Conformal Symmetry for Black Holes in Four Dimensions and Irrelevant Deformations

TL;DR

This work establishes a concrete bridge between four-dimensional, non-extremal black holes and a two-dimensional conformal field theory by using the subtracted geometry to expose an $AdS_3\times S^2$ structure upon uplift. It constructs an explicit interpolating flow between the original asymptotically flat black hole and the subtracted geometry, and shows that this flow corresponds to irrelevant deformations with $(h,\bar h)=(2,2)$ in the dual CFT, enabling quantitative assessment of when CFT computations are valid. The analysis provides a clear criterion for the CFT window in terms of the IR temperatures $T_L, T_R$ and the deformation parameters $\alpha_i$, clarifying the regime where holographic techniques yield reliable results and how corrections can be systematically computed. Overall, the paper gives a robust framework for applying AdS/CFT to asymptotically flat black holes and outlines future directions, including extensions to rotation and a detailed perturbative scheme in the dual CFT.

Abstract

It has been argued several times in the past that the structure of the entropy formula for general non-extremal asymptotically flat black holes in four dimensions can be understood in terms of an underlying conformal symmetry. A recent implementation of this idea, carried out by Cvetič and Larsen, involves the replacement of a conformal factor in the original geometry by an alternative conformal factor in such a way that the near-horizon behavior and thermodynamic properties of the black hole remain unchanged, while only the asymptotics or "environment" of the geometry are modified. The solution thus obtained, dubbed "subtracted geometry", uplifts to an asymptotically AdS$_{3}\times S^{2}$ black hole in five dimensions, and an AdS/CFT interpretation is then possible. Building on this intuition we show that, at least in the static case, the replacement of the conformal factor can be implemented dynamically by means of an interpolating flow which we construct explicitly. Furthermore, we show that this flow can be understood as the effect of irrelevant perturbations from the point of view of the dual two-dimensional CFT, and we identify the quantum numbers of the operators responsible for the flow. This allows us to address quantitatively the validity of CFT computations for these asymptotically flat black holes and provides a framework to systematically compute corrections to the CFT results.

Figures (3)

• Figure 1: Log plot of $\gamma(r) \equiv \frac{d\log(\Delta)}{d\log r}$ for the general solution \ref{['general solution 1']}. The bottom red curve with $\gamma(r\gg 2m)=1$ corresponds to the subtracted geometry ($a_{1}=a_{2}=a_{3}=0$), while the various curves with $\gamma(r\gg 2m)=4$ correspond to the original geometry with different values for $a_{1}=a_{2}=a_{3}\equiv 1/\sinh(\delta)$. The different curves have increasingly larger values of $\delta$ towards the right, so we see that the original and subtracted geometries agree over a broader range in $r$ as the magnetic charges $B\sim \sinh(2\delta)$ increase.
• Figure 2: The dashed red line represents the difference between the original and subtracted fields $\phi_1\,$. The solid black line is a linear function with slope $4e^{-2\delta_1}$. For this plot, we chose $\delta_0=\delta_1=\delta_2=\delta_3 = 15$, $m=1$, and the domain is $r \in [100,10^{-4} e^{2\delta_1}]\,$.
• Figure 3: The dashed red line represents the difference between the original and subtracted fields $\phi_0\,$. The solid black line is the function $4(e^{-2\delta_1} + e^{-2\delta_2} + e^{-2\delta_3}) \frac{(\Pi_c^2+\Pi_s^2) r - 2m \Pi_s^2}{(\Pi_c^2-\Pi_s^2) r + 2m \Pi_s^2}$. For this plot, we chose $\delta_0 = \delta_1=\delta_2=\delta_3 = 15$, $m=1$, and the domain is $r\in [100,10^{-4} e^{2\delta_1}]\,$.