Stringy ER=EPR

TL;DR

This work establishes explicit string-theoretic realizations of ER$= $EPR by performing Lorentzian continuations of Euclidean dualities, notably the FZZ duality for the two-dimensional cigar and its proposed AdS$_3$ uplift to Euclidean BTZ-like geometries. The authors develop a formalism where Euclidean time winding operators are treated via angular quantization and their Lorentzian continuation is encoded through a deformed moduli-contour in the worldsheet path integral, giving rise to a condensate of entangled folded strings that populate the EPR description. The connected, two-sided black hole in the Hartle–Hawking state is dual to an entangled superposition of disconnected spacetimes with a winding-string condensate on the EPR side, and similar structures hold for AdS$_3$ via the SL$(2,C)_k/$SU$(2)$/BTZ setups. The construction clarifies how entanglement and geometry map under holography at the level of worldsheet CFTs, and provides a concrete framework to explore microstate ensembles, perturbations, and the role of mutual locality in Lorentzian string amplitudes. Overall, the work deepens the connection between quantum information concepts and spacetime topology in string theory, offering a toolkit for analyzing ER–EPR dualities across 2D and 3D holographic setups.

Abstract

The ER = EPR correspondence relates a superposition of entangled, disconnected spacetimes to a connected spacetime with an Einstein-Rosen bridge. We construct examples in which both sides may be described by weakly-coupled string theory. The relation between them is given by a Lorentzian continuation of the FZZ duality of the two-dimensional Euclidean black hole CFT in one example, and in another example by continuation of a similar duality that we propose for the asymptotic Euclidean AdS3 black hole. This gives a microscopic understanding of ER = EPR: one has a worldsheet duality between string theory in a connected, eternal black hole, and in a superposition of disconnected geometries in an entangled state. The disconnected description includes a condensate of entangled folded strings emanating from a strong-coupling region in place of a horizon. Our construction relies on a Lorentzian interpretation of Euclidean time winding operators via angular quantization, as well as some lesser known worldsheet string theories, such as perturbation theory around a thermofield-double state, which we define using Schwinger-Keldysh contours in target space.

Figures (19)

• Figure 1.1: The $\mathrm{AdS}_3$ HH State and the Dual TFD. The conformal diagram of the asymptotic $\mathrm{AdS}_3$ two-sided black hole (or, rather, its top half) is shown on the left. Over each point is an additional circle, which is suppressed in the figure. The left and right asymptotic $\mathrm{AdS}_3$ regions are causally separated by the horizons, represented by the diagonal dotted lines. The future singularity is the zigzag line at the top of the diagram. The Hartle-Hawking state is prepared by halving the Euclidean continuation of the black hole, shown by the half-disk, and gluing it to the zero-time slice of the black hole on the horizontal dashed line. The Euclidean time periodicity is $\beta$, the inverse Hawking temperature of the black hole. The zero-time slice has the topology of an annulus and the halved Euclidean black hole resembles a half-bagel, obtained by revolving the dashed line and half-disk around the suppressed circle. Above the Hawking-Page temperature, this bulk state is dual to the thermofield-double state in two copies of the boundary CFT on a circle (times time), shown on the right. The yellow half-torus prepares the state on its two circle boundaries, which then evolve forward in Lorentzian time along the two blue cylinders. The Hartle-Hawking cap that prepared the bulk state corresponds to the solid half-torus obtained by filling in the interior of the yellow surface.
• Figure 1.2: Schematic of$\mathrm{ER=EPR}$. The connected, two-sided, asymptotically $\mathrm{AdS}$ black hole admits a dual description as an entangled superposition of disconnected spacetimes, represented schematically by the wedges on the left, each of which is dual to an energy eigenstate $\left| n \right>_\mathrm{L}, \left| n \right>_\mathrm{R}$ of the two copies of the boundary CFT. The $\mathrm{ER=EPR}$ correspondence asserts that this equivalence of entangled quantum states to connected spacetimes holds more generally. We find examples of string dualities of this type, relating a string in a connected target space to a string in an entangled superposition of disconnected targets.
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• Figure 1.5: The Two-Dimensional Black Hole. The Lorentzian continuation of the Euclidean cigar is a two-sided, eternal black hole. The horizons are the diagonal dotted lines, and the past and future singularities are the zigzag hyperbolas at the bottom and top. The geometry is invariant under time reflection about the dashed line, which enables the construction of the Hartle-Hawking state. The cigar, which has the topology of a disk, is cut in half and glued to the black hole along the fixed line of the reflection symmetry, similar to Fig. \ref{['fig:ads-hh']} but resembling the Schwarzschild causal diagram rather than BTZ. This Euclidean cap prepares the state on the dashed line, which is then evolved forward in Lorentzian time.