We study a conjectured correspondence between any codimension-two convex surface and a quantum state (SS-duality for short). By applying thermofield double formalism to the SS-duality, we show that thermal geometries naturally emerge as a result of hidden quantum entanglement between two boundary CFTs. We therefore propose a general framework to emerge the thermal geometry from CFT at finite temperature, without knowing many details about the thermal CFT. As an example, the case of 2d CFT is considered. We calculate its information metric and show that it is either BTZ black hole or thermal AdS as expected.
Gan, W. C., Shu, F. W., & Wu, M. H. (2017). Emergent geometry, thermal CFT and surface/state correspondence. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 772, 464–470. https://doi.org/10.1016/j.physletb.2017.07.011