Positive gravitational subsystem energies from CFT cone relative entropies

TL;DR

This paper explores a new gravitational energy concept linked to relative entropy in holographic CFTs, extending previous results to lightcone-bound regions and providing analytic extremal surface expressions in AdS.

Contribution

It introduces a novel gravitational energy associated with lightcone-bound regions in holographic CFTs, generalizing prior spherical region results and offering analytic extremal surface formulas.

Findings

Derived an analytic expression for extremal surfaces in AdS for lightcone regions.

Proved the Markov property of the CFT vacuum for lightcone-bounded regions holographically.

Extended the holographic proof of the Markov property to non-conformal theories with relevant perturbations.

Abstract

The positivity of relative entropy for spatial subsystems in a holographic CFT implies the positivity of certain quantities in the dual gravitational theory. In this note, we consider CFT subsystems whose boundaries lie on the lightcone of a point $p$. We show that the positive gravitational quantity which corresponds to the relative entropy for such a subsystem $A$ is a novel notion of energy associated with a gravitational subsystem bounded by the minimal area extremal surface $\tilde{A}$ associated with $A$ and by the AdS boundary region $\hat{A}$ corresponding to the part of the lightcone from $p$ bounded by $\partial A$. This generalizes the results of arXiv:1605.01075 for ball-shaped regions by making use of the recent results in arXiv:1703.10656 for the vacuum modular Hamiltonian of regions bounded on lightcones. As part of our analysis, we give an analytic expression for the extremal surface in pure AdS associated with any such region $A$. We note that its form immediately implies the Markov property of the CFT vacuum (saturation of strong subadditivity) for regions bounded on the same lightcone. This gives a holographic proof of the result proven for general CFTs in arXiv:1703.10656. A similar holographic proof shows the Markov property for regions bounded on a lightsheet for non-conformal holographic theories defined by relevant perturbations of a CFT.