Extremal black branes upon compactification in the near horizon throat region are known
to give rise to

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0$ automatically satisfies this, as was the generic case in~\cite{Kolekar-ml-2018sba}.\ For the case with simply one scalar field $\Psi$, the criteria for a stable $AdS_2$ critical point are \begin{equation} U_h=0\,, \qquad \frac{\partial U}{\partial\Psi}\bigg\rvert_h=0\,, \qquad \frac{\partial U}{\partial(\Phi^2)}\bigg\rvert_h=\frac{-2}{L^2}\,, \qquad \frac{\partial^2 U}{\partial\Psi^2}\bigg\rvert_h> -{\Phi_h^2\over 4L^2}\,. \end{equation} ]]>

0$ everywhere in the bulk as well. This then would imply that ${\tilde\Phi}(u)$ is a monotonically increasing function as $u$ increases and flows towards the boundary. A heuristic picture of the setup appears in figure~\ref{fig1}\ (see also the discussion on nonconformal D2-branes, section~\ref{cfnD2M2}, which exemplifies this). \paragraph{Step 2:} now we proceed to argue that ${\tilde\Phi}'$ is positive near the boundary for suitable boundary conditions, namely that the ultraviolet of the theory belongs in the hvLif family~(\ref{hvLifmetric}) that we have been focussing on (which also includes AdS for exponents $z=1, \theta=0$). The extremal branes we are considering here are excited states at finite charge density in these theories: the near boundary region corresponds to the high energy regime of the dual, well above the characteristic scales of the excited states. So it suffices to use the asymptotic (uncharged zero temperature) form of these backgrounds. Using~(\ref{hvLifBPhi}), we have $d_i=D-2$ and\ ${\tilde\Phi}=\Phi^{\frac{2}{d_i}}$. Then retaining only relevant factors, we have \bea \tilde{\Phi}&\sim& \bigg(z-\frac{2\theta}{d_i}\bigg)^{\frac{d_i-\theta}{zd_i-2\theta}}u^{\frac{d_i-\theta}{zd_i-2\theta}}\,, \qquad \tilde{\Phi}'\sim \bigg(z-\frac{2\theta}{d_i}\bigg)^{\frac{d_i-\theta}{zd_i-2\theta}}\frac{(d_i-\theta)}{(zd_i-2\theta)}\frac{1}{u^{\frac{d_i(z-1)-\theta}{zd_i-2\theta}}}\,, \nonumber \\ \tilde{\Phi}''&\sim& -\bigg(z-\frac{2\theta}{d_i}\bigg)^{\frac{d_i-\theta}{zd_i-2\theta}}\frac{(d_i-\theta)(d_i(z-1)-\theta)}{(zd_i-2\theta)^2}\frac{1}{u^{\frac{d_i(z-1)-\theta}{zd_i-2\theta}+1}}\,. \eea \looseness=-1 Then $\frac{\tilde{\Phi}''}{\tilde{\Phi}}\leq 0$ gives the null energy condition $(d_i-\theta)(d_i(z-1)-\theta)\geq 0$. A reasonable dual field theory requires positivity of specific heat if the theory is excited to finite temperature. Since the entropy for these theories scales as\ $S\sim V_{d_i} T^{d_i-\theta\over z}$, the positivity of the corresponding specific heat imposes\ ${d_i-\theta\over z}\geq 0$.\ This implies $d_i-\theta\geq 0$ since $z\geq 1$. Alongwith the null energy condition, this leads to\ $(d_i(z-1)-\theta)\geq 0$. These two conditions together imply \be\label{zdi-2theta} zd_i-2\theta = (d_i(z-1)-\theta)+(d_i-\theta)\geq 0\,. \ee Then we see that $\tilde{\Phi}'$ is positive in this near boundary region. Roughly, ${\tilde\Phi}\sim u^n$ and ${\tilde\Phi}'\geq 0$ and ${\tilde\Phi}'' \leq 0$ require $n\geq 0$ and $n(n-1)\leq 0$, \ie\ $0\leq n\leq 1$. We have argued that this is true if the null energy conditions and positivity of specific heat are satisfied. Thus finally, we have shown that for the ultraviolet data we are considering, ${\tilde\Phi}(u)=\Phi^{2/(D-2)}$ is monotonically decreasing as $u$ flows to the interior (lower energies). Since the exponent ${2\over D-2}$ is positive, this implies that $\Phi(u)$ satisfies the same monotonicity property. This proves that the holographic c-function~(\ref{cfn}) we propose in fact satisfies the c-theorem. At the IR $AdS_2$ horizon, ${\cal C}$ in~(\ref{cfn}) approaches the extremal black hole entropy~(\ref{extrEntropy}), which is the IR number of degrees of freedom controlling the number of black hole microstates, akin to a central charge for this subsector. In fact it is this requirement that ${\cal C}\ra S_{BH}$ at the IR $AdS_2$ fixed point which fixes the precise scaling of ${\cal C}$ in terms of $\Phi$\ (else any positive power of $\Phi$ is monotonic, from the above arguments). \looseness=-1 It is interesting to note that we have mainly used the first null energy condition in~(\ref{necconstraints2}) in the above arguments. The second null energy condition appears to be more a condition on the matter configurations: for instance, the second condition for hvLif backgrounds~(\ref{hvLifmetric}) gives $z\geq 1,\ d_i+z-\theta\geq 0$ in~(\ref{hvLifnec}), which follow from reality of the fluxes supporting the background, and also follows from specific heat positivity. to illustrate the condition in more generality, let us restrict to $D=4$ for simplicity: then the second condition in~(\ref{necconstraints2}) gives \be \frac{(\Phi^2)''}{\Phi^2}\ \leq\ \frac{(B^2)''}{B^2} \,, \ee which says that the dilaton ``acceleration'' is not greater than that of the 2-dim metric. As we approach the $AdS_2$ region, we have\ $B^2\sim (u-u_0)^2$ so this becomes\ $\frac{(\Phi^2)''}{\Phi^2} \lesssim {2\over (u-u_0)^2}$\ which is trivially satisfied as $u\ra u_0$ since the right hand side grows large. Thus the near $AdS_2$ region does not provide any additional constraint from this energy condition. However the near boundary region gives nontrivial constraints on the exponents defining the theory from this energy condition as we have seen. We will discuss this further later. One might be concerned that the null energy conditions (and the Einstein equations) are second order equations while renormalization group flow is first order. It is important to note in this regard that the boundary conditions we have imposed is on the first derivative ${\tilde\Phi}'$, which then automatically implies monotonicity. This physical boundary condition has effectively ruled out the other (growing) mode which would likely be singular in the interior. In explicit examples (\eg\ nonconformal branes redux, later), we can check this dilatonic c-function in fact has the right behaviour. Consider for instance an extremal brane in an hvLif theory where $B^2, \Phi^2$ near the boundary have the form~(\ref{hvLifBPhi}) while in the near $AdS_2$ region, $B^2\sim (u-u_0)^2$ and $\Phi\sim u^A$ globally, with $A={d_i-\theta\over zd_i-2\theta}$\,. Then using the arguments around~(\ref{zdi-2theta}), we see that $A\geq 0$ so that $\Phi^2(u)$ can be seen explicitly to monotonically decrease through the bulk as $u$ decreases flowing towards $AdS_2$. We also see that $A\leq 1$ so that $\Phi''\leq 0$ in accord with the first energy condition in~(\ref{necconstraints2}). The second energy condition in the near boundary region simply imposes the constraints on the exponents that we have seen, which are required of the theory. In the near $AdS_2$ region, $B^2\sim (u-u_0)^2$ and so as described above, the second energy condition is satisfied. This family includes AdS where $z=1, \theta=0$ and $\Phi^2=u^2$. From the point of dual 1-dim theories which flow to the $CFT_1$ dual to the $AdS_2$ bulk theory, the arguments above suggest that ${\cal C}$ in~(\ref{cfn}) is a candidate c-function. While spatial coarse-graining does not make sense in $0+1$-dim (no space!), the renormalization group defined in terms of integrating out high energy modes does make sense, \ie\ as a flow to lower energies (IR). In the present context, we have defined the holographic c-function ${\cal C}$ as essentially inherited from the higher dimensional theory that has been compactified: it would be interesting to understand the c-function from the dual 1-dim point of view. ]]>

0$ at the $dS_2$ critical point. However this violates the condition~\eqref{NEC2-UdU} which we expect must hold if we take the potential $U$ as arising from some higher dimensional reduction as we have discussed (implicitly taking $U$ to have a leading term arising from a negative cosmological constant as in known brane realizations followed by positive flux contributions). Of course there are rolling (time-dependent) scalar solutions, as \eg\ arises from reductions of $dS_4$\ (say with Poincare metric $ds^2={R_{dS}^2\over\tau^2}(-d\tau^2+dw^2+dx_i^2)$)\,. In 4-dim Einstein gravity with a positive cosmological constant $\Lambda>0$, the 2-dim potential simply becomes $U=2\Lambda\Phi>0$, and the 2-dim dilaton is $\Phi^2\sim {1\over\tau^2}$\,. The nature of such solutions (even in this simple classical sense) might be different from our $AdS_2$ discussions here and might be worth exploring (see \eg~\cite{Anninos-ml-2017hhn}). ]]>