Hessian potential for Fefferman-Graham metric

TL;DR

This paper establishes a connection between the Fefferman-Graham metric and Hessian geometry by identifying a Hessian potential that relates bulk deformations to boundary Fisher metrics without using the Ryu-Takayanagi formula.

Contribution

It introduces a Hessian potential framework that links the Fefferman-Graham metric to entanglement thermodynamics, bypassing traditional holographic entanglement entropy calculations.

Findings

Deformation of the bulk Hessian potential acts as a source for the boundary Fisher metric.

The deformation coincides with the Fefferman-Graham metric.

The approach does not require the Ryu-Takayanagi formula.

Abstract

The Fefferman-Graham metric is frequently used for derivation of the first law of the entanglement thermodynamics. On ther other hand, the entanglement thermodynamics is well formulated by the Hessian geometry. The aim of this work is to relate them with each other by finding the corresponding Hessian potential. We find that the deformation of the bulk Hessian potential for the pure AdS spacetime behaves as a source potential of the boundary Fisher metric, and the deformation coincides with the Fefferman-Graham metric. A peculiar feature different from related works is that we need not to use the Ryu-Takayanagi formula for the above derivation. The canonical parameter space in the Hessian geometry is a kind of the model parameter space, rather than the real classical spacetime in the usual setup of the AdS/CFT correspondence. However, the underlying mathematical structure is the same as that of the AdS/CFT correspondence. This suggests the presence of more global class of holographic transformation.