Bulk modified gravity from a thermal CFT by the conformal flow

Abstract

We construct a bulk spacetime from a boundary conformal field theory, 𝑂⁡(𝑁) free scalar model, at finite temperature using a smearing technique, called a conformal flow. The bulk metric is constructed from the thermal two-point function of the smeared boundary elementary field in such a way that it can be interpreted as an information metric associated with the boundary thermofield double state. Near the boundary (UV region), an asymptotically anti–de Sitter (AdS) spacetime is obtained with a leading order perturbation of scalar mode. Based on the falloff behavior of the perturbations and the 𝑂⁡(𝑁) symmetry in the conformal field theory, we argue that the corresponding bulk theory is a modified gravity with scalar mode such as 𝑓⁡(𝑅) gravity rather than Einstein's general relativity coupled minimally to matter fields. Moving to the Einstein frame, we show that the metric is asymptotically the same as the AdS black brane solution. On the other hand, deep in the bulk (IR region), the spacetime turns out to be conformally equivalent to the near horizon limit of AdS extremal black brane, though it is no longer a solution of 𝑓⁡(𝑅) gravity, and hence more general classes of modified gravity need to be considered.