Surface counterterms and regularized holographic complexity
HJE
Surface counterterms and regularized holographic
RunQiu Yang 0 2
Chao Niu 0 1
KeunYoung Kim 0 1
0 Gwangju 61005 , Korea
1 School of Physics and Chemistry, Gwangju Institute of Science and Technology
2 Quantum Universe Center, Korea Institute for Advanced Study
The holographic complexity is UV divergent. As a nite complexity, we propose a regularized complexity" by employing a similar method to the holographic renormalization. We add codimensiontwo boundary counterterms which do not contain any boundary stress tensor information. It means that we subtract only nondynamic background and all the dynamic information of holographic complexity is contained in the regularized part. After showing the general counterterms for both CA and CV conjectures in holographic spacetime dimension 5 and less, we give concrete examples: the BTZ black holes and the four and ve dimensional Schwarzschild AdS black holes. We propose how to obtain the counterterms in higher spacetime dimensions and show explicit formulas only for some special cases with enough symmetries. We also compute the complexity of formation by using the regularized complexity.
Gaugegravity correspondence; Holography and condensed matter physics

1 Introduction
2
Surface counterterms and regularized complexity
Coordinate dependence in discarding divergent terms
Surface counterterms in CA and CV conjectures 2.2.1 2.2.2
Surface counterterms in CA conjecture
Surface counterterms in CV conjecture
3
Examples for BTZ black holes
CA conjecture in nonrotational case
CA conjecture in rotational case
CV conjecture in BTZ black hole
Examples for Schwarzschild AdSd+1 black holes
Regularized complexity in CA conjecture
Regularized complexity in CV conjecture
Summary
j (tL; tR)i := e i(tLHL+tRHR)jTFDi
minimal number of simple gates from the reference state to a particular state [1{3]. The
quantum entanglement has also been found to play an important role in the quantum
gravity, especially for the study on the AdS/CFT correspondence. While most of recent works
have paid attention to the holographic entanglement entropy [4, 5], quantum complexity
in gravity was studied in [6{10]: by paying attention to the growth of the EinsteinRosen
bridge the authors found a connection between AdS black hole and quantum complexity in
the dual boundary conformal eld theory (CFT). In this study, they consider the eternal
AdS black holes, which are dual to thermo eld double (TFD) state [11]
jTFDi := Z 1=2 X exp[ E =(2T )]jE iLjE iR :
The states jE iL and jE iR are de ned in the two copy CFTs at the two boundaries of
the eternal AdS black hole (see gure 1) and T is the temperature. With the Hamiltonians
HL and HR at the left and right dual CFTs, the time evolution of a TFD state
t
L
WDW
r
h
t
At the two boundaries of the black hole, tL and tR stand for two states dual to the states in TFD.
rh is the horizon radius. At the left panel, B is the maximum codimensionone surface connecting
tL and tR. At the right panel, the yellow region with its boundary is the WDW patch, which is the
closure (inner region with the boundary) of all spacelike codimensionone surfaces connecting tL
and tR.
can be characterized by the codimensiontwo surface at xed times t = tL and t = tR at
the two boundaries of the AdS black hole [10, 11]. There are two proposals to compute the
complexity of j (tL; tR)i state holographically: CV(complexity=volume) conjecture and
CA(complexity= action) conjecture.
The CV conjecture [7, 12] states that the complexity of j (tL; tR)i at the boundary
CFT is proportional to the maximal volume of the spacelike codimensionone surface which
connects the codimensiontwo surfaces denoted by tL and tR, i.e.
HJEP09(217)4
CV =
max
V ( )
GN `
;
where GN is the Newton's constant.
is all the possible spacelike codimensionone
surfaces which connect tL and tR and ` is a length scale associated with the bulk geometry
such as horizon radius or AdS radius and so on. This conjecture satis es some properties
of the quantum complexity. However, there is an ambiguity coming from the choice of a
length scale `.
This unsatisfactory feature motivated the second conjecture: CA conjecture [9, 10]. In
this conjecture, the complexity of a j (tL; tR)i is dual to the action in the WheelerDeWitt
(WDW) patch associated with tL and tR, i.e.
The WDW patch associated with tL and tR is the collection of all spacelike surface
connecting tL and tR with the null sheets coming from tL and tR. More precisely it is the
domain of dependence of any spacelike surface connecting tL and tR (see the right panel of
gure 1 as an example). This conjecture has some advantages compared with the CV
conjecture. For example, it has no free parameter and can satisfy Lloyd's complexity growth
bound in very general cases [13{15]. However, the CA conjecture has its own obstacle in
(1.3)
(1.4)
CA =
IWDW
~
:
{ 2 {
computing the action: it involves null boundaries and joint terms. Recently, this problem
has been overcome by carefully analyzing the boundary term in null boundary [16, 17].
As both the CV and CA conjectures involve the integration over in nite region, the
complexity computed by the eqs. (1.3) and (1.4) are divergent. The divergences appearing
in the CV and CA conjectures are similar to the one in the holographic entanglement
entropy. It was shown that the coe cients of all the divergent terms can be written as the
local integration of boundary geometry [18, 19], which is independent of the bulk stress
tensor. This result gives a clear physical meaning of the divergences in the holographic
complexity: they come from the UV vacuum structure at a given time slice and stand for
the vacuum CFT's contribution to the complexity. One interesting thing is to consider the
HJEP09(217)4
contribution of excited state or thermal state to the complexity. As the divergent parts of
the holographic complexity is xed by the boundary geometry, the contribution of matter
elds and temperature can only appear in the nite term of the complexity. This gives us
a strong motivation to study how to obtain the nite term in the complexity.
The rst work regarding this
nite quantity is the \complexity of formation" [20],
which is de ned by the di erence of the complexity in a particular black hole space time
and a reference vacuum AdS spacetime. By choosing a suitable vacuum spacetime, we
can obtain a
nite complexity of formation. However, there are two somewhat ambiguous
aspects in using \complexity of formation" to study the
nite term of complexity. First,
we need to appoint additional spacetime as the reference vacuum background. In general
cases, it will not be obvious how to choose the reference vacuum spacetime. For example,
in ref. [20], the reference vacuum spacetime for the BTZ black hole is not the naive limit of
setting mass M = 0. Second, to make the computation about the di erence of complexity
at the
between two spacetimes meaningful, we need to appoint a special
coordinate and apply this coordinate to both spacetimes. For example, in the ref. [20], the
holographic complexity of two spacetime at the
nite cuto is computed in Fe
ermanGraham coordinate [21, 22]. It will be better if we can compute the complexity without
referring to a speci c coordinate system.
As the refs. [18, 19] have shown that the divergent terms have some universal
structures, a naive consideration is that, we can separate the divergent term and just discard
them. However, this may give a coordinate dependent result as we shows in the section 2.1.
In this paper, we will propose another method to obtain the nite term of the complexity,
which we will call \regularized complexity". Colsely following the method of the
holographic renormalization [23{26] we will add codimensiontwo surface counterterms for a
given dimension d + 1,1
1For holographic renormalization of entanglement entropy, we refer to [27{29]. In particular, our method
is similar to [29].
{ 3 {
to the complexity formula in the CV and CA conjectures (1.3) and (1.4) respectively.2 Here
B is the codimensiontwo surface of given time t = tL or tR at the cuto boundaries. `AdS
is the radius of the AdS space. g
is the induced metric at the cuto
boundaries, R
A
is the Ricci tensor from g , ij is the induced metric of the time slice tL or tR and Kij
is the extrinsic curvature of the time slice tL or tR embedded into the boundaries. F (2n)
V
and F (2n) are invariant combinations of R ; g ; ij and Kij with a mass dimension 2n,
A
so Vct is of volume dimension d and Ict is dimensionless. The concrete form of F (2n) and
V
F (2n) will be determined based on the divergent structure developed in [18, 19]. When the
bulk dimension is even (d is odd) a logarithmic divergence appears, and F (d 1) and F (d 1)
should be understood as a counterterm for the logarithmic divergence. The counter terms
are determined by the boundary metric alone and do not contain any boundary stress
tensor information.
The procedure to obtain the regularized complexity is similar to holographic
renormalization. However, there are two di erences. First, the surface counterterms we will
show are the codimensiontwo surface at the boundary rather than the codimensionone
surface. Because the complexity, as shown in the
gure 1, is de ned by the time slices
denoted by tL and tR, which are codimensiontwo surfaces, it is natural that the surface
counterterms should be expressed as the geometric quantities of these codimensiontwo
surfaces. Second, the surface counterterms can contain the extrinsic geometrical quantities
of the codimensiontwo boundary rather than only the intrinsic geometrical quantities
unlike in renormalizing free energy. One reason for this di erence is that free energy involves
the equations of motion and we need to keep the equations of motion invariant when we
renormalize the free energy but complexity has no directly relationship with the equation
of motion.
The organization of this paper is as follows. In section 2, we will give the surface
counterterms for both CA and CV conjectures. We rst show an example how the
coordinate dependence appears if we just discard the divergent terms, which will give the
inspiration on how to construct the surface counterterms. Then we will explicitly give the
minimal subtraction counterterms both for CA and CV conjectures up to the bulk
dimension d + 1
5. In the sections 3 and 4, we will use our surface counterterms to compute
the regularized complexity for the BTZ black holes and Schwarzschild AdS black holes for
both CA and CV conjectures. A summary will be found in section 5.
Surface counterterms and regularized complexity
Coordinate dependence in discarding divergent terms
To regularize the complexity we may try the same method as the entanglement entropy
case, for example, in refs. [30{32] i.e. nd out the divergent behavior and then just discard
2In this paper, the capital latin letters I; J;
xd = z. The Greek indices ; ;
surface and x0 = t. The little latin letters i; j;
at the xed z and t surface.
run from 0 to d, which stand for the all coordinates and
run from 0 to d
1, which stand for the local coordinate at the
xed z
run from 1 to d
1, which stand for the local coordinates
{ 4 {
all the divergent terms. However, in the following example for the CV conjecture, we will
show such a method is a coordinatedependent so ambiguous. Such coordinate dependence
can appear also in the CA conjecture, subregion complexity, and in the entanglement
entropy, if we just naively discard the divergent terms.
Let us rst consider a Schwarzschild AdS4 black brane geometry
where M is the parameter proportional to the mass density of the black hole,3 fx; yg are
dimensionless coordinates scaled by `AdS and the horizon locates at r = rh = (2M `2AdS)1=3.
For simplicity, we consider the complexity of a thermal state at tR = tL = 0. Because
of symmetry, the maximal surface is just the t = 0 slice in the bulk. The volume of this
2 is the area of 2dimensional surface spanned by x; y and rm ! 1 is the UV
cuto . As a dimensionless cuto , we introduce
following expansion for integration (2.3)
= `AdS=rm. When
! 0, we can nd the
with
slice is
3Mass density = M=(4 GN `2AdS)
4From here on we set GN = 1.
{ 5 {
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
the e ects of matter elds. If we do so, we obtain a nite result4
CV; nite =
F (M )`2AdS + F (M )O(`4AdS=r04)
1
;
The process from eq. (2.4) to eq. (2.5) can be de ned as a kind of background
subtraction, i.e.
A similar method was applied in refs. [30{32] to nd the nite term of the entanglement entropy.
hole. For example, we may use a new coordinate ft; r0; ; g by the following coordinate
and ft; r; ; g
when r
`AdS. Here F (M ) is an arbitrary function and F (0) = 0. We see that, from the
AdS/CFT viewpoint, there is not any physical di erence between coordinate ft; r0; ; g
. Because time t is not changed, the t = 0 slices in both coordinate are the
same surface, which means their volumes, as the geometry qualities, are independent of
the choice of coordinate. Let 0 = `AdS=r0m be the UV cuto in a new coordinate system.
The coordinate transformation (2.7) implies the following relationship between
and 0
In a new coordinate system, the volume of t = 0 slice reads
term is di erent! Now assume we don't know the result in the coordinate ft; r; ; g and
use coordinate ft; r0; ; g rst, then by the eq. (2.6), we nd that
CV0; nite =
In recent papers [18, 19], the authors analysed the divergent structure of the
complexity in the CV and CA conjecture in the Fe ermanGraham (FG) coordinate and, in our
example case, the rst term of (2.4) is shown as a divergent term. Naively, this divergent
term can be discarded to regularize the complexity. However, if we use another coordinate
system such as (2.7) di erent from the FG coordinate, we have to discard not only the
divergent term but also a nite piece, the second term of (2.9). Therefore, it will be better
if we can identify the divergence structure of the complexity in a coordinate independent
way, and subtract it to regulate the complexity. Another advantage of this coordinate
independent regularized complexity lies in the computation of the complexity of formation.
Unlike ref. [20] we do not need to worry about the coordinate dependence of the cuto .
To propose a well de ned subtraction for the regularized complexity, we follow the
procedure of the holographic renormalization [23{26]. In this procedure, the divergences
are canceled by adding covariant local boundary surface counterterms determined by the
nearboundary behaviour of bulk
elds. Inspired by [18, 19] we use the counterterms
expressed in terms of intrinsic and extrinsic curvatures. We will show that both for the CA
conjecture and CV conjecture, we can add suitable covariant local boundary counterterms
to cancel the divergences appearing in the complexity. For a resolution of the example in
this section see eqs. (2.61){(2.63) in section 2.2.2.
2.2
2.2.1
Surface counterterms in CA and CV conjectures
Surface counterterms in CA conjecture
In this subsection, we will rst consider the CA conjecture. For the CA conjecture, we
need to compute the action for the WDW patch. Since it has null boundaries one needs to
{ 6 {
consider appropriate boundary terms. It was proposed in refs. [16, 17, 19, 33] as
I =
where the rst line is the EinsteinHilbert action with the cosmological constant integrated
over the WDW region denoted by M, the second line is various boundary terms de ned
at the boundary of M and third line is the joint terms de ned on the corners of two
di erent boundaries. B stands for the timelike or spacelike boundary, N for the null
boundary, J for the joints connecting timelike or spacelike boundaries and J 0 for the
joints connecting boundaries, one or both of which are null surfaces. K is the
GibbonsHawkingYork extrinsic curvature and h is the determinant of the induced metric.
is a
parameter of the generator of the null boundary and
is the nona nity parameter of null
kJ .
cross section of constant
in null surface N .
is the determinant of the metric on the
is the induced metric at the joints. The
expression for
and a can be found in ref. [17]. As the joint terms J does not occur for
the WDW patches, we will not show
here. a is written as
a = <
8
:
ln(jnI kI j) ;
ln(jkI kI j=2) ;
where nI is the unit normal vector (outward/future directed) for nonnull intersecting
boundary, and kI is the other null normal vector (future directed) for null intersecting
boundary. The sign in the eq. (2.12) can be appointed as follows: \+" appears only when
the WDW patch appears in the future/past of null boundary component and the joint is at
the past/future end of null component. It was pointed by ref. [17] that the action (2.11),
in its form without I , depends on the parametrization of null generators. It rst appeared
in ref. [17] and was studied further in refs. [18, 20, 33]. Moreover, we will see later that the
divergent terms in this form cannot be canceled by adding covariant surface terms. Thus,
to make the action with the null boundaries to be invariant under the reparametrization
on the null normal vector eld,5 an additional boundary term(I ) at the null boundaries
is added [19]:
I =
5For the joint terms and boundary terms, there is still an ambiguity: we may add any term of which
variation vanishes. Because the variational principle does not determine the boundary term uniquely we
have a freedom to add any nondynamic term to the complexity without any physical e ects. However, if
lead some dynamic e ects. The physical meaning of this kind of additional freedom is not clear for us.
= p
Now let us analyze how to add the surface terms so that we can obtain a nite
complexity. The goal here is very similar to the case that we add some boundary terms to make
the total free energy
nite in holographic renormalization. However, there is an important
di erence. Our goal here is to make the complexity itself nite, so the surface terms do
not need to be invariant under the metric variation. This admits that the surface terms
can contain not only the intrinsic geometry but also the extrinsic geometry.
In the Fe ermanGraham (FG) coordinate system [21, 22], any asymptotic AdSd+1
spacetime can be written as6
ds2 = gIJ dxI dxJ =
2
`Az2dS [dz2 + g~ (z; x )dx dx ] ;
(2.15)
1; d denote the full sppacetime coordinates, ;
=
0; 2;
d
1 denote the coordinate labeled at the
xed z surface. We consider the case
in which the metric g~
along the boundary directions has a power series expansion with
respective to z when z ! 0:
g~ (z; x ) = g~(0)(x ) + z2g~(1)(x ) +
+ zdg~(d=2)(x ) + zdh~ (x ) ln z +
;
(2.16)
where the coe cient of logarithmic term is nonzero only if d is even. In fact, the expansion
structure and coe cients of eq. (2.16) may be deformed by a relevant operator (see ref. [34]
for example), which will not be considered in this paper for simplicity. The expansion
coe cients g~(n) with n < d=2 and h~
are completely determined by g~(0). The higher order
coe cients are not xed by g~(0) alone and they encode information of the expectation
value of the boundary energymomentum tensor [24, 25]. We will see that these higher
order terms are irrelevant in determining the counterterms.
At the UV cuto
z =
, the induced metric (denoted by g ) at the boundary
(codimensionone) surface is
g
=
2
`Az2dS g~ ;
and we use \ ~ " to denote the conformal boundary metric at the surface z = . Likewise,
in this paper, the notation \X~ " (indices are suppressed) means that it is computed by the
conformal metric g~
and we use g~
to raise and lower its indexes. For example, we will
decompose the metric g
as
1 and fx g = ft; yig and we may introduce
`tilde'variables
so
N~ 2 =
N 2 ;
z
2
6We introduce the dimensionless coordinate z; x scaled by `AdS so g~
is dimensionless and g (2.17)
has length dimension 2. All tildevariables in this subsection are dimensionless.
{ 8 {
Furthermore, the expansion for g~
(2.16) can give similar expansions for N~ , ~ij , and L~i:
where we can x N~ (0) = 1; L~i(0) = 0 and we can also de ne that
As another convention, in this paper, we will always use the notation X(n) to denote the
coe cient of z(2n) in the expansion of the eld X.
Let us consider the Ricci tensor R
and the Ricci scalar R for boundary metric g
and the extrinsic curvature tensor Kij for the t = 0 surface7 (codimensiontwo) embedded
in the z =
boundary surface (codimensionone). Then we nd that the conformal Ricci
tensor R~ , Ricci scalar R~ and extrinsic curvature K~ij are
~
R
= R ;
R~ =
2
`Az2dS R;
~
Kij =
`AdS Kij :
z
For later use, we de ne two projections from z =
surface to the z =
and t = 0 surface
^
Rij = hi hi R
;
~
Rij = h~i h~i R~
^
:
Like the metric, we can also expand the Ricci tensor and the extrinsic curvature and other
geometrical quantities with respective to z.
Next, we will show that the divergent terms in the action (2.11) at a given time t can
be reorganized as the following surface integrals
Ict =
Z
B
where B is the codimensiontwo surface at a given time t and xed z = . F (2n) is the
invariant combinations of R ; g ; ij and Kij . The maximum level of divergence of F (2n)
A
is 1= d 1 2n but F (2n) may also include less divergent terms than 1= d 1 2n. (It is explained
A
below eq. (2.40).) When the bulk dimension is even (d is odd) a logarithmic divergence
appears, and F d 1 should be understood as a counterterm for the logarithmic divergence.
We can de ne the regularized nite action, Ireg, as
7Here we set t = 0 just for convenience, we can set t to be any xed value.
Ireg
lim(I
!0
Ict,L
Ict,R) :
{ 9 {
(2.21)
(2.22)
t
t
r
h
t
panel: the null boundaries of the WDW patch are changed into the null sheets coming from the
boundary and there is a nullnull joint at the cuto . (here we only show the part near
tR. The part near tL is similar.) Right panel: the boundaries of the WDW patch are the same,
but, the original nullnull joints at the AdS boundary are sliced out by a time like boundary and
two nulltimelike joints are added. (here we only show the righttop part of quarter. The other
parts are similar.)
where I is the action (2.11) computed with the AdS boundary at the cuto surface z = .
Ict,L and Ict,R are the surface counterterms de ned by (2.25) at the left boundary and right
boundary, respectively.
Before discussing the surface counterterm Ireg let us rst explain how to compute I . It
needs to be regulated. As pointed by ref. [18], there are two di erent methods to regulate
the WDW patch as we show in
gure 2. At the left panel of gure 2, the boundaries of
the WDW patch are changed into the null sheets coming from the nite cuto
boundary
and there is a nullnull joint at the cuto . At the right panel of gure 2, the boundaries of
WDW patch is the same, however, original nullnull joints at the AdS boundary is sliced
out by a time like boundary, so the nullnull joint at the boundary disappears but there is
an additional GibbosHawkingYork boundary term and two nulltimlike joints. As the rst
approach is more convenient in analyzing the divergent behavior near the AdS boundary,
the term I in the eq. (2.26) is computed by this approach.
To nd Ict or F (2n), we rst need to analyze the divergent structure of (2.11). The
A
divergences come from the action near the boundary. We only need to analyze the divergent
behavior at the one side boundary since the other side is similar. We will analyze the
divergent behavior in the FG coordinate and show that the divergent term (the whole
divergent term rather than only the coe cients of divergent terms) in this coordinate can
be written as the codimensiontwo surface terms. After subtracting this codimensiontwo
surface terms, we end up with the
nite result. As the subtraction terms are written
in terms of the geometrical quantities of the codimensiontwo surface, the
nal result is
independent of the choice of coordinate. This means the result of eq. (2.26) is the same for
all the coordinate systems.
We rst consider the case that d + 1 is odd number. In this case there is no anomaly
divergent term. All the divergent terms in the FG coordinate system are in the form of
Idiv = IC(1A) + IC(2A) + O
IC(1A) =
`dAd1S ln(d
the power series of the cuto . At any side of the two boundaries, the rst two divergent
terms in the action (2.11) were obtained in ref. [19]:
where
for d
2 and8
IC(2A) =
First, let us consider the leading divergent term IC(1A) . It is expressed in terms of the
`tilde' variables (2.22) and can be rewritten in terms of real induced metric as
IC(1A) =
ln(d
1) Z
d
d 1xp (0) :
By the inversion of the expansion to elds (2.21)
p (0) = p
IC(1A) =
ln(d
4
4
1
1) Z
B
2
B
2 ij (1)
ij
d
d 1 p
x
Similarly, the subleading divergent term IC(2A) can be rewritten as
IC(2A) =
subleading divergent term. By the relationship (2.31) and the Einstein equations for the
8This is di erent from the results reported in the refs. [18, 19]. It seems that the null normal vectors
used in refs. [18, 19] are not a nely parameterized. If we take this into account we nd an additional
contribution to the subleading divergent terms. See the appendix A for details.
(2.27)
(2.28)
(2.29)
(2.30)
(2.31)
(2.32)
(2.33)
(2.34)
(2.35)
we nd that the subleading divergent term in the rst counterterm is
As a result, (2.34) and (2.36) together give the new subleading divergent term
In other words, because the counter term (2.32) already cancels the part of the subleading
divergence (the rst term in (2.29)), we only need to introduce the second line of (2.34) as
a new counterterm. Therefore, the second function FA;2 is
The general structure of divergences in the CA conjecture was suggested to be [18]
Idiv =
x
2n X c~i;n(d)[R~ (0); K~ (0)]i2n :
where we dropped the log term in ref. [18] by considering the null boundary term (2.13)
is a schematic expression indicating invariant combinations of R~(0); K~i(j0), g~
following ref. [19]. We recovered GN here to make clear Idiv is dimensionless. [R~ (0); K~ (0)]2n
i
(0) and ~ij
(0) with
a mass dimension of 2n. Thus, 2n[R~ (0); K~ (0)]i2n is dimensionless. The index i stands for
a di erent combination. For example, in eq. (2.28) there is only one term (say i = 1)
and one can read [R~ (0); K~ (0)]01 = 1 with c~1;0 = ln(d
invariant combinations [R~ (0); K~ (0)]i2=fK~ (0)2; K~i(j0)K~ (0)ij ; R~(0); R^~(0)g with the corresponding
coe cients c~i;1 which can be read from eq. (2.29). To be more concrete, c~i;1 was summarized
in table 1.
For some symmetry arguments for the divergence structure and pattern, we
1)=(4 ). In eq. (2.29) there are four
refer to ref. [18].
Once we obtain the divergent structures (2.39) for d 5, we can repeat the steps that we have done for d = 3; 4. Thus we propose that the following counter terms work for
d
5 as well as d = 3; 4.
Ict =
`AdS n=0
This is similar to eq. (2.39) in structure. in eq. (2.39) is absorbed to ~(0) and R~ (0), leaving
`AdS to take into account dimension. However, note that the structure of [R~ (0); K~ (0)]i2n and
[R; K]i2n are not the same as shown in eq. (2.29) and eq. (2.37). I.e. the expected level
of divergence of [R; K]i2n is equal to or less than 1= d 1 2n. To be more concrete, ci;1
was summarized in table 1. Finally, for notational convenience, we rewrite (2.40) as, with
(2.38)
(2.39)
(2.40)
c~1;0 and c1;0 are read from eq. (2.28) and eq. (2.32).
c~i;1 is the coe
cient of [R~ (0); K~ (0)]i2=fK~ (0)2; K~i(j0)K~ (0)ij ; R~(0); R^~(0)g in eq. (2.29) and ci;1 is the coe cient of
[R; K]i2=fK2; Kij Kij ; R; R^g in eq. (2.37). It is valid up to holographic spacetime dimension 5
HJEP09(217)4
or less.
GN = 1,
Ict =
Z
B
d
x p
In order to show the explicit formulas for higher dimensions than 5, we rst have to
obtain the divergence structure explicitly similar to Eq (2.29) following refs. [18, 19]. We think
it can be done straightforwardly but the
nal formulas will be very complicated. Thus,
it may not be so illuminating for the purpose of explaining the methodology. However, it
is possible to obtain the simple formulas for some special cases. This case is explained in
detail in appendix B.
is given by
For all the cases that d is odd integer greater than 1, there is a logarithmic divergent
term9 in the action (2.11). At the cuto
in any given coordinate system, the counterterm
Z
B
ln( =`AdS)
d
d 1 p
x
`AdS A
d 1 F (d 1) + O( ) :
9One should note that if we don't add I into the action (2.11), the additional logarithm divergent term
will appear [18] in any dimension. In general, it has following forms
Ilog;CA = ln(p
=`AdS)
c1
d 1 + d 3
c2
:
Here coe cients c1; c2;
are determined by the conformal boundary metric g~(0) but ; are arbitrary
constants depending on the choice of null normal vectors in null surfaces. As the
and
can not be
determined by theory itself, such terms cannot be written as the covariant geometrical quantities of the
boundary metric. This results show that it is necessary to add the term I into the action (2.11) to obtain
an covariant regularized complexity.
(2.41)
(2.42)
(2.44)
(2.43)
coordinateindependent.
d)F A(d 1)
de ne a \regularized complexity" as follows
. Note the integration term in eq. (2.44) is nite and
After we obtain the regularized form of the action in the WDW region, we propose to
CA,reg = lim
1
This is similar to the holographic renormalization of the onshell action for a free energy. In
the holographic renormalization of the free energy [23], the counterterms are only intrinsic
geometric quantities not to a ect the equations of motion. However, when we regularize
the complexity, this restriction may be relaxed and the extrinsic quantities may be
included. For both a free energy and the complexity, the relative value between two states is
important so a subtraction of the same value from two states are allowed. The complexity
describes the minimum number of quantum gates required to produce some state from a
particular reference state, so it does not matter if we add any constant value in complexity
to both states. As a reference state we can appoint any nondynamic quantum state. The
subtraction term Ict,R and Ict,R are de ned by the boundary metric and does not contain
any bulk dynamics and matter
elds information, so they are nondynamic subtraction
terms. Therefore, we can consider CA,reg as a well de ned \regularized compelxity" in the
CA conjecture.
2.2.2
Surface counterterms in CV conjecture
Similarly to the CA case, we can de ne the regularized complexity for the CV conjecture as
where V is the maximum value connecting tL and tR after we use a nite cuto z =
to
replace the real AdS boundary. Vct,L and Vct,R are the surface counterterms
Z
B
x p
at the left boundary (Vct,L) and right boundary (Vct,R) respectively. When the bulk
dimension is even (d is odd) a logarithmic divergence appears, and F d 1 should be understood
V
as a counterterm for the logarithmic divergence.
We rst consider the odd bulk dimensions. To nd Vct or FV;n we rst need to analyze
the divergent structure of (1.3). The rst two divergent terms in the volume (1.3) for d
2
were obtained in ref. [19]:
where
Vdiv = V (1) + V (2) + O
1
d 5
;
d
d
d 1xp~(0) 1 d
;
(2.46)
(2.47)
(2.48)
(2.49)
and
With these two equations, the rst volume divergent term reads
The subleading divergent term can be written as
V (2) =
2(d
3
`AdS
2)(d
Z
3) B
d
d 1 p
x
As the same as the CA conjecture, the rst surface counterterm has also contribution on
the subleading divergence
d
^
R
"
x
1
1
R
2
:
which leads that total subleading divergent term reads
2(d
3
`AdS
2)(d
d
d 1 p
x
d
2
1
(R^
R=2)
so we obtain
V
Such step can be continued for higher dimensional case, so we see that we can use
codimensiontwo surface terms as the counterterms to cancel all the divergences in the
volume (1.3).10
The general structure of divergences in the CV conjecture was suggested to be [18]
Vdiv = `dAdS
d
Z
B
x d 1
The structure is the same to the CA case (2.39) apart from the overall factor `dAdS
accounting for the dimension of volume.
However, the explicit expressions for ~ci;n and
[R~ (0); K~ (0)]2n are di erent from the CA case.
i
indicating invariant combinations of R~(0); K~i(j0), g~(0) and ~ij
[R~ (0); K~ (0)]2n is a schematic expression
i
(0) with a mass dimension of
10When we nished this paper, we noted two refs. [35, 36] which also developed a general regulated
volume expansion for the volume of a manifold with boundary. It will be interesting to study if this is
equivalent to our method when it is applied to the CV conjecture.
2(d 2)(d 3)
(d 1)(d 2)(d 3)
d
[R~(0); K~ (0)]i2=fR~^(0); R~(0); K~ (0)2g in eq. (2.50) and ci;1 is the coe cient of [R; K]i2=fR^; R; K2g in
eq. (2.54). It is valid up to holographic spacetime dimension 5 or less.
2n. Thus, 2n[R~ (0); K~ (0)]2n is dimensionless. The index i stands for a di erent
combii
nation. For example, in eq. (2.49) there is only one term (say i = 1) and one can read
[R~ (0); K~ (0)]01 = 1 with c~1;0 = 1=(d
[R~ (0); K~ (0)]i2=fR~^(0); R~(0); K~ (0)2
1). In eq. (2.50) there are three invariant combinations
g with the corresponding coe cients c~i;1 which can be read
from eq. (2.50). To be more concrete, c~i;1 was summarized in table 2. For some symmetry
arguments for the divergence structure and pattern, we refer to ref. [18].
Once we obtain the divergent structures (2.56) for d 5, we can repeat the steps that we have done for d = 3; 4. Thus we propose that the following counter terms work for
d
`AdS n=0
This is similar to eq. (2.56) in structure. in eq. (2.56) is absorbed to ~(0) and R~ (0), leaving
`AdS to take into account dimension. However, note that the structure of [R~ (0); K~ (0)]i2n and
[R; K]i2n are not the same as shown in eq. (2.50) and eq. (2.54). I.e. the expected level of
divergence of [R; K]i2n is equal to or less than 1= d 1 2n. To be more concrete, ci;1 was
summarized in table 2. Finally, for notational convenience, we rewrite (2.57) as
i
i
(2.57)
(2.58)
(2.59)
Z
B
d
X `2AndS FV(2n)(d; R ; g ; ij ; Kij ) ;
with F (2n)(d; R ; g ; ij ; Kij )
V
X ci;n(d)[R; K]i2n
which is (2.47).
To nd the explicit formulas for higher dimensions than 5, we rst have
to obtain the divergence structure explicitly similar to eq. (2.50) following refs. [18, 19]. We
think it can be done straightforwardly but the
nal formulas will not be so illuminating.
However, similarly to the CA case, it is possible to obtain the simple formulas for some
special cases. It is shown in detail in appendix B.
When the bulk dimension d + 1 is even, the logarithmic divergent term will appear,
which is similar to the case in the CA conjecture. The counterterm at the cuto
in any
Here F (d 1) = limd0!d(d0
V
and coordinate independent.
d)FV(d 1)
. Note that the integration in the equation is nite
As an example, let us compute the regularized complexity by the CV conjecture for
the example shown in section 2.1. The metric of the boundary and the codimensiontwo
surface at t = 0 are
HJEP09(217)4
The Ricci tensor is zero at the boundary at xed r = `AdS= and the extrinsic curvature
is also zero at the surface of t = 0 embedding in the boundary. So the subleading term in
eq. (2.55) is zero and there is only one term in the surface counterterm, which reads
Vct,L = Vct,R =
=
We see that this is just the value shown in eq. (2.3). So in this coordinate system, the
surface counterterm is as the same as the background subtraction term and we
regularized complexity is just as the same as one shown in eq. (2.5). Of course, we can also
compute the regularized complexity in the coordinate ft; r0; x; yg, where the relationship
between r and r0 is given by eq. (2.7). The surface counterterm at the cuto
0 then is
coordinate system reads
Z
B
ln( =`AdS)`dAdS
d
x
F (d 1) + O( ) :
V
dCreg =
dt
dC :
dt
3
2`AdS
Vct,L = Vct,R =
3
2`AdSF (M ) + O( 0):
We see that in this coordinate system, the counterterm is not proportional to the volume
of the pure AdS spacetime, as its value depends on mass M . However, one can nd that
the regularized complexity is still as the same as eq. (2.5), which is independent of the
choice of F (M ).
conjectures.
We want to stress that it is important to use (2.51) as a subtraction term rather
than (2.49). If we used (2.49) as a subtraction term, we would not have (2.63) so the
regularized complexity becomes coordinate dependent and ambiguous. In sections. 3 and 4,
we will give more examples for computing the regularized complexity for the CV and CA
Note also that our surface counterterms are nondynamic and have no relationship to
the bulk matter eld, so such subtraction keeps all the information of bulk matter eld in
the complexity. In addition, if there is asymptotic timelike Killing vector eld
at the boundary we have
This means the previous studies about the complexity growth, in fact, studied the behavior
of the regularized part of the whole complexity.
(2.60)
(2.62)
(2.63)
(2.64)
If we let
be any parameter in the system which has no e ect on the boundary
metric, and we can de ne a \complexity of formation" between two di erent states labeled
by
= 1 and
If is the temperature and 1 = T; 2 = 0, (2.65) gives the complexity of formation studied
In this section, we will give examples to compute the regularized complexity for both CA
and CV conjectures in the BTZ black holes. One form of the metric for the rotational BTZ
black hole is [37, 38],
HJEP09(217)4
ds2 =
with r 2 (0; 1), ' 2 [0; 2 ] and the function f (r) is described by
where M is the mass parameter11 and J is the angular momentum:
r+2 + r2
This black hole arises from the identi cations of points of the antide Sitter space by a
discrete subgroup of SO(2, 2). The surface r = 0 is not a curvature singularity but, rather,
a singularity in the causal structure if J 6= 0. Although the parameter M plays the role
of mass, it is possible to admit M to be negative when J = 0. In these cases, except for
M =
1, naked conical singularities appear, so these cases should be prohibited. In the
special case that J = 0 and M =
1, the conical singularity disappears. The con guration
is just the pure AdS3 solution with f (r) = r2=`2AdS + 1. For the case that J > 0, we need
that M
J to avoid the naked singularity. The BTZ black hole also has thermodynamic properties similar to those found in higher dimensions. We can de ne the temperature T , entropy S and angular velocity as
T =
r
2
+
r
2
2 r+
;
S =
r+
2
;
=
r
r+`AdS
:
3.1
CA conjecture in nonrotational case
We rst consider the case that J = 0. For the case M > 0 the WDW patch is shown in
the left panel of gure 3. We de ne rh
r+ = `AdS
M . For a special case of tL = tR = 0,
the null sheets coming from left boundary and right boundary just meet with each other
p
at r = 0.
11The physical mass for the BTZ black hole is M=8.
(3.1)
(3.2)
(3.3)
(3.4)
r
h
r = rm
=
0
0 is shown in the left panel, where the null sheets coming
from tR and tL meet each other at the surface r = 0. The case for M =
1 is shown in the right
panel, where the null sheets coming from tR and tL will meet each other at r = 0 and t =
=2.
In order to compute the regularized complexity in the CA conjecture, we rst need to
regularize the WDW patch, which is shown in the left panel of gure 3. Note that this
approach is di erent from the approach in ref. [20]. Taking the symmetry into account, we
only need to compute the bulk term, the boundary terms and joints at the green region.
Let us introduce the outgoing and infalling null coordinates u; v de ned by12
u(t; r) = t
r ;
v(t; r) = t + r (r) ;
Z
r (r) =
[r2f (r)] 1dr =
2
`AdS ln
2rh
r
r + rh
rh + v0 ;
(3.5)
where v0 is an integration constant. The null boundaries at the green region in left panel
of gure 3 is given by v = vm and u = um, where vm
r (rm). One can check that the dual normal vectors for such null boundaries
are kI = [(dt)I +r 2
f 1(dr)I ] and kI =
[(dt)I r 2
f 1(dr)I ]. Here we explicitly exhibit
the freedom of choosing dual normal vector by two arbitrary constants
and
. In the
green region of gure 3, there are a bulk integration term, two null boundary terms, a
nullnull joint term in the action.
Using the method similar to ref. [20], the bulk action is expressed as
where the factor 2 is multiplied to take the both sides (tL and tR) into account. It is
di erent from the result in ref. [20] because we used a di erent regularization method.
However, the nal results of the complexity will be the same. As the measurement of
nullnull joints at the corners r = 0 is zero, such joint term has no contribution to the action.
12Strictly speaking, this relationship for r and r can only be used when rh > 0. However, we can see
that it has a well de ned limit when rh ! 0+. So the M = 0 case can be regarded as the limit of rh ! 0+.
The joint term at the boundary is given by following expression 1 2
r ln
Since kI is a nely parameterized, only the null boundary term shown in eq. (2.13) has
contribution. The expansions of kI and kI are
= gIJ
r
;
= gIJ
In order to compute the value of I , we need to nd the a ne parameter
and
and kI , respectively. On the null boundary of the green region shown in the gure (
3
), the
coordinates t and r are the functions of , i.e., t = t( ) and r = r( ). By the equation
HJEP09(217)4
kI =
dr
d
=
we see that
= r= for kI . Similarly, we nd that
r= for kI . So we obtain that
I =
2
r
`AdS
r
+(
!
) =
rm ln
2
rm2
Ibulk =
rm
2
`AdS + O(1=rm):
r
:
I
;
(3.7)
(3.8)
for kI
(3.9)
(3.12)
(3.13)
Adding up all results, we have
Ireg(M
0) = Ibulk + Ijoint + I = 0 ) CA,reg(M
0) = 0;
(3.11)
so the regularized complexity is zero for all M
0. Note that the complexity is already
nite without any regularization in this case. Indeed, for d = 2, the counterterm we
derived in (2.32) is always zero so our computation here is consistent. We also see that
the regularized complexity is independent of the choice of
and , which is expected as
and
are gauge degrees of freedom in the choices of the dual normal vector for null
surface. Note that the UV divergent behavior shown in ref. [18] depends on these two
gauge parameters. However, in our formula, as the additional term I has been added into
the action (2.11), the nal result is independent of the gauge choices on the null normal
vector elds.
equation
When M =
1, the expression of r in the eq. (3.5) should be replaced by following
r =
`AdS arctan(`AdS=r) + v0:
In this case, we see that the null sheets coming from the r = 1; t = 0 will meet each other
at the position of r = 0 and t =
[r (0)
v0] =
=2 respectively (see the right panel
of gure 3). The computation of the regularized complexity is very similar to the case of
M > 0. Eq. (3.6) can still be used to compute the bulk term, but the result now becomes
(right panel) for the rotational BTZ black hole. The null sheets coming from tR = tL = 0 meet
each other at the surface r = r0 2 (r ; r+).
The joint term at r = rm and the null boundary term have the same expressions shown in
eq. (3.7) and (3.10). Therefore, without any counterterm
Using our regularized complexity, we can compute the complexity of formation (2.65):
Ireg(M =
1) = Ibulk + Ijoint + I =
) CA,reg(M =
1) =
`AdS
2
`AdS :
2~
m
r
+
∞
∞
=
=
r
r = r0
`AdS ;
2~
m
=
r
r−
r−
r−
r−
(3.14)
(3.15)
(3.17)
r
r
=
=
r
r
m
m
t
R
r ln
r0 + r
r0
r
r+ ln
r+
r0
= 0:
C = Creg(M
Creg(M =
1) =
which reproduces the result in ref. [20]. Because M =
1 has lower energy, it is the vacuum
solution rather than the case with the limit M ! 0.
3.2
CA conjecture in rotational case
For the case that J 6= 0, the mass M must be nonnegative value. There is an inner horizon
behind in the outer horizon. In this case, the Penrose diagram and the WDW patch is
shown in the left panel in
gure 4. As the same as the case of J = 0, we introduce the
infalling coordinate and outgoing coordinate u and v by the eq. (3.5), however, the function
r (r) then becomes
r =
Z
dr
=
r ln
r + r
j
r
r j
r+ ln
r + r+
j
r
r+j
:
(3.16)
In the region r 2 (r ; 1), the function r (r) has two monotonic regions, i.e., (r+; 1) and
(r ; r+). From eq. (3.16), we nd that r (1) = 0; r (r+) =
1 and r (r ) = 1. In this
case, the null sheets coming from the tL and tR meet each other at the inner region of event
horizon at nite radius r = r0 6= 0. The value of r0 can be determined by equation r (r) = 0
with the restriction r < r+. Then we obtain the following transcendental equation
The computation for the regularized action is very similar to what we have done at the
case of J = 0. The bulk term can be computed by the same formula shown in the eq. (3.6)
but the lower limit of the integration is r0, i.e.
The null boundary term shown in eq. (3.10) now reads
r+ ln
2
r+
r0
:
I =
rm ln
2
rm2
r0 ln
2
r
2
0
+ rm
r0:
As the nal result is independent on the choice of ; , we have xed
=
= 1. The
contribution of the joint terms at the boundary r = rm have the same formula as the J = 0
case but we need to add the joint term at r = r0 since they have nonzero values
Finally, combining the results in eqs. (3.18), (3.20) and (3.21), we nd that
Ireg = Ibulk + Ijoint + I =
=
2
r0 ln (r+2
r02)(r02
r2 )=r04
r+ ln
2
r+
r0
r+ ln
2
r0 ln
2
r+
r0
2
`AdSf (r0)
:
depends on only r =r+ =
such that Ireg=r+
^
I( `AdS). Or
where there is no surface counterterms since d = 2. This result goes to zero when r
(so r0 ! 0), which reproduces the case with J = 0 shown in (3.11). Because Ireg=r+
`AdS we can introduce an auxiliary dimensionless function I^(x)
Ireg(T; ) =
1
2`2AdS
I^( `AdS) =
2I^( `AdS) S:
where we used the expressions in (3.4) for r0. The value of I^( `AdS) can be computed only
numerically with r0 determined by (3.17).
Let us consider two special cases. First, for small momentum case ( `AdS
1)
I^( `AdS) = c0 `AdS ln( `AdS) +
1
2
I^( `AdS) =
ln(1
`AdS) +
;
:
where c0
1:19967
. Second, for low temperature case (1
`AdS
1),
Note that I^( `AdS) is less than zero for small `AdS but larger than zero for large
Using our regularized complexity, we can compute the complexity of formation (2.65),
C = Creg
1), which reproduces the result in ref. [20].
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
! 0
(3.23)
(3.24)
(3.25)
`AdS.
HJEP09(217)4
Now let us calculate the regularized complexity for the CV conjecture in the BTZ black
hole. For simplicity, we consider the complexity of a thermal state de ned on the time slice
tR = tL = 0. The maximal volume is just like eq. (2.3)
V = 4
Z rm
r+
1
dr =
Here we introduce a cuto at the boundary by r = rm = `AdS= . In this case, we only
need one surface term eq. (2.51),
Then the regularized complexity eq. (2.46) can be written as
Let us rst consider the case J = 0. For M
0, it turns out that
Vc(t1) = `AdS
=
2
2 `AdS :
Z
B
CV,reg = 0:
4 r+`AdS` 1:
(3.28)
For M =
1, there is no horizon and the regularized complexity is
CV,reg,vac = 4 ` 1
0
1
!
`AdS
dr =
4 `AdS` 1
2
;
where the subscript \vac" is added since M =
1 is the lowest energy state. Using our regularized complexity, we can compute the complexity of formation (2.65):
which reproduces the result in ref. [20] if we choose ` = `AdS.
Next, let us consider the case J 6= 0. The regularized complexity eq. (3.28) yields
`
`AdS
CV,reg = 4
0
r
2
r+2)(r2
r2 )
1
1A dr
4 r+:
(3.32)
Like (3.22), by introducing C^ = ``Ad1SCV,reg=r+, we have
`
`AdS
CV,reg(T; ) =
1
2 T `2AdS ^
2`2AdS
C
( `AdS) =
2 ^( `AdS) S ;
C
where the value of C^( `AdS) can be determined only numerically.
Let us consider two special cases. First, for small momentum case ( `AdS
1)
^
C
( `AdS) =
2
( `AdS)2 +
:
(3.27)
(3.29)
(3.30)
(3.31)
(3.33)
(3.34)
Second, for low temperature case (1
1), we nd
Interestingly, low temperature behaviour of the CA and CV conjectures are similar. Indeed
they are exactly the same if we choose ` = 4 2~`AdS.
We conclude this section by showing how to derive (3.35) in detail. First we consider
the leading behavior of C^( `AdS) at low temperature limit, i.e.,
`AdS ! 1. If we de ne
= r =r+ = `AdS the volume integral C^( `AdS) can be written as
Z a
1
Z a
1
0
p(x
dx +
x2 )
1
1A dx
!
1 dx
1
1
1
p(x
1)P (x; x )
+ nite term
1
where 0 < x < 1 and a(a > 1) is any constant. P (x; x ) is de ned as
(x
1)P (x; x )
x )(x + 1)(x + x )=x4
x )h(x; x ) ;
where the second line de nes another function h(x; x ) for convenience. To read o the
singular part of eq. (3.36) we de ne H(x ) as
H(x )
Z a
1
2
= p
=
h0
p
1
h0
p(x
ln(px
dx
1)(x
x )h0
1 + p
x
x )j1a
ln(1
x ) + nite term ;
where h0
h(1; x ) = 2(1 + x ). On the other hand, we have
H(x )
^( `AdS)
C
=
=
4
Z a
Z a
1
1
p(x
q
x )h(x; x )
h0
h(x; x )
1)(x
x )h(x; x )h0(ph0 + ph(x; x ))
dx
p(x
1)(x
x )h(1; x )
!
+ nite term
dx + nite term
(3.36)
(3.37)
(3.38)
(3.39)
Therefore, for the limit `AdS ! 1, we nd that
Examples for Schwarzschild AdSd+1 black holes
A general Schwarzschild AdSd+1 (d
3) black hole is given by following metric
with
Here ! is the `mass' parameter
ds2 = r2f (r)dt2 + r2f (r)
+ r2d 2d 1;k
dr2
k
`AdS
1
!d 2
rd :
!d 2 = rhd 2
h + k ;
d 2d 1;k = <>>> Xd1 dxi2;
>
>
8> d 2 + sin2 d 2d 2;
>
>>> i=1
>:> d 2 + sinh2 d 2d 2;
hyperbolic horizon. The (d
1)dimensional line element d 2d 1;k is given by
with the horizon position rh and k = f1; 0; 1g corresponding to spherical, planar and
d = 3; 4.
with
Here 2
d 2 is a line element of d
2 dimensional unit sphere. The dimensionless volume
of the spatial geometry will be denoted by
d 1;k. The horizon locates at r = rh. For
simplicity, we still consider the case tR = tL = 0 and try to nd the regularized complexity
in both the CA and CV conjectures. In this paper, we will only focus on the cases of
4.1
Regularized complexity in CA conjecture
Case of d = 3. In order to compute the regularized complexity in the CA conjecture,
let us rst introduce the outgoing and infalling null coordiantes u; v de ned by the same
manner shown in eq. (3.5), but the function r now should be changed as
r =
rh`2AdS
3rh2 + k`2AdS
0
j
r
rhj
ln @ qr2 + rrh + rh2 + k`2AdS
1
A +
0
2v1 arctan @ q3rh2 + 4k`2AdS
2r + rh
v
1 =
`2AdS(3rh2 + 2k`2AdS)
2(3rh2 + k`2AdS)q3rh2 + 4k`2AdS
:
(3.40)
(4.1)
(4.2)
(4.3)
(4.4)
1
A ;
(4.5)
(4.6)
in the Schwarzschild AdS black holes. The null boundaries of the WDW patch come from the
boundary and there is a nullnull joint at the cuto
r = rm. In addition, in order to
regularize the singularity, we need to use an additional cuto at r = " ! 0, so there are also some
new joints and spacelike boundaries.
In order not to make the computation too complicated, we assume rst rh > 2`AdS= 3
when k < 0.
gure 5.)
Similar to the case in BTZ black hole, there is a nullnull joint at the cuto r = rm.
When rm = 1, the null sheets coming from the boundaries will meet the singularity before
they meet each others. In order to regularize the singularity, we need to use an additional
cuto
at r = " ! 0, so there are also some new joints and spacelike boundaries. (See
The null boundaries at the green region in gure 5 is given by v = vm and u = um,
where vm
for such null boundaries are still given by kI =
[(dt)I + r 2
f 1(dr)I ] and kI = [(dt)I
r 2
f 1(dr)I ]. Here we still explicitly exhibit the freedom of choosing the dual normal
vector by two arbitrary constant
and .
The bulk action is expressed as
h
=
3 2;k Z rm
Here I0 is the nite term, which reads
I0 =
+
2;kd
2
( k2`4AdS + 3k`2AdSrh2 + rh4 ln
6(k`2AdS + 3rh2)
rh(k2`4AdS + 5k`2AdSrh2 + 3rh4) "
3(k`2AdS + 3rh2)q4k`2AdS + 3rh2 2
r (r)]r2dr
arctan
2
k 2;k`AdS ln(rm=`AdS) + I0 + O(`AdS=rm) :
r
4
h
3(k`2AdS + 3rh2) ln
rh
`AdS
rh
q
4k`2AdS + 3rh2
!#)
:
We see that a logarithm term appears in eq. (4.7). As the nullspacelike joints at the corners
r = 0 have no contributions on the action [20]. The joint term at the in nite boundary for
= gIJ
= gIJ
1)
:
By the similar method in eq. (3.10), we nd the null boundary term I is
I =
d 1;k
4
[2 ln(d
1) + ln(
`AdS=rm2) +
2
2
d
1
]rmd 1 :
An important di erence between the Schwarzschild black hole and the BTZ black hole
is that there is a spacelike curvature singularity at r = 0. We need to make a cuto
at
r = 0 so that the computation cannot touch the singularity. As a result, there are two
be given the similar method shown ref. [20].13 For the case, d = 3, it is
spacelike surface terms at r = " ! 0. The contribution of such terms on the action can
( 2rh(2k(k`2A`2AdSdS++3rrh2h2)) ln
r
2
h
(k`2AdS + 3rh2)q4k`2AdS + 3rh2
(k`2AdS + rh2)(2k`2AdS + 3rh2)rh arctan
s
r
2
h
9
;
= + O(`AdS=rm):
For the case that d > 2, the surface counterterm is nonzero. We see FA0 = ln(d 1)=(4 ).
It is easy to see that Kij = 0 and R = R^ = k(d
1)(d
2)=r2. Specializing that d = 3,
there is a logarithm counterterm in the subleading counterterm.
general d is given by following expression
d 1;k rd 1 ln(jk k j=2)
4
4
4
d 1;k rmd 1 ln
d 1;k rmd 1 ln
2
rm
2
rm
d 1;k rmd 1 ln
rm2
k2`4AdS +
2rm4
:
4
2
k 2;k`AdS ln(rm=`AdS) :
Thus, the surface counterterm for d = 3 reads
Ict,L = Ict,R =
4
2;k ln 2 rm2 +
4
2
k 2;k`AdS ln(rm=`AdS) :
13However, there is a little di erence between our result and the result in ref. [20]. In ref. [20], the null
sheets come from the boundary r = 1. Here the null sheets come from the cuto surface r = rm.
One can check that kI is still a ne parameterized, so we still nd that only the null
HJEP09(217)4
boundary term shown in eq. (2.13) has contribution. The expansions of k and kI in
general d are
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
1
~ rm!1
( 4
2;k
4 2~
+ rh(4k2`4AdS + 5k`2AdSrh2 + 3rh4) arctan
2;kk`2AdS :
4 2~
2k2`4AdS
2(k`2AdS + 3rh2)
rh2(rh2 + 3k`2AdS) ln
(k`2AdS + 3rh2)
`AdS
`AdS
=
;
As we expected, all the divergent terms have disappeared and the result is independent of
the values of
and
when we choose the null normal vectors for the null boundaries.
Though eq. (4.15) is obtained by the assumption rh > 2`AdS= 3 when k =
make an analytical extension to get the regularized complexity when `AdS < rh < 2`AdS= 3
by following analytical extension
q
3rh2
4`2AdS = i 4`2AdS
q
3rh2; arctan
3
p
r
2
4`2AdS = iarctanh
4`2AdS
3 :
(4.16)
Finally, we obtain the regularized complexity for d = 3
CA;reg =
lim (Ibulk + IGHY + Ijoint + I
Ict,L
(4.15)
1, we
p
(4.17)
(4.18)
On the other hand, by the following identity for arctanh function and logarithm function
when x > 1
arctanh(x) = [ln(1 + x)
x)] ;
one can check that the eq. (4.15) has well de ned limit at rh = `AdS and is analytical in
the neighbourhood of rh = `AdS + 0+. So the eq. (4.15) can extend into the whole region
of rh
`AdS when k =
1. By this analytical extension, it is easy to nd that the vacuum
regularized complexity for k = 0; 1(rh = 0) and k =
1(rh = `AdS) is
CA,reg,vac =
2;kk`2AdS :
4 2~
p
p
We plot the regularized complexity (4.15) and (4.18) in gure 6. They may not be
positive but their di erence, the complexity of formation, is always positive and the same
as the results in ref. [20].
We note that the complexity of formation for k =
1 and
`AdS < rh < 2`AdS= 3 has also been given by ref. [20] in a very implicit manner. In fact,
one can prove that it is just the same as the eq. (4.15) in the sense of analytical extension
shown in (4.16).
When `AdS= 3 < rh < `AdS and k =
1, i.e., the small black hole case in hyperbolic
black holes, the logarithm function and arctanh function become multiple values and, the
casual structure of such hyperbolic black hole is very di erent from what we have shown
in the
gure 5. In principle, we need an additional computation for this case. We leave
this case in future works.
When d = 4, we see that the logarithm term will not appear but the
subleading counterterm appears. By eq. (2.32) and (2.38), the total surface counterterm reads
The bulk can be computed by the same method shown in eq. (3.6) and the result is
4 3;k Z rm
And the contribution of boundary terms coming from the singularity is
The joint terms and null boundary terms I can be obtained by eq. (4.9) and (4.11)
with d = 4. Then we nd that all the divergent terms can be canceled with each other and
we obtain a nite regularized complexity
CA,reg =
4 ~
3;k (k`2AdS + rh2)3=2(rh2
k`2AdS) :
By this result, we can obtain the vacuum regularized complexity
We plot the regularized complexity (4.23) and (4.24) in gure 6. They may not be positive
but their di erence, the complexity of formation, is always positive and the same as the
results in ref. [20]. Similarly, the eq. (4.23) is valid when rh
`AdS in hyperbolic black
holes. The case of small black hole needs another computation.
4.2
Regularized complexity in CV conjecture
Now let us calculate the regularized complexity for the CV conjecture. For the case tL =
tR = 0, the maximal valume surface bounded by codimensiontwo surface tL and tR is just
the time slice of t = 0. Then volume of this codimensionone surface can be obtained by
the following integration
of which near boundary (r ! 1) behaviour is
= 2 d 1;k
rh
Z 1 r
d 2
Z 1 d 2
r
dr
k
`AdS
1
2 d 1;k`AdS
r
d 2
Z 1
r
d
h
rd
k
r
h
2 + 2
1
`AdS
2
k`2AdS rd 4 +
dr:
1=2
dr ;
We will give the regularized complexity in di erent dimension and k.
3
2
2
:
:
3;k
4
4
2
I0 =
CA,reg,vac =
k`2AdS + 2rh2
4 3;~k `3AdS k;1:
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
(4.26)
HJEP09(217)4
(a) CA conjecture: solid lines are (4.15) and (4.23). Dashed lines are (4.18) and (4.24).
HJEP09(217)4
(b) CV conjecture: solid lines are (4.34) and (4.39). Dashed lines are (4.35) and (4.40). For
k = 0 see (4.30).
of formation (solid line minus dashed line) is always positive, which agree to gure 4, 5, 10 and 11
in [20].
Planar geometry. In this case, k = 0 and the boundary is just a at spacetime, which
leads that R
= Kij = 0 at the boundary. We can obtain the results for general dimension.
Because the divergence structure (4.26) is
2 d 1;0`AdS
Z rm
r
d 2 +
d
2rh2 +
dr ;
the volume of the maximal surface at the cuto rm = `AdS= is
= 2 d 1;0`AdS
0
(d
d 1
`AdS
1) d 1 +
d 1
`AdS
1) d 1 +
Z rm
p (d
2(d
0
d 1
r2
rd
h
rd 2
!
r
A dr
r
d
d 1 1
h
The surface counterterms are
Vc(tn) = 0;
d
d 1 p
x
=
(d
d
d 1;k`AdS ;
1) d 1
(4.27)
(4.28)
(4.29)
r
CV,reg =
`AdS d 1;0
2Vc(t1))
p (d
(d
2 2;k`AdS
Z rm
k`2AdS +
2r
dr ;
and CV,reg,vac = 0. We plot the regularized complexity (4.30) for d = 3; 4 in gure 6. The
complexity of formation is always positive and the same as the results in ref. [20].
Spherical and hyperbolic geometries for d = 3. In this case, the divergence
strucso we have
V =2 2;k`AdS
Z rm
" 2
`2Ad2S +
k`2AdS ln
2
2
2
h +
2
k`2AdS ln(rh=`AdS)
q
k`2AdS + r2
rrh (k`2AdS + rh2)
r +
k`2AdS
2r
!
dr ;
where rm = `AdS= . Now we need the rst order surface counterterm and the subleading
logarithmic counterterm:
Vc(t1) =
Vc(t2) =
B
3
k 2;k`AdS ln :
d x
2 p
3
2;k`AdS ;
Thus, the regularized complexity can be written as
CV,reg = lim
!0 `
1
(V
= 2 2;k`AdS` 1
3
2Vc(t1)
0
2Vc(t2))
0
2x A dx
x
2
2
h +
k
2
1
lnxhA :
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
Here we de ne x = r=`AdS and xh = rh=`AdS. All the divergent terms have disappeared,
as expected. It is straightforward to nd that the vacuum regularized complexity is
CV,reg,vac =
k` 1 2;k`AdS ln2
3
1
2
:
We plot the regularized complexity (4.34) and (4.35) in gure 6. They may not be positive
but their di erence, the complexity of formation, is always positive and the same as the
results in ref. [20].
Spherical and hyperbolic geometries for d = 4. In this case, the divergence
struc2
Z rm
dr ;
x2 +
1
2 A dx
h +
second surface counterterms, which are
3
`AdS Z
B
4
B
d x
3 p
d x
3 p
4
3;k`AdS ;
3 3
2 (R^
3
R=2)
K2
=
4
4
k 3;k`AdS :
2
Then the regularized complexity can be written as
!0 `
(V
= 2 3;k`AdS` 1
4
2Vc(t1)
0
2Vc(t2))
0
All the divergent terms have disappeared and the vacuum regularized complexity is
8 4
4
CV,reg,vac = < 3` 3;k`AdS;
: 0; k =
1:
We plot the regularized complexity (4.39) and (4.40) in gure 6. They may not be
positive but their di erence, the complexity of formation, is always positive and the same
as the results in ref. [20].
5
In this paper, we studied how to obtain the nite term in a covariant manner from the
holographic complexity for both CV and CA conjectures when the boundary geometry
is not deformed by relevant operators. Inspired by the recent results that the divergent
terms are determined only by the boundary metric and have no relationship to the stress
tensor and bulk matter elds, we showed that such divergences can be canceled by adding
codimensiontwo boundary counterterms. If bulk dimension is even, a logarithmic
divergence appears. These boundary surface counterterms do not contain any boundary stress
k + x2
xx2h2 (k + x2 )
h
x2 +
1
2 A dx
h +
1
kxh
2 A :
(4.36)
kxh
2 5 ;
(4.37)
(4.38)
(4.39)
(4.40)
tensor information so they are nondynamic background and can be subtracted from the
complexity without any physical e ects. In the CA conjecture, with the modi ed boundary
term proposed by ref. [19] di erent from the framework in the refs. [18, 20], our regularized
complexity is also independent on the choice of the normalization of the a ne parameters
of the null normal vectors. We argue that the regularized complexity for both CV and
CA conjectures contain all the information of dynamics and matter elds in the bulk for
given time slices, and we can use them to study the dynamic properties of the holographic
complexity such as the growth rate and the complexity of formation.
We showed the minimal subtraction counterterms for both CA and CV conjectures up
to the dimension d + 1
5. By these surface conunterterms, we calculated the regularized
complexity for the nonrotational and rotational BTZ black holes and the Schwarzschild
AdS black holes in four and
ve dimensions with di erent horizon topologies. They also
directly show that the problem that the complexity depends on the choice of the
normalization about the null normal vectors in the CA conjecture will not appear in the regularized
complexity. As a check, we use our regularized complexity to compute the complexity of
formation in the BTZ black holes and the AdSd+1 black holes based on both CA and CV
conjectures and reproduced the same results shown in ref. [20]. However, unlike ref. [20]
we do not need to worry about the coordinate dependence of the cuto , because our
regularized complexity is de ned to be coordinate independent.
Using this regularized complexity, we can study the e ects of bulk matter elds and
thermodynamic conditions on the holographic complexity at a
xed dual boundary (the
codimensionone surface) geometry. There are many future works. For example, we can
study its behavior in holographic superconductor models to see if it can play a role of an
order parameter in phase transitions or if there is any interesting and special behavior at
zero temperature limit [39]. We also can directly compute the complexity at di erent time
slices and compute its derivative with respective to tL or tR rather than only the case
tL = tR shown in the examples in this paper and obtain the whole growth rate if (@=@t)
is a timelike Killing vector at the boundaries.
In this paper, it is assumed that the asymptotic boundary geometry has an expansion
shown in eq. (2.16). However, it can be deformed by a relevant operator, for example by
a scalar eld with negative mass.14 In such a deformed metric, the divergent structure
will depend also on the information of the matter eld, so our formalism cannot cancel all
the divergences in both CV and CA conjectures. It would be interesting to analyse the
UV divergent structures in this case and nd the counterterms. We are now investigating
this problem.
Acknowledgments
We would like to thank Rob Myers for valuable discussions and correspondence.
We
also thank YongJun Ahn for plotting
gure 6. The work of K.Y.Kim and C. Niu was
supported by Basic Science Research Program through the National Research
Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future
Planning(NRF14We thank the anonymous referee to draw our attention to this issue.
2017R1A2B4004810) and GIST Research Institute(GRI) grant funded by the GIST in
Cosmology" at Sun YatSen university in Zuhai, China for the hospitality during our visit,
where part of this work was done.
A
Subleading divergent terms in CA conjecture
is slightly di erent from theirs.15
In this appendix, we will show how to obtain the contribution of the null surface term
coming from the nonzero
in the action (2.11) in more detail. It seems that refs. [18, 19]
neglected an O(z3) order contribution from the null boundary contribution, so our result
Based on ref. [17], the null boundary term in the action can be written as
I
N =
8
N
d d2x :
kI = ( dz + n dx ) :
kI =
nI nI =
1 :
is the parameter of integral
curve of kI . Following the ref. [18], we assume k has the following form near the boundary
Here
is a constant but n is the function of z and x . Using the metric (2.15), we
nd that
where n
= g n . Because the eq. (A.2) is not an additional assumption we can always
write the normal vector for the null surface as eq. (A.3) in the FG coordinate system. The
null condition kI kI = 0 shows that n
must be a normalized unit timelike vector, i.e.
Now let us
nd the nona nity parameter
for this null normal vector.
Using
k r k = k we have
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
We will solve this equation order by order in z. One can see that under the gauge N~ (0) = 1
and L~i(0) = 0 in eq. (2.20), n
must have the following form
and
has the following series expansion with respective to z
n =
t + n (1)z2 + n (2)z4 +
;
= (0) + (1)z3 +
:
Here the coe cients fn (1); n (2);
g and f
(0); (1);
g are only the function of x .
15Recently, we learned that the authors of [18] obtained the same results as ours by a di erent method
and it would be updated in their revised version. We thank the authors of [18] for sharing the manuscript
before posting.
As the eq. (A.2) shows that dz=d
= kz and dx =d
= k on the integral curve of kI ,
we can use the following replacement when we compute the integral (A.1)
2
`AdS
+ n
2
`Azd2S dz :
Therefore,
dkz
dkt
d
I
N =
2 2z3
4
`AdS
2 2z3
4
`AdS
2
`AdS Z
8
4 2nt(1)
4
`AdS
N
z2 dzd2x :
z5 + O(z7);
The relevant components of connection I JL in eq. (A.5) are
zz =
t
tt = O(z2);
z =
t
tz =
z :
t :
up to the order of z5. These give
2 2nt(1)
4
`AdS
z5 =
I
N =
d
d 3
g~t(t1) =
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
2(g~t(t1) + nt(1)) 4
z5 =
[ (0)(nt(1)z2
1)
3 (1)] :
(0) = 0;
(1) =
nt(1) =
g~t(t1) :
2
Taking all these into account we evaluate the eq. (A.1) as
Using the equation we nd the null boundary term contributes to the subleading divergence
I
N =
8 (d
8 (d
d 3
3)(d
1
3)(d
d x
2 p~(0) R^~(0)
d x
2 p
R
2d
2(d
2d
2(d
1)
3 R~(0)
1)
R
+ O( d 5) :
z = 0;
z
2
`AdS
zg~t(t1) + O(z3);
[ (0) + z3 (1)];
2
2
`AdS
g~t(t1) ;
d 1 p
x ~g~t(t1) Z
3) z=
Z
2
Z
Z
2) B
2) B
t
tt =
zz = 0 :
+ O(z3);
z
2 ddz + O( d 5
d
d 1xp~(0)g~t(t1) + O( d 5) :
R~^(0)
2d
2(d
1)
3 R~(0) ;
With this additional term, the subleading divergent term for the CA conjecture in ref. [18]
should be modi ed as
16 2~ B
d 3(d
1)(d
2)(d
3)
5 (A.16)
so the rst two divergent terms in ref. [19] should be modi ed as
CA,div =
Z
B
2
d 1xp~(0) ln(d
1)K~i(j0)K~ (0)ij
2(d
1)(d
1) 1
3(d
2)(d
2(R^~(0)
R~(0)=2) !
2(d
2)
3)
1)R^~(0) 3
5 + O( 5 d) :
The result (A.17) can been obtained also by a di erent approach, in which we use the
a nely parameterized kI i.e.
= 0. We still can write kI in the form shown in eq. (A.2),
but we cannot demand that
is a constant. Instead, we assume
has the following series
expansion with respective to z
Here except for (0), the other coe cients are only functions of x . By this approach, the
null surface term is still zero but, unlike the results in refs. [18, 19], there is an additional
contribution from
(1). To see this, let us assume kI is the a nely parameterized null
normal vector for the other null surface at the joint. Then according to ref. [18]
2 1 +
(0) +
(1) !
2 + O( 4)
as eq. (A.21) is an exact result in the FG coordinate system.
where
has a similar series expansion to :
Thus the inner product of these two null vectors is16
Using the expression in eq. (2.12), we nd that
Ijoint =
`dAd1S Z
8 d 1
8 d 3
J 0
J 0
J 0
d
d 1 p
x ~ ln
d 1 p
x ~ ln
p (0) (0) ! Z
`AdS
d 1xp~(0)
2
( (0) (0)
J 0
(0) +
~dd 1
x
(1) !
(0) + (1)z2 +
kI = ( dz
n dx ) ;
(0) + (1)z2 +
kI kI = 2 2
z2 :
:
:
+ O( 5 d) :
(A.17)
(A.18)
(A.19)
(A.20)
(A.21)
(A.22)
The logarithmic term in the last line of eq. (A.22) is the same one in ref. [18], but there is
an additional subleading divergent term due to
(1) and
Now let us compute (1) and (1). Using the geodesic equation up to the order of z5,
(1)
(1)
(0) =
21 g~t(t1) :
By this result, one can check that the subleading term should be
d 1xp~(0) 4
2
2K~ (0)2 + 2K~i(j0)K~ (0)ij + (d2
4d + 1)R~(0)
d(d
3)R^~(0) 3
d 3(d
1)(d
2)(d
3)
which is di erent from the result shown in eq. (A.16). It is because the action (2.11)
without I depends on the parameterization of the null normal vector. Note that (1) and
(1) also have additional contributions to the subleading term in I . Using the method in
ref. [19], one can compute such additional contribution. If we take both of two additional
subleading contributions coming from Ijoint and I into account, we nd the subleading
divergent term is still the same as eq. (A.17).
B
The counterterms in higher dimension: examples in symmetric spaces
Although the universal counterterms in higher dimension are complicated in general, it is
possible to obtain simple formulas in some case: if the space has spherical, hyperbolic, or
planar symmetry and the time slices at the boundary are given at constant t (t is the orbit
of the timelike Killing vector eld at the boundary).
Let us consider the metric of the form
ds2 =
r2f (r)e
(r)dt2 +
r2f (r)
+ r2d 2d 1;k ;
where k = 1; 0; 1 for spherical, planar and hyperbolic space respectively. The Ricci tensor
at any cuto surface r = rm reads
R
= diag[0; (d
2)k=rm2;
; (d
2)k=rm2] ;
The projection of the Ricci tensor on the constant time slices tL or tR is R^ji = diag[(d
2)k=rm2;
; (d
2)k=rm2]. The extrinsic curvature of these two time slices at the
cuto surface vanish, i.e., Kij = 0. Thus, the scalar invariants Pi ci;n(d)[R; K]i2n are the
combination of the nth order scalar polynomials consisting of the contraction of R
or
R^ji . Furthermore, in our case with the metric (B.1), it is enough to consider the Ricci scalar
R, because any other scalar invariants are equivalent to R. Aa a result, FV(2n) or F (2n) can
A
X ci;n(d)[R; K]i2n = c1;n(d)Rnjr=rm ;
i
where we introduce c1;n without summation because we have only one kind of term in
[R; K]i2n, the Ricci scalar R. There is one exception which cannot be expressed as eq. (B.3):
(A.23)
5 ;
(A.24)
(B.1)
(B.2)
(B.3)
The divergent structure can be obtained by setting f (r) = 1=`2AdS + k=r2 and analysing
the asymptotic behavior of
2 d 1;k`AdS
rh
Z rm rd 2(1 + k`2AdS=r2) 1=2dr :
(for even d) ;
pnkn`2AndSrd 2 2ndr
pnkn`2n
1
Ad2Sn rmd 1 2n;
pnkn`2n
d 1 A2dnS rmd 1 2n +pnk(d 1)=2`dAd1S ln(rm=`AdS); (for odd d) ;
if d is odd and n = (d
1)=2, there is a logarithmic divergence, which should be treated
separately following eqs. (2.44) and (2.60).
Our goal in the following subsections is to nd the concrete expression of c1;n in both
CV and CA conjectures so to
V
A
nd F (2n) and F (2n). There are two factors simplifying
our analysis: i) the scalar curvature R is coordinate independent so c1;n(d) is also
coordinate independent. ii) with an assumption that the matter part does not contribute the
counterterms, c1;n(d) can be obtained from the vacuum solution.
CV conjecture. The maximal volume with the boundary time slices tL = tR = 0 is
Z rm rd 2
rh
dr :
The divergent part is
Vdiv = 2 d 1;k`AdS
>>>2 d 1;k`AdS
>>>2 d 1;k`AdS
X
[ d2 1 ] Z rm
n=0
[ d2 1 ]
X
[ d2 1 ] 1
X
where
where the coe cients pn are de ned by the following expansion
(1 + x) 1=2 = X pnxn; with jxj < 1 ;
pn =
(1=2)
(n + 1) (1=2
Rjr=rm =
1)(d
rm2
2)k :
Using the expression (B.2), we can write the counterterms shown in eq. (2.47) as
Z
B
Vct = `AdS
dd 1x p
X `2AndScn(d)Rn ;
cn(d) =
1
2n)[(d
1)(d
2)]n
[ d2 1 ]
n=0
pn
(B.4)
(B.6)
(B.7)
(B.8)
(B.9)
HJEP09(217)4
where pn is given in eq. (B.6). In other words, FV(2n) is written as
V
F (2n) =
( 1 )
2
(n + 1) (
1
n) (d
1
1)(d
2)]n
:
When d is odd and n = (d
( 1 )
CA conjecture. It is enough to nd out the divergent structure of the onshell action
for the vacuum solution. The general results for the joint term and the boundary term can
be obtained from the eqs. (4.9) and (4.11) with !d 2 = 0:
1) +
1
1
d 1;k rmd 1 ln 1 +
k`2AdS
rm2
:
The bulk term can be written as
Ibulk =
d 1;k Z Z
2
8 `AdS
p
gdd+1x R + d(d
1)=`2AdS =
WDW
(rd)0dtdr =
d 1;k I
2
8 `AdS @WDW
d 1;kd Z Z
2
8 `AdS
rddt :
WDW
r
d 1dtdr
and r satisfy dt
expressed as,
where \WDW" means the \WheelerDeWitt" patch. The divergent part in eq. (B.12)
comes from the near boundary of the WDW patch, @WDW, which are the infalling and
outgoing null geodesics coming from tL = tR = 0 and r = rm. At these null geodesics, t
dr=[r2f (r)] = 0. Then we can see that the divergent part of Ibulk can be
Ibulk,div =
2
d 1;k Z rm rd 2(1 + k`2AdS=r2) 1dr :
After combining the eqs. (B.11) and (B.13), we nd that,
1)=2, the counterterm is modi ed as (2.60), where F (d 1) =
(B.10)
Itotal,div =
1)
d 1;k X ( 1)nkn`2AndS n(d
4
1
2n) rm
d 1 2n : (B.14)
When d is odd, the last term in the summation in eq. (B.14) (2n = d 1) should be replaced
by a logarithmic term ( 1)(d 1)=2k(d 1)=2`dAd1S(d
1) ln(rm=`AdS). Comparing eq. (B.14)
with eq. (2.25) and noting the all the divergent terms in eq. (B.14) should be canceled by
2Ict, we nd that F (0) = ln(d
A
1)=(4 ) and,
A
F (2n) =
1
8 n(d
( 1)n(d
1
2n)[(d
1)Rn
1)(d
2)]n
for n > 0 :
(B.15)
1 ( 1)n(d
8 n[(d
1)(d
1)2R)]nn .
When d is odd and n = (d
Eqs. (B.10) and (B.15) are the counterterms for any dimensions d > 2. We used the
speci c coordinate but the
nal results are coordinate invariant. As a consistency check,
we con rmed that we can reproduce the eqs. (4.14), (4.19), (4.29), (4.33) and (4.38) from
eqs. (B.10) and (B.15).
1)=2, the counterterm is modi ed as (2.44), where F (d 1) =
n=1
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