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

TL;DR

This work analyzes the modular structure of time-interval algebras for conformal Generalized Free Fields in $0+1$ dimensions, revealing that for non-integer scaling dimensions the boundary modular conjugation/flow are non-geometric due to a Generalized Hilbert Transform (GHT). The GHT is shown to geometrize in the AdS$_2$ bulk as a local antipodal symmetry, restoring locality and Haag duality in bulk observables, and enabling a local $ ilde{PSL}(2,oldsymbol{R})$-covariant net on emergent AdS$_2$. The authors prove two complementary theorems—twisted modular inclusion and twisted modular intersection—that guarantee a representation of $ ilde{PSL}(2,oldsymbol{R})$ from boundary algebraic data, and they discuss implications for stringy AdS$_2$ geometries in large-$N$ theories without a large gap, as well as the appearance of two PSL$(2,oldsymbol{R})$ copies on the future and past horizons of higher-dimensional eternal black holes. The framework clarifies how bulk locality can emerge from nonlocal boundary modular data and provides a bridge between operator-algebraic structures and holographic spacetime geometry, with potential extensions to higher dimensions and more general spacetimes.

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 Poincaré 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.

Figures (16)

• Figure 2: Modular flow for the time interval algebra $\mathcal{A}_{(-1,1)}$: (a) The modular flow (green) is local for the operators inside $\mathcal{A}_{(-1,1)}$. For an operator outside the interval, the modular flow (blue) takes it to the past/future horizon in finite modular time, $s_0 = \frac{1}{2\pi} \log\left( \frac{t-q}{t-p}\right)$. The modular flow remains local until $s_0$. (b) The top plot shows the modular flow $t(s)$ (blue) for an operator inside $\mathcal{A}_{(-1,1)}$ starting at $t(0)=0$. The conformal factor (orange) remains positive for all modular time and the flow is local. The bottom plot shows the modular flow (blue) for an operator outside the interval starting at $t(0)=2$. The conformal factor (orange) becomes negative for negative modular flow at finite $s_0<0$ when the operator reaches the future horizon and there is a discontinuity. For $s<s_0$, $t(s)$ is mapped to $(-\infty, -1)$ with a frequency dependent phase. Hence the flow is non-local in general.
• Figure 3: For the time interval algebra $\mathcal{A}_{(-1,1)}$, the local anti-unitary map $\mathcal{R}_I$ maps $f(t) \to |t|^{2(\Delta-1)} f(1/t)$. The modular conjugation of operators in the interval $(-1,0)$ is the GHT of operators in $(-\infty,-1)$ and the modular conjugation of operators in $(0,1)$ is the GHT of operators in $(1,\infty)$ and vice versa.
• Figure 4: Haag's duality is restored in the bulk (a) The geometrization of the commutant relation in Theorem \ref{['thm-comm-half-line-general']}$(\mathcal{A}_{(0,\infty)})'=\mathcal{T}^\dagger \mathcal{A}_{(-\infty,0)}\mathcal{T}$. (b) The geometrization of the commutant relation in Theorem \ref{['thm-comm-half-line-general']}$(\mathcal{A}_{(-\infty,0)})'=\mathcal{T} \mathcal{A}_{(0,\infty)}\mathcal{T}^\dagger$.
• Figure 5: Haag's duality is restored in the bulk for time interval algebras. (a) The commutant $(\mathcal{A}_{(-1,1)})'$ in the bulk. (b) Theorem \ref{['thm-comm-I-general']} says that $(\mathcal{A}_{(-1,1)})'$ is the algebraic union of the two orange regions.
• Figure 6: Modular flow of half-infinite interval regions $\mathcal{A}_{(0,\infty)}$ and $\mathcal{A}_{(-\infty,0)}$ in the bulk.

Theorems & Definitions (80)

• Definition 2.1: Time interval algebras
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 3
• Definition 3.1
• Definition 3.2: Covariance
• Definition 3.3
• Definition 3.4: Haag's duality