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We compute the modular Hamiltonians of regions having the future horizonlying on a null plane. For a CFT this is equivalent to regions with boundary ofarbitrary shape lying on the null cone. These Hamiltonians have a localexpression on the horizon formed by integrals of the stress tensor. We provethis result in two different ways, and show that the modular Hamiltonians ofthese regions form an infinite dimensional Lie algebra. The corresponding groupof unitary transformations moves the fields on the null surface locally alongthe null generators with arbitrary null line dependent velocities, but act nonlocally outside the null plane. We regain this result in greater generalityusing more abstract tools on algebraic quantum field theory. Finally, we showthat modular Hamiltonians on the null surface satisfy a Markov property thatleads to the saturation of the strong sub-additive inequality for the entropiesand to the strong super-additivity of the relative entropy.

Modular Hamiltonians on the null plane and the Markov property of the vacuum state