Modular intersections, time interval algebras and emergent AdS2

Abstract

We compute the modular flow and conjugation of time interval algebras of conformal Generalized Free Fields (GFF) in 0 + 1-dimensions in the vacuum. For non-integer scaling dimensions, for general time intervals, the modular conjugation and the modular flow of operators outside the algebra are non-geometric. This is because they involve a Generalized Hilbert Transform (GHT) that treats positive and negative frequency modes differently. However, the modular conjugation and flows viewed in the dual bulk AdS₂ are local, because the GHT geometrizes as the local antipodal symmetry transformation that pushes operators behind the Poincaré horizon. These algebras of conformal GFF satisfy a Twisted Modular Inclusion and a Twisted Modular Intersection property. We prove the converse statement that the existence of a (twisted) modular inclusion/intersection in any quantum system implies a representation of the (universal cover of) conformal group PSL(2, ℝ), respectively. We discuss the implications of our result for the emergence of AdS₂ geometries in large N theories without a large gap. Our result applied to higher dimensional eternal AdS black holes explains the emergence of two copies of PSL(2, ℝ) on future and past Killing horizons.