9.1 Entanglement entropy in string theory

It is generally believed that the problem of the entanglement entropy of a black hole can and should be resolved in string theory. This was originally suggested by Susskind and Uglum [213Jump To The Next Citation Point]. Indeed, taking that string theory is UV finite, the corresponding entropy calculation should result in a finite quantity. More specifically, the effective action of a closed string can be decomposed into powers of string coupling g as 2(n−1) g, where n is the genus of the string-world sheet. The string configurations with spherical topology, n = 0, give a 1∕g2 contribution. In a low energy approximation this is exactly the contribution to Newton’s constant G ∼ g2. Thus, one may expect that taking into account just n = 0 closed string configurations, one will correctly reproduce both the entanglement entropy and Newton’s constant. In the Euclidean formulation, the prescription of [213] is to look at the zero genus string world sheet, which intersects the Killing horizon. In the Lorentzian picture this corresponds to an open string with both ends attached to the horizon. The higher genus configurations should give some corrections to the n = 0 result. This is a very attractive idea. However, a very little progress has been made in the literature to actually calculate the entanglement entropy directly in string theory. The reason is of course the technical complexity of the problem. Some support to the idea of Susskind and Uglum was found in the work of Kabat, Shenker and Strassler [151], where the entropy in a two-dimensional O(N ) invariant σ-model was considered. In particular, it was found that the state counting of the entropy in the UV regime may be lost if considered in the low energy (IR) regime. This type of behavior models the situation with the classical Bekenstein–Hawking entropy. Presumably this analysis could be generalized to the string theory σ-model considered either in optical target metric [9, 10] or in the Euclidean metric with a conical singularity at the horizon (as suggested in [33]). Possibly in the latter case the results obtained for strings on orbifolds [63] can be useful (see [50, 49, 51] for earlier works in this direction).

Another promising approach to attack the problem is to use some indirect methods based on dualities. For example, the AdS/CFT correspondence has been used in [25] to relate the entanglement entropy of a string propagating on a gravitational AdS background with a Killing horizon to the thermal entropy of field theory defined on a boundary of AdS and then, eventually, the thermal entropy to the Bekenstein–Hawking entropy of the horizon.

An interesting approach to the entanglement entropy of extremal black holes via AdS2/CFT1 duality is considered in [4, 192], where, in particular, one can identify the entanglement entropy and the microcanonical statistical entropy. This approach is based on the earlier work of Maldacena [168] in which the Hartle–Hawking state is identified with an entangled state of two copies of CFT, defined on two boundaries of the maximally-extended BTZ spacetime. In the accurately taken zero temperature limit, the reduced density matrix, obtained by tracing over the states of one copy of CFT, of the extremal black hole is shown to take the form

d(N) ρ = -1---∑ |k >< k|, (332 ) d(N ) k=1
which describes the maximally-entangled state in the two copies of the CFT1 living on the two boundaries of global AdS2. d(N ) is the dimension of the Hilbert space of CFT1. The corresponding entanglement entropy S = − Trρ ln ρ = lnd (N ) then is precisely equal to the micro-canonical entropy in the familiar counting of BPS states and thus is equal to the black-hole entropy [32].
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