On the Role of Counterterms in Holographic Complexity

We consider the Complexity=Action (CA) proposal in Einstein gravity and investigate new counterterms which are able to remove all the UV divergences of holographic complexity. We first show that the two different methods for regularizing the gravitational on-shell action proposed in \cite{Carmi:2016wjl} are completely equivalent, provided that one considers the Gibbons-Hawking-York term as well as new counterterms inspired from holographic renormalization on timelike boundaries of the WDW patch. Next, we introduce new counterterms on the null boundaries of the WDW patch for four and five dimensional asymptotically AdS spacetimes which are able to remove all the UV divergences of the on-shell action. Moreover, they are covariant and do not change the equations of motion. At the end, we calculate the null counterterms for an AdS-Schwarzschild black hole.


Introduction
The discovery of Ryu-Takayanagi [3,4,6] and HRT formulas [5,7] marked the beginning of a new era during which we have been encountering increasing evidence [3][4][5][6][7][8][9][10][13][14][15][16][17][18][19][20][21][22] that quantum gravity and quantum information theory are two indispensable topics. The connection is strong enough that one might introduce a dictionary between them and apply the AdS/CFT correspondence [23] to calculate diverse quantities such as entanglement entropy [3-5, 8, 9], mutual information [10][11][12] and entanglement of purification [13][14][15][16]. Another important concept in quantum information theory is computational complexity which is believed to be useful in understanding the interior of black holes [17][18][19][20]22]. In quantum field theory, computational complexity is defined as the minimum number of gates, i.e. simple unitary operations, needed to make a specific state from a reference state. For quantum states which are dual to black holes in AdS, computational complexity has interesting properties [17,22]: it is an extensive quantity and after the thermal equilibrium it increases linearly with time until it reaches its maximum value e S , where S is the thermal entropy of the black hole. Next, it fluctuates around this value for a long time which is of order the quantum recurrence time e e S , then it reduces to its minimal value. Moreover, its late time growth saturates the Lloyd's bound [20][21][22]24]. Recently, in the framework of AdS/CFT two proposals have been introduced to calculate complexity in the gravity side: the Complexity=Volume (CV) [17,18,22,25] and the Complexity=Action (CA) [20][21][22] proposals. According to the CA proposal, the holographic complexity C for a boundary state on a time slice Σ, is given by the on-shell gravitational action on a region of spacetime called Wheeler-De Witt (WDW) patch, as follows (1.1) The WDW patch is defined as the domain of dependence of a Cauchy slice in the bulk which assymptotically approaches the time slice Σ on the boundary. In the following, we consider eternal black holes for which the WDW is defined as the domain of dependence of a Cauchy surface started from the right boundary at time t R and ended on the left boundary at t = t L (see the left panel of Figure 1). Then the holographic complexity is associated to the quantum complexity of a state in the dual CFT at time τ = t L + t R .
On the other hand, it is well known that the on-shell action is a UV divergent quantity, and hence, one has to introduce a regularization method. In [1] two regularizations were suggested. In the first regularization shown in the left panel of Figure 1 the WDW patch is cut at r = δ. Therefore, the WDW patch has two timelike boundaries on the left and right sides. In the second regularization shown in the right panel of Figure 1, the spacetime is cut at r = δ and the null boundaries of the WDW patch start from r = δ. In [1] the structure of the UV divergences of holographic complexity have been studied. It was shown that the structure of the UV divergence are the same in both regularizations, though their coefficients are not equal. On the other hand, in [2], the null counterterm I ct (see Eq (2.8)) was considered, and shown that it removes the ambiguities of the null vectors and at the same time cancels the most divergent term in Eq (2.5), however, the coefficients are not equal again. The first aim of the paper is to show that the two regularization are equivalent. To do so, we notice that in the first regularization the WDW patch has two extra timelike boundaries in contrast to that of the second regularization. Indeed, these timelike boundaries are pieces of the boundaries of spacetime. Therefore, one might write some types of counterterms on the timelike boundaries of WDW patch, which are similar to those applied in holographic renormalization [33][34][35][36]. In section 3, we show that adding these counterterms will resolve the issue of the inequality of the coefficients in the two regularizations. The second aim of the paper is to seek counterterms in Einstein gravity to remove the UV divergences of holographic complexity. The first attempt in this regard has been done in [26], and the following counterterms were obtained by minimal subtraction here the integral is performed on the null-null joint points J located on the cutoff surface at r = δ (see the right panel of Figure 1). Moreover, g µν is the induced metric on the r = δ boundary of spacetime and R µν is the Ricci tensor made out of g µν . h ij and K ij are the induced metric and the extrinsic curvature tensor of the joint points, respectively. Furthermore, F (2n) A is a function of an invariant combinations of {R µν , g µν , h ij , k ij } and is of mass dimension 2n. In the following, we want to introduce new counterterms which are written on the boundaries of the WDW patch which are codimension-one null surfaces. Moreover, we want to write counterterms which are covariant and do not change the equations of motion. The organization of the paper is as follows: in section 2, we fix our notations. In Section 3, we consider two different methods for regularizing holographic complexity (see Figure 1), and argue they are completely equivalent. In other words, we show the structure of the UV divergences of the on-shell action as well as their coefficients are the same. In section 4, we discuss on the general form of the counterterms on null boundaries of the WDW patch in an asymptotically AdS spacetime. These counterterms are able to remove all the UV divergences of holographic complexity. In section 5, we calculate the null counterterms for an AdS-Schwarzschild black hole in Einstein gravity. At the end, in section 6, we conclude and discuss about charged black holes.

Setup
In this section, we fix our notations. For simplicity, in the following we restrict ourselves to an Einstein gravity in d + 1 dimensions whose action is written as follows is the cosmological constant in which L is the AdS radius of curvature. This action has an AdS-Schwarzschild black hole solution whose metric may be parametrized by where dΩ 2 d−1 is the metric of a unit d − 1 dimensional sphere, and r 0 is related to the radius of the horizon r h via Moreover, we define tortoise coordinate r * (r) as follows To calculate the holographic complexity, one needs to compute on-shell gravitational action on the WDW patch. The action is composed of different parts as follows [28] In the following, we will introduce each part. In general, the WDW patch has timelike, spacelike, and null boundaries, which are codimension-one hypersurfaces and we show them by T , S, N , respectively. The extrinsic curvature of the corresponding boundaries are denoted by K t , K s and K n , and one has to include a Gibbons-Hawking-York (GHY) term [29,30] for each boundary In the GHY term for null surfaces λ, is the coordinate on the null boundaries. In the following, we choose λ to be affine, hence the GHY action will be zero on null boundaries. There are also some joint points where two boundaries intersect each other. The joints shown by J are codimension-two hypersurfaces and their action is given in terms of the function a which is given by the logarithm of the inner product of the normal vectors to the corresponding boundaries [31,32] The sign of different terms in action, depend on the relative position of the boundaries and the bulk region of interest (see [28] for more details). Moreover, it was shown [28] that there is an ambiguity in the normalization of normal vectors to null boundaries, and one has to introduce a counterterm on the null boundaries as follows to remove the ambiguities Here, γ is the determinant of the induced metric and the quantity Θ is the expansion of the null generators and is defined as follows As we will see, this term together with other counterterms play a crucial role in order to get the desired results. It is worth noting that in general to write Eq. (2.8) one could use an undetermined length scaleL, though for simplicity we have fixed it to beL = L d−1 where L is the radius of AdS. Moreover, in the following, we set the time on the left and right boundaries as t L = t R = τ 2 to have a symmetric WDW patch shown in Figure 1. Moreover, we calculate holographic complexity at times t > t c , when the past light sheets from the left and right boundaries of spacetime do not touch the past singularity. In tis case, it is straightforward to show that the critical time t c is given by t c = 2 (r * (δ) − r * (r max )) [27].

Different Regularizations
In this section, we will study the UV divergences of the holographic complexity of an eternal two-sided AdS-Schwarzschild black hole in Einstein gravity by applying the CA proposal for two different regularizations shown in Figure 1. In the first regularization, we will cut the WDW patch at the radius r = δ (See the left panel of Figure 1), while in the second regularization, we will cut the spacetime at r = δ (See the right panel of Figure 1). The aim is to verify that the two different regularizations are equivalent, in the sense that holographic complexity have the same UV divergence structure with the same coefficients in both of them. It should be pointed out that, these regularizations have already been studied for global AdS d+1 spacetimes in [1,2]. Indeed, it has been shown that the structure of the UV divergences in the two regularizations are the same but their coefficients are different. Looking at the two WDW patches in Figure 1, one observes that in the first regularization, the WDW patch have two extra timelike boundaries at r = δ (one on the left hand side and the other on the right hand side of the WDW patch). Here we want to show that by adding some types of counterterms (See Eq. (3.15)) similar to those applied in holographic renormalization, and the corresponding GHY term for these two timelike boundaries, not only the structure of the UV divergences, but also their coefficients become exactly the same. In the following, we consider the AdS-Schwarzschild solution (2.2). It is evident that the UV divergence structure of holographic complexity comes from the asymptotic behavior of the solution, and if one considers a pure AdS spacetime instead of (2.2), one should obtain the same result.
Having fixed our notations, now we calculate the holographic complexity for the geometry given in Eq. (2.2) using the two regularizations. To proceed let us first write the equations for the null boundaries B i of the corresponding WDW patches. It is straightforward to check that for the first regularization, one has while for the second regularization, one has With this notation, and using the fact that for this metric, one has the contribution of the bulk action in the first and second regularizations are given by Here V d−1 is the volume of a unit d − 1 dimensional sphere. Since the second terms in the above expressions are finite, and we are only interested in comparing the divergent structure of the holographic complexity, we just need to consider the first terms in these expressions. Indeed, by adding and subtracting r * (δ) to the divergent part of the first regularization, one finds Therefore, as far as the bulk term is concerned, there is a difference between the UV divergences of the two regularizations. Now we consider the contribution of the joint points to the action. In the first regularization, there are two timelike-null joints b and d on the right timelike boundary (see the left panel of figure (2)) whose actions are as follows where k 1 and k 2 are the normal vectors to the null surfaces B 1 and B 2 in Eq (3.1), and are given by we choose the normalization of the normal vectors such that k i .t = c, in whicht = ∂ t and c is a positive constant [28]. On the other hand, s is the spacelike outward-directed normal vector to the timelike boundary at r = δ.
It is straightforward to show that in the first regularization we have On the other hand, in the second regularization there is a null-null joint e, on the right hand side of the WDW patch, whose action is given by here k 1 and k 2 are the normal vectors to the null boundaries B 1 and B 2 in Eq (3.2), respectively. It is evident that k 1 = k 1 and k 2 = k 2 . Therefore, we have Now we consider the counterterm I (0) ct . One can find the the expansions Θ i and affine parameters λ i of the null boundaries in the first regularization as follows Moreover, since in the two regularizations the normal vectors, and hence their null expansions Θ i are the same, one can conclude that the counterterms I (0) ct are equal in these regularizations.
Of course, this is not the whole story. Indeed, as said before, in the first regularization the WDW patch has two extra timelike boundaries at r = δ, in comparison to the WDW patch in the second regularization (See Fig (1)). Therefore, in the first regularization, one should consider the corresponding action for each of these timelike boundaries. Naturally one can write a GHY term, Eq. (3.16), on each of them. On the other hand, from holographic renormalization one can write the following counterterms on the whole boundary of spacetime at r = δ [33-36] 1 .
where h is the determinant of the induced metric on the r = δ surface and R is the corresponding Ricci scalar. Moreover, the logarithmic counterterm exists for even d, and its coefficient a d is related to the conformal anomaly of the dual CFT [34,37]. One should note that the timlelike boundary of the WDW patch in the first regularization is a finite piece of the whole boundary of the spacetime. Therefore, inspired by holographic renormalization, one might consider the following counterterms on the timelike boundaries of the WDW patch in the left panel of Figure 1 I Here since we do not have any logarithmic divergent terms in the action I (See Eq. (2.5)), we do not apply the logarithmic counterterms in Eq. (3.14). As we will see in the following, it is crucial to include the above timelike counterterms to show that the coefficients of the UV divergences of the on-shell action in the two regularizations are exactly the same. Now we calculate the divergent parts of the GHY term (3.16) and timelike counterterms (3.15) on r = δ surfaces, respectively. The GHY term is given by where the factor of two is included to account for the contributions of the left and right timelike boundaries at r = δ, h is the determinant of the induced metric on the timelike boundary, and K is its extrinsic curvature. One can write Note that in Eq. (3.16) the integral on the time coordinate is taken on an interval from the past to the future null boundaries which is given by Then it is straightforward to compute the divergent part of the GHY term, Now we consider the timelike counterterms (3.15). By applying one can easily write here a factor of two is included to consider the contributions of the left and right timelike boundaries at r = δ. By adding Eq. (3.19) to Eq. (3.21), one has It is then evident that these divergent terms cancel those coming from the bulk term in Eq. (3.5), and leads to 2 Therefore, from Eq. (3.11), (3.13) and Eq. (3.23), one can conclude that the divergent parts of the total action in the two regularizations are equal to each other, Therefore, in both regularizations the structure and coefficients of the UV divergences of holographic complexity are exactly the same, provided that one takes into account all surface terms including the counterterms (3.15) inspired by holographic renormalization. To the best of our knowledge, the holographic renormalization counterterms have never been considered before in the literature of holographic complexity. Moreover, we used them on a small time interval of the AdS boundary, which is one of the boundaries of the WDW patch. Therefore, it seems that in the calculation of the on-shell action in any region of spacetime, it is necessary to consider the role of counterterms on all boundaries of that region. Since, the calculation of holographic complexity in the second regularization is easier, in the rest of the paper, we apply it.

General Form of Null Counterterms
In the previous section, we discussed the important role of counterterms in the equivalence of the regularizations. The aim of this section is to explore new types of counterterms on null boundaries of the WDW patch which are able to remove all the UV divergences of holographic complexity. We should also emphasize that using the minimal subtraction scheme, certain counterterms have been introduced in [26], which could make the on-shell action finite. However, those counterterms are written on joint points of the WDW, and are not on the codimension-one boundaries of the WDW.
In what follows, we would like to revisit the procedure and find new types of counterterms which are: covariant, written on the null boundaries, and do not change the equations of motion. Our strategy is to first extract the UV divergent terms of the on-shell action, and then to rewrite them in terms of the intrinsic and extrinsic properties of the null boundaries. Next, we apply the minimal subtraction scheme and introduce the appropriate counterterms. In [1,2] these divergent terms have been calculated for an asymptotically AdS d+1 spacetime in Fefferman-Graham coordinates. To study the general form of the counterterms, we consider an asymptotically AdS geometry whose metric in the Fefferman-Graham coordinates is as follows where [34,35] g ij (z, x) = g (0) ij (x) + z 2 g (1) ij (x) + · · · (4.2) Here z is the radial coordinate and the boundary is located at r = δ. Moreover, R ij and R are Ricci tensor and Ricci scalar constructed out of g ij . Since we are interested in computing the on-shell action on a subspace (e.g. WDW patch) that could contain several null, spacelike, timelike boundaries as well as their intersections, we will have to consider several codimension-one and codimension-two boundaries. It is then crucial to write the final action in a covariant way to make sure that the new counterterms will not alter the variational principle. To proceed it is useful to decompose the coordinates x i into t and σ a for a = 1 · · · d − 1. Assuming g where R ab is the Ricci tensor of the joint points where two null boundaries intersect. On the joint points the coordinates are given by σ a . Moreover, one gets such that where R = g (0)ab R ab is the Ricci scalar of the joint point. In what follows, it is also useful to expand the determinant of the asymptotic metric around t = 0. Indeed, applying Eq. (4.2) one has ij (t = 0, σ) + · · · , (4.8) where h ij (σ) = g ij (t = 0, σ). Therefore, one obtains which can be recast into the following form [1] det with the identifications of The last ingredient we need to compute the on-shell action in the WDW patch is the extrinsic curvature along the null boundaries. In our coordinate system, the induced metric on a null boundary N may be written as follows with the assumptions that ab + · · · . (4.13) In the following, we work with the second regularization and set t L = t R = 0, then near the asymptotic boundary at z = δ, the future B 1 and past B 2 null boundaries (See Figure 2) are given by [1] tt + · · · , for t ≥ 0, tt + · · · , for t ≤ 0 (4.14) and their normal vectors are given by k F = α(dt − dt + ) and k P = −β(dt − dt − ). Here α and β are constants that appear due to the ambiguity in the normalization of the normal vector to null surface N . Moreover, we need to calculate the affine parameter λ of the null surfaces. For future null boundary, the affine parameter to the order that we are interested in here, is given by Eq.
In this notation the object we are looking for may be defined as follows In what follows, we will have to deal with the trace and inner product of the extrinsic curvature tensor Θ i j which are defined as Θ = Θ i i and Θ · Θ = Θ i j Θ j i , respectively. We note, however, that since the metric component g ta starts at order O(z 2 ), the component Θ t t starts at order O(z 5 ). Therefore, up to the order O(z 3 ) that we are interested in here, it is sufficient to work with Θ = Θ a a and Θ · Θ = Θ a b Θ b a . It is then straightforward to compute these objects using the asymptotic behavior of the metric. In particular, for the future null boundary we have If one plugs Eq. (4.17) into Eq. (4.9), one has Next, one can find (4.19) Then it is straightforward to write as well as (4.21) Moreover, if one applies Eq. (4.11), the above expressions may be recast into the following forms We have now all the ingredients to compute the on-shell action and find the corresponding divergent terms. To proceed, we will only consider the contribution of different terms to the action near r = δ as shown in Figure 2. Moreover, since the two regularizations are the same, we choose the second regularization. Actually using the expressions we have presented so far, it is straightforward to show that for one-half of the WDW patch, the divergent part of the on-shell action to order

O(δ d−3 ) is given by
where P and F stand for passed and future null boundaries whose normal vectors are also denoted by k p and k F , respectively. Furthermore, by applying Eq. (4.5) the leading divergent term of the on-shell action is given by It is worth mentioning that we have already fixed the undetermined length scaleL in the counterterm One can show that this choice removes the most divergent term of holographic complexity which is at order 1 δ d−1 . Therefore, the most divergent term that is remained in Eq. (4.24) is at order 1 δ d−3 . Now the aim is to add proper counterterms to remove these UV divergences. We note, however, that using minimal subtraction, new counterterms have been studied in [26] that is essentially the above terms with a minus sign. Of course, since eventually we would like to have a covariant action, it is curtail to make sure that adding any terms would not alter the variational principle. Therefore in what follows, we would like to introduce new counterterms defined on the null boundaries of the WDW patch which remove the above divergent terms. Actually the counterterms should be written in terms of the induced metric on the null boundary and possibly its derivative. To write the corresponding counterterms, one may apply Eq. (4.22) and obtain the following expressions.
It is then straightforward to see that the divergent terms in Eq.(4.24) may be written as follows where the integration is over future and past null surfaces of one side of the WDW patch. It is then easy to write the corresponding counterterms that are essentially the above expressions with a minus sign, i.e.

I
(1) (4.27) and we have to calculate it for each null boundary N of the WDW patch. It should be pointed out that the above counterterms work for d = 3 and 4. When d = 2 the holographic complexity is finite and no counterterms are needed (see also [26]). Moreover, for higher dimensions, it seems that higher powers of R and R ij would appear in the above expression. Furthermore, it is evident that for black branes these null counterterms are zero. In the next section, we compute the above counterterms for an AdS-Schwarzschild black hole.

Holographic Complexity
In this section, we calculate the holographic complexity for the AdS-Schwarzschild solution (2.2) at time t > t c . As mentioned above, the two methods of regularization are the same, hence we apply the second one. In this case, the WDW patch is given by the right panel of Figure 1. The bulk action is as follows Then, the above integrals can be rewritten as follows Now by integration by parts, the bulk action can be recast into There is a GHY term for the future singularity at r = r max , By adding it to the bulk action, one has On the other hand, there are four joint points and their contributions are given by The counterterm I ct for the four null boundaries is as follows From Eq. (5.6) and Eq. (5.7), it is evident that the ambiguities α and β are canceled in the action, and hence one can write Now to extract the UV divergent terms in Eq. (5.8), we expand it around δ = 0. When d is even, we have Now we calculate the new counterterm I (1) ct introduced in Eq. (4.27). The induced metric on the null surfaces is given by then for the null boundary B 1 , from Eg. (3.12), (4.16) and (4.21), one obtains Therefore, one has and the first term in Eq. (4.27) vanishes. In other words, for each null boundary I (1) ct is given by when d = 3, for the four null boundaries, we have Therefore, the total action is given by In the following, we want to study the behavior of I tot when δ → 0 and r max → ∞. Since, our null counterterms (4.27), are valid for d < 5, we consider the cases for which d = 3, 4.

d=4
For d = 4, if we take the limit r max → ∞, then Eq. (5.9) is simplified as follows Moreover, form Eq. (5.13) we can write the I (1) ct for the four null surfaces as follows Now one can see that the UV divergent terms in Eq (5.17) and Eq (5.16) cancel each other, and the total action, is convergent in the limit r max → ∞. Therefore, the counterterm introduced in Eq. (4.27) removes all the UV divergences of the on-shell action. Using the tortoise coordinate and taking the integrals in Eq. (5.18), one can find the finite part of the on-shell action. However, the result is very complicated and we do not write it here.

d=3
Now we study the case of d = 3. When r max → ∞, from Eq. (5.8) we have The only UV divergent term in the above expression is log δ. On the other hand, from Eq. (5.13) the new counterterm for d = 3 is given by Therefore, the new counterterm Eq. (4.27) removes all the UV divergences in the on-shell action.

New Counterterm on the Singularity
Now we look at the behavior of the total action in the limit of r max → ∞. Since, in the third line of Eq. (5.19) we have rmax r 2 dr f (r) = −r 3 0 log r max , (5.22) this log-term cancels the log r max term in the first line of Eq. (5.19), and hence the action I tot is convergent in this limit. However, in Eq. (5.20) there is such a term, and one can see that this log-term remains in the total action. Therefore, the total action is divergent when r max → ∞, and this is very problematic. It seems that one might resolve the issue by applying the proposals of [38,39,43], in which it has been shown that the cutoff r max near the singularity is related to the UV cutoff δ near the asymptotic boundary of spacetime. In particular, for AdS-Schwarzschild black holes one has [38] here r h is the radius of the horizon. Moreover, It was suggested [38] that the r = r max cutoff can be interpreted as a UV cutoff, and action counterterms were added on this surface. Motivated by this proposal, one might apply Eq. (5.23), and rewrite the logarithmic divergent term in Eq (5.22), as follows Now our aim is to find a new type of counterterm which can cancel the above term. One possibility would be to add the following counterterm on each of the null-spacelike joint points 4 , which are the intersection of the null boundaries B 1 and B 4 with the future singularity at r = r max . These points are denoted by e and f in the right panel of Figure 1.
here J is the null-spacelike joint point on the future singularity, h is the determinant of the induced metric on it, and R is the Ricci scalar of the joint point 5 . Moreover, It should be emphasized that the new counterterm Eq. (5.25) breaks the diffeomorphism invariance of the action for odd d, and might introduce a type of anomaly in the dual CFT. At the moment, we have no idea about this anomaly, and one either has to find the source of the anomaly or give another recepie to resolve the issue. Another important point is that one could add some counterterms on the future singularity [38,42] similar to Eq. (3.14). However, in the limit of r max → ∞ they go to zero and hence do not have any significance. Therefore, to extract the finite part of the on-shell action, one might add a counterterm on each boundary of the WDW patch: such that one applies Eq.
(4.27) on null surfaces, Eq. (3.15) on timelike surfaces. In this manner, one also has to add the counterterm Eq. (5.25) on null-spacelike joint points for odd d.

Growth Rate of Holographic Complexity
Now we can talk about the rate of growth of holographic complexity. From Eq. (3.2), one can find the location of the null-null joint point m, which is the intersection of the null boundaries B 2 and B 3 , as follows here t = t L + t R , then one has The rate of growth of the on-shell action I is obtained as follows [27], Putting everything together, one obtains the rate of growth of holographic complexity as follows here the second term in the parenthesis is the contribution of the null counterterms I (1) ct . From the above expression, it is evident that at late times when r m → r h , we have

Discussion
In this paper, we first examined the equivalence of the two methods of regularization proposed in [1] for an AdS-Schwarzschild solution in Einstein gravity (See Figure 1). The two methods have been already studied in [1,2], and it has been proved that the structure of the UV divergences of holographic complexity are the same in both regularizations. However, their coefficients do not match on both sides. From Figure 1, it is evident that in the first regularization the WDW patch has two extra timelike boundaries at r = δ, in contrast to the WDW in the second regularization. Here, we observed that after adding timelike counterterms (3.15) which are inspired by holographic renormalization as well as the Gibbons-Hawking-York term for these extra timelike boundaries of the WDW patch, the coefficients of the UV divergences of holographic complexity in the two regularizations become exactly the same. Therefore, one can conclude that the two methods of regularization are completely equivalent. Then we introduced new types of counterterms on null boundaries of the WDW patch for an asymptotically AdS spacetime in four and five dimensions which are covariant and are able to remove all the UV divergences of holographic complexity. To do so, we first studied the UV divergences of the gravitational on-shell action for an asymptotically AdS spacetime in the second regularization. This step has already been taken in [1] and [2]. However, in [1,2] the UV divergences have been written in terms of the extrinsic and intrinsic curvatures of the joint points. Since we were interested to find covariant counterterms on the null boundaries of the WDW patch, here we first rewrote the divergent terms in terms of the extrinsic curvature of the null boundaries, i.e. Θ and Θ.Θ. Next, by applying the minimal subtraction scheme, we found counterterms on the null boundaries. The result is presented in Eq. (4.27). It should be pointed out that some type of counterterms were also introduced in [26] which remove all the UV divergences of holographic complexity. However, those counterterms are written on the joint points which are not the boundaries of the WDW patch. On the other hand, the counterterms found here are defined on the null boundaries of the WDW patch and are covariant. It was also observed that these null counterterms modify the early time behavior of the growth rate of holographic complexity, although they become zero at late times when r m → r h . However, the null counterterms suffer from a log r max divergence for odd d, in which r = r max is the location of the future singularity. To resolve the problem we applied the proposals of [38,39,42,43], in which it was argued that putting a UV cutoff at r = δ, might introduce a cutoff behind the horizon and near the singularity at r = r max , such that they are related to each other by Eq. (5.23). If this proposal works, one can rewrite the log r max term in holographic complexity as log δ. Next, to remove this logarithmic divergent term, one might add a counterterm such as Eq. (5.25) on the null-spacelike joints located at r = r max surface.
Moreover, one can consider charged black hole solutions [21,27,[40][41][42] such as a dyonic black hole in four dimensions whose holographic complexity is calculated in [44]. In this case, the metric is given by where Q e,m are related to the electric and magnetic charges of the solution. For d = 3, one can show that the divergent part of the action (2.5) is logarithmic On the other hand, the divergent part of the null counterterms (4.27) is given by Therefore, the null counterterms I (1) ct are also able to remove all the UV divergences of the holographic complexity for charged black holes.