text:
• Pan
Directed by Joe Wright. Starring Cara Delevingne, Amanda Seyfried, Hugh Jackman, Rooney Mara, Garrett Hedlund, Paul Kaye, Nonso Anozie, Levi Miller, Bronson Webb, Julian Seager, Kathy Burke, Adeel Akhtar, Jack Lowden, Deborah Rosan, Amy Morgan, Harry Lister Smith, Jamie Beamish, Jamie Wilson, Aaron Monaghan. Written by Jason Fuchs, from characters created by J.M. Barrie.
"Such tremendous effort, Mick…" Michael Caine's retired composer whispers to Harvey Keitel's still-working film director towards the middle of Youth, the latest Fellini-esque trip from Paolo Sorrentino.
"Such tremendous effort, for such modest results."
Caine's character is speaking about all the little things he's intentionally done while raising his daughter (played by Rachel Weisz) over the years – all the things he's done for her to remember him by – and the ultimate effect he now feels they have.
It's a poignant moment. And one, I couldn't help feel, that somewhat describes the film that surrounds it. There's so much care and beauty and note-perfect detail in Sorrentino's presentation here. But what's the ultimate result?
Youth is a little bit of a mess – and certainly not as on-target as the director's previous film, the Oscar-winning The Great Beauty – but there are so many striking sequences and memorable scenes that I can't help but greatly admire the film anyway.
Caine is Fred Ballinger, a famous composer who opens the film by refusing to come out of retirement to perform at the Queen's request. His reasons for doing so – spelled out later on – form the film's unexpectedly touching crux.
He's staying (living?) at a luxurious mountainside retreat at the foot of the Swiss Alps, alongside the kind of oddball characters you expect from a Fellini film: a morbidly overweight Hispanic man who was once a famed soccer celebrity, a couple who don't say a word to each other during meals, a young escort who services the older guests, and so on.
And then there's Mick Boyle (Keitel), Fred's longtime friend and a film director, working with a team of writers while at the hotel to finish of the script for his upcoming film.
That film is set to star famed actress Brenda Morel, who Boyle brought to fame a long time ago in his earlier films. Jane Fonda shows up for little more than a single scene as Brenda towards the end of the film, but it's the kind Hollywood-bitter, can't-take-your-eyes-off her powerhouse performance that elevates the entire film.
Fonda will likely get a Supporting Actress Oscar out of her five minutes of screentime in Youth, and it's the best sequence in the movie, even though it only tangentially relates to the main storyline.
Also on hand: Fred's daughter, played by Weisz, who has just been left by her fiancée (who happens to be Mick's son), and Jimmy Tree (played by Paul Dano), a famous actor preparing for his next role but best remembered for his lead in a blockbuster action franchise. I think the Dano character is inspired by Shia LaBeouf, but the characterization veers awfully close to Birdman territory.
What's Youth all about? There's a lot of reminiscing about the past and ruminating about the future, and the movie seems to go down the same road that Sorrentino went down in The Great Beauty. But there are unexpected moments of depth and clarity and, indeed, great beauty, and the filmmaking craft and satiric bite make this palatable for audiences looking for more than two old guys chatting about the meaning of it all.
And there are individual scenes that just stick with you: that shot of Miss Universe in the swimming pool, an unlikely K2 mountain climbing story told by a supporting character, a hilarious pop music video dream sequence starring Paloma Faith as herself.
There are also two unforgivably poor moments of CGI that take you out of the movie: one involving a tennis ball that looks about as convincing as an episode of Tim and Eric Awesome Show, Great Job, and the other is a laughably green-screened five-second mountaineering shot.
But Youth has enough going for it to overcome the occasional scene that doesn't work. Sorrentino's second English-language feature (after the Sean Penn-starring This Must Be the Place) retains his trademark curiosities, but it's his most accessible to date.
Now in Cinemas: Reviews for April 10, 2008
Now in Cinemas: Reviews for March 6, 2008
The Kids Are All Right | The Good Heart


text:
\section{Introduction}

Recently, via the AdS/CFT correspondence, \cite{Maldacena:1997re}, following the grounding works of \cite{Cacciatori:2009iz, Benini:2015noa, Benini:2015eyy}, there has been remarkable development in microscopic counting of microstates of supersymmetric $AdS$ black holes via twisted or superconformal indices of dual field theories. It was shown that entropy functional extremizes to give the Bekenstein-Hawking entropy of the black holes. In the dual field theory, this extremization procedure is understood as the $\mathcal{I}$-extremization where the Witten index of $1d$ superconformal quantum mechanics is extremized to give the entropy. $1d$ quantum mechanics is dual to the $AdS_2$ horizon geometry of extremal black hole solutions. See \cite{Zaffaroni:2019dhb} for a review and references.

Therefore, it is natural to try to understand the $\mathcal{I}$-extremization principle from two-dimensional supergravity point of view. Indeed, $a$-maximization of $4d$ SCFTs \cite{Intriligator:2003jj}, $c$-extremization of $2d$ SCFTs, \cite{Benini:2012cz, Benini:2013cda}, and $\tau_{RR}$-extremization of $3d$ SCFTs, \cite{Barnes:2005bm}, were understood from five-\cite{Tachikawa:2005tq, Szepietowski:2012tb}, three- \cite{Karndumri:2013iqa, Karndumri:2013dca, Karndumri:2015sia} and four-\cite{Amariti:2015ybz} dimensional gauged supergravity theories by identifying the corresponding quantities of $a$, $c$, and $\tau_{RR}$, respectively. On the other hand, even though supergravity in two dimensions has been studied long enough, \cite{Nicolai:1998gi, Paulot:2006zp, Samtleben:2007an, Ortiz:2012ib, Anabalon:2013zka, Bossard:2018utw}, our understanding is elementary and two-dimensional supergravity models are rare compare to the other dimensional theories.

For domain wall backgrounds of gravity theories in dimensions higher than two, by introduction of superpotential, the second order equations of motion often reduce to first order flow equations, \cite{Freedman:1999gp, Skenderis:1999mm, DeWolfe:1999cp}. The flow equations arises naturally, as they are the Hamilton-Jacobi equations of gravity-scalar theories, \cite{deBoer:1999tgo}. In supergravity, the flow equations, in fact, reproduces the BPS equations obtained from the supersymmetry variations, \cite{Freedman:1999gp}. They also provide the holographic $c$-theorem, \cite{Freedman:1999gp, Skenderis:1999mm, Girardello:1998pd}. Solutions of the flow equations are known to be non-perturbatively stable, \cite{Freedman:2003ax}. See section 9 and 10 of \cite{DHoker:2002nbb} for a review. However, the flow equations are known in dimensions higher than two.

In this work, we extend the known results of holographic renormalization group flows in dimensions higher than two and derive the flow equations in two-dimensional dilaton gravity. We begin by considering general two-dimensional dilaton gravity coupled to a scalar field. By introducing a superpotential, we derive the first order flow equations from the equations of motion. We find a quantity which decreases monotonically along flows and give some comments on holographic $c$-theorem.

As examples, we show that recently studied supersymmetric $AdS$ black holes generically dimensionally reduce to two-dimensional dilaton gravity. To be specific, we consider the supersymmetric black holes in $AdS_4$, \cite{Cacciatori:2009iz}, $AdS_6$, \cite{Suh:2018tul}, and $AdS_5$, \cite{Nieder:2000kc}. Their microstates are counted by topologically twisted index of $3d$, \cite{Benini:2015noa, Benini:2015eyy}, $5d$, \cite{Hosseini:2018uzp, Crichigno:2018adf}, and, $4d$, \cite{Bae:2019poj}, dual field theories, respectively. We present the flow equations of higher-dimensional $AdS$ black holes in two dimensions. We also show that from the dilaton at the $AdS_2$ fixed point, we can obtain the Bekenstein-Hawking entropy of higher-dimensional $AdS$ black holes, $e.g.$, \cite{Myers:1994sg} and \cite{Gegenberg:1994pv}.

In section 2, we review the flow equations of gravity in dimensions higher than two. In section 3, we consider two-dimensional dilaton gravity and derive their flow equations. We find a quantity which monotonically decrease along flows and give some comments on holographic $c$-theorem. In section 4, as examples, we obtain two-dimensional dilaton gravity from supersymmetric $AdS_4$, $AdS_6$, and $AdS_5$ black holes. We present their flow equations. We conclude in section 5.

\section{Review of gravity in dimensions higher than two}

For domain wall backgrounds in dimensions higher than two, the second order equations of motion reduce to first order flow equations, \cite{Freedman:1999gp, Skenderis:1999mm, DeWolfe:1999cp}. The flow equations arise naturally, as they are the Hamilton-Jacobi equations of dynamical system of gravity-scalar theories, \cite{deBoer:1999tgo}. In supergravity, the first order equations reproduce the BPS equations obtained from the supersymmetry variations, \cite{Freedman:1999gp}. Solutions of the equations are known to be non-perturbatively stable, \cite{Freedman:2003ax}. In this section, we review section 9 of \cite{DHoker:2002nbb}. We consider the gravity coupled to a scalar field,
\begin{equation}
S\,=\,\frac{1}{4\pi{G}_N^{(d+1)}}\int{d}^{d+1}x\sqrt{+g}\left[-\frac{1}{4}R+\frac{1}{2}\partial_\mu\phi\partial^\mu\phi+V\right]\,,
\end{equation}
in Euclidean spacetime. The equations of motion are
\begin{align}
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}\,&=\,2\left[\partial_\mu\phi\partial_\nu\phi-g_{\mu\nu}\left(\frac{1}{2}\partial_\rho\phi\partial^\rho\phi+V\right)\right]\,, \notag \\
\frac{1}{\sqrt{g}}\partial_\mu\left(\sqrt{g}g^{\mu\nu}\partial_\nu\phi\right)-\frac{dV}{d\phi}\,&=\,0\,.
\end{align}
Therefore, at critical points, the scalar potential satisfies
\begin{equation} \label{fpoint}
\left.\frac{dV}{d\phi}\right|_*\,=\,0\,.
\end{equation}

\subsection{Flow equations}

We consider the domain wall background,
\begin{equation} \label{flatdw}
ds^2\,=\,e^{2A}ds_{d}^2+dr^2\,.
\end{equation}
The equations of motion reduce to
\begin{align} \label{eqeq}
(d-1)A''+\frac{d(d-1)}{2}A'^2\,=&\,-\phi'^2-2V\,, \notag \\
\frac{d(d-1)}{2}A'^2\,=&\,\phi'^2-2V\,, \notag \\
\phi''+dA'\phi'\,=&\,\frac{dV}{d\phi}\,,
\end{align}
where the primes denote the derivative with respect to $r$. However, the first equation is obtained from the last two equations. Hence, there are only two independent equations of motion,
\begin{align}
A'^2\,=&\,\frac{2}{d(d-1)}\left(\phi'^2-2V\right)\,, \notag \\
\phi''+dA'\phi'\,=&\,\frac{dV}{d\phi}\,.
\end{align}
Miraculously, by employing a superpotential, $W$, the second order equations reduce to first order flow equations,
\begin{align} \label{firstordersystem}
\frac{d\phi}{dr}\,=&\,\frac{dW}{d\phi}\,, \notag \\
\frac{dA}{dr}\,=&\,-\frac{2}{d-1}W\,,
\end{align}
where the superpotential generates the scalar potential,
\begin{equation}
V\,=\,\frac{1}{2}\left(\frac{dW}{d\phi}\right)^2-\frac{d}{d-1}W^2\,.
\end{equation}
This result trivially extends to theories with multiple scalar fields. A large class of solutions in supergravity falls into the flat domain wall backgrounds we considered, like, holographic RG flows, and easily extended to wrapped branes, and black hole solutions. For more complicated backgrounds, like Janus solutions, \cite{Freedman:2003ax, Clark:2005te, Suh:2011xc, Bobev:2013yra}, which are $AdS$ domain walls, and also sphere domain walls, the basic structure of the flow equations stays the same, but gets more involved.

\subsection{Holographic $c$-theorem}

The, so-called, monotonicity theorems provide measures of degrees of freedom which monotonically decreases along the renormalization group flows in field theories. There are $c$-theorem, \cite{Zamolodchikov:1986gt}, $a$-theorem, \cite{Cardy:1988cwa, Komargodski:2011vj}, and $F$-theorem, \cite{Jafferis:2011zi, Casini:2012ei}, in two-, four-, and three-dimensional field theories, respectively. There are attempts at $a$-theorem in six-dimensional field theories, $e.g$, \cite{Elvang:2012st, Grinstein:2014xba, Cordova:2015fha}.

One of the immediate applications of holographic RG flows is the holographic $c$-theorem, \cite{Girardello:1998pd, Freedman:1999gp, Skenderis:1999mm} and \cite{Myers:2010xs, Myers:2010tj}. See section 10 of \cite{DHoker:2002nbb} for a review. Though the following argument is valid in $d>1$ dimensions, we will restrict to the AdS$_5$/CFT$_4$ correspondence as it provides a holographic $c$-function which is indeed dual to the $c$-function of four-dimensional field theories, \cite{Freedman:1999gp}. For the domain wall background, from the difference of the first and the second equations in \eqref{eqeq}, we find that
\begin{equation} \label{pos}
A''\,=\,-\frac{2}{d-1}\phi'^2\,\leq\,0.
\end{equation}
We define a function,
\begin{equation}
C(r)\,=\,\frac{\pi}{8G_N}\frac{1}{A'^3}\,.
\end{equation}
Due to \eqref{pos}, derivative of the function is always nonnegative,
\begin{equation}
C'(r)\,=\,\frac{\pi}{8G_N}\frac{-3A''}{A'^4}\,\geq\,0\,.
\end{equation}
Therefore, this function naturally introduces a holographic $c$-function. Moreover, from the flow equations, \eqref{firstordersystem}, we have, \cite{Freedman:1999gp},
\begin{equation}
C(r)\,\sim\,-\frac{1}{W^3}\,.
\end{equation}
See \cite{Szepietowski:2012tb} also for a relation of holographic $a$-theorem and $a$-maximization.

\section{Two-dimensional dilaton gravity}

We consider two-dimensional dilaton gravity coupled to a scalar field,
\begin{equation}
S\,=\,\frac{1}{16\pi{G}_N^{(2)}}\int{d}^2x\sqrt{-g}e^{2\Phi}\left[R+2(2\alpha+1)\partial_\mu\Phi\partial^\mu\Phi-\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-e^{-2\Phi}V(\Phi, \phi; \alpha)\right]\,,
\end{equation}
where $\Phi$ is the dilaton and $\phi$ is the scalar field. In two dimensions, we cannot go to Einstein frame by performing conformal transformations, and they just transform to other string frames. Choosing value of the constant parameter, $\alpha$, is equivalent of conformal transformations. When we study examples from higher dimensions latter in \eqref{alphaa}, it will be clear why we have chosen the factor to be $(2\alpha+1)$. See \cite{Grumiller:2002nm} for a review of two-dimensional dilaton gravity.

The equations of motion are
\begin{align} \label{eomtwo}
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}-e^{-2\Phi}\left(\nabla_\mu\nabla_\nu{e}^{2\Phi}-g_{\mu\nu}\nabla_\rho\nabla^\rho{e}^{2\Phi}\right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \notag \\
=\,-2(2\alpha+1)\partial_\mu\Phi\partial_\nu\Phi+(2\alpha+1)g_{\mu\nu}\partial_\rho\Phi\partial^\rho\Phi+\frac{1}{2}\partial_\mu\phi\partial_\nu\phi-\frac{1}{4}g_{\mu\nu}\partial_\rho\phi\partial^\rho\phi-\frac{1}{2}e^{-2\Phi}Vg_{\mu\nu}&\,, \notag \\
2(2\alpha+1)\frac{1}{\sqrt{-g}}\partial_\mu\left(\sqrt{-g}g^{\mu\nu}\partial_\nu\Phi\right)-2(2\alpha+1)\partial_\mu\Phi\partial^\mu\Phi-\frac{1}{2}\partial_\mu\phi\partial^\mu\phi+R-\frac{1}{2}e^{-2\Phi}\frac{\partial{V}}{\partial\Phi}\,=&\,0\,, \notag \\
\frac{1}{\sqrt{-g}}\partial_\mu\left(\sqrt{-g}g^{\mu\nu}\partial_\nu\phi\right)+2\partial_\mu\Phi\partial^\mu\phi-e^{-2\Phi}\frac{\partial{V}}{\partial\phi}\,=&\,0\,.
\end{align}
On the left hand side of the Einstein equation in \eqref{eomtwo}, in addition to the Einstein tensor, there are derivative terms of the dilaton. In two dimensions, the Einstein tensor vanishes identically,
\begin{equation}
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}\,=\,0\,.
\end{equation}

There are two classes of solutions: i) solutions with constant dilaton and scalar field and ii) solutions with linear dilaton. See, for instance, around (12) and (13) of \cite{Grumiller:2014oha}. The first class of solutions are maximally symmetric, $i.e.$, Minkowski, de Sitter, and anti-de Sitter, and we will concentrate on the first class of solutions. For the first class of solutions, $\Phi\,=\,\Phi_*$ and $\phi\,=\,\phi_*$, from the equations of motion, we obtain
\begin{equation} \label{fpoint}
V|_*\,=\,0\,, \qquad R-\left.\frac{1}{2}e^{-2\Phi}\frac{\partial{V}}{\partial\Phi}\right|_*\,=\,0\,, \qquad \left.\frac{\partial{V}}{\partial\phi}\right|_*\,=\,0\,.
\end{equation}
Curvatures of the solutions are determined by the second equation in \eqref{fpoint}. For the solutions satisfying the conditions, $i.e.$, constant dilaton and scalar field with maximally symmetric background, we refer to them as critical points.

\subsection{Flow equations}

We consider the domain wall background,
\begin{equation} \label{background}
ds^2\,=\,e^{2A}\left(-dt^2+dr^2\right)\,.
\end{equation}
The equations of motion reduce to
\begin{align} \label{fos}
2\left(\Phi''+2\Phi'\Phi'\right)-2A'\Phi'-(2\alpha+1)\Phi'\Phi'+\frac{1}{4}\phi'\phi'+\frac{1}{2}e^{-2\Phi+2A}V\,&=\,0\,, \notag \\
-2A'\Phi'-(2\alpha+1)\Phi'\Phi'+\frac{1}{4}\phi'\phi'-\frac{1}{2}e^{-2\Phi+2A}V\,&=\,0\,, \notag \\
(2\alpha+1)\left(\Phi''+\Phi'\Phi'\right)+A''+\frac{1}{4}\phi'\phi'+\frac{1}{4}e^{-2\Phi+2A}\frac{\partial{V}}{\partial\Phi}\,&=\,0\,, \notag \\
\phi''+2\Phi'\phi'-e^{-2\Phi+2A}\frac{\partial{V}}{\partial\phi}\,&=\,0\,,
\end{align}
where the primes denote the derivative with respect to $r$.

From the sum of the first and the second equations in \eqref{fos}, we obtain a relation between derivatives of functions without the scalar potential,
\begin{equation} \label{eomdifference}
\left(\Phi''+2\Phi'\Phi'\right)-2A'\Phi'-(2\alpha+1)\Phi'\Phi'+\frac{1}{4}\phi'\phi'\,=\,0\,.
\end{equation}
Starting from this relation, we look for the first order flow equations by trial and error. By introducing a superpotential, $W$, we obtain the flow equations,
\begin{align} \label{2flow}
\frac{d\Phi}{dr}e^{-A}\,=&\,W\,, \notag \\
\frac{d\phi}{dr}e^{-A}\,=&\,-4\frac{\partial{W}}{\partial\phi}\,, \notag \\
\frac{dA}{dr}e^{-A}\,=&\,\frac{\partial{W}}{\partial\Phi}+2W-(2\alpha+1)W\,.
\end{align}
Therefore, at critical points, the superpotential satisfies
\begin{equation} \label{fpoint2}
W|_*\,=\,0\,, \qquad \left.\frac{\partial{W}}{\partial\phi}\right|_*\,=\,0\,.
\end{equation}
The superpotential produces the scalar potential by
\begin{equation}
V\,=\,e^{2\Phi}\left[8\left(\frac{\partial{W}}{\partial\phi}\right)^2-4W\frac{\partial{W}}{\partial\Phi}-8W^2+2(2\alpha+1)W^2\right]\,.
\end{equation}
This result trivially extends to theories with multiple scalar fields.{\footnote {Similar first order equations in two-dimensional dilaton gravity were also considered in \cite{Castro:2018ffi}, see (5.14) and appendix A.1 therein.}}

\subsection{Comments on holographic $c$-theorem}

In this subsection, we analogously follow the derivation of holographic $c$-theorem in dimensions higher than two in section 2.2. From the difference of two equations of motion, \eqref{eomdifference}, we obtain
\begin{equation}
\left(e^{-2A}\Phi'\right)'\,=\,\left((2\alpha+1)-2\right)e^{-2A}\Phi'\Phi'-\frac{1}{4}e^{-2A}\phi'\phi'\,\leq\,0\,,
\end{equation}
where we choose the free parameter, $\alpha$, to be{\footnote{At $\alpha\,=\,1/2$, the kinetic term of the dilaton field is canonically normalized. As we can freely perform conformal transformations, we can choose $\alpha$ to be at any value.}}
\begin{equation} \label{alhalf}
\alpha\,\leq\,\frac{1}{2}\,.
\end{equation}
From the supersymmetry equations, \eqref{2flow}, we also find that
\begin{equation}
e^{-2A}\Phi'\,=\,e^{-A}W\,.
\end{equation}
Thus, finally, we find a quantity,
\begin{equation}
C(r)\,=\,e^{-A}W\,,
\end{equation}
whose derivative is negative or zero,
\begin{equation} \label{md}
C'(r)\,\leq\,0\,.
\end{equation}

In fact, as $C(r)$ vanishes at critical points by \eqref{fpoint2}, this quantity may not play the role of holographic $c$-function. However, whether it is a holographic $c$-function or not, there is an intriguing interpretation of the quantity. From \eqref{fpoint2} and \eqref{md}, we have
\begin{align} \label{cd1}
W|_*\,=\,0\,, \\  \label{cd2}
(e^{-A}W)'\,\leq\,0\,.
\end{align}
These conditions mean that $C(r)$ vanishes at critical points, \eqref{cd1}, and $C(r)$ decreases monotonically along the flows, \eqref{cd2}. Therefore, it implies that there can be $only$ one critical point.

In the following section, we will see that the only critical point corresponds to the IR fixed point from higher dimensions, which is the near horizon in the case of black hole solutions. However, there is no UV fixed point in two-dimensional gravity. The running dilaton in the UV means that the two-dimensional description breaks down and one must uplift to higher dimensions.{\footnote{We would like to thank the anonymous referee for comment on this.}} An example of such a flow is discussed in section 2.3 of \cite{Cvetic:2016eiv}, in which case, the UV fixed point is the BTZ black hole in $AdS_3$.

\section{Two-dimensional dilaton gravity from gauged supergravity}

\subsection{Supersymmetric $AdS_4$ black holes}

We review the supersymmetric $AdS_4$ black hole solutions of \cite{Cacciatori:2009iz, DallAgata:2010ejj, Hristov:2010ri} from gauged $\mathcal{N}\,=\,2$ supergravity. We employ the conventions of appendix A in \cite{Benini:2015eyy}. Their microstates are counted by topologically twisted index of $3d$ SCFTs, \cite{Benini:2015noa, Benini:2015eyy}. The action is given by{\footnote {In \cite{Benini:2015eyy} the scalar fields are denoted by $\phi_1\,=\,\phi_{12}$, $\phi_2\,=\,\phi_{13}$, $\phi_3\,=\,\phi_{14}$.}}
\begin{equation}
S\,=\,\frac{1}{16\pi{G}_N^{(4)}}\int{d^4x}\sqrt{-g_4}\left[R_4-\sum_{i=1}^3\frac{1}{2}\partial_\mu\phi_i\partial^\mu\phi_i-\sum_{a=1}^4\frac{1}{2}L_a^{-2}F_a^2-V_4\right]\,,
\end{equation}
where the scalar potential is
\begin{align}
V_4\,=&\,-4\tilde{g}^2\left(\cosh\phi_1+\cosh\phi_2+\cosh\phi_3\right)\,, \notag \\
=&\,-2\tilde{g}^2\left(L_1L_2+L_3L_4+L_1L_3+L_2L_4+L_1L_4+L_2L_3\right)\,,
\end{align}
If we define the superpotential by
\begin{equation}
W_4=\,-\frac{1}{4}\left(L_1+L_2+L_3+L_4\right)\,,
\end{equation}
then, the scalar potential is obtained from
\begin{equation}
V_4\,=\,8\sum_{i=1}^3\left(\frac{\partial{W_4}}{\partial\phi_i}\right)^2-6W_4^2\,.
\end{equation}
We set the gauge coupling constant to be
\begin{equation}
\tilde{g}\,=\,\frac{1}{\sqrt{2}}\,.
\end{equation}
We introduced a parametrization of three real scalar fields,
\begin{align}
L_1\,&=\,e^{-\frac{1}{2}\left(\phi_1+\phi_2+\phi_3\right)}\,, \qquad L_2\,=\,e^{-\frac{1}{2}\left(\phi_1-\phi_2-\phi_3\right)}\,, \notag \\
L_3\,&=\,e^{-\frac{1}{2}\left(-\phi_1+\phi_2-\phi_3\right)}\,, \qquad L_4\,=\,e^{-\frac{1}{2}\left(-\phi_1-\phi_2+\phi_3\right)}\,,
\end{align}
where $L_1L_2L_3L_4\,=\,1$. The field strength of four $U(1)$ gauge fields are
\begin{equation}
F_a\,=\,dA_a\,.
\end{equation}

For the supersymmetric $AdS_4$ black hole solutions of \cite{Cacciatori:2009iz}, we consider the background of
\begin{equation}
ds^2\,=\,e^{2f}\left(-dt^2+dr^2\right)+e^{2g}ds^2_\Sigma\,,
\end{equation}
where $\Sigma$ denotes the Riemann surfaces of curvatures, $k\,=\,\pm1$. The field strength of the gauge fields are
\begin{equation}
F_a\,=\,-\frac{a_a}{\sqrt{2}}Vol_\Sigma\,,
\end{equation}
where the magnetic charges, $a_a$, are constant and $Vol_\Sigma$ is the unit volume form. The first order BPS equations are obtained by solving the supersymmetry variations of the fermionic fields,{\footnote{We suspect that the signs in front of $e^{-2g}$ should be flipped in (A.27) of \cite{Benini:2015eyy}.}}
\begin{align} \label{ads4susy}
f'e^{-f}\,=\,-\frac{1}{4}\left(L_1+L_2+L_3+L_4\right)-\frac{1}{4}e^{-2g}\left(\frac{a_1}{L_1}+\frac{a_2}{L_2}+\frac{a_3}{L_3}+\frac{a_4}{L_4}\right)\,, \notag \\
g'e^{-f}\,=\,-\frac{1}{4}\left(L_1+L_2+L_3+L_4\right)+\frac{1}{4}e^{-2g}\left(\frac{a_1}{L_1}+\frac{a_2}{L_2}+\frac{a_3}{L_3}+\frac{a_4}{L_4}\right)\,, \notag \\
\phi_1'e^{-f}\,=\,-\frac{1}{2}\left(L_1+L_2-L_3-L_4\right)-\frac{1}{2}e^{-2g}\left(\frac{a_1}{L_1}+\frac{a_2}{L_2}-\frac{a_3}{L_3}-\frac{a_4}{L_4}\right)\,, \notag \\
\phi_2'e^{-f}\,=\,-\frac{1}{2}\left(L_1-L_2+L_3-L_4\right)-\frac{1}{2}e^{-2g}\left(\frac{a_1}{L_1}-\frac{a_2}{L_2}+\frac{a_3}{L_3}-\frac{a_4}{L_4}\right)\,, \notag \\
\phi_3'e^{-f}\,=\,-\frac{1}{2}\left(L_1-L_2-L_3+L_4\right)-\frac{1}{2}e^{-2g}\left(\frac{a_1}{L_1}-\frac{a_2}{L_2}-\frac{a_3}{L_3}+\frac{a_4}{L_4}\right)\,,
\end{align}
and there is a twist condition on the magnetic charges,
\begin{equation}
a_1+a_2+a_3+a_4\,=\,\frac{k}{\tilde{g}^2}\,.
\end{equation}
One can solve the BPS equations and obtain the supersymmetric black hole solutions which are interpolating between the $AdS_4$ boundary and the $AdS_2$ horizon.

There is an $AdS_2\times\Sigma$ solution of the horizon. We introduce another parametrization of the scalar fields,
\begin{equation}
z_{1,2,3}\,=\,\frac{L_{1,2,3}}{L_4}\,,
\end{equation}
or, equivalently,
\begin{equation}
L_1^4\,=\,\frac{z_1^3}{z_2z_3}\,, \qquad L_2^4\,=\,\frac{z_2^3}{z_3z_1}\,, \qquad L_3^4\,=\,\frac{z_3^3}{z_1z_2}\,, \qquad L_4^4\,=\,\frac{1}{z_1z_2z_3}\,.
\end{equation}
The horizon is at
\begin{align} \label{ads4ads211}
z_1\,=\,\frac{2(a_2+a_3)(a_1-a_4)^2-(a_1+a_4)[(a_2-a_3)^2+(a_1-a_4)^2]+4(a_4-a_1)\sqrt{\Theta}}{2a_4(a_4-a_1+a_2-a_3)(a_4-a_1-a_2+a_3)}\,, \notag \\
z_2\,=\,\frac{2(a_1+a_3)(a_2-a_4)^2-(a_2+a_4)[(a_1-a_3)^2+(a_2-a_4)^2]+4(a_4-a_2)\sqrt{\Theta}}{2a_4(a_4+a_1-a_2-a_3)(a_4-a_1-a_2+a_3)}\,, \notag \\
z_3\,=\,\frac{2(a_1+a_2)(a_3-a_4)^2-(a_3+a_4)[(a_1-a_2)^2+(a_3-a_4)^2]+4(a_4-a_3)\sqrt{\Theta}}{2a_4(a_4+a_1-a_2-a_3)(a_4-a_1+a_2+a_3)}\,,
\end{align}
and
\begin{align} \label{ads4ads222}
e^{2f}\,=&\,\frac{1}{r^2}\frac{\Pi}{\sqrt{2}\Theta}\left(F_2+\sqrt{\Theta}\right)^{1/2}\,, \notag \\
e^{2g}\,=&\,\frac{1}{\sqrt{2}}\left(F_2+\sqrt{\Theta}\right)^{1/2}\,,
\end{align}
where
\begin{align}
\Theta\,=&\,F_2^2-4a_1a_2a_3a_4\,, \notag \\
F_2\,=&\,\frac{1}{4}\left(a_1+a_2+a_3+a_4\right)^2-\frac{1}{2}\left(a_1^2+a_2^2+a_3^2+a_4^2\right)\,, \notag \\
\Pi\,=&\,\frac{1}{8}(a_1+a_2-a_3-a_4)(a_1-a_2+a_3-a_4)(a_1-a_2-a_3+a_4)\,.
\end{align}

Now we dimensionally reduce the action on the background of supersymmetric $AdS_4$ black holes. The reduction ansatz for the metric is
\begin{equation}\label{alphaa}
ds^2_4\,=\,e^{2\alpha{g}}ds_2^2+e^{2g}ds^2_\Sigma\,,
\end{equation}
where $\alpha$ is a constant parameter. The reduced action is two-dimensional dilaton gravity,
\begin{equation}
S\,=\,\frac{vol_\Sigma}{16\pi{G}_N^{(4)}}\int{d^2x}\sqrt{-g_2}e^{2g}\left[R_2+2(2\alpha+1)\partial_\mu{g}\partial^\mu{g}-\sum_{i=1}^3\frac{1}{2}\partial_\mu\phi_i\partial^\mu\phi_i-e^{-2g}V_2\right]\,,
\end{equation}
where the scalar potential is
\begin{align}
V_2\,=\,e^{2(\alpha+1){g}}\Big[&-2\tilde{g}^2\left(L_1L_2+L_3L_4+L_1L_3+L_2L_4+L_1L_4+L_2L_3\right) \notag \\
&+e^{-4g}\left(\frac{a_1^2}{L_1^2}+\frac{a_2^2}{L_2^2}+\frac{a_3^2}{L_3^2}+\frac{a_4^2}{L_4^2}\right)-2ke^{-2g}\Big]\,.
\end{align}
The scalar potential satisfies the relations, \eqref{fpoint}, at the $AdS_2$, critical point in \eqref{ads4ads211} and \eqref{ads4ads222}.

We consider the background,
\begin{equation}
ds^2_2\,=\,e^{2f}\left(-dt^2+dr^2\right)\,.
\end{equation}
We obtain the flow equations which satisfy the equations of motion of two-dimensional dilaton gravity,
\begin{align}
f'e^{-f}\,=&\,W+\frac{\partial{W}}{\partial{g}}\,, \notag \\
g'e^{-f}\,=&\,W\,, \notag \\
\phi'_ie^{-f}\,=&\,-4\frac{\partial{W}}{\partial\phi_i}\,,
\end{align}
where the superpotential is given by
\begin{equation}
W=\,-\frac{1}{4}\left(L_1+L_2+L_3+L_4\right)+\frac{1}{4}e^{-2g}\left(\frac{a_1}{L_1}+\frac{a_2}{L_2}+\frac{a_3}{L_3}+\frac{a_4}{L_4}\right)\,.
\end{equation}
The superpotential satisfies the relations, \eqref{fpoint2}, at the $AdS_2$ critical point in \eqref{ads4ads211} and \eqref{ads4ads222}. The scalar potential is obtained from
\begin{equation}
V_2\,=\,e^{2g}\left[8\sum_{i=1}^3\left(\frac{\partial{W}}{\partial\phi_i}\right)^2-4W\frac{\partial{W}}{\partial{g}}-6W^2\right]\,.
\end{equation}
By introducing the superpotential, we note that the flow equations are merely a rewriting of the BPS equations in $AdS_4$, \eqref{ads4susy}, as it should. Moreover, we could reproduce the flow equations from the general analysis of section 2 by replacing
\begin{equation}
\Phi\,\rightarrow\,g\,, \qquad \phi\,\rightarrow\,\phi_i\,, \qquad A\,\rightarrow\,f\,, \qquad W\rightarrow\,W\,, \qquad V\,\rightarrow\,V_2\,,
\end{equation}
and setting
\begin{equation}
\alpha\,=\,0\,.
\end{equation}

Finally, we reproduce the Bekenstein-Hawking entropy of the supersymmetric $AdS_4$ black holes obtained in \cite{Benini:2015eyy},
\begin{equation}
S_{\text{BH}}\,=\,\left.\frac{e^{2g}vol_{\Sigma}}{4G_N^{(4)}}\right|_*\,,
\end{equation}
where $vol_\Sigma$ is the area of Riemann surfaces of unit radius, in this case. We make an observation that the two-dimensional dilaton, $\Phi\,\rightarrow\,g$, contains the information of the Bekenstein-Hawking entropy of higher dimensional $AdS$ black holes.

\subsection{Supersymmetric $AdS_6$ black holes}

We review the supersymmetric $AdS_6$ black hole solutions of pure $F(4)$ gauged supergravity in \cite{Suh:2018tul}, which were generalized to solutions of matter coupled $F(4)$ gauged supergravity in \cite{Hosseini:2018usu, Suh:2018szn}. Their microstates are microscopically counted by topologically twisted index of $5d$ SCFTs, \cite{Hosseini:2018uzp}. See also \cite{Crichigno:2018adf}.

The bosonic field content of pure $F(4)$ gauged supergravity, \cite{Romans:1985tw}, consists of the metric, $g_{\mu\nu}$, a real scalar, $\phi$, an $SU(2)$ gauge field, $A^I_\mu$, $I\,=\,1,\,2,\,3$, a $U(1)$ gauge field, $\mathcal{A}_\mu$, and a two-form gauge potential, $B_{\mu\nu}$. The field strengths are defined by
\begin{align}
\mathcal{F}_{\mu\nu}\,=&\,\partial_\mu\mathcal{A}_\nu-\partial_\nu\mathcal{A}_\mu\,, \notag \\
F^I_{\mu\nu}\,=&\,\partial_\mu{A}^I_\nu-\partial_\nu{A}^I_\mu+\tilde{g}\epsilon^{IJK}A^J_\mu{A}^K_\nu\,, \notag \\
G_{\mu\nu\rho}\,=&\,3\partial_{[\mu}B_{\nu\rho]}\,, \notag \\
\mathcal{H}_{\mu\nu}\,=&\,\mathcal{F}_{\mu\nu}+mB_{\mu\nu}\,.
\end{align}
The action is given by
\begin{align}
S\,=\,\frac{1}{4\pi{G}_N^{(6)}}\int{d^6x}\sqrt{-g_6}&\left[-\frac{1}{4}R+\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{4}e^{-\sqrt{2}\phi}\left(\mathcal{H}_{\mu\nu}\mathcal{H}^{\mu\nu}+F^I_{\mu\nu}F^{I\mu\nu}\right)+\frac{1}{12}e^{2\sqrt{2}\phi}G_{\mu\nu\rho}G^{\mu\nu\rho}\right. \notag \\
&\left.-\frac{1}{8}\epsilon^{\mu\nu\rho\sigma\tau\kappa}B_{\mu\nu}\left(\mathcal{F}_{\rho\sigma}\mathcal{F}_{\tau\kappa}+mB_{\rho\sigma}\mathcal{F}_{\tau\kappa}+\frac{1}{3}m^2B_{\rho\sigma}B_{\tau\kappa}+F^I_{\rho\sigma}F^I_{\tau\kappa}\right)-V_6\right]\,,
\end{align}
where $\tilde{g}$ is the $SU(2)$ gauge coupling constant and $m$ is the mass parameter of the two-form field. The scalar potential is
\begin{equation}
V_6\,=-\frac{1}{8}\left(\tilde{g}^2e^{\sqrt{2}\phi}+4\tilde{g}me^{-\sqrt{2}\phi}-m^2e^{-3\sqrt{2}\phi}\right)\,,
\end{equation}
If we define the superpotential by
\begin{equation}
W_6\,=\,-\frac{1}{4\sqrt{2}}\left(\tilde{g}e^{\frac{\phi}{\sqrt{2}}}+me^{-\frac{3\phi}{\sqrt{2}}}\right)\,,
\end{equation}
then, it gives the scalar potential by
\begin{equation}
V_6\,=\,2\left(\frac{{\partial}W_6}{\partial\phi}\right)^2-5W_6^2\,.
\end{equation}
We employ the mostly-minus signature.

For the supersymmetric $AdS_6$ black hole solutions, we consider the background of
\begin{equation}
ds^2\,=\,e^{2f}\left(dt^2-dr^2\right)-e^{2g_1}\left(d\theta_1^2+\sinh^2\theta_1d\phi_1^2\right)-e^{2g_2}\left(d\theta_2^2+\sinh^2\theta_1d\phi_2^2\right)\,.
\end{equation}
The gauge fields are
\begin{equation}
A^3\,=\,a_1\cosh\theta_1{d}\phi_1+a_2\cosh\theta_2{d}\phi_2\,,
\end{equation}
where the magnetic charges, $a_1$ and $a_2$, are constant, and the $U(1)$ gauge field is $\mathcal{A_\mu}\,=\,0$.  The two-form field is given by
\begin{equation}
B_{tr}\,=\,-\frac{2}{m^2}a_1a_2e^{\sqrt{2}\phi+2f-2g_1-2g_2}\,,
\end{equation}
and the three-form field strength of the two-form gauge potential vanishes identically. The first order BPS equations are obtained by solving the supersymmetry variations of the fermionic fields,
\begin{align} \label{ads6susy}
f'e^{-f}\,=&\,-\frac{1}{4\sqrt{2}}\left(\tilde{g}e^{\frac{\phi}{\sqrt{2}}}+me^{-\frac{3\phi}{\sqrt{2}}}\right)+\frac{k}{2\sqrt{2}\tilde{g}}e^{-\frac{\phi}{\sqrt{2}}}\left(e^{-2g_1}+e^{-2g_2}\right)-\frac{3}{\sqrt{2}\tilde{g}^2m}e^{\frac{\phi}{\sqrt{2}}-2g_1-2g_2}\,, \notag \\
g_1'e^{-f}\,=&\,-\frac{1}{4\sqrt{2}}\left(\tilde{g}e^{\frac{\phi}{\sqrt{2}}}+me^{-\frac{3\phi}{\sqrt{2}}}\right)-\frac{k}{2\sqrt{2}\tilde{g}}e^{-\frac{\phi}{\sqrt{2}}}\left(3e^{-2g_1}-e^{-2g_2}\right)+\frac{1}{\sqrt{2}\tilde{g}^2m}e^{\frac{\phi}{\sqrt{2}}-2g_1-2g_2}\,, \notag \\
g_2'e^{-f}\,=&\,-\frac{1}{4\sqrt{2}}\left(\tilde{g}e^{\frac{\phi}{\sqrt{2}}}+me^{-\frac{3\phi}{\sqrt{2}}}\right)-\frac{k}{2\sqrt{2}\tilde{g}}e^{-\frac{\phi}{\sqrt{2}}}\left(3e^{-2g_2}-e^{-2g_1}\right)+\frac{1}{\sqrt{2}\tilde{g}^2m}e^{\frac{\phi}{\sqrt{2}}-2g_1-2g_2}\,, \notag \\
\frac{1}{\sqrt{2}}\phi'e^{-f}\,=&\,\frac{1}{4\sqrt{2}}\left(\tilde{g}e^{\frac{\phi}{\sqrt{2}}}-3me^{-\frac{3\phi}{\sqrt{2}}}\right)-\frac{k}{2\sqrt{2}\tilde{g}}e^{-\frac{\phi}{\sqrt{2}}}\left(e^{-2g_1}+e^{-2g_2}\right)-\frac{1}{\sqrt{2}\tilde{g}^2m}e^{\frac{\phi}{\sqrt{2}}-2g_1-2g_2}\,,
\end{align}
and there are twist conditions on the magnetic charges,
\begin{equation}
a_1\,=\,-\frac{k}{\lambda\tilde{g}}\,, \qquad a_2\,=\,-\frac{k}{\lambda\tilde{g}}\,,
\end{equation}
where $k\,=\,\pm1$ are curvatures of the Riemann surfaces and $\lambda\,=\,\pm1$. The $AdS_2\,\times\,\Sigma_{\mathfrak{g}_1}\times\Sigma_{\mathfrak{g}_2}$ horizon solution is given by
\begin{equation} \label{ads6ads2}
e^f\,=\,\frac{2^{1/4}}{g^{3/4}m^{1/4}}\frac{1}{r}\,, \qquad e^{g_1}\,=\,e^{g_2}\,=\,\frac{2^{3/4}}{g^{3/4}m^{1/4}}\,, \qquad e^{\frac{\phi}{\sqrt{2}}}\,=\,\frac{2^{1/4}m^{1/4}}{g^{1/4}}\,,
\end{equation}
where $\mathfrak{g}_1>1$ and $\mathfrak{g}_2>1$ are genus of the Riemann surfaces. The full black hole solutions are interpolating between the $AdS_6$ boundary and the $AdS_2$ horizon.

Now we dimensionally reduce the action on the background of supersymmetric $AdS_6$ black holes. The reduction ansatz for the metric is
\begin{equation}
ds^2_6\,=\,ds_2^2-e^{2g}\left(ds^2_{\Sigma_{\mathfrak{g}_1}}+ds^2_{\Sigma_{\mathfrak{g}_2}}\right)\,.
\end{equation}
The reduced action is two-dimensional dilaton gravity,
\begin{align}
S\,=\,\frac{vol_{\Sigma_{\mathfrak{g}_1}}vol_{\Sigma_{\mathfrak{g}_2}}}{4\pi{G}_N^{(6)}}\int{d^2x}\sqrt{-g_2}e^{2g_1+2g_2}&\left[-\frac{1}{4}R_2-\frac{1}{2}\partial_\mu{g_1}\partial^\mu{g_1}-\frac{1}{2}\partial_\mu{g_2}\partial^\mu{g_2}-2\partial_\mu{g_1}\partial^\mu{g_2}\right. \notag \\
&\left.+\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-e^{-2g_1-2g_2}V_2\right]\,,
\end{align}
where the scalar potential is
\begin{align}
V_2\,=e^{2g_1+2g_2}&\left[-\frac{1}{8}\left(\tilde{g}^2e^{\sqrt{2}\phi}+4\tilde{g}me^{-\sqrt{2}\phi}-m^2e^{-3\sqrt{2}\phi}\right)-\frac{k_1}{2}e^{-2g_1}-\frac{k_2}{2}e^{-2g_2}\right. \notag \\
&\left.+\frac{1}{2\tilde{g}^2}e^{-\sqrt{2}\phi}\left(e^{-4g_1}+e^{-4g_2}\right)+\frac{2}{\tilde{g}^4m^2}e^{\sqrt{2}\phi-4g_1-4g_2}\right]\,.
\end{align}
The scalar potential satisfies the relations, \eqref{fpoint}, at the $AdS_2$, critical point, \eqref{ads6ads2}. This reduction was previously performed in \cite{Kim:2019fsg}.

We consider the background,
\begin{equation}
ds^2_2\,=\,e^{2f}\left(dt^2-dr^2\right)\,.
\end{equation}
We only consider the case of
\begin{equation}
g\,\equiv\,g_1\,=\,g_2\,>\,1\,, \qquad k\,\equiv\,k_1\,=\,k_2\,=\,-1\,.
\end{equation}
We obtain the flow equations which satisfy the equations of motion of two-dimensional dilaton gravity,
\begin{align}
f'e^{-f}\,=&\,W+\frac{\partial{W}}{\partial{g}}\,, \notag \\
g'e^{-f}\,=&\,W\,, \notag \\
\frac{1}{\sqrt{2}}\phi'e^{-f}\,=&\,-\sqrt{2}\frac{\partial{W}}{\partial\phi}\,,
\end{align}
where the superpotential is given by
\begin{equation}
W=\,-\frac{1}{4\sqrt{2}}\left(\tilde{g}e^{\frac{\phi}{\sqrt{2}}}+me^{-\frac{3\phi}{\sqrt{2}}}\right)-\frac{k}{\sqrt{2}\tilde{g}}e^{-\frac{\phi}{\sqrt{2}}-2g}+\frac{1}{\sqrt{2}\tilde{g}^2m}e^{\frac{\phi}{\sqrt{2}}-4g}\,.
\end{equation}
The superpotential satisfies the relations, \eqref{fpoint2}, at the $AdS_2$ critical point, \eqref{ads6ads2}. The scalar potential is obtained from
\begin{equation}
V_2\,=\,e^{4g}\left[2\left(\frac{\partial{W}}{\partial\phi}\right)^2-2W\frac{\partial{W}}{\partial{g}}-5W^2\right]\,.
\end{equation}
By introducing the superpotential, we note that the flow equations are merely a rewriting of the BPS equations in $AdS_6$, \eqref{ads6susy}, as it should.

Finally, we reproduce the Bekenstein-Hawking entropy of the supersymmetric $AdS_6$ black holes obtained in \cite{Suh:2018tul},
\begin{equation}
S_{\text{BH}}\,=\,\left.\frac{e^{4g}vol_{\Sigma_{\mathfrak{g}_1}}vol_{\Sigma_{\mathfrak{g}_2}}}{4G_N^{(6)}}\right|_*\,.
\end{equation}
We make an observation that the two-dimensional dilaton, $\Phi\,\rightarrow\,g$, contains the information of the Bekenstein-Hawking entropy of higher dimensional $AdS$ black holes.

\subsection{Supersymmetric $AdS_5$ black holes}

We review the supersymmetric $AdS_5$ black hole solutions of gauged $\mathcal{N}\,=\,4$ supergravity in five dimensions in \cite{Nieder:2000kc}. See also \cite{Dao:2018xya}. The microstates are microscopically counted by topologically twisted index of $4d$ $\mathcal{N}\,=\,4$ super Yang-Mills theory, \cite{Bae:2019poj}.

The bosonic field content of $SU(2)\times{U(1)}$-gauged $\mathcal{N}\,=\,4$ supergravity in five dimensions, \cite{Romans:1985ps}, consists of the metric, $g_{\mu\nu}$, a real scalar, $\varphi$,{\footnote{In \cite{Romans:1985ps} the scalar field is parametrized by $\varphi\,=\,\sqrt{\frac{2}{3}}\phi$.}}  an $SU(2)$ gauge field, $A^I_\mu$, $I\,=\,1,\,2,\,3$, a $U(1)$ gauge field, $a_\mu$, and two-form gauge potentials, $B^\alpha_{\mu\nu}$. The field strengths are defined by
\begin{align}
f_{\mu\nu}\,=&\,\partial_\mu{a}_\nu-\partial_\nu{a}_\mu\,, \notag \\
F^I_{\mu\nu}\,=&\,\partial_\mu{A}^I_\nu-\partial_\nu{A}^I_\mu+g_2\epsilon^{IJK}A^J_\mu{A}^K_\nu\,.
\end{align}
The action is given by
\begin{align}
S\,=\,\frac{1}{4\pi{G}_N^{(5)}}\int{d^5x}\sqrt{-g_5}&\left[-\frac{1}{4}R+3\partial_\mu\varphi\partial^\mu\varphi-\frac{1}{4}e^{-4\varphi}f_{\mu\nu}f^{\mu\nu}-\frac{1}{4}e^{2\varphi}\left(F^I_{\mu\nu}F^{I\mu\nu}+B^\alpha_{\mu\nu}B^{\alpha\mu\nu}\right)\right. \notag \\
&\left.+\frac{1}{4}\epsilon^{\mu\nu\rho\sigma\tau}\left(\frac{1}{g_1}\epsilon_{\alpha\beta}B^\alpha_{\mu\nu}D_\rho{B}^\beta_{\sigma\tau}-F^I_{\mu\nu}F^I_{\rho\sigma}a_\tau\right)-V_5\right]\,,
\end{align}
where $g_1$ and $g_2$ are the $U(1)$ and $SU(2)$ gauge coupling constants, respectively, and we define
\begin{equation}
\tilde{g}\,\equiv\,\sqrt{2}g_1\,=\,g_2\,.
\end{equation}
The scalar potential is
\begin{equation}
V_5\,=-\frac{1}{8}g_2\left(g_2e^{-2\varphi}+2\sqrt{2}g_1e^\varphi\right)\,,
\end{equation}
If we define the superpotential by
\begin{equation}
W_5\,=\,\frac{\tilde{g}}{6\sqrt{2}}\left(2e^{-\varphi}+e^{2\varphi}\right)\,,
\end{equation}
then, it gives the scalar potential by
\begin{equation}
V_5\,=\,\frac{3}{4}\left(\frac{{\partial}W_5}{\partial\varphi}\right)^2-3W_5^2\,.
\end{equation}
We employ the mostly-minus signature.

For the supersymmetric $AdS_5$ black hole solutions, we consider the background of
\begin{equation}
ds^2\,=\,e^{2f}\left(dt^2-dr^2\right)-e^{2g}ds^2_{H_3}\,,
\end{equation}
where
\begin{equation}
ds^2_{H_3}\,=\,d\phi^2+\sinh^2\phi{d}\theta^2+\sinh^2\phi\sin^2\theta{d}\psi^2\,.
\end{equation}
The gauge fields are
\begin{equation}
A^1_\theta\,=\,a\cosh\phi\,, \qquad A^2_\psi\,=\,b\cos\theta\,, \qquad A^3_\psi\,=\,c\sin\theta\cosh\phi\,,
\end{equation}
where the magnetic charges, $a$, $b$, and $c$, are constant. The $U(1)$ gauge field and two-form field are vanishing. The first order BPS equations are obtained by solving the supersymmetry variations of the fermionic fields,
\begin{align} \label{ads5susy}
f'e^{-f}\,=&\frac{\tilde{g}}{6\sqrt{2}}\left(2e^{-\varphi}+e^{2\varphi}\right)+\frac{\sqrt{2}}{\tilde{g}}e^{\varphi-2g}\,, \notag \\
g'e^{-f}\,=&\frac{\tilde{g}}{6\sqrt{2}}\left(2e^{-\varphi}+e^{2\varphi}\right)-\frac{\sqrt{2}}{\tilde{g}}e^{\varphi-2g}\,, \notag \\
\varphi'e^{-f}\,=&\frac{\tilde{g}}{3\sqrt{2}}\left(e^{-\varphi}-e^{2\varphi}\right)+\frac{\sqrt{2}}{\tilde{g}}e^{\varphi-2g}\,,
\end{align}
and there are twist conditions on the magnetic charges,
\begin{equation}
a\,=\,\frac{1}{\tilde{g}}\,, \qquad b\,=\,\frac{1}{\tilde{g}}\,, \qquad c\,=\,-\frac{1}{\tilde{g}}\,.
\end{equation}
The $AdS_2\,\times\,H_3$ horizon solution is given by
\begin{equation} \label{ads5ads2}
e^{2f}\,=\,\frac{1}{4^{4/3}}\frac{1}{r^2}\,, \qquad e^{2g}\,=\,\frac{1}{4^{1/3}}\,, \qquad e^{3\varphi}\,=\,4\,,
\end{equation}
where $\tilde{g}\,=\,2\sqrt{2}$. The full black hole solutions are interpolating between the $AdS_5$ boundary and the $AdS_2$ horizon.

Now we dimensionally reduce the action on the background of supersymmetric $AdS_5$ black holes. The reduction ansatz for the metric is
\begin{equation}
ds^2_5\,=\,ds_2^2-e^{2g}ds^2_{H_3}\,.
\end{equation}
The reduced action is two-dimensional dilaton gravity,
\begin{equation}
S\,=\,\frac{vol_{H_3}}{4\pi{G}_N^{(5)}}\int{d^2x}\sqrt{-g_2}e^{3g}\left[-\frac{1}{4}R_2-\frac{3}{2}\partial_\mu{g}\partial^\mu{g}+3\partial_\mu\varphi\partial^\mu\varphi-e^{-3g}V_2\right]\,,
\end{equation}
where the scalar potential is
\begin{equation}
V_2\,=e^{3g}\left[-\frac{1}{8}\tilde{g}^2\left(e^{-2\varphi}+2e^\varphi\right)+\frac{3}{2}e^{-2g}+\frac{3}{2\tilde{g}^2}e^{2\varphi-4g}\right]\,.
\end{equation}
The scalar potential satisfies the relations, \eqref{fpoint}, at the $AdS_2$, critical point, \eqref{ads5ads2}.

We consider the background,
\begin{equation}
ds^2_2\,=\,e^{2f}\left(dt^2-dr^2\right)\,.
\end{equation}
We obtain the flow equations which satisfy the equations of motion of two-dimensional dilaton gravity,
\begin{align}
f'e^{-f}\,=&\,W+\frac{\partial{W}}{\partial{g}}\,, \notag \\
g'e^{-f}\,=&\,W\,, \notag \\
\varphi'e^{-f}\,=&\,-\frac{\partial{W}}{\partial\varphi}\,,
\end{align}
where the superpotential is given by
\begin{equation}
W=\,\frac{\tilde{g}}{6\sqrt{2}}\left(2e^{-\varphi}+e^{2\varphi}\right)-\frac{\sqrt{2}}{\tilde{g}}e^{\varphi-2g}\,.
\end{equation}
The superpotential satisfies the relations, \eqref{fpoint2}, at the $AdS_2$ critical point, \eqref{ads6ads2}. The scalar potential is obtained from
\begin{equation}
V_2\,=\,e^{3g}\left[\frac{3}{4}\left(\frac{\partial{W}}{\partial\varphi}\right)^2-\frac{3}{2}W\frac{\partial{W}}{\partial{g}}-3W^2\right]\,.
\end{equation}
By introducing the superpotential, we note that the flow equations are merely a rewriting of the BPS equations in $AdS_5$, \eqref{ads5susy}, as it should.

Finally, we reproduce the Bekenstein-Hawking entropy of the supersymmetric $AdS_5$ black holes obtained in \cite{Bae:2019poj},
\begin{equation}
S_{\text{BH}}\,=\,\left.\frac{e^{3g}vol_{H_3}}{4G_N^{(5)}}\right|_*\,.
\end{equation}
We make an observation that the two-dimensional dilaton, $\Phi\,\rightarrow\,g$, contains the information of the Bekenstein-Hawking entropy of higher dimensional $AdS$ black holes.

\section{Conclusions}

In this paper, we derived the flow equations in two-dimensional dilaton gravity. We commented on holographic $c$-theorem by introducing a quantity which monotonically decreases along the flow. As we have seen in the examples, two-dimensional dilaton gravity is ubiquitous from dimensional reduction of $AdS$ black hole solutions in string and M-theory. Their dynamics and entropy could be understood from two-dimensional perspective.

This opens several intriguing directions we may pursue. Following the studies of higher-dimensional holographic renormalization group flows, \cite{Freedman:1999gp, Skenderis:1999mm, DeWolfe:1999cp}, it is natural to further investigate the flow equations in two-dimensional dilaton gravity. There could be the Hamilton-Jacobi formulation origin of the flow equations analogous to the higher-dimensional flows, \cite{deBoer:1999tgo}. It is also interesting to study the non-perturbative stability of solutions, \cite{Freedman:2003ax}.

There are more supersymmetric black holes in $AdS_4$, \cite{Guarino:2017eag, Guarino:2017pkw, Azzurli:2017kxo, Hosseini:2017fjo, Benini:2017oxt, Bobev:2018uxk}, $AdS_6$, \cite{Suh:2018tul, Hosseini:2018usu, Suh:2018szn}, and $AdS_7$, \cite{Gauntlett:2000ng, Gauntlett:2001jj}. Furthermore, in addition to the magnetically charged static black holes we studied in this work, there is a different class of $AdS$ black holes with electric charges and rotations, $e.g.$, \cite{Hosseini:2017mds, Hosseini:2018dob, Choi:2018fdc}. It would be interesting to consider them from two dimensions.

In relation of $AdS$ black holes from two-dimensional perspective, further understanding of dual $1d$ superconformal quantum mechanics could be pursued, $e.g.$, appendix B of \cite{Benini:2015eyy}.

Our construction provides examples of two-dimensional dilaton gravity from string and M-theory. Although we did not attempt to show by dimensional reduction of higher dimensional gravity, such consistent truncations have been obtained in \cite{Gouteraux:2011qh} for generic dimensions and in \cite{Cvetic:2016eiv} and \cite{Castro:2018ffi} for $AdS_3$ and $AdS_5$ gravity, respectively. See also \cite{Li:2018omr}. It would be interesting to understand the physics of two-dimensional theories along the line of \cite{Almheiri:2014cka, Jensen:2016pah, Maldacena:2016upp, Engelsoy:2016xyb}.

\vspace{1.8cm}

\medskip
\bigskip
\leftline{\bf Acknowledgements}
\noindent We are grateful to Nakwoo Kim and Myungbo Shim for helpful discussions and to Daniel Z. Freedman and Henning Samtleben for instructive communications. Also we would like to thank the anonymous referee for clarifying comments on interpretations of some results. This research was supported by the National Research Foundation of Korea under the grant  NRF-2019R1I1A1A01060811.

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New iPad Pro 2021 could land soon with a mini LED screen
By James Rogerson 06 January 2021
Possibly coming early this year
We've been hearing for a while that Apple might upgrade the screen on a future iPad model, offering a mini LED display rather than an LCD one, and it now looks like that day could come sooner rather than later.
DigiTimes claims that it expects Apple will "possibly" launch a 12.9-inch iPad Pro with a mini LED screen in the first quarter of 2021 (so by the end of March).
We would however take this with a pinch of salt, as the site doesn't sound very certain, and it appears to have come to this conclusion based on claims from Epistar's company president, Fan Chin-yung, that "shipments for mini LED chips will begin to contribute to consolidated revenues in first-quarter 2021."
These are the best tablets
And these are the best iPads
The Samsung Galaxy S21 is launching very soon
It might land later
Epistar produces LEDs, so it presumably wouldn't make this comment if it didn't have customers lined up, though there's no confirmation that Apple is one of them, and – according to DigiTimes - Fan Chin-yung additionally stated that large-volume shipments of mini LED chips wouldn't begin until the second or third quarter, and that revenues from them wouldn't peak until then either.
So based on that it sounds to us like a mini LED iPad Pro might not arrive until later in 2021, assuming it launches this year at all.
Still, it's not the first time that we've heard there could be a mini LED iPad Pro at some point in 2021, and it's a change we'd like to see, as mini LED can improve contrast and color reproduction, as well as being more energy-efficient than an LCD screen.
The iPad Pro 2020 launched back in March, so we may well see an iPad Pro 2021 this coming March, whether or not it uses this new screen tech. If such a slate is landing that soon we'd expect plenty more leaks and rumors in the coming weeks and months, so stay tuned to TechRadar for all the updates.
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2020 was a record year for phishing sites
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Currently there are 317 children at the St. John Vianney orphanage and School in Tanzania. The need for safe drinking water is desperate. The frequent dry seasons are a true threat to the lives of the children and those who help care for them. Sr. Mary J Wandia and her Adorer Missionary Sisters of the Poor are afraid of a cholera outbreak. Please help secure water for them!
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Forum Studios is situated in the heart of Rome, wedged among the typical parks and monuments of one of the most beautiful cities in the world. Just a 5 minute walk connects Forum Studios to the famous cultural centre, Auditorium Parco della Musica, that presents a very interesting schedule of international artists every day.
Other nearby sites and attractions include the world-famous Olimpic Stadium and the huge and amazing park of Villa Borghese. Here we have picked out a selection of recommendable places to visit in your free time..
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Auditorium Parco della Musica was designed by Italian architect Renzo Piano. Three large concert halls (Sala Petrassi, in memory of Goffredo Petrassi, about 700 seats; Sala Sinopoli, in memory of Giuseppe Sinopoli, about 1200 seats; Sala Santa Cecilia, about 2800 seats) are structurally separated to ensure soundproofing, though joined at the base by a continuous lobby. A fourth "concert hall" called Cavea is an open air theater recalling ancient Greek and Roman theaters. The fan-shaped layout is formed around the central piazza. The blobs, beetles, scarabs, turtles, insect carapaces, computer mice — all are names given to these structures with thin red Roman brick and weighty dramatic lead roofs.
The Parco della Musica was formally inaugurated on 21 December 2002. After a few years it became the world's most visited music facility with over one million spectators a year.
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As a State institution the Museum was born in 1883 and as such belongs to a significant historical moment when newly unified Italy undertook the founding of national institutions. The government did not fail to show a timely and appropriate interest in supporting a 'national' art.
The National Gallery of Modern Art has thus acquired the status of "mother museum", that is to say the museum of the 19th and 20th centuries.
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Villa Borghese is a large landscape garden in the naturalistic English manner in Rome, containing a number of buildings, museums (see Galleria Borghese) and attractions. It is the second largest public park in Rome (80 hectares or 148 acres) after that of the Villa Doria Pamphili. The gardens were developed for the Villa Borghese Pinciana ("Borghese villa on the Pincian Hill"), which was built by the architect Flaminio Ponzio who developed sketches by Scipione Borghese. Borghese used it as a 'villa suburbana' (a party villa) at the edge of Rome, and to also house his art collection. The gardens as they are now were remade in the early nineteenth century.
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The Colosseum, or the Coliseum, originally the Flavian Amphitheatre, is an elliptical amphitheatre in the centre of the city of Rome, Italy, the largest ever built in the Roman Empire. It is considered one of the greatest works of Roman architecture and Roman engineering.
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The Papal Basilica of Saint Peter, officially known in Italian as Basilica Papale di San Pietro in Vaticano and commonly known as Saint Peter's Basilica, is a Late Renaissance church located within the Vatican City. Saint Peter's Basilica has the largest interior of any Christian church in the world. St. Peter's is famous as a place of pilgrimage, for its liturgical functions and for its historical associations. It is associated with the papacy, with the Counter-reformation and with numerous artists, most significantly Michelangelo.
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Piazza Navona is a city square in Rome, Italy. It is built on the site of the Stadium of Domitian, built in 1st century AD, and follows the form of the open space of the stadium. The ancient Romans came there to watch the agones ("games"), and hence it was known as 'Circus Agonalis' (competition arena). It is believed that over time the name changed to 'in agone' to 'navone' and eventually to 'navona'.
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The piazza is at the foot of the Capitoline Hill and near the Roman Forum.
The Pantheon (an adjective meaning "to every god") is a building in Rome, commissioned by Marcus Agrippa as a temple to all the gods of Ancient Rome, and rebuilt by Emperor Hadrian in about 126 AD. The nearly-contemporary writer (2nd–3rd centuries AD), Cassius Dio, speculated that the name comes either from the statues of so many gods placed around this building, or else from the resemblance of the dome to the heavens.
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To Eat & Drink
Alla Rampa
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Alla Rampa restaurant satisfies even the most fastidious tastes for more than 40 years. It is situated in the centre of the Old Town of Rome, between Spanish Steps and Mignanelli Square tourist attractions.
Ristorante Vecchia Roma
Piazza Campitelli 18 - Rome http://www.ristorantevecchiaroma.com
The Restaurant is situated in Piazza Campitelli, near the Campidoglio, in an elegant building. Since 1870 the restaurant, owned by the Palladino brothers, has been famous for its delicateness and sophistication. They are glad to delight their clients with their gastronomic offers.
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Could you have been lied to about your loved one's death?
On Behalf of B. L. Jensen, L.P. | Apr 13, 2017 | Wrongful Death
How would you feel if you found out — nearly two years after the fact — that your elderly mother had died at the hands of an inept doctor doing an unauthorized procedure instead of pneumonia like you were told?
Would it make you feel even worse to learn that the hospital forged documents with your name on them, giving consent to the procedure?
That's the dark reality for the family of a Texas woman, who only found out that their mother died a painful death, covered in blood, while a doctor mishandled a catheter in a large vein in her neck thanks to the efforts of a whistleblower. The odds are good, had the family known of the medical facility's previous problems — which included letting an intensive care patient die after his breathing machine became disconnected because nurses didn't even recognize what the alarm on the unit was signaling — they might have taken their mother elsewhere in the first place.
As it is, all they can do now is file a wrongful death claim against the physician and the medical center in hopes that it will bring to light the deceptions happening in many of our nation's hospitals and nursing homes.
Here are some fast facts that should alarm anybody:
— Preventable errors may be the number one cause of the death in hospitals.
— More than seven out of 10 euphemistically known "adverse events," or medical errors, were the result of negligence.
— Over 90 percent of those errors were entirely preventable.
— "Near misses," where the patient could have been harmed, but the mistake was prevented at the last minute, occur 300 times more often than actual errors.
— Most hospital leaders would prefer that patients only learn of errors by request and feel that the state shouldn't release information at all under certain circumstances.
— You can trust nurses to report errors far more often than physicians. In some facilities, more than 93 percent of errors were reported by nurses.
If you learn that a fatal medical error was hidden from you at the time of your loved one's death, contact an attorney about a wrongful death claim as soon as you learn the truth. It may not be too late to pursue a claim if you were purposefully deceived.
Source: Houston Chronicle, "She believed her mom died peacefully. Two years later, a nurse wrote to say what really happened.," Mike Hixenbaugh, April 07, 2017


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Home / Bitcoin / Robinhood Review 2022: An Investing App to Avoid
Robinhood Review 2022: An Investing App to Avoid
Byadmin May 15, 2022 August 22, 2022
Robinhood Review: We want to help you make informed decisions. Some links on this page clearly marked may direct you to a partner website and may result in us earning a referral commission.
Founded in 2013, Robinhood is a free platform that makes stock market trading easy — too easy, according to some experts and regulators. And it doesn't offer the tax-advantaged investment accounts essential for building wealth over the long term.
The app has 21 million customers and over $102 billion in assets and strives to be "easy, friendly and approachable" for its customers. The average age of users is 31 and about 50% of them are new investors, a Robinhood spokesperson said.
While its low cost and usability will certainly captivate users, investors should be wary of Robinhood.
In recent years, Robinhood has faced fines, public scrutiny, and lawsuits. In 2020, Robinhood was hit with a $65 million fine by the Securities and Exchange Commission (SEC) for misleading customers. (A representative at the time said the settlement "relates to historic practices that do not reflect Robinhood today.") In June 2021, Robinhood was fined $70 million by the Financial Industry Regulatory Authority (FINRA) for malfunctioning and misleading its customers. (In response, Robinhood said it had "invested heavily in improving platform stability, enhancing our educational resources, and building our customer support and legal and compliance teams.")
Earlier in 2021, Robinhood settled a lawsuit for wrongful death about the death of a 20-year-old user who committed suicide after seeing an incorrect negative balance of $730,000 on the app. (A Robinhood spokesperson said: "We were shocked by the death of Alex Kearns. Since June, we have made improvements to our range of options… We remain committed to making Robinhood a place to learn and invest responsibly. .")
For a list of cheap and accessible alternatives to Robinhood, see NextAdvisor's Best Online Brokers of 2022. Or read on to see some more pros and cons of the Robinhood app.
Pros and Cons of Robinhood
Free platform: the account is free to use and users can create an account in minutes.
ractional Shares Available: Ability to buy partial shares of a company's stock.
No costs to trading cryptocurrencies: Robinhood offers 7 coins in 48 states.
Minimum $0 account Easy to use
New pension or tax-advantaged accounts – Robinhood only offers taxable brokerage accounts, so if investors are looking for tax-advantaged investment accounts like Roth or traditional IRAs, they should look elsewhere.
Gamify investing and trading — Regulators have accused Robinhood of pushing new investors into "unnecessary trading risks" with features like digital confetti intended to celebrate trade. (Robinhood retired the confetti feature prior to the IPO.)
Lack of learning resources — Educational resources for new investors are limited to a newsletter and some articles on the Robinhood website. In-depth reports are only available with a Robinhood Gold membership ($5 monthly fee).
Lack of some diversified investment products — No mutual funds or bonds, which can make it difficult to build a well-rounded portfolio.
Robinhood at a glance
Self-directed investment app Two main accounts available: brokerage and cash management Available investments include more than 5,000 stocks and ETFs listed on US exchanges, more than 650 stocks on global exchanges via ADRs, options, gold, crypto Gold price offers Level II market data, in-depth research reports from Morningstar, larger instant deposits Educational articles on the basics of investing in the "Learn" section No minimum account opening or account maintenance fees 24/7 customer support phone line
Who is Robinhood best for?
So is Robinhood any good? Well, it can be attractive to newer investors without a lot of money to invest with. With no trading or commission fees, no account minimums or account maintenance fees, it's an inexpensive way to get wet with the stock market.
However, the app's gamified approach to investing makes it too easy to trade quickly and often. Robinhood users are nine times more likely to trade than users of other low-cost brokers like E*Trade, according to a November 2020 survey by behavioral finance experts. other studies have shown that passive investing – also known as buy and hold – builds more wealth in the long run than taking active trading decisions. So Robinhood's ease of trading can work against you.
Another reason to look elsewhere? Robinhood does not offer Roth IRAs or traditional IRAs. Financial experts love these accounts because they help protect you from taxes while building wealth. Other discount brokers allow you to make the same investments you could with Robinhood, except within a tax-advantaged retirement account.
Robinhood may be a good option for people looking to invest in cryptocurrency, although it lacks some of the more advanced features you get with cryptocurrency exchanges. Financial experts advise investors to limit their crypto holdings to about 5% of their total portfolio and achieve other financial goals first, such as eliminating high-yield debt, building an emergency fund, and investing in a traditional retirement account.
While some discount brokers offer in-depth reports and detailed research, online workshops, and virtual trading to help newer investors learn the basics of investing, Robinhood's educational resources are limited to an in-app Learning section. While it is a robust library of content, novice investors could benefit from accessing more learning resources for free.
Finally, those looking to get AI-powered guidance through a robo-advisor, or have their portfolios managed by a professional, will also have to look elsewhere.
Open an account with Robinhood
1. Research discount broker platforms and apps. If you are looking for an inexpensive way to start trading, start by doing your homework and looking at various discount brokerage platforms. What kind of services, products and investments do they offer?
2. Compare rates. Look beyond commission trading fees and see if there are any account maintenance fees, account opening minimums, or other fees. Consider the expense ratio of each stock or fund you plan to invest in.
3. Submit an application through the Robinhood app. You must provide personal information such as your tax ID and financial information. Within a few days you will receive an email approving your application or asking for more information. Should Robinhood need to verify your identity, the process could take another five to seven days.
During the application process you have the option to preload your first deposit. That way, the money will be ready to use when your application is approved.
What is the monthly fee for Robinhood?
Robinhood has no monthly fees, no account maintenance fees, and no trading or commission fees for stocks, ETFs, and cryptocurrency.
Is Robinhood good for beginners?
Robinhood makes it easy for new investors to get started. There are no minimum amounts to open an account, nor commission fees. But younger investors will benefit most from tax-advantaged investment accounts like the Roth IRA, which isn't available on Robinhood, and a buy-and-hold strategy to maximize compound interest, which goes against Robinhood's incentives to trade often.
How do I get my money out of Robinhood?
The limits for withdrawing funds from your Robinhood account are as follows:
Up to 5 withdrawals per business dayWithdrawal up to $50,000 per business day
When you deposit money, it can take up to five business days for the transaction to be processed. Once the transaction is completed, you can withdraw the funds.
After a sale, there is a settlement period from the trading date, plus two trading days before you can withdraw funds.
Is it safe to buy on Robinhood?
Robinhood is a registered brokerage and member of the SIPC meaning that as an account holder your assets will be protected if the brokerage becomes insolvent. It also protects against unauthorized trade or theft. In these cases, assets in your Robinhood account are covered up to $500,000 per customer. This includes $250,000 in cash.
There are also standard security measures, such as two-factor authentication on the app or SMS, and encryption of sensitive information.
There are always risks associated with investing and there is no protection against the ups and downs of the market.
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Williamsburg Neighbors Rally "Where's Our Park?" On the Spot It Was Promised to Be!
"Promises, promises! I'm all through with promises now!" is a line from a Burt Bacharach song that could be the musicalized feelings of those fighting for the full 28 acres of the Bushwick Inlet Park. The promise was originally made by Mayor Bloomberg 10 years ago in order to push through changes in rezoning that opened the door to luxury towers on Williamsburg's waterfront. The rezoning went through and quickly up rose the skyline along the East River. In counterpoint to the speed with which affluent dwellings were developed the promised park's progress was stalled. Mayor Bloomberg's administration put off purchasing the entire park's parcel of land for various reasons, one of them being price. Rather predictably as time passed, the real estate prices rose exponentially along this waterfront. The passing of time also passed the problem of completing the park to a new Mayor. The bottom line is: it's been 10 years and still no park.
"There are children here today that weren't even born yet when we were first promised this park," said Jens Rasmussen a longtime proponent of this park at the Friends of Bushwick Inlet Park's Rally on Saturday, May 16th. The rally took place on an asphalt alley alongside the City Storage facility, properties that would more than make up the missing acres that would complete the Bushwick Inlet Park as promised.
Children were a large part of this rally. Standing side by side, behind the electeds and community leaders who took to the stage, was a line of kids each holding a letter or punctuation mark that spelled out #Where's Our Park? all in a rainbow of colors, plus other groups of kids were holding up banners. It should go without saying that children need green spaces. I said it should go without saying, but when North Brooklyn holds the brass ring for having the least amount of green space per capita in NYC going on decades, "Children need green spaces" needs to be said over and over again by hundreds of voices.
"Where's Our Park?" was chanted over a hundred times during the rally by the estimated over 300 to 500 people who were there.
(top left) Council Member Stephen Levin has been and still is front and center in the fight to complete Bushwick Inlet Park; (top right) "A park is progress!" says AM Joe Lentol; (lower left) State Senator Daniel Squadron, "I think we're going to stay out here until they give us a park!"; (lower right) State Senator Martin Dilan, "I was here for the rezoning and I asked then: How can we believe you?" regarding the decade old promise for the park.
Council Member Stephen Levin said, "You don't make a promise you can't keep. So keep your promises! There are a lot of kids here, and I was told as a kid to keep my promises." You could hear 'What is this broken promise teaching these kids?' loud and clear underneath his words.
"I was there 10 years ago when Bushwick Inlet Park was promised and I asked 'How can we believe you?' Today, as new rezoning proposals spring up throughout my district this incomplete park weighs heavily over that process. We can't plan in good faith knowing something as significant as Bushwick Inlet Park, the one condition of a rezoning plan that created the North Brooklyn we know today, was never met," said Senator Martin Malavé Dilan.
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The fourth generation BMW M3 is the largest, fattest M3 ever created.Yet it 's also the fastest and most tenacious while being entirely livable as a daily driver. In creating this new generation of M3, BMW has built a car that is as close to everything to everyone as anything on the market. Whether it be at the track, on the highway or making a coffee run, the M3 handles it all with ease. But can a car that is all things to all people be a true M3?
The moment I pushed the start button and laid eyes on the M menu in the iDrive I knew that this M3 would be dramatically different than all those that came before it. For better or worse, a different kind of technology is at play with this car. From M Dynamic mode, to the much celebrated M Differential Lock, BMW has given the new version of this iconic car the tools to help the typical M formula of performance reach new heights.
Our day started out with some track time at the famed Laguna Seca racetrack. The track is home to one of the most famous turns (or series of turns) in the world. The corkscrew (as it 's reverently known) drops six stories in the matter of seconds all while trying to send you through the seat of the car. If you 've driven it on a video game or even the reproduction at Ascari, you haven 't really driven it. Nothing really prepares you for the combination of braking, dropping and accelerating that it demands.
Yet the M3 made it deceptively easy. With the M Differential, EDC and DSC set to their highest levels, the car flatters even the most ham-fisted driver. A 414 hp car doesn 't have a right to be as easy to drive and forgiving as the M3 can be. Reigning in the car from a drift in turn four was effortless. And when I found myself pushing the car through turn five a little too hard (a turn much different than any video game portrays it) the M3 used its processors to bring the car back into line so subtly that no one but the driver knew a mistake had been made.
Some purists may scoff at the notion of fine tuning suspension and throttle settings via a large knob in the center of the console but the results speak for themselves. Through this, the M3 was able to power out of Laguna Seca 's corkscrew as easily as it was able to soak up road imperfections on the road. Cars equipped without the optional technology package (including EDC and a host of other ways to fine tune your M3) didn 't feel composed in the corners and thus wasn 't quite as fast around the track.
The M3 now features a 4.0L V8 rather than the inline six found in the last two generations. But don 't expect the wizards at M to create a burbling V8 like everyone else. High revving and exceptionally light weight (at 331 lbs it 's lighter than the inline it replaces), the engine has exceptional character. While it 's down on torque to its rivals, the power builds to a crescendo at redline that is nothing less than addicting.
Managing it and all the other technology is a new CPU that was developed in house at the M division. The V8 also has double Vanos and individual throttle bodies – both slightly different in concept than the previous engine 's same technology. And to help the engine get the enormous amount of oxygen required, BMW has created a number of scoops and air-inlets (including one in the hood itself) that help deliver 600 ft of air a minute.
The car is also differentiated from the more pedestrian 3 Series (and even the mighty 335i) by a suspension that, save for one component, is entirely new and made from aluminum. Elsewhere the M3 has added structural bracing and, in turn, more torsional rigidity for better control, response and performance. Specifically M has added a reinforced strut plate and thrust plate to the front of the car and a V strut in the rear. This is not just a 335i with a different engine. The M3 is a completely new car focused on ultimate performance.
Outside the car only carries over the doors, glass and rear light cluster from the 3 Series. The effect is an M3 that 's a little less subtle than the E46 M3 without looking too boy-racer like.
In order to save weight (and to prevent the first 4000lbs M3 coupe or sedan) the M3 coupe uses several key weight saving measures. The most noticeable is a carbon fiber and plastic reinforced roof that saves 12 lbs, helping to bring the center of gravity lower for improved handling. The car also has a lighter aluminum hood, lighter brakes and wheels and an engine that is moved even further back than the 335i.
On the aero side of things the bottom of the M3 is completely covered helping to reduce lift at high speeds. Even the redesigned mirrors lend a hand producing 3lbs of downforce on each side.
All this helps the M3 differentiate itself from the already excellent 300hp 335i. So extensive are the modifications to the M3 that comparing the two cars is almost pointless on the face of it. Where the 335i is meant to appeal to the public at large, the M3 is a much more focused machine. The M3 revs quicker, shifts quicker, steers more precisely and overall feels much more special. While the 335i is a quick car that handles well (it is 130 lbs lighter), the M3 is a fast car that handles exceedingly well.
And the sound. The glorious sound of the M3 's 4.0L V8 coming down the front stretch of Laguna Seca is tattooed on my brain for eternity. The V8 's soundtrack is addicting to the point of absurdity behind the wheel. And the sound isn 't like a typical V8. Instead it has a motorsport like mechanical quality to it that gets increasingly frenetic has the revs swell to redline.
The M3 isn 't without fault however. It 's 3700 lbs for God-sake. And while the brakes had begun to fade towards the end of my track time with the car, I don 't consider that too surprising considering the cars were all using street pads and stock fluid. However the most noteworthy issue beyond the excess weight is a slight (most won 't notice it) loss of steering feel as compared to the E46 M3. And, as much as I love the new V8, there was just something overtly visceral about the previous inline six made for such a great package. However antiquated it might be, I 'll miss it. All of this adds up to the new M3 feeling slightly less involved than the previous iterations.
But these downsides are forgotten the moment you go from 3rd to 2nd while blipping the throttle. This car, on the right road, is like an automotive symphony and you 're conducting. You 're in control of everything and there are no surprises. The car is always there and ready for more. More grip, more power, more brakes and what seems like a super computer behind it all invisibly helping you look like the next Lewis Hamilton.
The M3 's limits are so high that you quickly come to terms with the fact that you can 't possibly approach them on public roads. Instead, it 's best to just head for your favorite road and settle in for what will surely be a profound experience.
My time in the M3 driving down West Carmel Valley Rd. just east of Laguna Seca was one of those kinds of experiences. A beautiful day on an exceptional road with no traffic can be a rare thing. And in a car like the M3, it can be nothing less than moving.
Bringing it all back to earth is the price however. The four door rings in at $54k while the two door (the lighter of the two) is just a hair over $58k. That 's two well equipped MINI Cooper S 's. Or one nice 135i plus a used E36 M3. But this new M3 almost defies those kind of comparisons. Instead you should think of it as a 911 competitor with more power, two extra seats and a real trunk.
And it 's when you consider it in that regard it begins to make sense. This new M3 is one of those rare cars that is as at home on the track as it is on the road. It 's exceptionally comfortable without being soft. It 's wickedly fast while still being accessible to drivers of all skill levels. And on top of it all, it still has the intangibles that makes an M3 the car that so many lust after.
can't wait to see it in ALMS. so long at it beats the P-cars, i'll be happy. if it squeezes by a ferrari or two that wouldn't be bad either.
Nice job M, think they have another winner in this car. Can totally agree Gabe that the sound from that V8 is addicting.
I'd love to see a comparison of the M3, 335xi with xDrive, and 135i.
I can tell you right now that anything with xDrive doesn't belong with the two other cars you mentioned. BMW doesn't believe (rightly so IMHO) that an AWD vehicle should be meant for sporty driving so they don't offer even the normally optional sport suspension on those cars. A RWD 335i would be an interesting comparison but I can tell you that the M3 is a much more focused performance car and would easily be my pick of the two if money wasn't an object.
The reason why I am considering the 335xi for my next purchase is because I have seen regular rear-wheel drive 335i's spin their inner rear tire hopelessly during autocrosses. They produce so much torque down low, but they lack a limited slip differential, so they simply can't put all that power down onto the pavement effectively. I was hoping xDrive would make the 335 a better autocrosser.
Nice review. Can't wait to test drive one and see how much better the next gerneration has become. Its awfully heavy and steering number than the E46 is a scary thought. I thought my wife's old 528 steered much better.
However, having come from an M roadster and E46 M-3 to a 911, I really don't see the cars as competitors. If you need the storage space, and the extra seats and want to save some rather serious $$ difference, the M-3 is an outstanding choice.
But the 911 is a sports car designed to be one from the ground up. P has honed that chassis over 40 years and its obvious within the first few miles behind the wheel. The 911 is a surgical driving instument. Amazing brakes, steering and balance in all its inputs.
BMW will sell a ton ( or many tons LOL) of M-3s and the inevitable comparisons to the 911 will continue just like when the E 46 M-3 came out.
IMHO….the M-3 is amazing sports sedan BUT the 911 is an amazing sports car.
Wow, those things sound like something that should be in NASCAR. Those engines make the sound of POWER.
Gabe- You are converting me from a MINI fan to a BMW fanatic. Great article.
Quick question for you. The M3 looks impressive, but for ~$2000 you can have Dinan reprogram the ECU of a 335i and give you 384 hp and 421 lb/ft of torque and still have $ leftover. Which would you recommend?
NIce review! Sounds like this car is a giant bag of technology and plenty of ponies… which will definitely set it apart from the competition and its family members.
That is something that boggles my mind. Why wouldn't BMW at least offer LSD as an option on the 335i? That is a HUGE handicap for a car with such a large and immediate level of torque.
The philosophy is that only the M-cars come with limited-slip differentials. The lesser cars are less sport-oriented and more luxury-oriented, so they don't get them. An LSD is one thing that sets an M-car apart from a non-M-car.
I agree, an LSD should be an option, just like it's an available option for the Mini Cooper S. One doesn't have to get a full-blown JCW kit to get an LSD; it's available separately as a stand-alone option.
It's true that the 335xi doesn't have the sports suspension, but that is at least more easily rectified by changing the stock suspension. Adding on 4wd to a regular 335i is a much more difficult task. 🙂 That's why I'm leaning more towards a 335xi, whose suspension I can mod later.
Gabe, Money not being an issue, Which would you prefer; the new M3 or the M5?
All of those videos are awesome. Looks like you had a fun time. I know that I would have.
BMW doesn't believe (rightly so IMHO) that an AWD vehicle should be meant for sporty driving so they don't offer even the normally optional sport suspension on those cars.
Gabe, I was under the impression that BMW couldn't offer a sports suspension on their AWD cars because their AWD set-up didn't allow it.
I was told their AWD design limits the car from being lowered sufficiently for a sports-suspension. Something about the axle being in the way.
The reason it sounded plausible is that it's sort of puzzling that BMW would intententionally deny customers a true AWD sports suspension for philosophical reasons, as you suggest, and grant Audi its one and only claim to fame over BMW for winter drivers, which seems to pretty much keep them Audi business.
Is BMW hesitant to do AWD cars "too well" and dull/confuse their RWD heritage? Is it a must-offer for the market, but not a main focus?
I've always wondered about this, and your comment touched off this flurry of questions–really, the same question. Thanks for reading.
Got to drive one Thursday. 12 miles, city streets. Mind-boggling. Sounds wonderful, driver's position and controls perfect, shifter super-smooth, steering precise. Floored it from a red light in first and bang – 8400-rpm limiter in incredibly short time. In all, maybe the best drive for the buck on the planet.
Then drove home in my year-old E93 335i. Compared well, still a superb car … but it ain't an M.
Keep your eye out for BMW Drive For The Cure at nearby dealership. The fleet touring Florida has the M-3 to drive (and an X6 to look at – The Incredible Hulk). Had a 1-series, but was pulled from fleet to keep from overshadowing model's launch.
Here are some great videos of BMW Rahal Letterman from Road Atlanta. Youtube: Dirk Mueller discusses his first stint charge from 7th to 3rd….