Bulk-boundary entanglement correspondence and the Ryu-Takayanagi conjecture in an $Ad{S}_2/CF{T}_1$ setup

Abstract

Using recent developments in expressing one-loop partition functions in Euclidean $Ad{S}_2$ space-times in terms of character integrals, we relate the one-loop effective action for a free field theory in $Ad{S}_2$ (comprised of a massless scalar field and a massless Majorana fermion field) to the partition function of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics (CQM) models on the two global $Ad{S}_2$ boundaries. The equal number of bosonic and fermionic degrees in the field theory guarantee that the one-loop calculation is free of all UV divergences except a logarithmic one consistent with the expected entanglement entropy behaviour in a CQM. Via a thermofield double representation, we compute the entanglement entropy between two copies of the $CF{T}_1$ (CQM), each living near one of the two boundaries of global $Ad{S}_2$, in a state at global time $\tau \to -\infty$. This entanglement entropy is expressed in terms of the logarithm of the regularised length of a closed particle trajectory infinitesimally near the rim of the Euclidean $Ad{S}_2$ disc. We view this relation between boundary quantum entanglement and a bulk geometrical quantity as the $Ad{S}_2/CF{T}_1$ version of the Ryu-Takayanagi conjecture in our setup. The boundary entanglement entropy is equal to 4 times the thermodynamic entropy read off from the regularised one-loop effective action in $Ad{S}_2$. Further, we compute the bulk entanglement entropy associated with black hole horizons in Lorentzian $Ad{S}_2$ and show that it precisely matches the boundary entanglement entropy.

v2:typographical corrections and assorted improvements; references added, 35 pages and 1 figure,

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