Bulk-boundary entanglement correspondence and the Ryu-Takayanagi conjecture in an $AdS_2/CFT_1$ setup

Using recent developments in expressing one-loop partition functions in Euclidean $AdS_2$ space-times in terms of character integrals, we relate the one-loop effective action for a free field theory in $AdS_2$ (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 $AdS_2$ 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 $CFT_1$ (CQM), each living near one of the two boundaries of global $AdS_2$, in a state at global time $\tau \rightarrow - \infty$. 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 $AdS_2$ disc. We view this relation between boundary quantum entanglement and a bulk geometrical quantity as the $AdS_2/CFT_1$ 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 $AdS_2$. Further, we compute the bulk entanglement entropy associated with black hole horizons in Lorentzian $AdS_2$ and show that it precisely matches the boundary entanglement entropy.