Application of Solving Inverse Scattering Problem in Holographic Bulk Reconstruction

TL;DR

This paper develops a unified inverse-scattering framework to reconstruct bulk AdS geometries from boundary two-point functions, extending previous two-field scalar methods to a single-operator approach through momentum dependence and applying it to scalar and gauge-field probes. The key advancement is formulating momentum-dependent potentials $V_k$ and $V_{\text{eff}}$ that enable extraction of metric components $z(\rho)$, $h(\rho)$, and, in anisotropic cases, $g_{tt}$, $g_{ii}$, and $g_{zz}$ via Gel'fand-Levitan-Marchenko-type inversions and related relations. The authors demonstrate robust reconstructions for BTZ, Schwarzschild-AdS, and RN-AdS backgrounds and implement a KK-filtering scheme to mitigate measurement noise using Hilbert transforms to enforce causality, highlighting experimental viability. While the framework excels for static backgrounds and scalar/gauge probes, challenges remain for fully coupled charged backgrounds and rotating or dynamical geometries, pointing to future hybrid approaches and broader applicability in holographic sensing through boundary observables.

Abstract

We investigate the problem of bulk metric reconstruction in holography by leveraging the inverse scattering framework applied to boundary two-point correlation functions. We generalize our previous work of scalar field and show that reconstruction can be achieved using a single operator rather than a pair. We also apply this method into reconstruction of static homogeneous anisotropic black holes and the reconstruction using correlation function of gauge field. In addition, we analyze the method's robustness under measurement noise and propose filtering strategies to improve reconstruction accuracy. This work advances data-driven bulk reconstruction by providing a concrete, experimentally viable pathway to recover spacetime geometry from field-theoretic observables.

Figures (12)

• Figure 1: The steps about how to reconstruct the geometry with two operator of two different conformal dimensions.
• Figure 2: Comparison between the reconstructed metric components and their exact values with our method. Here it uses two-point functions of $k_1=0$ and $k_2=1$ to recover the metric.
• Figure 3: The steps on how to reconstruct the geometry with a single operator.
• Figure 4: Left panel: correlation functions for $(\Delta,k_x,k_y)=$(4,0,0)(red), (4,1,0)(blue), (4,0,1)(black) and (3,0,0)(green). The solid lines correspond to the imaginary part of the correlation functions and the dashed lines correspond to the real part of the correlation functions. Right panel: the reconstructed components of metric and the comparison to their exact results.
• Figure 5: The conductivity of Schwarzschild-AdS black hole, there are real and imaginary parts. Here we choose $k=0.5$. The pole in the imaginary part at $\omega=0$ corresponds to a $\delta$-function in the real part.