Surface counterterms and regularized holographic complexity

Journal of High Energy Physics, Sep 2017

The holographic complexity is UV divergent. As a finite complexity, we propose a “regularized complexity” by employing a similar method to the holographic renor-malization. We add codimension-two boundary counterterms which do not contain any boundary stress tensor information. It means that we subtract only non-dynamic back-ground 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 five 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.

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Surface counterterms and regularized holographic complexity

HJE Surface counterterms and regularized holographic Run-Qiu Yang 0 2 Chao Niu 0 1 Keun-Young 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 codimension-two boundary counterterms which do not contain any boundary stress tensor information. It means that we subtract only non-dynamic 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. Gauge-gravity 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 non-rotational 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 Einstein-Rosen 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 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. can be characterized by the codimension-two 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 space-like codimension-one surface which connects the codimension-two 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 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 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 Wheeler-DeWitt (WDW) patch associated with tL and tR, i.e. 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 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 space-time. By choosing a suitable vacuum space-time, 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 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. [20], 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 di erence of complexity at the 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. [20], the holographic complexity of two space-time at the nite cut-o 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 codimension-two 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 codimension-two surface of given time t = tL or tR at the cut-o boundaries. `AdS is the radius of the AdS space. g is the induced metric at the cut-o 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 codimension-two surface at the boundary rather than the codimension-one surface. Because the complexity, as shown in the gure 1, is de ned by the time slices denoted by tL and tR, which are codimension-two surfaces, it 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 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 coordinate-dependent 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 2-dimensional surface spanned by x; y and rm ! 1 is the UV cuto . As a dimensionless cut-o , 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 cut-o 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 erman-Graham (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 cut-o . 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 near-boundary 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 Einstein-Hilbert 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 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 GibbonsHawking-York extrinsic curvature and h is the determinant of the induced metric. is a parameter of the generator of the null boundary and is the non-a 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 non-null 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 non-dynamic 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 erman-Graham (FG) coordinate system [21, 22], any asymptotic AdSd+1 space-time can be written as6 ds2 = gIJ dxI dxJ = 2 `Az2dS [dz2 + g~ (z; x )dx dx ] ; (2.15) 1; d denote the full sppace-time 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 energy-momentum tensor [24, 25]. We will see that these higher order terms are irrelevant in determining the counterterms. At the UV cut-o z = , the induced metric (denoted by g ) at the boundary (codimension-one) 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 tilde-variables 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 (codimension-two) embedded in the z = boundary surface (codimension-one). 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 codimension-two 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 null-null joint at the cut-o . (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 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.) where I is the action (2.11) computed with the AdS boundary at the cut-o 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 cut-o boundary and there is a null-null joint at the cut-o . At the right panel of gure 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 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 codimension-two surface terms. After subtracting this codimension-two surface terms, we end up with the nite result. As the subtraction terms are written in terms of the geometrical quantities of the codimension-two 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 cut-o . 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 cut-o 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) coordinate-independent. 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 on-shell 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 non-dynamic 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 non-dynamic 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 cut-o 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 codimension-two 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 cut-o 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 codimension-two 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 cut-o 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 space-time, 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 non-dynamic 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 time-like 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 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. 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 non-rotational 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 null-null 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 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 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 cut-o 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 null-null joint at the cut-o r = rm. In addition, in order to regularize the singularity, we need to use an additional cut-o at r = " ! 0, so there are also some new joints and space-like 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 null-null joint at the cut-o 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 cut-o at r = " ! 0, so there are also some new joints and space-like 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 null-spacelike 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 space-like curvature singularity at r = 0. We need to make a cut-o 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 space-like 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 cut-o 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 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 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 space-time, 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 cut-o 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 codimension-two 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 non-dynamic 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 non-rotational 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 cut-o , 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 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 [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 Yong-Jun 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 Yat-Sen 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 time-like vector, i.e. Now let us nd the non-a 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 cut-o 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 n-th 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 on-shell 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 \Wheeler-DeWitt" 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 This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] J. Watrous, Quantum computational complexity, arXiv:0804.3401. [2] S. Gharibian, Y. Huang, Z. Landau and S.W. 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Run-Qiu Yang, Chao Niu, Keun-Young Kim. Surface counterterms and regularized holographic complexity, Journal of High Energy Physics, 2017, 42, DOI: 10.1007/JHEP09(2017)042