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 [130Jump To The Next Citation Point]. 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 Z2 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 ρ = 𝜖2 from the anti-de Sitter boundary, one obtains that there is a dynamic gravity induced on the brane, with the induced Newton’s constant
2 2 1∕GN = 2N ∕(π𝜖 ), (324 )
where N is the number of colors in the superconformal SU (N ) Yang–Mills theory. N 2 in this case plays the role of the number of species. We notice that, according to the AdS/CFT dictionary, the parameter 𝜖, 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
A-(Σ-) N-2A(Σ-) SBH = 4GN = 2π𝜖2 = Sent. (325 )
As Hawking, Maldacena and Strominger [130Jump To The Next Citation Point] suggested, the right hand side of Eq. (325View Equation) can be interpreted as the entanglement entropy of N 2 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. (317View Equation) and noting that, in a Z2 brane configuration, this result should be multiplied by a factor of 2, we get exactly the right-hand side of Eq. (325View Equation). In [130] one considers de Sitter spacetime (so that Eq. (325View Equation) 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 𝒩 = 4 SU (N ) 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 R does not appear at all in the effective action produced by the weakly coupled 𝒩 = 4 superconformal gauge multiplet.


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