renormalized ERE is computed in the same way as in eq. (1.6). Also, a renormalized
version of the total action can be derived by evaluating the renormalized Einstein-AdS
action on the conically singular bulk manifold that considers the brane as a singular source
in the Riemann curvature. Then, the contribution due to the cosmic brane can be rewritten
as a renormalized version of the NG action.
The organization of the paper is as follows: in section 2, we explicitate the relation
between renormalized volume and renormalized Einstein-AdS action, making contact with
the mathematical literature in the four [4] and six [5] dimensional cases. We also give
our conjecture for the general relation in the 2n−dimensional case, relating it to Albin’s
prescription [7]. In section 3, we exhibit the emergence of the renormalized total action, in
agreement with Dong’s cosmic brane prescription [35], from evaluating the I
ren
EH
action on
the conically singular manifold considering the brane as a source. We also comment that the
inclusion of the renormalized NG action for the brane can be considered as a one-parameter
family of deformations in the definition of the renormalized bulk volume. In section 4, we
obtain the renormalized modular entropy
e
S
ren
m
and the renormalized ERE S
ren
m
starting
from the renormalized total action, and we compare the resulting modular entropy with
the existing literature for renormalized areas of minimal surfaces [6]. In section 5, we
consider the computation of S
ren
m
for the particular case of a ball-like entangling region
at the CFT
2n−1
, and we check that the usual result for S
ren
EE
is recovered in the m → 1
limit. Finally, in section 6, we comment on the physical applications of our conjectured
renormalized volume formula, on our topological procedure for computing renormalized
EREs and on future generalizations thereof.
2 Renormalized Einstein-AdS action is renormalized volume
The standard EH action, when evaluated on an AAdS Einstein manifold, is proportional
to the volume of the manifold, which is divergent. We propose that for 2n−dimensional
spacetimes, the renormalized Einstein-AdS action I
ren
EH
is also proportional to the corre-
sponding renormalized volume of the bulk manifold. To motivate this conjecture, we first
introduce I
ren
EH
, and we then compare it with known formulas for the renormalized volume
of AAdS Einstein manifolds in four and six-dimensions. Finally, we give a concrete formula
for the renormalized volume in the general 2n−dimensional case, and we comment on its
properties.
There are different but equivalent prescriptions for renormalizing the action. For ex-
ample, the standard Holographic Renormalization scheme [14–20] and the Kounterterms
procedure [23–27]. The equivalence between the two renormalization schemes, for the case
of Einstein-AdS gravity, was explicitly discussed in refs. [26, 27], so using either one or
the other is a matter of convenience. However, as discussed in the introduction, we con-
sider the Kounterterms-renormalized action as it can be readily compared with the existing
renormalized volume formulas.
We consider the 2n− dimensional Einstein-AdS action I
ren
EH
as derived using the Koun-
terterms prescription [23], which is given by
I
ren
EH
[M
2n
] =
1
16πG
Z
M
2n
d
2n
x
√
G (R − 2Λ) +
c
2n
16πG
Z
∂M
2n
B
2n−1
, (2.1)
– 4 –