Stringy ER = EPR

Abstract

The ER = EPR correspondence asserts that quantum entanglement and geometric spatial connection are closely related. A two-sided black hole, for example, is equivalent to an entangled superposition of disconnected geometries. In this dissertation, we construct perturbative string dualities that give explicit examples of this equivalence.

The string dualities are obtained by continuation in the sense of the target time coordinate of CFT dualities for the SL(2,R)/U(1) and Z\SL(2,C)/SU(2) coset WZW models. For large k, these CFTs admit a weakly-coupled description as a string in a Euclidean black hole target space of two-dimensional dilaton-gravity and three-dimensional AdS gravity, respectively. They also admit dual, strongly-coupled descriptions with a non-contractible target Euclidean time circle and a condensate of winding strings that wrap it. The latter description is the sine-Liouville background of Fateev, Zamolodchikov, and Zamolodchikov in the case of SL(2,R)/U(1), and a similar dual that we propose for Z\SL(2,C)/SU(2). By continuing the target Euclidean time coordinate on both sides of these dual backgrounds, we obtain Lorentzian dualities relating an ER description of a string in a connected black hole and an EPR description in a disconnected target with a condensate of entangled strings.

The first part of the dissertation is devoted to a study of the SL(2,R)/U(1) CFT itself in the semi-classical limit. We construct the saddle-point expansion for the functional integral that computes the reflection coefficient of the CFT. To do so requires that we complexify the target space and sum over complex-valued saddles, which, remarkably, include configurations that hit the singularity of the Lorentzian black hole within the complexified target.

The second part of the dissertation proposes the sine-Liouville dual description of the Z\SL(2,C)/SU(2) CFT, and develops the tools necessary to establish the string dualities for ER = EPR by continuation. Part of the construction relies on understanding string perturbation theory in different spacetime states---the Hartle-Hawking state in the connected ER description of the black hole and the thermofield-double state in the disconnected EPR description. The state dependence is encoded in the choice of Schwinger-Keldysh contour for the worldsheet functional integral. We show that the sine-Liouville Euclidean time winding condensate leads in the Lorentzian continuation to a condensate of pairs of entangled folded strings, one on each side of the disconnected target and emanating from a strong-coupling region in place of a horizon. Each pair of strings is prepared in the worldsheet thermofield-double state in the sense of angular quantization, and a related angular deformation of the string moduli contour of integration is required to define string perturbation theory in the thermal EPR microstates. Finally, we discuss an infinitesimal interpretation of the dualities that gives equivalent semi-classical descriptions of a conformal perturbation that shifts the mass of the black hole.