### 7.4 Holographic entanglement entropy of higher dimensional black holes

In higher dimensions there is no known exact solution similar to Eq. (305). However, a solution in the
form of a series expansion in is available. In the rest of this section we focus on the case of boundary
dimension 4. Then one finds [131]
where is the boundary metric and coefficient [131]
is the local covariant function of the boundary metric. Coefficient is not expressed as a local
function of the boundary metric and is related to the stress-energy tensor of the boundary
CFT [60], which has an essentially non-local nature. is a local, covariant, function of the
boundary metric and is obtained as a variation of the integrated conformal anomaly with respect to
the boundary metric [60]. Its explicit form was computed in [60]. is a traceless tensor
and in mathematics literature it is known as the “obstruction” tensor [122]. The explicit form
of or is not important if one wants to compute the UV divergence terms in the
entropy.
One may choose the boundary metric describing a static spherically-symmetric black hole to take the
form

The minimal surface lies in the hypersurface of the constant of 5-dimensional spacetime (310). The
induced metric on the hypersurface takes the form
where functions and have a -expansion due to Eq. (310). The
minimal surface can be parameterized by . Instead of the radial coordinate , it
is convenient to introduce the coordinate so that, near the horizon, one has
. The coordinate measures the invariant distance along the radial direction.
By spherical symmetry, the area to be minimized is
where is the saddle point. In the vicinity of the horizon (), we can neglect the dependence of
the functions and on the coordinate . The minimization of the area of the surface
gives the equation
The area of the minimal surface is then given by the integral
Using Eq. (310) we find that . Substituting this expansion into Eq. (316) we
find that the first two terms produce divergences (when goes to zero), which, according to our proposal,
are to be interpreted as UV divergences of the entanglement entropy. At the black hole horizon, one has the
relation . Putting everything together and applying proposal (304), one finds for
the divergent part
where is the horizon area.
The logarithmic term in Eq. (317) is related to the logarithmic divergence as (calculated holographically
in [131])

in
the quantum effective action of boundary CFT. This relation is a particular manifestation of the general
formula (280) that relates the logarithmic term in the entropy to the conformal anomalies of type A
and B. One notes that in the superconformal gauge theory one has that
.
It should be noted that the UV finite terms and their dependence on the size of the box can be
computed in the limit of small . This calculation is given in [202]. In particular, in any even
dimension one finds a universal term in the entropy that takes the form (up to numerical factor)
and is proportional to the value of the “obstruction tensor” on the black
hole horizon. The direct calculation of such terms in the entanglement entropy on the CFT side is not yet
available.