Constructing AdS_2 flow geometries

TL;DR

The paper addresses how interior flow geometries that are asymptotically $AdS_2$ can be encoded in boundary observables and how a solvable microscopic RG flow, modeled by a two-flavour SYK system with a relevant deformation, maps to a deformed dilaton-gravity description. It develops a macroscopic framework based on energy-condition constraints and a boundary-to-bulk map that reconstructs the full bulk metric from a boundary two-point function of a nearly massless scalar, including time dependence and thermal extensions. Microscopically, it constructs an exact large-$N$ RG flow in a deformed SYK model, computes boundary correlators across the flow, and identifies two near-conformal regimes connected by the bulk dilaton dynamics, with thermodynamics guiding the bulk potential $U( extPhi)$. Holographic considerations then relate these flows to a 2D dilaton-gravity theory, reconstructing the bulk dilaton potential and couplings, and showing the flow interpolates between two (near) $AdS_2$ geometries, with speculative extensions to $dS_2$ interiors. Overall, the work provides explicit mechanisms to connect microscopic RG data to macroscopic bulk geometries in $AdS_2$, offering a concrete laboratory for holographic RG flow and dilaton-matter couplings in low dimensions.

Abstract

We consider two-dimensional geometries flowing away from an asymptotically AdS$_2$ spacetime. Macroscopically, flow geometries and their thermodynamic properties are studied from the perspective of dilaton-gravity models. We present a precise map constructing the fixed background metric from the boundary two-point function of a nearly massless matter field. We analyse constraints on flow geometries, viewed as solutions of dimensionally reduced theories, stemming from energy conditions. Microscopically, we construct computationally tractable RG flows in SYK-type models at vanishing and non-vanishing temperature. For certain regimes of parameter space, the flow geometry holographically encoding the microscopic RG flow is argued to interpolate between two (near) AdS$_2$ spacetimes. The coupling between matter fields and the dilaton in the putative bulk is also discussed. We speculate on microscopic flows interpolating between an asymptotically AdS$_2$ spacetime and a portion of a dS$_2$ world.

Figures (7)

• Figure 1: Entropy as a function of the temperature (in logarithmic scale) to leading order in the large $N$ and $q$ expansion. Different curves correspond to different values of $s^2=10^{-6},10^{-5}, 10^{-4},10^{-3}, 10^{-2}, 10^{-1}, 10^{0}$, from left to right.
• Figure 2: Entropy as a function of the temperature in the two linear regimes. The blue dots correspond to the exact evaluation of (\ref{['entropy']}), while the solid yellow lines correspond to the approximations in (\ref{['series_deepIR']}) and (\ref{['series_intIR']}). In both cases we fix $s^2=10^{-6}$.
• Figure 3: (a) Dilaton potential for $s=10^{-5,-4,-3,-2}$, from bottom to top. The almost vertical regimes in this logarithmic plot correspond to regimes of linear dilaton potential, that we zoom in (b). In a physical setting, the potential will be cut-off in the UV at a value of $\Phi$ where the potential is still linear, so the curvy features in the upper-right corner of the plot will not be of interest. (b) We zoom close to the linear regimes in the deep and intermediate IR regions for the yellow curve with $s=10^{-4}$. The dashed lines correspond to the approximations given in (\ref{['potential_deep']}) and (\ref{['potential_int']}). The brown line corresponds to keeping only the linear term, while in light blue we also included the first correction.
• Figure 4: (a) Correction to the boundary two-point function for the centaur geometry in blue and the AdS$_2$ black hole in yellow ($e^{2\gamma(z)} = \sinh^{-2} z$). We see both curves rapidly overlapping as $\omega$ grows larger, while differences appear for small-$\omega$. $z_c=0.1$ in the plot. (b) The metric of the centaur geometry as a function of bulk coordinate $z$. The blue dots are the numerical metric reconstruction from the boundary two-point function in (\ref{['centaur_G']}), while the solid curve is the exact plot of the metric in (\ref{['centaur_metric']}).
• Figure 5: (a) The free energy as a function of the inverse temperature for $s^2=0.01$. The dots correspond to numerically integrating (\ref{['two_app']}) for different temperatures, while the solid line corresponds to the analytic result at this value of the coupling. (b) The coefficient $f_2$ as a function of $s^2$ in a logarithmic scale. The dots correspond to numerically integrating the free energy at each $s^2$ and fitting by a function of the form of (\ref{['free_low_temps']}). The solid line is the analytic result $f_2 = \pi^2 \sqrt{4s^2+1} /4s^2$.