State Operator Correspondence and Entanglement in AdS_2/CFT_1

TL;DR

This paper articulates a state–operator correspondence for AdS$_2$/CFT$_1$ with two AdS$_2$ boundaries, proposing that bulk string states on AdS$_2\times K$ are twisted Hartle–Hawking states generated by a large ${\rm U}(N)$ symmetry. It demonstrates how black hole entropy can be interpreted as both the ground-state degeneracy and the entanglement entropy between two copies of the CFT$_1$, and outlines a concrete bulk–boundary dictionary in which CFT$_1$ operators map to bulk twisted states via boundary twists and (orbifold) symmetries. The work shows that conformal invariance is preserved in this twisted framework and provides holographic prescriptions for computing entanglement measures that reproduce the Wald entropy in the appropriate limit, while highlighting the role of enhanced near-horizon symmetries. Finally, it speculates on the origin and realization of the proposed $U(N)$ symmetry, including possible connections to discrete symmetries at special moduli and structures like $M_{24}$, suggesting rich directions for understanding the microstate structure of extremal black holes in AdS$_2$ geometries.

Abstract

Since euclidean global AdS_2 space represented as a strip has two boundaries, the state / operator correspondence in the dual CFT_1 reduces to the standard map from the operators acting on a single copy of the Hilbert space to states in the tensor product of two copies of the Hilbert space. Using this picture we argue that the corresponding states in the dual string theory living on AdS_2 x K are described by twisted version of the Hartle-Hawking states, the twists being generated by a large unitary group of symmetries that this string theory must possess. This formalism makes natural the dual interpretation of the black hole entropy, -- as the logarithm of the degeneracy of ground states of the quantum mechanics describing the low energy dynamics of the black hole, and also as an entanglement entropy between the two copies of the same quantum theory living on the two boundaries of global AdS_2 separated by the event horizon.

Figures (3)

• Figure 1: Global AdS$_2$ and the location of the horizon(s). The two vertical solid lines label the two boundaries of AdS$_2$ at $\sigma=-\pi$ (left) and $\sigma=0$ (right). The dashed lines label the locations of the event horizons of the original black hole.
• Figure 2: Conformal map from unit disk to the strip. The left boundary of the strip is at $\sigma=-\pi$ and the right boundary is at $\sigma=0$.
• Figure 3: Generating a state in string theory on AdS$_2$ from a state $W_{ab}|a\rangle_{(1)} |b\rangle_{(2)}$ in the two copies of the Hilbert space of CFT$_1$. The thick semi-circular line is the boundary of AdS$_2$ whereas the thin diameter is the line on which the string fields, appearing in the argument of $f_W$, live. The dashed line reaching the boundary of AdS$_2$ denotes a cut which relates the field configurations on the right of the cut to those on the left of the cut by a transformation by $W$.