6.2 Logarithmic terms in 6-dimensional conformal field theory
In six dimensions, omitting the total derivative terms, the conformal anomaly is a combination of four
different conformal invariants 
where , and are cubic in the Weyl tensor
and is the Euler density (60)
As was shown in  in a free conformal field theory with scalars, Dirac fermions and 2-form fields, one
Applying the formulas (55) to , and and using the relation (62) for the Euler number, one
finds for the logarithmic term in the entanglement entropy of 4-dimensional surface in a 6-dimensional
conformal field theory
is the Euler number of the surface , and
where tensors with Latin indices are obtained by contraction with components of normal vectors
. Note that in Eq. (295) we used for brevity the notation .
Eqs. (293), (294), and (295) agree with the results obtained in .
Let us consider some examples.
6-dimensional Schwarzschild black hole. The 6-dimensional generalization of the Schwarzschild solution
where is a metric of unit 4-sphere. The area of horizon is . The Euler number of the
horizon . This metric is Ricci flat so that only the Riemann tensor contributes to the Weyl
tensor. The logarithmic term in this case is
It is interesting to note that this term vanishes in the case of the interacting conformal theory,
which is dual to supergravity on AdS7. Indeed, in this case one has [131, 17]
so that . This is as expected. The Riemann tensor does not appear in the conformal anomaly of
the strongly interacting theory so that the anomaly identically vanishes if the spacetime is Ricci
flat. This property is not valid in the case of the free tensor multiplet  so that the logarithmic
term of the free multiplet is non-vanishing.
Conformally-flat extreme geometry. In conformally-flat spacetime the Weyl tensor so that
terms (293), (294) and (295) identically vanish. The logarithmic term (291) then is determined by the
anomaly of type A only. In particular, this is the case for the extreme geometry with equal radii
of two components. One has
for this extreme geometry. This geometry is conformal to flat spacetime and the logarithmic term (299) is
the same as for the entanglement entropy of a round sphere in flat 6-dimensional spacetime. This generalizes
the result discussed in Section 5.3.2 for the entropy of a round sphere due to a conformal scalar field. The
result (299), as is shown in [39, 181], generalizes to a spherical entangling surface in a conformally-flat
spacetime of any even dimension.