Abstract
We define a generalized entanglement measure in the context of the AdS/CFT correspondence. Compared to the ordinary entanglement entropy for a spatial subregion dual to the area of the Ryu-Takayanagi surface, we take into account both entanglement between spatial degrees of freedom as well as between different fields of the boundary theory. Moreover, we resolve the contribution to the entanglement entropy of strings with different winding numbers in the bulk geometry. We then calculate this generalized entanglement measure in a thermal state dual to the BTZ black hole in the setting of the D1/D5 system at and close to the orbifold point. We find that the entanglement entropy defined in this way is dual to the length of a geodesic with non-zero winding number. Such geodesics probe the entire bulk geometry, including the entanglement shadow up to the horizon in the one-sided black hole as well as the wormhole growth in the case of a two-sided black hole for an arbitrarily long time. Therefore, we propose that the entanglement structure of the boundary state is enough to reconstruct asymptotically AdS3 geometries up to extremal surface barriers.
Highlights
The growth of the area of a Cauchy slice through the wormhole asymptoting to a fixed time t on the two boundaries is only captured by the RT formula for a short period in t [8], which has lead to the proposal that features other than the entanglement entropy of the boundary state are needed to describe the part of the wormhole geometry behind the horizon [9,10,11]
The generalization of the RT formula to geodesics with non-zero winding number found in this publication considerably strengthens the “entanglement builds geometry” proposal
10For the high temperature phase, this argument assumes that the projection onto the Sn subset in the deformed theory does not project out exactly those twisted sectors that are responsible for the leading order contribution to the thermal partition function at high temperatures
Summary
We begin by introducing the SN orbifold theory and reviewing the relevant features of this CFT which we will need in the following. A twisted sector is specified by the number nm of cycles of length m. The total length of the cycles is given by the number of copies of the seed theory,. The thermal partition function of this theory is determined from the following recursion formula (see appendix A), ZN (τ ) =. Where Z(τ ) is the partition function of the seed theory. The contribution of a single cycle of length m is given by. A twisted sector contains multiple cycles which in total give a contribution to the partition function of. The total partition function (2.2) is given by summing over all sectors. This can be seen explicitly by comparing (2.6) with the dominant contributions to (2.2).
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