the computations of Appendix \ref{Appendix.A0}, in general D-dimensions read
\begin{equation}\label{eq21a}
 \tilde{G}_\alpha = \frac{1}{\alpha!} \tilde{G}_{R_1 R_2 ... R_\alpha} dX^{R_1 R_2 ... R_\alpha}, \quad dX^{R_1 R_2 ... R_\alpha} \equiv dX^{R_1} \wedge dX^{R_2} \wedge ... \wedge dX^{R_\alpha},
\end{equation}
\begin{equation}\label{eq21b}
 \ast_D \tilde{G}_\alpha = \frac{\sqrt{g_D}}{(D- \alpha)!\ \alpha!} \varepsilon_{R_1 R_2 ... R_{\alpha} R_{\alpha +1} ... R_{D- \alpha}} \tilde{G}^{R_1 R_2 ... R_\alpha} dX^{R_{\alpha +1} ... R_{D- \alpha}},
\end{equation}
\begin{equation}\label{eq21c}
 \tilde{G}_\alpha \wedge \tilde{H}_\beta = \frac{1}{\alpha!} \frac{1}{\beta!} \tilde{G}_{R_1 R_2 ... R_\alpha} \tilde{H}_{S_1 S_2 ... S_\beta}\ dX^{R_1 R_2 ... R_\alpha S_1 S_2 ... S_\beta},
\end{equation}
\begin{equation}\label{eq21d}
 \tilde{G}_\alpha \wedge \ast_D \tilde{G}_\alpha = \frac{\sqrt{g_D}}{\alpha!} \tilde{G}_{R_1 R_2 ... R_D} \tilde{G}^{R_1 R_2 ... R_D}\ dX^{1 2 ... D},
\end{equation}
\begin{equation}\label{eq21e}
 dX^{1 2 ... D} = \frac{1}{D!}\varepsilon_{R_1 R_2 ... R_D} dX^{R_1 R_2 ... R_D},
\end{equation}
\begin{equation}\label{eq21f}
 \varepsilon^{R_1 R_2 ... R_\alpha S_1 S_2 ... S_{D-\alpha}}\  \varepsilon_{R_1 R_2 ... R_\alpha T_1 T_2 ... T_{D-\alpha}} = \alpha! (D-\alpha)!\ \delta^{[S_1...}_{T_1 ...} \delta^{S_{D-\alpha}]}_{T_{D-\alpha}},
\end{equation}
for the $\alpha$-form $\tilde{G}$ and $\beta$-form $\tilde{H}$. It is also notable that we have used
\begin{equation}\label{eq22}
 \tilde{G}^{PQRS} = \frac{1}{\sqrt{g_{11}}\ 7!}\, \varepsilon^{PQRS R_1 R_2 ... R_7} \tilde{G}_{R_1 R_2 ... R_7},
\end{equation}
as a result of (\ref{eq21b}) with $D=11$.

\end{appendices}

redpajama_set_name: RedPajamaArXiv


text:
Handcar Park
Whistle Stop Playground
Vendor Details
What is N Scale
What is NTRAK
What is TTRAK
HO Group
HO Group Layouts
HO Scale Modular Layout
What is O Guage and O Scale?
AustNtrak Model Railroad Club Located in and around Austin, Texas, this N Scale club celebrated it's 20th anniversary on Jan 11th, 2006. Membership to AustNtrak costs $20 for a lifetime membership and they have participated in the Temple Train show for a number of years. Many of Centr
Houston N'Crowd The first NTrak group to organize in southeast Texas in 1978. They graciously donated some of their old NTrak modules to the club when they moved away the NTrak standard.
Tex-N Modular Railroad Club Formed in 1988, this N Scale club is noted for its fully scenicked modules and uncommon layout configurations. They have participated in the Temple Train show for a number of years.
Anthracite Railroads Historical Society, Inc. Over 1,800 authors, historians, railroad employees, photographers, model railroaders, and just plain railfans share an interest in the railroads that served the anthracite coal mining region of eastern Pennsylvania. These include the Central Railroad of N
Conrail Historical Society This society was established to encourage the writing, collection and preservation of historical and contemporary material associated with Conrail in particular, and the American Railroad Systems in general, and to preserve the heritage of Conrail and pre
Conrail Historical Society, Inc. This society was established to encourage the writing, collection and preservation of historical and contemporary material associated with Conrail in particular, and the American Railroad Systems in general, and to preserve the heritage of Conrail and pre
Friends of the Burlington Northern Railroad This historical society was incorporated in the state of Idaho on January 3, 1993. It was chartered to gather, preserve, and share information about the history and the current operations of the Burlington Northern Railroad.
Penn Central Railroad Historical Society Their mission is to preserve the history of the railroad known as the Penn Central Transportation Company, including its people and equipment, its heritage before its formation in 1968, its all-too-brief life, and its slow fade into Conrail.
The Friends of the Burlington Northern Railroad (FOBNR) This historical society was incorporated in the state of Idaho on January 3, 1993. It was chartered to gather, preserve, and share information about the history and the current operations of the Burlington Northern Railroad.
The Santa Fe Railway HIstorical & Modeling Society This is an independent non-profit corporation devoted to preserving, studying and sharing information about all aspects of the former Santa Fe Railway, from its rich history and fascinating operations to its distinctive equipment and facilities.
Alkem Scale Models Photoetched kits and detail parts for HO and N Scale model railroaders.
Atlas Model Railroad Company A pioneer and leader in the model railroad industry for more than 50 years.
BLMA Models Some of the best Z, N, and HO Scale Locomotive and Scenic detail parts available today.
CVP Products Home of the simple and elegant EasyDCC and Texas-based DCC manufacturer. The club's N Scale layouts use EasyDCC.
DeLuxe Innovations Since 1993, DeLuxe has been dedicated to offering good running and accurate as possible N Scale products for enthusiasts. If you model N Scale intermodal, you know about DeLuxe!
Digitrax One of if not THE industry leader in Digital Command Control systems. The club's HO Scale layout is provisioned for wireless Digitrax Command Control.
Gold Medal Models Gold Medal Models was established in 1985 to produce exhibition quality detail parts for builders of plastic model ship kits. In 1992, GMM added N-scale and HO-scale model railroad detail accessories to its range and their model railroad products now tota
KATO, USA One of the leading manufacturers of precision railroad products for the modeling community. KATO's parent company, Sekisui Kinzoku Co., Ltd., is headquartered in Tokyo, Japan. Ganbatte Hiroshi!
Micro-Trains Line Co. Manufacturer of the industry standard Magne-Matic coupler and some of the best, prototypically accurate N Scale freight cars.
Microscale Industries The industry leader for screen printed decals in N, HO, and O Scales.
Plano Model Products Providing Photo Etched details for the Railroad Modeler since 1989. Great N and HO scale train and vehicle detail parts produced just up the road in Richardson, Texas.
Wm. K. Walthers, Inc. Founded in 1932, Walthers has become the world's largest distributor of model railroad equipment worldwide.
Modeler Forums
The Railwire This site has a number of forums: N Scale, Z Scale, HO/HOn3/O/On3, Prototype Photography, Modeling Tutorials and Hints, Weathering, Detailing, and Scratchbuilding, (home to monthly weathering challenges) DCC/Electronics The N Scale forum is the most acti
The Railwire The N Scale forum is the most active followed by Weathering, Detailing, and Scratchbuilding and Prototype Photography. Lots of excellent modeling and tips are on display.
Modeling References
The Model Train Magazine Index This site contains an index to Model Railroad magazines from 1933 to the present. These include: Model Railroader, Railroad Model Craftsman, Narrow Gauge and Short Line Gazette, Railmodel Journal, Railroad Modeler, Mainline Modeler, Model Railroading, Gre
ATSF Resource (Qstation.org) A great link resource for all things ATSF.
BN Resource (QStation.org) A great link resource for all things BN.
BNSF Resource (QStation.org) A great link resource for all things BNSF.
Conrail Cyclopedia The essential Conrail website with photos and features on locomotives, freight cars, and more for railfanning or model railroading.
Penn Central Dedicated to PC on the net. Numerous reference, photos and resources.
Southern Pacific – my Espee Modeler's Archive One of the web's most comprehensive sites for California's railroad to the U.S. from 1861-1996.
Fallen Flags and Other Railroad Photos One of the most visited railroad related web sites run by George Elwood. The site operates on donations only, so if you find it helpful, you might donate to support the site as other club members have.
GIMP The GNU Image Manipulation Program is a freely distributed piece of software for such tasks as photo retouching, image composition and image authoring. It works on many operating systems, in many languages.
RailcarPhotos.com Thousands of photographs of North American railroad cars covering a variety of models and paint schemes. Visitors are encouraged to contribute photos and information making the site more complete and always changing.
RailPictures.net This site offers rail photographers a streamlined, efficient, and easy-to-use online database in which to display their photos, free of charge. They have more than 152,000 photos online.
The NERAIL North American Railroad Photo Archive Created in late 1997, this archive contains 100,000+ photographs from all over North America, including New England, its original focus. Many different road names and equipment types are represented, with most photographs dating from the last few decades.
Archives Select Month December 2016 June 2016 November 2014 August 2014 February 2013 August 2011 October 2009 September 2009 August 2009 June 2009 May 2009 April 2009 March 2009 February 2009 January 2009 November 2008 October 2008 September 2008 August 2008 July 2008 June 2008 May 2008 April 2008 March 2008 February 2008 January 2008 December 2007 November 2007 October 2007 September 2007 August 2007 July 2007 June 2007 May 2007 April 2007 March 2007 February 2007 January 2007 November 2006 October 2006 September 2006 August 2006 July 2006 June 2006 May 2006 April 2006 March 2006 February 2006 January 2006 December 2005 November 2005 October 2005 September 2005 August 2005 July 2005 June 2005 April 2005 March 2005 February 2005 January 2005 December 2004 November 2004 October 2004 September 2004 August 2004 July 2004 June 2004 May 2004 April 2004 March 2004 February 2004 January 2004 November 2003 October 2003 September 2003 August 2003 June 2003 May 2003 April 2003 March 2003 February 2003 January 2003 September 2002 August 2002 July 2002 June 2002 March 2002 September 2001 August 2001
AustNtrak Model Railroad Club
Houston N'Crowd
Tex-N Modular Railroad Club
Anthracite Railroads Historical Society, Inc.
Conrail Historical Society
Conrail Historical Society, Inc.
Friends of the Burlington Northern Railroad
Penn Central Railroad Historical Society
Temple Railroad and Heritage Museum
The Friends of the Burlington Northern Railroad (FOBNR)
The Santa Fe Railway HIstorical & Modeling Society
Alkem Scale Models
Atlas Model Railroad Company
CVP Products
KATO, USA
Micro-Trains Line Co.
Wm. K. Walthers, Inc.
The Railwire
The Model Train Magazine Index
ATSF Resource (Qstation.org)
BN Resource (QStation.org)
BNSF Resource (QStation.org)
Conrail Cyclopedia
Penn Central
Southern Pacific – my Espee Modeler's Archive
Fallen Flags and Other Railroad Photos
RailcarPhotos.com
The NERAIL North American Railroad Photo Archive
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text:
Kit Harington endures heartbreak in new Testament of Youth trailer
The Game of Thrones actor is swapping high fantasy for wartime suffering as Vera Brittain's fiancé Roland Leighton
By Ellie Walker-Arnott
Monday, 10th November 2014 at 10:03 am
Kit Harington swaps the snow-capped mountains of Westeros for the trenches of the First World War in the latest trailer for Testament of Youth.
The Game of Thrones actor and Alicia Vikander star in the moving two minute clip as Vera Brittain and her fiance, poet and soldier Roland Leighton.
The trailer sees Roland (a well-to-do and clean-cut Harington) meet and fall for Vera Brittain, before the Great War changes the course of their lives.
"Life, love and you. Think what they have meant to me," Harington can be heard saying over footage from the film. "Let's agree. No more fear."
Published in 1933, Brittain's Testament of Youth is the first instalment of her memoirs, recording her experiences during the war. The film will also star Emily Watson, Hayley Atwell, Colin Morgan, Alexandra Roach, Anna Chancellor, Dominic West and Miranda Richardson.
Testament of Youth will be in UK cinemas on 16th January 2015
Anna Chancellor
Game of Thrones star Kit Harington in Testament of Youth trailer
The new Macbeth trailer looks like an episode of Game of Thrones
All about Game of Thrones
Downton Abbey's Ed Speelers and Game of Thrones' Alfie Allen star in heist movie Plastic – trailer
Jude Law goes cockney and takes his clothes off in trailer for Dom Hemingway


• Solution of Carboxymethyl cellulose (CMC).
• Stabilizes against tartrate precipitation over long-term and inhibits the formation, growth and precipitation of potassium bitartrate crystals.
"Tartaric stability has always been a challenge in winemaking and sometimes is a source of tough decisions right before bottling. Last year, we started encouraging and assisting our customers to use in-line dosing following the final filter; Cellogum L for white wine and Maxigum and Cellogum L for red wine. This practice implied considerable savings in terms of economy and time.


text:
Featured+
2016 Black Diamond Awards: Those who make a difference
February 17, 2016 February 26, 2016 Vermont Sports 4985 Views Brian Mohr, Catamount Trail Association, Ember Photo, Emily Johnson, Kingdom Trails, Tim Tierney, Vermont Mountain Bike Association
What makes Vermont a great place to recreate? It's the landscape, for sure, but it's also the people and organizations who have helped to clear it, shape it, share it and ensure we have access to it.
Best Outdoor Organization: Catamount Trail Association
What started as an organization dedicated to preserving one of Vermont's best-loved trails has turned into the 300-mile winter use trail into backbone of backcountry skiing in this state. The CTA not only helps to organize events along its trails but has helped backcountry skiers gain access to new terrain and fostered events around the state ranging from ski mountaineering races, to training clinics to tours. The CTA has also been integral in galvanizing efforts to create new glade skiing via the Vermont Backcountry Alliance. Runners-up: 2. Fellowship of the Wheel, 3. Kingdom Trails, 4. Vermont Adaptive Ski & Sports, 5. Rochester Area Sports Trails Alliance (RASTA).
Greatest Contribution to Vermont's Outdoor Community: VMBA
Over the years, the Vermont Mountain Bike Association has worked closely with local groups and landowners to make our state a place that people come to ride from around the U.S. This past year, VMBA published the result of many years of hard work: a statewide trail map showing more than 16 networks. VMBA also worked closely with local groups to help protect 133 acres in the Kingdom Trails network and helped the Trust for Public Land acquire land around Ascutney Mountain. Runners-up: 2. Outdoor Gear Exchange, 3. Rochester Area Sport Trails Alliance (RASTA), 4. RunVermont. 5. Green Mountain Club
Best Outdoor Photographer: Brian Mohr/Emily Johnson, Ember Photo
Vermont Sports contributing editor Brian Mohr and his wife and partner Emily Johnson not only document the best of Vermont's outdoor world, they live it every day and have been active voices in preserving and growing our backcountry trails. The couple document their snow/bike/surf/camping adventures with their two young daughters, Lil Maiana Snow and Lenora Sky and in a way that just makes you want to be them. Runners-up 2. Herb Swanson, 3. Jeb Wallace-Brodeu , 4. Hubert Schriebl, 5. Tristan Von Duntz
Vermont Outdoors Person of the Year: Tim Tierney at Kingdom Trails
Over his past 12 years as executive director, Tim Tierney has quietly built the Kingdom Trail Association into one of the great outdoor sports success stories in the country. Tierney and his team have worked with more than 50 private landowners to build a network of more than 100 miles of single and double track. The trail system now gets more than 70,000 visits per year from people across the United States and Canada, brining an annual economic impact to the region of more than $6.5 million. This past year, Tierney helped the organization raise more than $150,000 from 870 donors and secured another $150,000 grant from the Vermont Housing and Conservation Board to preserve the legendary Sidewinder trail and an adjacent 133 acres.
← 2016 Black Diamond Awards: The Seven Best Events in Vermont
Speak Up: A Plan for Change →
Yoga Cross-Training for Skiers
February 3, 2016 Sophia Light Barsalow 0
Get strong: climb indoors
January 20, 2016 Evan Johnson 0
Nordic racing: Get the most out of your mass start
January 21, 2016 Vermont Sports 0


Students who choose not to take the full IB Diploma can select courses from either of the two IB Course Packages options according to students' own strengths and interests. There is no requirement of a certain number of higher level and standard level courses. Students electing for this option do not complete an Extended Essay or CAS. IB Diploma students and IB Course students are taught together in the same classroom and sit the same IB examinations at the end of the courses. Students should seek the advice of their teachers, parents, and IB Coordinator and carefully consider the decision whether to take the full IB Diploma or a program of IB Courses.
Note: All courses are subject to availability, sufficient enrolment and timetable constraints.


Schools for Fair Funding members voted unanimously to reopen the Montoy v. State of Kansas lawsuit, in an attempt to force the state to provide more funding for public schools.
The group conducted a voice vote at a meeting Friday in Salina. USD 410 Superintendent Steve Noble said nobody in attendance voted against reopening the suit. USD 410 and USD 398 are member districts of the group.
Schools for Fair Funding has 74 member districts, representing a full-time enrollment of 168,018, General Counsel John Robb said Tuesday. Of the member districts, 51 attended the meeting in Salina.
The lawsuit is based on two sections from the Kansas Constitution, he said. One requires that the state provide for improvement in education. Another requires that Kansas adequately fund public schools.
Public schools are one of only four institutions with such constitutional guarantees, Noble said. The others are services for disabled citizens, courts, and corrections. That indicates where founding fathers' priorities were, he said: people who can't fend for themselves and law and order.
The Kansas Supreme Court dismissed the lawsuit in 2006 after Kansas legislators drafted a three-year funding plan. For the 2009-10 school year, the plan called for districts to receive $4,492 per full-time equivalent student, with an adjustment for inflation. Including inflation, the amount for 2009-10 would have been $4,597 per pupil.
The money to provide funding was supposed to be untouchable. Noble said it was put in a figurative lock box.
Lost tax revenues because of the recession caused the state to reduce per-pupil aid several times, eventually settling at $4,012 per pupil.
"Somebody found the key to the lock box," Noble said Monday.
Schools for Fair Funding will ask the Kansas Supreme Court to reopen the Montoy case, claiming the state legislature is violating the agreement that was the basis for dismissing the lawsuit originally.
Schools for Fair Funding is treading new ground, because the Kansas Supreme Court has never reopened a case, he said.
Reopening the case is preferable to filing a new lawsuit because a new suit would have to go through several levels of courts to reach the Supreme Court. The Montoy lawsuit was filed in 1999, and took seven years to reach a conclusion. Starting at the Supreme Court would save money and time, Noble said.


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Home · Fashion
Balmain's Olivier Rousteing Pays Tribute to Michael Jackson With Latest Menswear Collection
By Avon Dorsey	· June 29, 2018 October 23, 2020
"If you're thinkin' about my baby, it don't matter if you're black or white!"
Strong lyrics from the late-great Michael Jackson's hit song "Black or White" rang true as inspiration for Balmain's recent menswear fashion show in Paris on June 25.
As a fitting tribute on the anniversary of Jackson's death, Balmain's creative director, Olivier Rousteing, channeled the King of Pop with a collection that included black leather loafers and white tube socks, intricately detailed leather jackets (a la Jackson's "Thriller" and "Beat It" videos), and a runway soundtrack that delivered some of Jackson's most beloved tunes.
The above mentioned lyrics also played a part in Rousteing's designing of the collection — he's a Black designer who was raised by white adoptive parents in France. In addition, the 32-year-old designer is vocal about his opinions on Black Lives Matter and diversity (in fashion and beyond). Rousteing has been known to make fashion statements and we recently witnessed this via his exclusive designs for Beyoncé during her Coachella wardrobe that honored HBCU's and the Black homecoming experience.
As a 80s baby, Rousteing would inevitably witness, feel and reinterpret the significance of Jackson's contributions to the world (both musically and aesthetically), and that effect was on full display at his runway presentation. Pay close attention to the Balmain "Bad" t-shirt that mimics Jackson's eponymous script font.
These designs are sure to be a big hit at retail. Take a look!


I've thought a great deal about dedication lately – as one in particular drew me 6,000 miles away to a quiet, windswept corner of Tel Aviv. I wasn't sure what to expect when we were invited to witness the naming of a beautiful new street in my father's honor. Members of Israel's cultural elite were present, as was the mayor of Tel Aviv and a number of musicians. As family and friends watched, my mother gently pulled aside a velvet cover to reveal a simple sign: "Distinguished violinist, humanitarian and lover of Israel." We cried, they clapped and it was soon over.
A short concert followed, and several people spoke about my dad's role in the nascent country's cultural beginnings. And during the course of the many speeches, I realized that the dedication wasn't about the ceremony at all. It was really about the intense commitment one man – my father – had for a country the world didn't always support as passionately as he did.
Some people take the first plane out of Israel when fighting erupts. Others take the first plane back, because they know, deep within themselves, that they must bring healing in whatever way they can. My father brought his music to be a salve for those who were afraid, or anxious, or wounded, and when he played, people stopped and listened and understood he was there just for them.
My dad – alternatively described as an American, a Russian immigrant, a citizen of the world – was really an Israeli at heart. And when he landed here, he knew he was home. Long ago he promised our mother, Vera, his partner in life, that she would always be connected to this land if she agreed to marry him and move to New York. It was a promise he always kept.
Dedication is not about a ceremony; it is about what lies in one's heart.
The same can be said as we light our candles for Hanukkah. It is not about the act of kindling the hanukiyyah; it is all about the story that was the architect of the nine branches. It is all about a small group of Jews who risked everything for the freedom to be Jewish. Not rich or famous or beloved: just Jewish.
2,000 years later, we still remember the struggle that brought down an empire and began a nation that still persists today. How amazing is that!
And it became clear to my family that it was very fitting that the place that will carry my dad's name exists surrounded by people who feel as passionately about the country as he did. As passionate as the Maccabees who fought for that country and made it enduring, vibrant and vital. Just like him.
May his memory and his dedication always be an inspiration as well as a blessing.


text:
Infected war eine brasilianische Thrash-Metal-Band aus Santo André, die 2002 gegründet wurde und sich 2016 auflöste.

Geschichte
Die Band wurde Anfang 2002 von dem Sänger und Gitarristen Rodrigo "Infectz" Costa gegründet. Bis Juli 2005 änderte sich die Besetzung um ihn mehrfach, ehe die Band inaktiv wurde. Im Februar des Folgejahres belebte Costa sie mit dem Bassisten Bruno "Mad Butcher" Tarelov wieder. Als Schlagzeuger stieß Rodrigo Ramos hinzu. Im Dezember 2006 spielten sie ein Konzert in der Bar Arena Metal in Osasco mit den ebenfalls gebuchten Jackhammer, Guillotine, Sounder, Conquistadores und Violator. Im April und Mai 2007 wurde ein erstes Demo aufgenommen, wonach Ramos die Besetzung verließ. Als Ersatz kam Hugo Golon dazu, ehe Henrique Perestrelo als zweiter Gitarrist dazu stieß. Diese Besetzungswechsel ereigneten sich im September und Oktober 2007, ehe das erste Konzert in veränderter Besetzung im Club Hangar 110 in São Paulo folgte, an dem auch die Gruppen Jackhammer, Side Effectz, Breakdown, Evokers und Total Mayhem teilnahmen. Im September 2008 unterzeichnete die Band einen Plattenvertrag bei Mutilation Records, worüber im Dezember das Debütalbum Who Is Not? erschien. Im selben Jahr war die Gruppe zudem auf dem Night of Living Thrashers Festival zu sehen. 2009 hielt die Band diverse Auftritte ab, wobei im Juli der Schlagzeuger Golo aus persönlichen Gründen die Besetzung verließ und durch Rodrigo Silva ersetzt wurde. Im Februar 2016 gab die Gruppe ihre Auflösung bekannt.

Stil
Laut classicthrash.com ist auf Who Is Not? klassischer Thrash Metal zu hören, der zwar energiegeladen sei, doch nicht so sehr, als dass er einen langanhaltenden Eindruck hinterlasse. Die Musik unterscheide sich nur im Detail von anderen Veröffentlichungen des Genres. Auch thethrashmetalguide.com ordnete die Band dem klassischen Thrash Metal zu. Die Songs seien schnell und intensiv und ließen Einflüsse von Vertretern des Genres aus Deutschland und der San Francisco Bay Area erkennen. Der raue Gesang klinge dem Hardcore Punk entliehen. In den Liedern verarbeite die Band schnelle Riffs. In The Damned Land könne man Anleihen aus dem Speed Metal heraushören und Possessed und Violent Reaction seien nicht schlechter als die besten Lieder von Vio-lences Eternal Nightmare. Thrash Attack und Death for Us All würde Destruction-Riffs verarbeiten.

Diskografie
 2007: Infected (Demo, Eigenveröffentlichung)
 2008: Who Is Not? (Album, Mutilation Records)
 2010: Radioactive Beer (Demo, Eigenveröffentlichung)
 2010: Live at Seven Beer (DVD, Eigenveröffentlichung)
 2013: Beerstalkers (Demo, Eigenveröffentlichung)
 2014: Infecção Urbana (Split mit Atrito Urbano, Eigenveröffentlichung)

Weblinks
 Infected bei Facebook

Einzelnachweise

Brasilianische Band
Thrash-Metal-Band

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After too many years, i've finally moved to a bigger place! Annabel and I upgraded to a 1 bedroom 1 bath with a big study. We can't wait to get back into the groove and start making arts and crafts once we've settled in and unpacked!