### 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 [213]. 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 as , where is the genus of the string-world sheet. The
string configurations with spherical topology, , give a contribution. In a low
energy approximation this is exactly the contribution to Newton’s constant . Thus,
one may expect that taking into account just 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 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 invariant -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
-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 AdS_{2}/CFT_{1} 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

which describes the maximally-entangled state in the two copies of the CFT_{1} living on the
two boundaries of global AdS_{2}. is the dimension of the Hilbert space of CFT_{1}. The
corresponding entanglement entropy 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].