Symmetries Near the Horizon

TL;DR

This work identifies and constructs three exact, gauge-invariant SL(2) generators that act on the combined gravity–matter system near the horizon of nearly-AdS$_2$ spacetimes, revealing how bulk matter can be translated relative to dynamical boundaries. In the semiclassical regime these generators realize SL(2) transformations on the physical boundary time and yield an operator–state map for NCFT$_1$, with direct parallels to SYK in the Schwarzian limit. The authors connect these exact generators to approximate, yet physically insightful, forms in SYK, relate them to the size operator and maximal chaos, and show how they enable bulk probing, including behind-horizon dynamics and the inner horizon, while clarifying the limitations at finite $N$ and with topology changes. The work further links the gravity construction to SYK via boundary dressing, modular/Rindler decompositions, and traversable-w wormhole physics, offering a unified view of horizon symmetries, bulk reconstruction, and chaotic dynamics with potential broad impacts for holography and quantum gravity insights.

Abstract

We consider a nearly-AdS$_2$ gravity theory on the two-sided wormhole geometry. We construct three gauge-invariant operators in NAdS which move bulk matter relative to the dynamical boundaries. In a two-sided system, these operators satisfy an SL(2) algebra (up to non-perturbative corrections). In a semiclassical limit, these generators act like SL(2) transformations of the boundary time, or conformal symmetries of the two sided boundary theory. These can be used to define an operator-state mapping. A particular large N and low temperature limit of the SYK model has precisely the same structure, and this construction of the exact generators also applies. We also discuss approximate, but simpler, constructions of the generators in the SYK model. These are closely related to the "size" operator and are connected to the maximal chaos behavior captured by out of time order correlators.

Figures (12)

• Figure 1: The three Killing vectors: boost $B$ (blue), momentum $P$ (pink), and global energy $E$ (green) given in (\ref{['Engen']}) in the coordinate system (\ref{['Coord']}). The shaded regions delineate different orbits of the symmetries.
• Figure 2: Geometrical action of the gauge invariant charges $\tilde{P}, \tilde{B}, \tilde{E}$. The points which are fixed by the symmetry generators are also indicated.
• Figure 3: (a) By performing euclidean evolution over time $\beta/2$ and inserting an operator at some point during the euclidean evolution we create a state. This defines a map between operators and states. These are states of a wormhole or states living in the Hilbert space of two copies of the dual boundary quantum system. (b) The same for the bra. (c) We can take the inner product and add the action of a charge $G^A(\pi,0)$, represented by the black dots. In the semiclassical regime, these charges act as $SL(2)$ generators on the states or the operators.
• Figure 4: Geometric action of the generators in Euclidean AdS$_2$ . Here the charges are inserted at the black points, at $\varphi = 0$ and $\varphi=\pi$. Blue lines follow the boost Killing vectors $\tilde{B}$ ; pink lines, the momentum ${\color{pi} \tilde{P}}$ ; and green lines, the global energy vectors ${\color{dg} \tilde{E}}$.
• Figure 5: (a) We consider the action of the charges. We have matter fields propagating from the bottom to the top indicated in red. These cause some backreaction on the boundary positions. These are summarized by the coupling to $\epsilon$ at the insertion points of operators. The definition of the charges involves computing a distance, which implicitly, or more explicitly (in the generators $\hat{G}^A$ in (\ref{['BoostSum']}), (\ref{['MQEn']}),(\ref{['POper']})), involve the propagation of other matter fields. The interesting terms come from correlators between these $\epsilon$ insertions. We only have two point functions of $\epsilon$, only one of which is indicated in the diagram by a doted line. Other diagrams contain a dotted line between black points and $\varphi_{t,b}$. (b) In some specific models we might get contractions between the fields in the definition of the charges and the insertions. We want to suppress this type of diagrams. They are indeed suppressed relative to those in (a) in the SYK model.