# Propagation of singularities on AdS spacetimes for general boundary conditions and the holographic Hadamard condition

## TL;DR

This paper studies how singularities propagate in solutions to the Klein-Gordon equation on asymptotically anti-de Sitter spacetimes with various boundary conditions, establishing results relevant for quantum field theory.

## Contribution

It extends the propagation of singularities results to general boundary conditions and introduces a holographic Hadamard condition for quantum fields in AdS.

## Key findings

- Propagation of singularities along generalized broken bicharacteristics.
- Uniqueness of parametrices with prescribed b-wavefront set.
- Validation of a holographic Hadamard condition for two-point functions.

## Abstract

We consider the Klein-Gordon equation on asymptotically anti-de Sitter spacetimes subject to Neumann or Robin (or Dirichlet) boundary conditions, and prove propagation of singularities along generalized broken bicharacteristics. The result is formulated in terms of conormal regularity relative to a twisted Sobolev space. We use this to show the uniqueness, modulo regularising terms, of parametrices with prescribed b-wavefront set. Furthermore, in the context of quantum fields, we show a similar result for two-point functions satisfying a holographic Hadamard condition on the b-wavefront set.