Modular Intersections, Time Interval Algebras and Emergent AdS$_2$

TL;DR

This paper analyzes the modular flow of conformal free fields in one dimension, revealing non-geometric features in the boundary theory that become geometric in the dual AdS$_2$ bulk, with implications for emergent spacetime structures.

Contribution

It introduces the concept of twisted modular inclusion and intersection in conformal GFFs, linking these properties to representations of the conformal group and bulk geometric emergence.

Findings

Modular conjugation involves a Generalized Hilbert Transform for non-integer dimensions.

In the dual AdS$_2$, modular flow becomes local and geometrizes as antipodal symmetry.

Existence of twisted modular properties implies a conformal group representation.

Abstract

We compute the modular flow and conjugation of time interval algebras of conformal Generalized Free Fields (GFF) in $(0+1)$-dimensions in 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 AdS$_2$ are local, because the GHT geometrizes as the local antipodal symmetry transformation that pushes operators behind the Poincar\'e horizon. These algebras of conformal GFF satisfy a $\textit{Twisted Modular Inclusion}$ and a $\textit{Twisted Modular Intersection}$ 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 $PSL(2,\mathbb{R})$, respectively. We discuss the implications of our result for the emergence of Stringy AdS$_2$ 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 $PSL(2,\mathbb{R})$ on future and past Killing horizons.