Surface Counterterms and Regularized Holographic Complexity

Prepared for submission to JHEP

arXiv:1701.03706v1 [hep-th] 13 Jan 2017

Surface Counterterms and Regularized Holographic Complexity

Keun-Young Kim,a Chao Niu,a Run-Qiu Yangb a

School of Physics and Chemistry, Gwangju Institute of Science and Technology, Gwangju 500-712, Korea b Quantum Universe Center, Korea Institute for Advanced Study, Seoul 130-722, Korea

E-mail: [email protected], [email protected], [email protected] Abstract: The holographic complexity is UV divergent. As a finite complexity, we propose a “regularized complexity” by employing a similar method to the holographic renormalization. We add codimension-two boundary counterterms which do not contain any boundary stress tensor information, which means that we subtract only non-dynamic background. All the dynamic information of holographic complexity is contained in the regularized part. After showing the general minimal subtraction counterterms for both CA and CV conjectures we give examples for the BTZ and Schwarzschild AdS black holes and compute the complexity of formation for them by using regularized complexity.

Contents 1 Introduction

1

2 Surface counterterms and regularized complexity 2.1 Coordinate dependence in discarding divergent terms 2.2 Surface counterterms in CA and CV conjectures 2.2.1 Surface counterterms in CA conjecture 2.2.2 Surface counterterms in CV conjecture

4 4 6 6 14

3 Examples for BTZ black holes 3.1 CA conjecture in non-rotational case 3.2 CA conjecture in rotational case 3.3 CV conjecture in BTZ black hole

17 17 19 21

4 Examples for Schwarzschild AdSd+1 black holes 4.1 Regularized complexity in CA conjecture 4.2 Regularized complexity in CV conjecture

23 24 28

5 Summary

30

A Subleading divergent terms in CA conjecture

31

1

Introduction

In recent years, the quantum entanglement and information gave us new viewpoints to see quantum gravity and black holes. One of the very interesting results in this aspect is the quantum complexity and its gravity dual description. Roughly speaking, the quantum complexity characterizes how difficult it is to obtain a particular quantum state from an appointed reference state. In a discrete system, such as a quantum logic circuit, it’s the 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], another aspect of entanglement in gravity was studied in [6–10]: by paying attention to the growth of the Einstein-Rosen bridge the authors found a connection between AdS black hole and quantum complexity in the dual boundary conformal field theory (CFT). In this study, they consider the eternal AdS black holes, which are dual to thermofield double (TFD) state, X |TFDi := Z −1/2 exp[−Eα /(2T )]|Eα iL |Eα iR . (1.1) α

–1–

tR

WDW

B tL

tL

rh

tR

rh

Figure 1. Penrose diagram for Schwarzschild AdS black hole and complexity in two conjectures. 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 codimension-one 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 space-like codimension-one surfaces connecting tL and tR .

The states |Eα iL and |Eα iR are defined in the two copy CFTs at the two boundaries of the eternal AdS black hole (see Fig. 1) and T is the temperature. A TFD state can be characterized by the codimension-two surface at fixed times t = tL and t = tR at the two boundaries of AdS black hole. There are two proposals to compute the complexity of a TFD state holographically: CV(complexity=volume) conjecture and CA(complexity= action) conjecture. The CV conjecture states that the complexity of a TFD state at the boundary CFT is proportional to the maximal volume of the space-like codimension-one surface which connects the codimension-two surfaces donated by tL and tR , i.e. [7, 11], V (Σ) CV = max . (1.2) ∂Σ=tL ∪tR ` Here Σ is all the possible space-like codimension-one 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 satisfies 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 TFD state is dual to the action in Wheel-DeWitt (WDW) patch associated with tL and tR , i.e., IWDW CA = . (1.3) π~ The WDW patch associated with tL and tR is the collection of all space-like surface connecting tL and tR with the null sheets coming from tL and tR . More precisely it is the domain of dependence of any space-like surface connecting tL and tR (see the right panel of Fig. 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 [12–14]. However, the CA conjecture has its own obstacle in computing

–2–

the action: it involves null boundaries and joint terms. Recently, this problem has been overcome by carefully analyzing the boundary term in null boundary [15, 16]. As both the CV and CA conjectures involve the integration over infinite region, the complexity computed by the Eqs. (1.2) and (1.3) 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 coefficients of all the divergent terms can be written as the local integration of boundary geometry [17, 18], which is independent of the bulk stress tensor. This result gives a clear physical meaning of the divergences in 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 contribution of excited state or thermal state to the complexity. As the divergent parts of the holographic complexity is fixed by the boundary geometry, the contribution of matter fields and temperature can only appear in the finite term of the complexity. This gives us a strong motivation to study how to obtain the finite term in the complexity. The first work regarding this finite quantity is the “complexity of formation” [19], which is defined by the difference of the complexity in a particular black hole space time and a reference vacuum AdS space-time. By choosing a suitable vacuum space-time, we can obtain a finite complexity of formation. However, there are two somewhat ambiguous aspects in using “complexity of formation” to study the finite term of complexity. First, we need to appoint additional space-time as the reference vacuum background. In general cases, it will not be obvious how to choose the reference vacuum space-time. For example, in Ref. [19], the reference vacuum space-time for the BTZ black hole is not the naive limit of setting mass M = 0. Second, to make the computation about the difference of complexity at the finite cut-off between two space-times meaningful, we need to appoint a special coordinate and apply this coordinate to both space-times. For example, in the Ref. [19], the holographic complexity of two space-time at the finite cut-off is computed in FeffermanGraham coordinate [20, 21]. It will be better if we can compute the complexity without referring to a specific coordinate system. As the Refs. [17, 18] 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 finite term of the complexity, which we will call “regularized complexity”. We will add codimension-two surface counterterms to the formulas of the complexity in the CV and CA conjectures to cancel the divergences. Such counterterms are determined by the boundary geometry and don’t contain any boundary stress tensor information. When the bulk dimension is even, a logarithmic divergence appears, so we need an anomaly counterterm to cancel it. We propose two different anomaly counterterms, both of which yield pure gauge transformation of the finite complexity and have no physical effects. The procedure to obtain the regularized complexity is similar to holographic renormalization. However, there are two differences. First, the surface counterterms we will show are the codimension-two surface at the boundary rather than the codimension-one surface. Because the complexity, as shown in the Fig. 1, is defined by the time slices denoted by tL and tR , which are codimension-two surfaces, it

–3–

is natural that the surface counterterms should be expressed as the geometric quantities of these codimension-two surfaces. Second, the surface counterterms can contain the extrinsic geometrical quantities of the codimension-two boundary rather than only the intrinsic geometrical quantities in renormalizing free energy. One reason for this difference 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 first 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.

2

Surface counterterms and regularized complexity

2.1

Coordinate dependence in discarding divergent terms

Both holographic complexity and entanglement entropy are related with the quantum entanglement in the dual field theory, so it seems that we can use the same method in the Refs. [22–24] to analyze the finite term, i.e. find out the divergent behavior and then just discard all the divergent terms. However, in the following example, we will show such a method is coordinate-dependent so ambiguous. For example, we will use the CV conjecture to show such kind of coordinate dependence. From the example we will show, one can easily find that such coordinate dependence can also appear in the CA conjecture and subregion complexity, and also in the entanglement entropy, if we just naively discard the divergent terms. Let us first consider a Schwarzschild AdS4 black brane geometry ds2 = −r2 f (r)dt2 + with f (r) =

dr2 + r2 (dx2 + dy 2 ) , r2 f (r) 1 `2AdS

−

2M , r3

(2.1)

(2.2)

where M is the mass of black hole, {x, y} are dimensionless coordinates scaled by ÀdS and the horizon locates at r = rh = (2M `2AdS )1/3 . For simplicity, we consider the complexity of a thermal state defined by the time slice tR = tL = 0. Because of symmetry, the maximal surface is just the t = 0 slice in the bulk. The volume of this slice is Z rm r p V (BH) = 2Ω2 dr . (2.3) f (r) rh

–4–

Here Ω2 is the area of 2-dimensional surface spanned by x, y and rm → ∞ is the UV cut-off. As a dimensionless cut-off, we introduce δ = ÀdS /rm . When δ → 0, we can find following expansion for integration (2.3), √ Ω2 `3AdS Ω2 πΓ(4/3) V (BH) = + ÀdS rh2 + O(δ) . δ2 2Γ(5/6)

(2.4)

In fact, as shown in Ref. [17], such leading divergent structure is universal and determined by the vacuum UV boundary. The effects of matter field can only effect the finite term. It seems that we can just directly discard this divergence and use the finite term to study the effects of matter fields. If we do so, we obtain a finite result, √ ÀdS Ω2 πΓ(4/3) 2 CV,finite = rh . (2.5) ` 2Γ(5/6) The process from Eq. (2.4) to Eq. (2.5) can be defined as a kind of background subtraction, i.e., Ω2 `3AdS 1 . (2.6) CV,finite = lim V − ` δ→0 δ2 A similar method was applied in Refs. [22–24] to find the finite term of the entanglement entropy. However, the metric (2.1) is not the only form of Schwarzschild AdS black hole. For example, we may use a new coordinate {t, r0 , θ, φ} by the following coordinate transformation −1 F (M )`2AdS 0 4 04 r =r 1+ , (2.7) + F (M )O(ÀdS /r ) r02 when r ÀdS . Here F (M ) is an arbitrary function and F (0) = 0. We see that, from the AdS/CFT viewpoint, there isn’t any physical difference between coordinate {t, r0 , θ, φ} and {t, r, θ, φ}. Because time t isn’t 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 δ 0 = ÀdS /r0 m be the UV cut-off in a new coordinate system. The coordinate transformation (2.7) implies the following relationship between δ and δ 0 , δ = δ 0 (1 + f (M )δ 02 + · · · ) . In a new coordinate system, the volume of t = 0 slice reads, √ Ω2 `3AdS Ω2 πΓ(4/3) 3 V (BH) = − 2Ω ` ÀdS rh2 + O(δ 0 ) . F (M ) + 2 AdS δ 02 2Γ(5/6)`

(2.8)

(2.9)

As expected, the leading divergent term is just as the same as Eq. (2.4). However, the finite term is different! Now assume we don’t know the result in the coordinate {t, r, θ, φ} and use coordinate {t, r0 , θ, φ} first, then by the Eq. (2.6), we find that, √ Ω2 πΓ(4/3) 0 −1 3 CV,finite = ` [−2Ω2 ÀdS F (M ) + ÀdS rh2 ] . (2.10) 2Γ(5/6)

–5–

As there is no physical restriction on F (M ), we see that such “background subtraction” can make the regularized complexity shows arbitrary behavior by choosing the form of F (M ). In addition, as M is the mass of the black hole, the different choice of F (M ) can lead that all physical results depending on M depend on an arbitrary function F (M ). In recent papers [17, 18], the authors proposed some results to write the coefficient of each divergent term into the boundary integration which only involves the intrinsic and extrinsic curvatures. However, such technology cannot avoid the ambiguity shown in this example, as the the coordinate transformation (2.7) changes the behavior of bulk radius coordinate. In the viewpoint of holographic duality, the background subtraction used in Eq. (2.6) is a kind of “renormalization” method and the radial coordinate stands for the scale of RG flow [25, 26]. A well proposed renormalization method needs the physical results to be independent of the choice of different qualities as energy scale. Result in Eq. (2.10) shows that the naive background subtraction (2.6) isn’t a well defined renormalization method when we compute the complexity. From the gravity viewpoint, the coordinate-dependent result shown in Eq. (2.10) is easily understood: the divergent term we take out from the volume is in its coordinate dependent form. Even in the forms of Refs. [17, 18], the form of UV cut-off is still in its coordinate dependent form, as it depends on a very special radial coordinate. To propose a well defined subtraction on the calculation of the regularized complexity, it’s natural to learn the lessons from the holographic renormalization on the free energy [26, 27]. In such case, the divergences are canceled by adding covariant local boundary surface counterterms determined by the near-boundary behavior of bulk fields. Now we are going to 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. 2.2 2.2.1

Surface counterterms in CA and CV conjectures Surface counterterms in CA conjecture

In this subsection, we will first 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 consider appropriate boundary terms. It was proposed in [15, 16, 18, 28] as Z 1 d(d − 1) d+1 √ I= d x −g R + 16π M `2AdS Z Z p 1 1 √ (2.11) + dd x |h|K − dd−1 xdλ γκ + Iλ 8π B 8π N Z Z √ √ 1 1 + dd−1 x ση + dd−1 x σa . 8π J 8π J 0 where the first line is the Einstein-Hilbert action with the cosmological constant integrated over the WDW region denoted by M, the second line is various boundary terms defined at the boundary of M and third line is the joint terms defined on the corners of two different boundaries. B stands for the time-like or space-like boundary, N for the null boundary, J for the joints connecting time-like or space-like boundaries and J 0 for the joints connecting boundaries, one or both of which are null surfaces. K is the Gibbons-Hawking-York extrinsic

–6–

curvature and h is the determinant of the induced metric. λ is a parameter of the generator of the null boundary and κ is the surface gravity of this null boundary with respective to the null normal vector k I = (∂/∂λ)I , i.e., k I ∇I k J = κk J .1 γ is the determinant of the metric on the cross section of constant λ in null surface N . σ is the induced metric at the joints. The expression for η and a can be found in [16]. As the joint terms J does not occur for the WDW patches, we will not show η here. a is written as ( ± ln(nI kI ) , a= (2.12) ± ln(k I k¯I ) , where nI is the other unit normal vector (outward/future directed) for non-null intersecting boundary, and k¯I 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. [16] that the action (2.11), in its form without Iλ , depends on the parametrization of null generators. It first appeared in Ref. [16] and was studied further in Refs. [17, 19, 28]. Moreover, we will see later that divergent terms in this form can’t be canceled by adding covariant surface terms. Thus, to make the action with the null boundaries diffeomorphism invariant,2 a coordinate dependent boundary term(Iλ ) at the null boundaries is added [18]: Z 1 √ Iλ = ∓ γΘ ln(ΘÀdS )dλdd−1 x , (2.13) 8π N where −(+) appears if N lies to the future (past) of M and √ 1 ∂ γ Θ= √ , γ ∂λ

(2.14)

Now let’s analyze how to add the surface terms so that we can obtain a finite complexity. The goal here is very similar to the case that we add some boundary terms to make the total free energy finite in holographic renormalization. However, there is an very important difference. Our goal here is to make the complexity itself finite, so the surface terms don’t 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. The procedure to find the surface counterterms are that we first write down the divergent terms in a suitable 1

In this paper, the capital Latin letters I, J, · · · run from 0 to d, which stand for the full coordinates and xd = z. The Greek indices µ, ν, · · · run from 0 to d − 1, which stand for the local coordinate at the fixing z surface and x0 = t. The little Latin letters i, j, · · · run from 1 to d − 1, which stand for the local coordinates at the fixing z and t surface. 2 For the joint terms and boundary terms, we still have a kind of ambiguity by adding any term which keeps invariant when we impose variation. This freedom is the subcase of freedom at general relativity: variation principle can’t determine the boundary term uniformly. Corresponding, we have a freedom to add any non-dynamic term into the complexity itself without any physical effects. However, it may leads some dynamic effects if we adding some additional terms at the null boundaries or the joints which aren’t at the AdS boundary, so the this kind of freedom doesn’t correspond to the freedom in complexity completely. It’s not very clear the physical meaning of this kind of additional freedom.

–7–

coordinate, then show that the whole such terms (not only the coefficients) can be canceled by a codimension-two boundary integration. In the Fefferman-Graham coordinate system [20, 21], any asymptotic AdSd+1 spacetime can be written as ds2 = gIJ dxI dxJ =

`2AdS 2 [dz + g˜µν (z, xµ )dxµ dxν ] , z2

(2.15)

where the indices I, J = 0, 1, 2, · · · , d − 1, d denote the full sppace-time coordinates, µ, ν = 0, 2, · · · d − 1 denote the coordinate labeled at the fixed z surface. The metric g˜µν along the boundary directions has a power series expansion with respective to z when z → 0: (0) µ (1) µ ([d/2]) µ ˜ ([d/2]) (xµ ) ln z + · · · , g˜µν (z, xµ ) = g˜µν (x ) + z 2 g˜µν (x ) + · · · + z 2[d/2] g˜µν (x ) + z d h µν (2.16)

where the coefficient of logarithmic term is nonzero only if d is even and [d/2] means the (0) ([d/2]) integer part of d/2. All the expansion coefficients in (2.16) from g˜µν (xµ ) to g˜µν are (0) µ determined by g˜µν (z, x ) by solving the vacuum Einstein’s equations. The higher order coefficients are related with the expectation value of the boundary energy-momentum tensor [26, 29]. However, we will see that those higher order terms are irrelevant in determining the counterterms. At the UV cut-off z = , the induced metric (denoted by gµν ) at the boundary (codimension-one) surface is `2 gµν = AdS g˜µν , (2.17) z2 and we use “ ˜ ” to stand for the conformal boundary metric at the surface z = . Likewise, ˜ (indices are suppressed) means that it is computed by the in this paper, the notation “ X” conformal metric g˜µν and we use g˜µν to raise and lower its indexes. For example, we will decompose the metric gµν as gµν dxµ dxν = −N 2 (z, t, y i )dt2 + σij (z, t, y i )(dy i − Li dt)(dy j − Lj dt) ,

(2.18)

where the indices i, j = 1, 2, · · · d − 1 and {xµ } = {t, y i }, we may introduce ‘tilde’-variables, 2 ˜2 = z N2 , N `2AdS

σ ˜ij =

z2 `2AdS

σij ,

˜ i = Li , L

(2.19)

so `2AdS ˜ 2 (z, t, y i )dt2 + σ [−N ˜ij (z, t, y i )(dy i − Li dt)(dy j − Lj dt)] z2 `2 = AdS g˜µν dxµ dxν . z2

gµν dxµ dxν =

(2.20)

˜ and σ Furthermore, the expansion for g˜µν (2.16) can give similar expansions for N ˜ij : ˜ =N ˜ (0) + z 2 N ˜ (1) + · · · + z 2[d/2] N ˜ ([d/2]) + · · · , N (0)

(1)

([d/2])

σ ˜ij = σ ˜ij + z 2 σ ˜ij + · · · + z 2[d/2] σ ˜ij

+ ··· ,

Li = Li(0) + z 2 Li(1) + · · · + z 2[d/2] Li([d/2]) + · · · ,

–8–

(2.21)

˜ (0) = 1, Li(0) = 0 and we can also define that where we can fix N N (n) =

ÀdS ˜ (n) N , z

(n)

σij =

`2AdS (n) σ ˜ . z 2 ij

(2.22)

As another convention, in this paper, we will always use the notation X (n) to denote the coefficient of z (2n) in the expansion of the field X. Let us consider the Ricci tensor Rµν and the Ricci scalar R for boundary metric gµν and the extrinsic curvature tensor Kij for the t = 0 surface3 (codimension-two) embedded in the z = boundary surface (codimension-one). Then we find that the conformal Ricci ˜ µν , Ricci scalar R ˜ and extrinsic curvature K ˜ ij are tensor R ˜ µν = Rµν , R

2 ˜ = z R, R `2AdS

˜ ij = K

z Kij . ÀdS

(2.23)

For later use, we define two projections from z = surface to the z = and t = 0 surface ˜ µ = ∂xµi . For example, the projections of the Ricci tensors are defined as by hi µ = h i ∂y ˆ ij = hi µ hi ν Rµν , R

˜ˆ ˜ ν˜ ν ˜ R ij = hi hi Rµν .

(2.24)

Similarly to 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, [d/2] Z

Ict =

X

n=1

B

√ 2n−2 dd−1 x σÀdS FA,n (Rµν , gµν , σij , Kij ) + Ianomaly ,

(2.25)

so that we can define the regularized finite action, Ireg , Ireg ≡ lim (I − Ict,L − Ict,R ) . →0

(2.26)

Here B is the codimension-two surface at a given time t and fixed z = and I is the value action (2.11) computed when AdS boundary is setting at cut-off surface, i.e., the surface z = . FA,n is the function of Rµν , gµν , σij and Kij . Ict,L and Ict,R are the surface counterterms defined by (2.25) at the left boundary and right boundary, respectively. Ianomaly is anomaly term which appears only in the even bulk dimensions. Before we go on giving the proof and how to compute the surface ounterterms, it’s necessary to clarify what’s the meaning of I . As pointed by Ref. [17], there are two different methods to regulate the WDW patch as we show in the Fig. 2. At the left panel of Fig. 2, the boundaries of the WDW patch are changed into the null sheets coming from the finite cut-off boundary and there is a null-null joint at the cut-off. At the right panel of Fig. 2, the boundaries of WDW patch is the same, however, original null-null joints at the AdS boundary is sliced out by a time like boundary, so the null-null joint at the boundary disappears but there is an additional Gibbos-Hawking-York boundary term and two null-timlike joints. As the first one approach is more convenient in analyzing the

–9–

tL

r = rm rh

r = rm

tR

tL

r = rm rh

r = rm

tR

Figure 2. Two different approaches in computing the action at the finite cut-off boundaries. Left panel: The null boundaries of WDW patch are changed into the null sheets coming from the finite cut-off boundary and there is a null-null joint at the cut-off. (here we only show that part near the tR . The part near tL is similarity.) Right panel: The boundaries of WDW patch are the same, however, original null-null joints at the AdS boundary are sliced out by a time like boundary and two null-timelike joints are added. (here we only show the right-top part of quarter. The other parts are similar.)

divergent behavior near the AdS boundary, the term I in the Eq. (2.26) is computed by this approach. To find Ict and FA,n , we first need to analyze the divergent structure of (2.11). Obviously, the divergences come from the action near the boundary. It only needs to analyze the divergent behavior at the one side boundary. The other side has a similar result. We will analyze the divergent behavior at FG coordinate and show that the divergent term (the whole divergent term rather than only the coefficients of divergent terms) in this coordinate can be written as the codimension-two surface terms. After subtracting this codimensiontwo surface terms, the remainder result then is finite. As the subtraction terms is in terms of the geometrical objects which only depend on geometrical quantities of the codimensiontwo surface, the final result is independent of the choice of coordinate. This means the result of Eq. (2.26) is the same for all the coordinate systems. Odd bulk dimensions We first consider the case that d + 1 is odd number. In this case there is no anomaly divergent term. All the divergent terms in FG coordinate system are in the form of power series of cut-off. At any side of the two boundaries, the first two divergent terms in the action (2.11) for d ≥ 3 were obtained in Ref. [18]: 1 (1) (2) Idiv = ICA + ICA + O d−5 , (2.27) where (1) ICA 3

`d−1 ln(d − 1) = AdS d−1 4π

Z

dd−1 x

p σ ˜ (0) ,

B

Here we set t = 0 just for convenience, we can set t to be any fixed value.

– 10 –

(2.28)

and4 (2) ICA

`d−1 AdS = 4π(d − 3)d−3

p (d − 3) ln(d − 1) ˜ (0) ˜ˆ (0) (0) d x σ ˜ (R /2 − R ) 2(d − 2) B  ˜ (0)ij − 3(d − 1)R ˜ (0) + 2(d − 1)R ˆ˜ (0) ˜ (0)2 + 2(d − 1)K ˜ (0) K dK ij . − 2(d − 1)(d − 2)(d − 3) Z

d−1

(2.29)

(1)

First, let us consider the leading divergent term ICA . It is expressed in terms of ‘tilde’ variables (2.22) and can be rewritten in terms of real induced metric as (1)

ICA =

ln(d − 1) 4π

Z

dd−1 x

p σ (0) .

(2.30)

B

By the inversion of the expansion to fields (2.21), p √ 2 (1) σ (0) = σ 1 − σ ij σij + O(5−d ) , 2

(2.31)

√ dd−1 x σ + O(3−d ) ,

(2.32)

ln(d − 1) . 4π

(2.33)

we find ln(d − 1) = 4π

(1) ICA

Z B

so FA,1 =

(2)

Similarly, the subleading divergent term ICA can be rewritten as (2) ICA

`2 = AdS 4π −

dK 2

Z

√

d−1

x σ

d B

ln(d − 1) ˆ (R/2 − R) 2(d − 2)

# ˆ + 2(d − 1)Kij K ij − 3(d − 1)R + 2(d − 1)R + O(5−d ). 2(d − 1)(d − 2)(d − 3)

(2.34)

However, the first order counterterm based on Eq. (2.33) also has contribution on the subleading divergent term. By the relationship (2.31) and the Einstein equations for the conformal metric g˜µν at the order of 2 [26, 29] (1) g˜µν

1 =− d−2

(0)

g˜µν (0) ˜ µν R − R(0) 2(d − 1)

! ,

(2.35)

we find that the subleading divergent term in the first counterterm is −

`2AdS 4π

√ ln(d − 1) ˆ + O(5−d ) . (R/2 − R) dd−1 x σ 2(d − 2) B

Z

4

(2.36)

This is different from the results reported in the Refs. [17, 18]. It seems that the null normal vectors used in Refs. [17, 18] is not affinely parameterized. If we take this into account we find an additional contribution on the subleading divergent terms. See the appendix A for details.

– 11 –

As a result, (2.34) and (2.36) together give the new subleading divergent term, `2 − AdS 4π

# 2 + 2(d − 1)K K ij − 3(d − 1)R + 2(d − 1)R ˆ dK ij dd−1 x σ + O(5−d ), (2.37) 2(d − 1)(d − 2)(d − 3) B

Z

√

"

so the second function FA,2 is FA,2 = −

ˆ 1 dK 2 + 2(d − 1)Kij K ij − 3(d − 1)R + 2(d − 1)R . 4π 2(d − 1)(d − 2)(d − 3)

(2.38)

For higher dimension when d ≥ 5, such steps can continue, so we see that we can use a codimension-two surface term as the counterterm to cancel all the divergences in the action (2.11). As the counterterm is constructed in terms of the geometric quantities of codimension-two boundary surface, it’s invariant under the coordinate transformation. Anomaly counterterm in even bulk dimension For all the case that d is odd integer and larger than 1, the last divergent term in the action (2.11) is a logarithmic divergent term 5 , which needs special consideration. In the FG coordinate system, it has following form, Z √ d−1 √ d−1 ∗ σd x σÀdS FA,(d+1)/2 + O() . (2.40) − ln(/ÀdS ) B

∗ Here FA,(d+1)/2 = limd0 →d (d0 − d)FA,(d+1)/2 . Note the integration term in Eq. (2.40) is finite. There are two methods to cancel this anomaly divergent term. The first method is just to take the anomaly counterterm Ianomaly as this form,6 (I) Ianomaly

Z = ± ln δcut B

√

√ ∗ σdd−1 x σ`d−1 AdS FA,(d+1)/2 .

(2.41)

Here δcut is the dimensionless cut-off at any coordinate and we assume that the boundary locates at δcut = 0 or δcut → ∞. If the AdS boundary is at the δcut = 0, the sign in the outside of integration is “−”, otherwise we should use “+”. Though this method is enough to cancel the divergent term in all the coordinate systems, it leads the the finial result is coordinate dependence. To see this, let’s consider that the AdS boundary locates at 5

One should note that if we don’t add Iλ into the action (2.11), the additional logarithm divergent term will appear [17] in all the dimension cases. In general, it has following forms c p c2 1 Ilog,CA = ln( αβ/ÀdS ) d−1 + d−3 · · · . (2.39) (0)

Here coefficients c1 , c2 , · · · are determined by the conformal boundary metric g˜µν but α, β are arbitrary constants depending on the choice of null normal vectors in null surfaces. As the α and β can’t be determined by theory itself, such terms cannot be written as the covariant geometrical quantities of boundary metric. This results show that it is necessary to add the term Iλ into the action (2.11) to obtain an covariant regularized complexity. 6 We can multiple a constant factor into the logarithmic term. However, this constant term can be obsorbed into the definition of %1 in the Eq. (2.42).

– 12 –

δcut = 0. Assume in a coordinate system, the cut-off δcut has following relationship between it and the cut-off in FG coordinate system, δcut = `−1 AdS

∞ X

%n n ,

(2.42)

n=1

where %n can be function of other coordinates and %1 is positive which can only depend on time t. One can easy check this relationship leads the difference regularized action in these two coordinate is, Z √ d−1 √ d−1 ∗ ∆Ireg = ln %1 σd x σÀdS FA,(d+1)/2 . (2.43) B

Choosing different coordinate system may lead different value of %1 and so leads different value for regularized action. To fix this freedom, we need to impose a coordinate condition at the cut-off surface to fix %1 , i.e., appointing a value for %1 . As the integration term in Eq. (2.41) is finite and non-dynamic, the different choice on %1 just gives a gauge transformation Ireg → Ireg + Ireg,0 with a non-dynamic term Ireg,0 , which has no effect on the physical results. The other method is to use following expression as the anomaly counterterm Z Z √ d−1 √ d−1 √ d−1 ∗ 1 (II) 1−d Ianomaly = ln ςÀdS σd x σd x σÀdS FA,(d+1)/2 . (2.44) d−1 B B Here ς is any positive value which only depends in time t. By this anomaly counterterm, the logarithmic divergence can be canceled and the final result is independent of the choice of coordinate systems. However, there is still an additional parameter ς which needs to be appointed. As a result this method has a scalar freedom in the regularized complexity and so it in fact is equivalent to using (2.41) as the anomaly counterterm. Again, different choice on the parameter just leads to a gauge transformation and doesn’t give any difference on physics. After appointing the coordinate condition (appoint a value for %1 ) for method (I) in the Eq. (2.41) and fixing the parameter ς in method (II) in Eq. (2.44), the counterterms in these two methods may be different. However, it’s easy to find that such difference is also a pure gauge transformation and has no physical difference. Taking these into account, we can choose any of these two anomaly counterterms in even bulk dimensions case and fix %1 or ς to be any positive value. After we obtain the regularized form of action in WDW region, we propose definition named “regularized complexity” as follows Ireg 1 (I − Ict,L − Ict,R ) = . →0 π~ π~

CA,reg = lim

(2.45)

This regularized complexity is very similar to the regularization on on-shell action in the holographic renormalization for free energy. The absolute values both of free energy and complexity have no physical meanings. By this, we can use some subtractions to obtain a finite value. In the renormalization for free energy [30], we need that its form with the

– 13 –

counterterms will give the equations of motion, so the counterterms can only contain the intrinsic geometrical quantities. However, this restriction isn’t needed when we regularize the complexity. By the definition, the physical meaning of complexity describes the minimum number of quantum gates required to produce state |ψi from a particular reference state, so it doesn’t matter if we add any constant value in complexity for both them. For a system with many different states |ψI i, in order to define complexity for very state, we can appoint any non-dynamic quantum stats as the reference state. This means that CA conjecture in fact have a freedom to adding any non-dynamic term. The subtraction term Ict,R and Ict,R are defined by the boundary metric which needs to be fixed as the boundary condition, so it is a non-dynamic and covariant subtraction term. Then we can treat CA,reg is a well defined “regularized compelxity”. As the subtracted terms does not contain the bulk dynamic and matter fields information, we don’t lose any information compared with the full complexity. 2.2.2

Surface counterterms in CV conjecture

Similarly to the CA case, we can define the regularized complexity for the CV conjecture as 1 CV,reg = lim (V − Vct,L − Vct,R ) , (2.46) →0 ` where Vct,L and Vct,R are the surface counterterms, [d/2] Z

Vct =

X

n=1

B

√ dd−1 x σ`2n−1 AdS FV,n (Rµν , gµν , σij , Kij ) + Vanomaly ,

(2.47)

at the left boundary (Vct,L ) and right boundary (Vct,R ) respectively. V is the maximum value connecting tL and tR after we use finite cut-off z = to replace the real AdS boundary. Vanomaly is the anomaly counterterm which is nonzero only when the bulk dimension is even. We first consider the odd bulk dimensions. To find Vct or FV,n we first need to analyze the divergent structure of (1.2). The first two divergent terms in the volume (1.2) for d ≥ 2 were obtained in Ref. [18]: 1 (1) (2) Vdiv = V + V + O d−5 , (2.48) where V

(1)

`d = AdS d−1

Z

p dd−1 x σ ˜ (0) 1−d ,

(2.49)

B

and V (2)

`dAdS (d − 1) =− 2(d − 2)(d − 3)d−3

" # p (0) 2 ˜ R (d − 2) ˜ˆ (0) ˜ (0)2 . ˜ (0) R − − K dd−1 x σ 2 2 (d − 1) B

Z

With these two equations, the first volume divergent term reads, Z √ ÀdS (1) V = dd−1 x σ + O(3−d ) , d−1 B

– 14 –

(2.50)

(2.51)

so we have

1 . d−1 The subleading divergent term can be written as, Z 2 √ `3AdS ˆ − R − (d − 2) K 2 + O(5−d ) . dd−1 x σ R V (2) = − 2(d − 2)(d − 3) B 2 (d − 1)2 FV,1 =

(2.52)

(2.53)

As the same as the CA conjecture, the first surface counterterm has also contribution on the subleading divergence, Z `3AdS R d−1 √ ˆ− d σ R + O(5−d ) , 2(d − 2)(d − 1) B 2 which leads that total subleading divergent term reads, Z 2 `3AdS 2 (d − 2) d−1 √ 2 ˆ − R/2) − d x σ − (R K + O(5−d ) , 2(d − 2)(d − 3) B d−1 (d − 1)2

(2.54)

so we obtain, FV,2 = −

2 1 2 ˆ − R/2) − (d − 2) K 2 . (R 2(d − 2)(d − 3) d − 1 (d − 1)2

(2.55)

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 (1.2). When bulk dimension d + 1 is even, the logarithmic divergent term will appear, which is similar to the the case in CA conjecture. The logarithmic divergent term for this case in FG coordinate system reads, Z √ ∗ d dd−1 x σFV,(d+1)/2 Vlog = ln(/ÀdS )ÀdS + O() . (2.56) B

∗ Here FV,(d+1)/2 = limd0 →d (d0 − d)FA,(d+1)/2 . Note the integration in this equation is also finite. As the same as the Eqs. (2.41) and (2.44) in CA conjecture, we have two methods to cancel this anomaly divergent term, Z √ ∗ (I) d Vanonmaly = ±ÀdS ln δcut dd−1 x σFV,(d+1)/2 (2.57) B

or (II) Vanonmaly

1 =− ln(ς`1−d AdS d−1

Z

d−1

d

x

√

σ)`dAdS

B

Z B

√ ∗ dd−1 x σFV,(d+1)/2 .

(2.58)

Here δcut is the dimensionless cut-off at any coordinate and we assume that the boundary locates at δcut = 0 or δcut → ∞. If the AdS boundary is at the δcut = 0, the sign in the outside of integration is “+”, otherwise we should use “−”. Similarly, there is a scalar degree of freedom in choosing the coordinate condition %1 in Eq. (2.42) for the anomaly subtracted term (2.57) or choosing the additional parameter ς in Eq. (2.58). Different choices on these two anomaly counterterms or the values of %1 and ς only lead to a gauge transformation on regularized complexity and have no physical effect.

– 15 –

As a very trivial example, let’s try to compute the regularized complexity by CV conjecture for the example shown in section 2.1. The metrics of the boundary and t = 0 codimension-two surface are, gµν dxµ dxν = −r2 f (r)dt2 + r2 (dx2 + dy 2 ),

σij dxi dxj = r2 (dx2 + dy 2 ).

(2.59)

It is easy to see that the Ricci tensor is zero for the boundary at fixed r = ÀdS /δ and the extrinsic curvature is also zero for 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, Z √ Ω2 `3AdS ÀdS Vct,L = Vct,R = d2 x σ = . (2.60) 2 2δ 2 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 find the 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 {t, r0 , x, y}, where the relationship between r and r0 is given by Eq. (2.7). The surface counterterm at the cut-off δ 0 then is, Vct,L = Vct,R =

Ω2 `3AdS − Ω2 `3AdS F (M ) + O(δ 0 ). 2δ 02

(2.61)

We see that in this coordinate system, the counterterm isn’t proportional to the volume of pure AdS space-time, as its value depends on mass M . However, one can find that the regularized complexity is still as the same as Eq. (2.5), which is independent of the choice of F (M ). In the following sections, we will give more examples for computing the regularized complexity for the CV and CA conjectures. Here it’s worth to emphasizing that our surface counterterms are non-dynamic and have no relationship to the bulk matter field, so such subtraction keep all the information of bulk matter field in the complexity. In addition, if there is asymptotic time-like Killing vector field ξ = (∂/∂t)µ at the boundary, then we have, dCreg dC = . dt dt

(2.62)

This means the previous studies about complexity growth, in fact, indeed studied the behavior of regularized part of whole complexity. Assume γ to be any parameter in the system which has no effect on the boundary metric, and we can define a kind of “complexity of formation” between two different states labeled by γ = γ1 and γ = γ2 as, ∆C = Creg (γ1 ) − Creg (γ2 ) .

(2.63)

Specializing that γ stands for the temperature and setting γ1 = T, γ2 = 0, then (2.63) gives the complexity of formation studied in Ref. [19].

– 16 –

3

Examples for BTZ black holes

In this section, we will give examples to compute the regularized complexity for both CA and CV conjectures in BTZ black holes. One form of the metric for the rotational BTZ black hole is [31, 32], dr2 Jdt 2 2 2 2 2 ds = −r f (r)dt + 2 , (3.1) + r dϕ − 2 r f (r) 2r with r ∈ (0, ∞), ϕ ∈ [0, 2π] and the function f (r) described by f (r) =

1 `2AdS

−

M J2 1 + = 2 [1 − (r+ /r)2 ][1 − (r− /r)2 ] , 2 4 r 4r ÀdS

(3.2)

where M is mass parameter7 and J is the angular momentum: M=

2 + r2 r+ − , `2AdS

J=

2r− r+ . ÀdS

(3.3)

This black hole arises from identifications of points of anti-de 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. Though parameter M plays the role of mass, it’s 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 configuration 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 define the temperature T , entropy S and angular momentum Ω as, T = 3.1

2 − r2 r+ − , 2πr+

S=

πr+ , 2

Ω=

r− . r+ ÀdS

(3.4)

CA conjecture in non-rotational case

We first consider the case that J = 0. Let’s consider the case M > 0. In this case, the √ WDW patch is shown in the left panel of Fig. 3. We define rh = r+ = ÀdS M . For the special case that tL = tR = 0, the null sheets coming from left boundary and right boundary just meet with each other at r = 0. In order to compute the regularized complexity in CA conjecture, we first need to regularize the WDW patch, which is shown in the left panel of Fig. 3. Note that this approach is different from the approach in Ref. [19]. Taking the symmetry into account, we only need to compute the bulk term, the boundary terms and joints at the green region. Let’s introduce the outgoing and infalling null coordiantes u, v defined by8 Z `2AdS r − rh ∗ ∗ ∗ 2 −1 ln + v0 . (3.5) u = t − r , v = t + r (r), with r (r) = [r f (r)] dr = 2rh r + rh 7

The physical mass for BTZ black hole is 8M . Strictly speaking, this relationship for r∗ and r can only be used when rh > 0. However, we can see that it has a well defined limit when rh → 0+ . So the case that when M = 0 can be regarded as the limit of rh → 0+ . 8

– 17 –

t = π/2

r=0

r = rm rh

r = rm

tL

tR

r = rm

r=0

tL

r = rm

tR

t = −π/2

r=0

Figure 3. Penrose diagram and regularized WDW patch for tR = tL = 0 in BTZ black holes when J = 0. The case for M ≥ 0 is shown in the left panel, where the null sheets coming from tR and tL meet with 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 = ±π/2.

Here v0 is integration constant. The null boundaries at the green region in left panel of Fig. 3 is given by v = v(0, rm ) = r∗ (rm ) = vm and u = u(0, rm ) = −r∗ (rm ) = um . One can check that the dual normal vectors for such null boundaries are kI = α[(dt)I + r−2 f −1 (dr)I ] and k¯I = β[(dt)I − r−2 f −1 (dr)I ]. Here we explicitly exhibit the freedom of choosing dual normal vector by two arbitrary constants α and β. In the green region of Fig. 3, there are bulk integration term, two null boundary terms, a null-null joint term in the action. Using the method similar to Ref. [19], the bulk action is expressed as (have taken the exponents 2 into account), Z rm 2 Ibulk = − 2 [vm − r∗ (r)]rdr = −rm + O(1/rm ). (3.6) ÀdS 0 We see that it’s different from the result in Ref. [19]. It’s no surprising because we use different regularization method. As the measurement of null-null joints at the corners r = 0 is zero, such joint term has no contribution to the action. The joint term at the boundary is given by following expression, 2 1 1 r f (r) I¯ Ijoint = − r ln(|k kI |/2) = r ln . (3.7) 2 2 αβ r=rm r=rm Since kI is affinely parameterized, only the null boundary term shown in Eq. (2.13) has contribution. The expansions of kI and k¯I are, Θ = g IJ ∇I kJ =

β α ¯ , Θ = g IJ ∇I k¯J = − . r r

(3.8)

¯ for k I and In order to compute the value of Iλ , we need to find the affine parameter λ and λ I k¯ , respectively. On the null boundary of green region shown in the Fig. (3), the coordinates t and r are both the function of λ, i.e., t = t(λ) and r = r(λ). By the equation, I ∂ dt ∂ I dr ∂ I I k = = + , ∂λ dλ ∂t dλ ∂r

– 18 –

(3.9)

we see that λ = r/α for k I . Similarly, we find that λ = −r/β for k¯I . So we obtain that, Iλ = −2·

1 8π

2π

Z

Z

0

dϕ 0

rm

α−1 drr

α ln r

αβ`2AdS rm ÀdS α +(α → β) = +rm . (3.10) ln 2 r 2 rm

For the case d = 2, only first order counterterm needs to be computed. However, we see this term is zero coincidentally. So the surface counterterm is zero. Then we finally reach, Ireg (M ≥ 0) = Ibulk + Ijoint + Iλ = 0 ⇒ CA,reg (M ≥ 0) = 0.

(3.11)

We see that regularized complexity is the same for all the case of M ≥ 0. We also see that the regularized complexity is independent of the choice of α and β, which is expected as α and β are gauge degree of freedom in the choices of dual normal vector for null surface. Note that UV divergent behavior shown in Ref. [17] depends on these two gauge parameters. However, in our formula, as the additional term Iλ has been added into the action (2.11), the final result is independent of the gauge choices on the null normal vector fields. When M = −1, the expression of r∗ in the Eq. (3.5) should be replaced by following equation, r∗ = −ÀdS arctan(ÀdS /r) + v0 . (3.12) In this case, we see that the null sheets coming from the r = ∞, t = 0 will meet each other at the position of r = 0 and t = ∓[r∗ (0) − v0 ] = ±π/2 respectively(see the right panel of Fig. 3). The computation about regularized complexity is very similar to the case that M > 0. Eq. (3.6) can still be used to compute the bulk term, but the result now becomes, Ibulk = −rm −

πÀdS + O(1/rm ). 2

(3.13)

The joint term at r = rm and the null boundary term have the same expressions shown in Eq. (3.7) and (3.10). Then still obtain the result that, Ireg (M = −1) = Ibulk + Ijoint + Iλ =

πÀdS ÀdS ⇒ CA,reg (M = −1) = . 2 2~

(3.14)

As the solution with M = −1 has lower energy, it’s better to treat this solution is the vacuum solution rather than the case with the limit M → 0. Using Eqs. (3.11) and (3.21), we can also obtain the complexity of formation shown in the Ref. [19]. 3.2

CA conjecture in rotational case

For the case that J 6= 0, the mass M must be non-negative 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 Fig. 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, Z `2AdS dr r + r− r + r+ ∗ r = = . (3.15) 2 − r 2 ) r− ln |r − r | − r+ ln |r − r | r2 f (r) 2(r+ − + −

– 19 –

V

U r = r0

r−

r+

r = rm

r = rm

tL

r=∞

rm

r=

r=

rm

r−

tR

r+

r=∞

r−

r = r0

r−

Figure 4. WDW patch in Penrose diagram (left panel) and Kruskal-type coordinate for rotational BTZ black hole. The null sheets coming from tR = tL = 0 meet with each other at the surface r = r0 ∈ (r− , r+ ).

In the region r ∈ (r− , ∞), the function r∗ (r) has two monotonic regions, i.e., (r+ , ∞) and (r− , r+ ). From Eq. (3.15), we find that r∗ (∞) = 0, r∗ (r+ ) = −∞ and r∗ (r− ) = ∞. In this case, the null sheets coming from the tL and tR meet with each other at the inner region of event horizon with finite 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 following transcendental equation, r0 + r− r0 + r+ r− ln − r+ ln = 0. (3.16) r0 − r− r+ − r0 Now the computation for 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 integration inferior limit now is r0 , i.e., Z rm 2 Ibulk = − 2 [vm − r∗ (r)]rdr = −rm + I0 (r0 ) + O(1/rm ), (3.17) ÀdS r0 where I0 is defined by r+ I0 (r0 ) = r0 − ln 2

r0 + r+ r+ − r0

.

The null boundary term shown in Eq. (3.10) now reads, 2 2 ÀdS ` rm r0 ln − ln AdS + rm − r0 . Iλ = 2 2 rm 2 r02

(3.18)

(3.19)

As the final result is independent on the choice of α, β, we have fixed α = β = 1. The contribution of joint terms at the boundary r = rm have the same formula compared with J = 0 but we need to add the joint terms at r = r0 as they have nonzero values, 1 1 2 Ijoint = rm ln[rm f (rm )] − r0 ln[−r02 f (r0 )]. 2 2

– 20 –

(3.20)

As the surface counterterms are still zero, combining the results in Eqs. (3.17), (3.19) and (3.20), we can find that, r+ r0 r0 + r+ Ireg = Ibulk + Ijoint + Iλ = − ln − ln[−`2AdS f (r0 )] 2 r+ − r0 2 (3.21) r r + r+ r0 + 0 2 2 2 2 4 . ln[(r+ − r0 )(r0 − r− )/r0 ] − ln = 2 2 r+ − r0 This result recover the Eq. (3.21) when r− → 0. For general case that 0 < r− < r+ , we need to solve the transcendental equation (3.16). We see that Ireg /r+ only depends on the ˆ AdS ) by introducing an auxiliary value of r− /r+ = ΩÀdS , i.e., we can write Ireg /r+ = I(Ω` ˆ dimensionless function I(x) . More detailed, we have, Ireg (T, Ω) =

ˆ 2πT ˆ AdS ) = 2I(ΩÀdS ) S. I(Ω` 2 π 1 − Ω2 ÀdS

(3.22)

ˆ AdS ) can only be determined numerically. For the case that ΩÀdS 1 The value of I(Ω` ˆ AdS ) can be expressed as, or 1 − ΩÀdS 1, we can find the leading term of I(Ω` ˆ AdS ) = − 1 ln(1 − ΩÀdS ) + · · · I(Ω` 2

ˆ AdS ) = −c0 ΩÀdS ln(ΩÀdS ) + · · · , I(Ω`

(3.23)

ˆ AdS ) is less than zero for small ΩÀdS but larger than with c0 ≈ 1.19967 · · · . Note that I(Ω` zero for large ΩÀdS . 3.3

CV conjecture in BTZ black hole

Now let’s calculate the regularized complexity for CV conjecture in BTZ black hole. For simplicity, we consider the complexity of a thermal state defined on the time slice tR = tL = 0. The maximal volume is just like Eq. (2.3),

Z

rm

V = 4π r+

4π`2AdS p dr = + 4π δ f (r) 1

Z

rm

r+

1

p − ÀdS f (r)

! dr − 4πÀdS r+ .

(3.24)

Here we introduce a cut-off at the boundary by r = rm = ÀdS /δ. In this case, we only need one surface term Eq. (2.51), (1)

√ 2π`2AdS dϕ σ = . δ B

Z

Vct = ÀdS

Then the regularized complexity Eq. (2.46) can be written as ! Z ∞ 1 1 (1) −1 p CV,reg = lim (V − 2Vct ) = 4π` − ÀdS dr − 4πr+ ÀdS `−1 . δ→0 ` f (r) r+

(3.25)

(3.26)

For J = 0 and M ≥ 0, we have CV,reg = 0.

– 21 –

(3.27)

When J = 0, the lowest energy state is M = −1. In this case, there is no horizon, the regularized complexity for this solution is, CV,reg,vac = 4π`−1

∞

Z

1 p

0

f (r)

! − ÀdS

dr = −4π`2AdS `−1 .

(3.28)

It’s easy to find that the Eq. (3.27) and Eq. (3.28) give the same complexity of formation shown in Ref. [19]. For J 6= 0, the regularized complexity Eq. (3.26) yields, Z

` ÀdS

∞

CV,reg = 4π r+



r2



q − 1 dr − 4πr+ . 2 )(r 2 − r 2 ) (r2 − r+ −

(3.29)

Similarly, for the case 0 < r− < r+ , by introducing Cˆ = ``−1 AdS CV,reg /r+ , we have, ` ÀdS

CV,reg (T, Ω) =

ˆ AdS ) 2πT `2AdS ˆ 2C(Ω` C(ΩÀdS ) = S. 2 2 π 1 − Ω ÀdS

(3.30)

ˆ AdS ) can only be determined numerically. For the case that ΩÀdS 1, The value of C(Ω` ˆ AdS ) can be expressed as, we can find the leading term of C(Ω` ˆ AdS ) = π 2 (ΩÀdS )2 + · · · . C(Ω`

(3.31)

ˆ AdS ) at the low temperature limit, i.e., Now let’s consider the leading behavior of C(Ω` ΩÀdS → 1. We define x = r/r+ and x− = r− /r+ = ΩÀdS , then the volume integral ˆ AdS ) can be written as C(Ω`   2 ˆ AdS ) Z ∞ C(Ω` x q = − 1 dx − 1 4π 1 (x2 − 1)(x2 − x2− ) ! Z ∞ 1 p − 1 dx − 1 = (x − 1)P (x, x− ) 1 ! Z a Z ∞ 1 p = dx + dx −1 −1 (x − 1)P (x, x− ) 1 a Z a dx p + finite term = (x − 1)P (x, x− ) 1

(3.32)

where 0 < x− < 1 and a is any constant with a > 1. Then we can write P (x, x− ) as the following form, (x − 1)P (x, x− ) ≡ (x − 1)(x − x− )(x + 1)(x + x− )/x4 ≡ (x − 1)(x − x− )h(x, x− ).

– 22 –

(3.33)

by introducing a function h(x, x− ). Now we take h0 = h(1, x− ) = 2(1 + x− ) and introduce an integration, Z a dx p H(x− ) ≡ (x − 1)(x − x− )h0 1 √ √ 2 (3.34) = √ ln( x − 1 + x − x− )|a1 h0 1 = − √ ln(1 − x− ) + finite term. h0 On the other hand, we have ˆ AdS ) C(Ω` − H(x− ) 4π ! Z a dx dx p = −p + finite term (x − 1)(x − x− )h(x, x− ) (x − 1)(x − x− )h(1, x− ) 1 Z a h0 − h(x, x− ) q = dx + finite term p √ 1 (x − 1)(x − x− )h(x, x− )h0 ( h0 + h(x, x− ))

(3.35)

=finite value. So for the limit ΩÀdS → 1, we find that, ˆ AdS ) √ C(Ω` 1 = − ln(1 − ΩÀdS ) + ln(2 2) − 1 + · · · . 4π 2

(3.36)

The interesting thing is that this result shows the leading terms of regularized complexity for the case r− → r+ have the similar behavior in CA and CV conjectures and they are exactly the same if we choose ` = 4π 2 ~ÀdS .

4

Examples for Schwarzschild AdSd+1 black holes

A general Schwarzschild AdSd+1 (d ≥ 3) black hole is given by following metric, ds2 = −r2 f (r)dt2 + with f (r) =

dr2 + r2 dΣ2d−1,k r2 f (r)

k 1 ωd−2 + 2 − d 2 r r ÀdS

(4.1)

(4.2)

Here ωd−2 is mass parameter and k = {1, 0, −1}, which describes the different topology of horizon and (d-1)-dimensional line element dΣ2d−1,k is given by,   dθ2 + sin2 θdΩ2d−2 , k = 1;     d−1 X 2 dΣd−1,k = (4.3) dx2i , k = 0;   i=1     2 dθ + sinh2 θdΩ2d−2 , k = −1.

– 23 –

r=0 r=ε

tL

r = rm

r = rm

tR

rh

r=0

Figure 5. Penrose diagram and regularized approach of WDW patch for tR = tL = 0 in Schwarzschild AdS black holes. The null boundaries of WDW patch come from the finite cutoff boundary and there is a null-null joint at the cut-off r = rm . In addition, in order to regularize the singularity, we need to use an additional cut-off at r = ε → 0, so there are also some new joints and space-like boundaries.

Here Ω2d−2 is line element of d − 2 dimensional unit sphere. For convention, we will define R Σd−1,k := dΣd−1,k so that r2 Σd−1,k is the volume of the surface by fixed t and r. We assume the horizon locates at r = rh . For simplicity, we still consider the case tR = tL = 0 and try to find the regularized complexity in both of CA and CV conjectures. In this paper, we will only focus on the cases of d = 3, 4. 4.1

Regularized complexity in CA conjecture

Case of d = 3 In order to compute the regularized complexity in CA conjecture, let’s first introduce outgoing and infalling null coordiantes u, v defined by the same manner shown in Eq. (3.5), but the function r∗ now should be changed as,     2 rh ` |r − rh | + 2v∞ arctan  q 2r + rh  , (4.4) r∗ = 2 AdS2 ln  q π 3rh + kÀdS 2 2 2 2 2 r + rr + r + k` 3r + 4k` h

with v∞ =

h

AdS

π`2AdS (3rh2 + 2k`2AdS ) q . 2 2 2(3rh + kÀdS ) 3rh2 + 4k`2AdS

h

AdS

(4.5)

√ In order not to make the computation too complicated, we assume first rh > 2ÀdS / 3 when k < 0. Similar to the case in BTZ black hole, there is a null-null joint at the cut-off r = rm . When rm = ∞, 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 cut-off at r = ε → 0, so there are also some new joints and space-like boundaries.

– 24 –

The null boundaries at the green region in the Fig. 5 is given by v = v(0, rm ) = = vm and u = u(0, rm ) = −r∗ (rm ) = um . The dual normal vectors for such null boundaries are still given by kI = α[(dt)I + r−2 f −1 (dr)I ] and k¯I = β[(dt)I − r−2 f −1 (dr)I ]. Here we still explicitly exhibit the freedom of choosing dual normal vector by two arbitrary constant α and β. The bulk action is expressed as, Z rm 3Σ2,k Ibulk = − [vm − r∗ (r)]r2 dr 2π`2AdS 0 (4.6) Σ2,k 2 kΣ2,k `2AdS =− r + ln(rm /ÀdS ) + I0 + O(ÀdS /rm ) . 4π m 2π

r∗ (rm )

Here I0 is the finite term, which reads, 2 2 Σ2,k d k 2 `4AdS + 3k`2AdS rh2 + rh4 rh4 k ÀdS + rh2 rh ln − ln I0 = − 2π ÀdS 6(k`2AdS + 3rh2 ) `2AdS 3(k`2AdS + 3rh2 )     rh (k 2 `4AdS + 5k`2AdS rh2 + 3rh4 )  π rh    q + − arctan q .  3(k`2 + 3r2 ) 4k`2 + 3r2 2 4k`2 + 3r2 AdS

h

AdS

AdS

h

h

(4.7) We see that a logarithm term appears in Eq. (4.6). As null-spacelike joints at the corners r = 0 have no contributions on the action [19]. The joint term at the infinite boundary for general d is given by following expression, 2 Σd−1,k d−1 Σd−1,k d−1 rm f (rm ) µ¯ r ln(|k kµ |/2) = r ln Ijoint = 4π 4π m αβ r=rm 2 2 Σd−1,k d−1 Σd−1,k d−1 kÀdS ωd−2 rm (4.8) = r ln 2 + r ln 1 + 2 − d 4π m 4π m rm r ÀdS αβ 2 Σd−1,k d−1 Σd−1,k d−1 k`2AdS k 2 `4AdS rm = r ln 2 + r − + ··· . 2 4 4π m 4π m rm 2rm ÀdS αβ One can check that kI is still affine parameterized, so we still find that only the null boundary term shown in Eq. (2.13) has contribution. The expansions of kµ and k¯I in general d are, Θ = g IJ ∇I kJ =

(d − 1)α ¯ (d − 1)β , Θ = g IJ ∇I k¯J = − . r r

By the similar method in Eq. (3.10), we find the null boundary term Iλ is, Σd−1,k 2 2 2 d−1 Iλ = [2 ln(d − 1) + ln(αβÀdS /rm ) + ]r . 4π d−1 m

(4.9)

(4.10)

Then important difference between Schwarzschild black hole and BTZ black hole is that there is a space-like curvature singularity at r = 0. We need to make a cut-off at r = 0 so that the computation can’t touch the singularity. As a result, there are two space-like

– 25 –

surface terms at r = ε → 0. The contribution of such terms on the action can be given the similar method shown Ref. [19]9 . For the case, d = 3, it is, 2 2 3Σ2,k 3Σ2,k rh2 (k`2AdS + rh2 ) k ÀdS + rh2 ∗ IGHY = ω1 (vm − r (0)) = ln 4π 4π 2(k`2AdS + 3rh2 ) rh2  s (4.11)  4k`2AdS (k`2AdS + rh2 )(2k`2AdS + 3rh2 )rh q + O(` /r ). + 3 + arctan AdS m  rh2 (k`2 + 3r2 ) 4k`2 + 3r2 AdS

AdS

h

h

For the case that d > 2, the surface counterterm is nonzero. We see FA,1 = ln(d − 1)/(4π). ˆ = k(d−1)(d−2)/r2 . Specializing that d = 3, there It is easy to see that Kij = 0 and R = R is a logarithm counterterm in the subleading counterterm. We use method (I) to obtain the subleading and fix the coordinate condition by setting %1 = 1, which gives, (I)

Ianomaly =

kΣ2,k `2AdS ln(rm /ÀdS ) . 4π

(4.12)

Then the surface counterterm for d = 3 reads, Ict,L = Ict,R =

Σ2,k ln 2 2 kΣ2,k `2AdS rm + ln(rm /ÀdS ) . 2π 4π

(4.13)

Finally, we obtain the regularized complexity for d = 3, 1 lim (Ibulk + IGHY + Ijoint + Iλ − Ict,L − Ict,R ) π~ rm →∞ Σ2,k rh4 − 2k 2 `4AdS − 3k`2AdS rh2 rh2 rh2 (rh2 + 3k`2AdS ) rh = 2 ln k + 2 − ln 4π ~ ÀdS 2(k`2AdS + 3rh2 ) ÀdS (k`2AdS + 3rh2 )  s  4k`2AdS rh (4k 2 `4AdS + 5k`2AdS rh2 + 3rh4 ) q arctan + + 3  rh2 (k`2 + 3r2 ) 4k`2 + 3r2

Creg =

AdS

+

Σ2,k k`2AdS 4π 2 ~

h

AdS

h

. (4.14)

As we expected, all the divergent terms have disappeared and the result is independent of the values of α and β when we choose the null normal vectors for the null boundaries. √ Though Eq. (4.14) is obtained by the assumption rh > 2ÀdS / 3 when k = −1, we √ make an analytical extension to get the regularized complexity when ÀdS < rh < 2ÀdS / 3 by following analytical extension, s s q q 2 4` 4`2AdS 3rh2 − 4`2AdS = i 4`2AdS − 3rh2 , arctan 3 − AdS = iarctanh − 3 . (4.15) rh2 rh2 On the other hand, by that following identity for arctanh function and logarithm function when x > 1, 1 arctanh(x) = [ln(1 + x) − ln(1 − x)] , (4.16) 2 9

However, there still a little difference between our result and the result in Ref. [19]. In Ref. [19], the null sheets come from the boundary r = ∞. Here the null sheets come from the cut-off surface r = rm

– 26 –

one can check that the Eq. (4.14) has well defined limit at rh = ÀdS and is analytical in the neighbourhood of rh = ÀdS + 0+ . So the Eq. (4.14) can extend into the whole region of rh ≥ ÀdS when k = −1. By this analytical extension, it’s easy to find that the vacuum regularized complexity for k = 0, 1(rh = 0) and k = −1(rh = ÀdS ) is, CA,reg,vac =

Σ2,k k`2AdS . 4π 2 ~

(4.17)

One can check that the difference of regularized complexity between black hole(brane) and vacuum AdS space-time, i.e., the complexity of formation, are the same results given by √ Ref. [19]. We note that the complexity of formation for k = −1 and ÀdS < rh < 2ÀdS / 3 has also been given by Ref. [19] in a very implicit manner. In fact, one can prove that it’s just as the same as the Eq. (4.14) in the sense of analytical extension shown in (4.15). One can check, if we choose method (II) as the counterterm or different value for %1 , the value of Creg and Creg,vac may be different, but the complexity of formation is the same. √ When ÀdS / 3 < rh < ÀdS 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 different from what we have shown in the Fig. 5. In principle, we need an additional computation for this case. We leave this case in future works. Case of d = 4 When d = 4, we see that the logarithm term will not appear but the subleading counterterm appears. By the Eq. (2.38), the total surface counterterm reads, Ict,L = Ict,R =

Σ3,k 3 3 (ln(3)rm + krm `2AdS ) . 4π 2

(4.18)

The bulk then can be computed the same method shown in Eq. (3.6), the result then is, Z rm Σ3,k 3 kΣ3,k `2AdS 4Σ3,k ∗ 2 Ibulk = − [v − r (r)]r dr = − r + rm + I0 + O(ÀdS /rm ), m 6π m 2π 2π`2AdS 0 (4.19) where, Σ3,k (k`2AdS + rh2 )5/2 I0 = − . (4.20) 4 k`2AdS + 2rh2 And the contribution of boundary terms coming from the singularity is, IGHY =

Σ3,k rh2 (rh2 + k`2AdS )3/2 . 2 2r2 + k`2AdS

(4.21)

The joint terms and null boundary terms can still be obtained by Eq. (4.8) and (4.10) with d = 4. Then we find that all the divergent terms still can cancel with each other and we obtain a finite regularized complexity, CA,reg =

Σ3,k (k`2AdS + rh2 )3/2 (rh2 − k`2AdS ) . 4π~ k`2AdS + 2rh2

– 27 –

(4.22)

By this result, we can obtain the vacuum regularized complexity, Σ3,k 3 CA,reg,vac = − ` δk,1 . (4.23) 4π~ AdS By the Eqs. (4.22) and (4.23), it’s easy to reproduce the result about complexity of formation shown in the Ref. [19]. Similarly, the Eq. (4.22) is valid when rh ≥ ÀdS in hyperbolic black holes. The case of small black hole needs another computation. 4.2

Regularized complexity in CV conjecture

Now let’s calculate the regularized complexity for CV conjecture. For the case that tL = tR = 0, the maximal valume surface bounded by codimension-two surface tL and tR is just the time slice of t = 0. Then volume of this codimension-one surface can be obtained by following integration, Z ∞ d−2 r p dr V = 2Σd−1,k f (r) rh −1/2 Z ∞ rhd k 1 1 k d−2 (4.24) + − + 2 dr = 2Σd−1,k r r2 `2AdS rd rh2 ÀdS rh Z ∞ k`2AdS d−4 d−2 = 2Σd−1,k ÀdS r − r + · · · dr. 2 rh We will give the regularized complexity in different dimension and k. Planar Geometry In this case, k = 0 and the boundary is just a flat space-time, which leads that Rµν = Kij = 0 at the boundary. We can obtain the results for general dimension. The volume of maximal surface under the cut-off rm = ÀdS /δ is, Z rm rhd d−2 r − 2 + · · · dr V = 2Σd−1,0 ÀdS 2r rh     Z rm d−1 d−1 d−1 r ` r AdS q = 2Σd−1,0 ÀdS  − rd−2  dr − h  + (4.25) d d−1 (d − 1)δ d−1 r rh h r2 − rd−2 ! √ 1 π(d − 2)Γ(1 + ) `d−1 d−1 AdS d + r . = 2Σd−1,0 ÀdS (d − 1)δ d−1 2(d − 1)Γ( 12 + d1 ) h The surface counterterms are, (1) Vct

ÀdS = d−1

(n) Vct

= 0,

√ Σd−2,k `dAdS dxd−2 σ = , (d − 1)δ d−1 B

Z

(4.26)

n > 1.

Then the regularized complexity can be written as ` ÀdS

(1)

CV,reg = `−1 lim (V − 2Vct ) AdS δ→0 √ Σd−1,0 π(d − 2)Γ(1 + d1 ) d−1 = rh . (d − 1)Γ( 12 + d1 )

– 28 –

(4.27)

Spherical and Hyperbolic Geometries for d = 3 In this case, we have Z V =2Σ2,k ÀdS

rm

k`2AdS r− + · · · dr 2r

rh `2AdS 2δ 2

r2 k`2 k`2 =2Σ2,k ÀdS + AdS lnδ − h + AdS ln(rh /ÀdS ) 2 2 2    Z rm 2 2 k` r q − r + AdS  dr . + r 2r 2 2 2 rh kÀdS + r2 − rh (kÀdS + rh )

(4.28)

Now we need the first order surface counterterm and the subleading logarithmic counterterm, which reads, Z √ Σ2,k `3AdS ÀdS (1) Vct = dx2 σ = 2 2δ 2 B (4.29) kΣ2,k `3AdS (I) Vanomaly = lnδ . 2 Here we still use the method (I) as the logarithmic counterterm and set %1 = 1 to fix the coordinate gauge freedom. Then the regularized complexity can be written as 1 (1) (I) CV,reg = lim (V − 2Vct − 2Vanomaly ) δ→0 `     Z ∞ 2 2 x x k  k q −x+ = 2Σ2,k `3AdS `−1  dx − h + lnxh  . x 2x 2 2 xh k + x2 − xh (k + x2h ) (4.30) Here we define x = r/ÀdS and xh = rh /ÀdS . All the divergent terms have disappeared, as expected. It is straightforward to find that the vacuum regularized complexity is 1 − ln2 . (4.31) CV,reg,vac = k`−1 Σ2,k `3AdS 2 Spherical and Hyperbolic Geometries for d = 4 In this case, we have Z ∞ k`2AdS 2 V = 2Σ3,k ÀdS r − + · · · dr 2 rh   Z ∞ 1 k x3 q = 2Σ3,k `4AdS  3 − + − x2 + 3δ 2δ x2h xh 2 2 k + x − x2 (k + xh )



x3h



k kxh  dx − + . 2 3 2 (4.32)

Similarly, we define x = r/ÀdS and xh = rh /ÀdS . In this case we need the first and second surface counterterms, which are, Z √ Σ3,k `4AdS ÀdS (1) Vct = dx3 σ = 3 3δ 3 B (4.33) Z kΣ3,k `4AdS `3AdS 2 ˆ 4 2 (2) 3√ Vct = − dx σ (R − R/2) − K = − . 4 3 9 2δ B

– 29 –

Then the regularized complexity can be written as 1 (1) (2) CV,reg = lim (V − 2Vct − 2Vct ) δ→0 `   Z ∞ r3 q − x2 + = 2Σ3,k `4AdS `−1  2 x xh k + x2 − xh2 (k + x2h )

  x3h kxh  k dx − + . 2 3 2 (4.34)

All the divergent terms have disappeared, as expected. It is straightforward to find that the vacuum regularized complexity is   4 Σ `4 , k = 1, 3,k AdS CV,reg,vac = 3` (4.35)  0, k = −1. One can check that these regularized complexity can give the same complexity of formation shown in the Ref. [19].

5

Summary

In this paper, we studied how to obtain the finite term in a covariant manner from the holographic complexity for both CV and CA conjectures. 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 fields, we showed that such divergences can be canceled by adding codimension-two boundary counterterms. These boundary surface counterterms do not contain any boundary stress tensor information so they are non-dynamic background and can be subtracted from the complexity without any physical effects. If bulk dimension is even, a logarithmic divergence appears. We proposed two methods to cancel this anomaly divergent term. Both methods yield gauge transformations of the regularized complexity with free parameters so do not lead any difference in physics. In the CA conjecture, with the modified boundary term proposed by Ref. [18] different from the framework in the Ref. [17, 19], our regularized complexity is also independent on the choice of the normalization 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 fields in the bulk, 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 non-rotational and rotational BTZ black holes and the Schwarzschild AdS black holes in four and five dimensions with different horizon topologies. These examples also explicitly show that the different methods to choose the anomaly counterterm are linked by gauge transformations and give the same physical results. 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

– 30 –

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. [19]. Using this regularized complexity, we can study the effects of bulk matter fields and thermodynamic conditions on the holographic complexity at a fixed dual boundary (the codimension-one 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 [33]. We also can directly compute the complexity at different 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, to obtain the whole growth rate.

A

Subleading divergent terms in CA conjecture

In this appendix, we will give the details about how to obtain the contribution of the null surface term coming from the nonzero κ in the action (2.11). It seems that Refs. [17, 18] neglected an O(z 3 ) order contribution from the null boundary contribution, so our result is slightly different from theirs. Based on Ref. [16], the null boundary term in the action can be written as, Z √ 1 IN = − σκdλd2 x . (A.1) 8π N Here k I = (∂/∂λ)I is the normal vector of the null surface, λ is the parameter of integral curve of k I . Following the Ref. [17], we assume kµ has following form near the boundary, kI = α(−dz + nµ dxµ ) .

(A.2)

Here α is a constant but nµ is the function of z and xµ . Using the metric shown in Eq. (2.15), we can find that, αz 2 k I = 2 (−∂z + nµ ∂µ ) , (A.3) ÀdS where nµ = g µν nν . The Eq. (A.2) isn’t an additional assumption, we can always write the normal vector for null surface into this form in FG coordinate system. The null condition kI k I = 0 shows that nµ must be a normalized unit time-like vector, i.e., nI nI = −1 . Now let’s find the surface gravity κ for this null normal vector. k µ ∇µ k ν = κk ν , we have following equation, dk I + ΓI JK k J k K = κk I . dλ

(A.4) Using the equation

(A.5)

We will solve this equation by the order of z. One can see that under the gauge about the boundary metric, the value of nµ must have following form, nµ = −δtµ + nµ(1) z 2 + nµ(2) z 4 + · · ·

– 31 –

(A.6)

and κ has following series expansion with respective to z, κ = κ(0) + κ(1) z 3 + · · · .

(A.7)

Here the coefficients {nµ(1) , nµ(2) , · · · } and {κ(0) , κ(1) , · · · } are only the function of xµ . As the Eq. (A.2) shown that on the integral curve of k I , we have dz/dλ = k z and µ dx /dλ = k µ , we can use following replacement when we compute the integration (A.1), `2 ∂ d αz 2 ∂ − dz . (A.8) = 2 + nµ µ , dλ = − AdS dλ ∂z ∂x αz 2 ÀdS This give following results, dk z 2α2 z 3 , = 4 dλ ÀdS dk t 2α2 z 3 4α2 nt(1) 5 = 4 − z + O(z 7 ), dλ ÀdS `4AdS Z √ κ `2 σ 2 dzd2 x . IN = AdS 8πα N z

(A.9)

To finish the Eq. (A.5), we need to compute the connection. Using the formula for ΓI JL , we find the relevant components of connection are, 1 1 Γz zz = − , Γz µz = Γz zµ = 0, Γz tt = − + O(z 3 ), z z 1 (1) Γt tt = O(z 2 ), Γt tz = − − z˜ gtt + O(z 3 ), Γt zz = 0 . z

(A.10)

Up to the order of z 5 , Eq. (A.5) gives following relevant equations, z: t:

αz 2 (0) 2α2 nt(1) 5 z = − [κ + z 3 κ(1) ], `4AdS `2AdS (1) −2(˜ gtt

+n

t(1)

αz 2 α2 ) 4 z 5 = 2 [κ(0) (nt(1) z 2 − 1) − z 3 κ(1) ] . ÀdS ÀdS

(A.11)

These give, (1)

κ(0) = 0,

κ(1) =

α˜ gtt , 2 ÀdS

(1)

nt(1) = −

g˜tt . 2

Take these into the Eq. (A.1), we obtain, Z Z √ (1) 1 d−1 IN = d x σ˜ gtt zdz 8π Z √ (1) Z 2−d 1 = dd−1 x σ ˜ g˜tt z dz + O(d−5 ) 8π Z p d−3 (1) = dd−1 x σ ˜ (0) g˜tt + O(d−5 ) . 8π(d − 3) z=

(A.12)

(A.13)

Now use the equation, (1) g˜tt

1 2d − 3 ˜ (0) ˜ˆ (0) =− R − R , d−2 2(d − 1)

– 32 –

(A.14)

we find the null boundary term contributes the subleading divergence, Z p 2d − 3 ˜ (0) d−3 ˜ˆ (0) 2 (0) d x σ R − R IN = − ˜ 8π(d − 3)(d − 2) B 2(d − 1) Z 1 2d − 3 2 √ ˆ =− d x σ R− R + O(d−5 ) . 8π(d − 3)(d − 2) B 2(d − 1)

(A.15)

With this addition term, the subleading divergent term for CA conjecture in Ref. [17] should be modified as,   Z ˜ˆ (0) ˜ (0)2 + 4K ˜ (0) K ˜ (0)ij − (3d + 1)R ˜ (0) + 2(d + 1)R p 4 K `d−1 ij  (A.16) dd−1 x σ ˜ (0)  − AdS 16π 2 ~ B d−3 (d − 1)(d − 2)(d − 3) and the first two orders of divergent terms in Ref. [18] should be modified as, ! " ˜ˆ (0) ˜ (0) p 2 (R − R /2) d x σ CA,div ˜ (0) ln(d − 1) 1 − 2(d − 2) B  ˜ˆ (0) ˜ (0)2 + 2(d − 1)K ˜ (0) K ˜ (0)ij − 3(d − 1)R ˜ (0) + 2(d − 1)R d K ij  + O(5−d ) . − 2 2(d − 1)(d − 2)(d − 3) `d−1 = 2AdSd−1 4π ~

Z

d−1

(A.17) The result (A.17) can also been obtained by a different approach, in which we insist the k I is affinely parameterized, i.e., κ = 0. We still can write k I into the form as shown in the Eq. (A.2), however, we can’t demand that α is a constant. Instead, we assume α has following series expansion with respective to z, α = α(0) + α(1) z 2 + · · · .

(A.18)

Here except for α(0) , the other coefficients are only functions of xµ . By this approach, the null surface term is still zero, but differing from the results in Refs. [17, 18], there is an additional contribution from α(1) . To see that, let’s assume k¯I is the affinely parameterized null normal vector for the other null surface at the joint. Then according to Ref. [17], it has following form k¯I = β(−dz − nµ dxµ ) , (A.19) where β has a similar series expansion to α, β = β (0) + β (1) z 2 + · · · ,

(A.20)

hence the inner product of these two null vectors is,10 k I k¯I = 2

αβ 2 z . `2AdS

10

(A.21)

In Ref. [17], there is an order O(z 6 ) correction in Eq. (A.21). However, such correction isn’t necessary, as Eq. (A.21) is an exact result in FG coordinate system.

– 33 –

Using the expression in Eq. (2.12), we can find that, Ijoint

`d−1 = − AdS 8πd−1

Z

`d−1 = − AdS 8πd−1

Z

=−

`d−1 AdS 4πd−1

−

`d−1 AdS 8πd−3

d−1

d

√

x σ ˜ ln

J0

αβ 2 `2AdS

" ! #) (1) (1) (0) β (0) α β α 2 1 + + (0) 2 + O(4 ) dd−1 x σ ˜ ln 2 (0) 0 ` α β J AdS !Z p (A.22) √ d−1 α(0) β (0) ln σ ˜d x ÀdS J0 ! Z p (1) (1) β α d−1 + (0) + O(5−d ) . d x σ ˜ (0) (0) 0 α β J √

(

The logarithmic term in the last line of Eq. (A.22) is just the logarithmic term in the Ref. [17], but there is an additional subleading divergent term causing by α(1) and β (1) . Now let’s compute α(1) and β (1) . Using the geodesic equation up to the order of z 5 , we can find that, α(1) β (1) 1 (1) = = g˜tt . (A.23) 2 α(0) β (0) By this result, one can check that the subleading term in Ref. [17] should be modified as,   (0) ˜ (0)ij ˜ˆ (0) (0)2 + 2K 2 − 5d + 8)R (0) + (d − 3)(d − 2)R d−1 Z ˜ ˜ ˜ p 2 K K − (d ` ij , dd−1 x σ ˜ (0)  − AdS 8π 2 ~ B d−3 (d − 1)(d − 2)(d − 3) (A.24) which is different from the result shown in Eq. (A.16). It’s no surprising as the action (2.11) without Iλ depends on the parameterization of null normal vector. Note that α(1) and β (1) also have additional contributions on the subleading term in Iλ . Using the method in Ref. [18], one can compute such additional contribution. If we take the both of two additional subleading contributions coming from Ijoint and Iλ into account, then we can find the subleading divergent term is still as the same as Eq. (A.17).

Acknowledgments We would like to thank Rob Myers for valuable discussions and correspondence. 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(NRF- 2014R1A1A1003220) and the GIST Research Institute(GRI) in 2016. We also would like to thank “11th Asian Winter School on Strings, Particles, and Cosmology” at Sun Yat-Sen university in Zuhai, China for the hospitality during our visit, where part of this work was done.

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Surface Counterterms and Regularized Holographic Complexity

Surface Counterterms and Regularized Holographic Complexity

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