Light-ray moments as endpoint contributions to modular Hamiltonians

Abstract

We consider excited states in a CFT, obtained by applying a weak unitary perturbation to the vacuum. The perturbation is generated by the integral of a local operator $J^{(n)}$ of modular weight $n$ over a spacelike surface passing through $x=0$. For $|n|\ge 2$ the modular Hamiltonian associated with a division of space at $x=0$ picks up an endpoint contribution, sensitive to the details of the perturbation (including the shape of the spacelike surface) at $x=0$. The endpoint contribution is a sum of light-ray moments of the perturbing operator $J^{(n)}$ and its descendants. For perturbations on null planes only moments of $J^{(n)}$ itself contribute.

LaTeX, 35 pages. v2: references added, version to appear in JHEP