### 8.3 Entropy in brane-world scenario

An interesting example where the Bekenstein–Hawking entropy is apparently induced in the correct way
is given in [130]. This example is closely related to the AdS/CFT correspondence discussed in Section 7. In
the Randall–Sundrum set-up [187] one may consider the regularized boundary, which appeared in our
discussion of Section 7, as a 3-brane with symmetry in an anti-de Sitter spacetime. In the framework
of the AdS/CFT correspondence this brane has a description in terms of CFT coupled to gravity at a UV
cutoff [124]. If the brane is placed at the distance from the anti-de Sitter boundary, one obtains
that there is a dynamic gravity induced on the brane, with the induced Newton’s constant
where is the number of colors in the superconformal Yang–Mills theory. in this case
plays the role of the number of species. We notice that, according to the AdS/CFT dictionary, the
parameter , which is an infrared cut-off on the anti-de Sitter side, is, in fact, a UV cut-off on the CFT
side. Now consider a black hole on the 3-brane. The Bekenstein–Hawking entropy can then be represented
as follows
As Hawking, Maldacena and Strominger [130] suggested, the right hand side of Eq. (325) can be
interpreted as the entanglement entropy of fields. This interpretation turns out to be the right one, if
one uses the holographic entanglement entropy discussed in Section 7. Indeed, taking the leading divergent
term in Eq. (317) and noting that, in a brane configuration, this result should be multiplied by a
factor of 2, we get exactly the right-hand side of Eq. (325). In [130] one considers de Sitter spacetime (so
that Eq. (325) is the entropy of the de Sitter horizon in this case) on the brane since it is the simplest
brane configuration one can construct in an anti-de Sitter spacetime. In [83] this proposal was extended to
the holographic entanglement entropy of a black hole on a 2-brane solution found in [84]. The
two-dimensional black hole is considered in [107]. Entropy of a generic black hole on the 3-brane was
considered in [202].
However, there are certain open questions regarding this example. First of all, we should note that the
weakly coupled supermultiplet contains the Yang–Mills fields (gauge bosons),
conformally-coupled scalars and the Weyl fermions [167]. Thus, it is a bit of a mystery how the
entanglement entropy of these, mostly non-minimally coupled, fields (gauge bosons and scalars) has
managed to become equal to the Bekenstein–Hawking entropy, when recalling the problems with the
non-minimal coupling we have discussed in Section 3.16. A part of this mystery is the fact that the
holographic regularization (which corresponds to infrared cut-off on the anti-de Sitter side) does not
have a clear analog on the boundary side. Indeed, if we take, for example, a standard heat
kernel regularization, we find that the term linear in the scalar curvature does not appear
at all in the effective action produced by the weakly coupled superconformal gauge
multiplet.