Complexity-action of subregions with corners

In the past, the study of the divergence structure of the holographic entanglement entropy on singular boundary regions uncovered cut-off independent coefficients. These coefficients were shown to be universal and to encode important field theory data. Inspired by these lessons we study the UV divergences of subregion complexity-action (CA) in a region with corner (kink). We develop a systematic approach to study all the divergence structures, and we emphasize that the counter term that restores reparameterization invariance on the null boundaries plays a crucial role in simplifying the results and rendering them more transparent. We find that a general form of subregion CA contains a part dependent on the null generator normalizations and a part that is independent of them. The former includes a volume contribution as well as an area contribution. We comment on the origin of the area term as entanglement entropy, and point out that its presence constitutes a robust difference between the two prescriptions to calculate subregion complexity (-action v.s. -volume). We also find universal $\log\delta$ divergence associated with the kink feature of the subregion. Similar flat angle limit as the subregion-CV result is obtained.


Introduction
The holographic principle equates entanglement entropy in a field theory to a geometrical object, called holographhic entanglement entropy, in the dual spacetime. This relation implies that quantum entanglement encodes information about the geometry of the dual space and plays a crucial role in the program of reconstructing spacetime from boundary (field theory) data. However, recently we have understood that entanglement entropy cannot be the only ingredient involved in spacetime reconstruction. After a black hole is formed, the interior grows for an exponentially large time but the holographic entanglement entropy fails to reproduce this growth [1,2]. New developments point to quantum complexity as the missing ingredient [3,4]. One way to think about complexity in quantum mechanical systems is as the minimum number of "simple" operations needed to go from a reference state to a target state. Quantum complexity is an active area of research in quantum information but not much is known about complexity in quantum field theories. The first steps in this direction were taken in [5,6] where the authors investigate circuit complexity in a free scalar quantum field theory.
There are currently several proposals for the geometrical construction dual to complexity. Two of these proposals are considered more promising and have been thoroughly explored: complexity-volume (CV) [7] and complexity-action(CA) [8,9]. These proposals for holographic complexity aim to capture the circuit -or gate-complexity of the corresponding dual state. In the gravity side it is natural to also define a complexity not of the whole state but of a region of space, i.e. subregion complexity [10][11][12]. The subregion complexity-volume proposal, subregion-CV, identifies the subregion complexity with the spatial volume bounded by the Ryu-Takayanagi and the boundary region. On the other hand, the subregion complexityaction proposal, subregion-CA, associates the complexity of a boundary region with the action evaluated on the intersection of the Wheeler-DeWitt (WDW) patch and the entanglement wedge of the given region [12]. Recently, in [13] the authors proposed several definitions for subregion complexity in a field theory and compare their properties to the holographic proposals advanced in []. They found a promising agreement of purification complexity and subregion-CA. However, the issue of which of the holographic proposals is the correct one, or if they correspond to different definitions of complexity, is not settled yet.
In the past, understanding the divergence structure of entanglement entropy in singular boundary regions was quite fruitful. A singular region is characterized by an opening angle 0 < θ < 2π. Cut-off independent coefficients, a(θ), arising from such regions were studied in a variety of quantum field theories [14][15][16] (free scalars, free fermions, interacting scalars) and in holographic models [17][18][19][20] as well. It was found that these coefficients represent an effective measure of the degrees of freedom of the underlying CFT. Furthermore, it was shown that the ratio a(θ → π)/C T , where C T is the central charge associated with the stress tensor T µν , is universal for any 3D CFTs.
Inspired by these lessons, in this paper we study the UV divergenge structure of subregion complexity-action of a boundary region with a geometric singularity or kink. Our goal is to take a first step towards understanding if geometrical singularities in the boundary region also encode cutoff independent and universal contributions to subregion complexity. To calculate subregion complexity-action we have to evaluate the action in the spacetime region determined by the intersection of the entanglement wedge and the Wheeler DeWitt patch. The calculation in the case of singular subregions is technically involved. We discuss the appropriate way to define the infrared cutoff and develop a systematic approach to calculate all the divergences. We uncover a divergence structure that is much richer than in the subregion complexityvolume case [21]. We find that the leading divergence is proportional to the volume of the boundary region and identify new contributions coming from the singularity to the cut-off independent coefficients of log and log 2 terms. As in the case of subregion complexity-action for a smooth subregion, there are subleading divergences that depend on the normalization of null normals. We comment on different choices of normalizations for these null vectors. In particular, we find that taking the scale α,β to be associated with the size of the region, R, instead of the UV cutoff, δ, leads to a desirable complexity behavior. We also point out a relation similar to the Bousso bound that is obeyed in the case of a kink singularity.
The systematic approach to study the divergence structure that we develop here can be easily extended to higher dimensions and more general geometric singularities. Furthermore, our detailed results can serve as a benchmark for proposals of subregion complexity in field theory.
This paper is organized as follows. In Section 2 we review the relevant ideas of entanglement entropy in singular subregions and their significance. In Section 3 we review the definitions of subregion-CV and subregion-CA and the results for subregion-CV for a region with a kink [21]. Sections 4 and 5 constitute the main parts of this paper. In Section 4 we setup the problem, point out some subtleties and outline the steps of the calculation. In Section 5 we present the final result and discuss its various properties. Section 6 contains the conclusions and future directions. All the the technical details of the calculations are presented in two appendices.

Entanglement entropy in singular subregions
Spatial subregions in the boundary theory that contain geometric singularities are known to have interesting contributions to the entanglement entropy. In quantum field theory, the entanglement entropy has an area law behavior. But the coefficients of the leading order area law contribution depend on the UV regularization of the theory. On the other hand, there are subleading contributions that are independent of the UV regularization and thus, contain unambiguous information about the boundary theory [22]. These contributions were later shown to be universal for a large class of CFTs [14]. When the boundary has sharp features or singularities it was found in [18,19] that there are additional contributions that are cut-off independent and universal . In this section we will review some results related to entanglement entropy in singular regions.
The metric of d + 1 dimensional AdS space in Poincare patch is, where n = d − 3. A cone is an example of a singular region on the boundary. In general, cones in different dimensions can be parametrized as, where ρ IR is an IR cutoff that has to be taken to infinity at the end of the calculation. The cone c n has a scaling symmetry along the radial direction. Due to this symmetry, the Hubeny-Rangamani-Takayanagi (HRT) surface [23] has the form The function h(θ) has a maximum at θ = 0 that we wil denote h 0 , h(0) = h max ≡ h 0 , and vanishes at the boundary, h(Ω) = 0. The entanglement entropy of c n is given by the area The extremality condition of the area determines the shape function h(θ). And since h(θ) specifies the HRT surface (2.3), this function will play an important role in the complexity calculations of the following sections.
In this paper we focus in d = 3 , i.e. n = 0. This is a c 0 cone which we will refer to as a kink, k. In this case the integrand is independent of θ and the area functional has an integration constant The opening (half) angle, Ω can then be written as a function of h 0 , For our purposes it is interesting to note the asymptotic behavior at small or large angles, The large angle or smooth limit is Ω = π/2, indicating a flat opening angle, i.e. no kink. In this limit the contributions coming from the singular point should vanish. An opening angle 2Ω larger than π indicates a concave corner. For the kink we consider here, the HRT surface is the same for Ω = Ω 0 and Ω = π − Ω 0 , but the notion of "in" and "out" of this HRT surface is reversed.
In [20] the authors found that cone regions contribute to universal terms in the entanglement entropy. These contributions introduce new log or log 2 terms that are cutoff independent, n (Ω) log 2 (R/δ) d even. (2.9) The functions a (d) n (Ω) are functions of the opening angle Ω. Since we are dealing with a pure state, a (d) n (π −Ω). An additional restriction on a d n (Ω) comes from the fact that when Ω → π/2 we are in the smooth limit, with no singularity, and therefore a (d) n (Ω = π/2) = 0. These constraints imply that in the large anlge limit a d n (Ω) is of the form (2.10) Thus, the conical singularity introduces a set of coefficients σ d n that encode regulator-independent information about the CFT. Remarkably, this same behavior for σ (d) n was found for field theory calculations of entanglement in regions with sharp corners [16]. Furthermmore, holographically, it can be shown that σ (d) n is purely determined by the boundary stress tensor charge C T . As mentioned in the Introduction, the motivation of the present work is to understand if similar cutoff independent and possibly universal contributions are present in the case of subregion complexity-action.

Subregion complexity
Currently there are two proposals for for holographic subregion complexity, subregion-CV and subregion-CA. Both of these proposals recover the original holographic state complexity in the limit when the region is the whole boundary. It is also required that they only include information from reasonably portion of the spacetime, i.e., the entanglement wedge.
In the CV approach, one takes the maximal spatial volume bounded by the boundary region and its HRT surface. Clearly, if we take the subregion to be the whole boundary we recover the original CV-complexity. Subregion-CV complexity was investigated in [10] for smooth subregions and in [21] for singular regions. In particular, for a 3 dimensional kink the subregion-CV complexity is [21], Interestingly, besides the regular volume contribution, a new term with log divergence appears as the kink contribution. The coefficient has the limiting behavior In [12] the authors proposed that the subregion-CA complexity is given by the action evaluated on the intersection of the entanglement wedge of the subregion and the WdW patch of the boundary time slice. Fig 1 schematically shows the upper half of the relevant bulk region. The spatial region is represented by a red line, the HRT surface by a blue curve and the light sheet associated to it, i.e. the boundary of the entanglement wedge, is the pink surface. The green surface represents the light sheet associated to the boundary interval and is, therefore, the boundary of the WdW patch. In addition to the region depicted in 4 Complexity-action of a region with 3d kink Figure 1: The intersection of the entanglement wedge and WDW patch. For clarity, we only show the t > 0 part of the region, V + . This is the upper half of the full region, V.The lower half V − , is symmetric with t → −t. In the δ → 0 limit, the red line represents both, the boundary region A and the surface W on the cutoff surface. The blue curve represents the HRT surface E. The green and pink surfaces represent the null hypersurfaces W + and E + , and their intersection at the green curve is the surface J + .

Setup
To compute the holographic complexity of a 3d kink region A in CA approach, one has to first define the bulk region V on which we compute the action. According to the description in the last section, this region is delimited by a boundary that consists of 4 hypersurfaces: (4.1) where W ± are the null boundaries of the WdW patch, and E ± are the null boundaries of the entanglement wedge. The HRT entangling surface is E = E + ∩ E − . The position of E in terms of Cartesian coordinates in Poincare is, The E ± are generated by null normal vectors V µ ± of E: The superscripts represent the orientation of E, since the kink with Ω and π − Ω share the same HRT surface, but differ by its orientation. For convex kinks with Ω < π/2, one takes V < ; for concave kinks with Ω > π/2, one takes V > . Due to the conformal flatness in Poincare patch, the light rays are straight lines, with linear coefficients f (λ). This reparameterization function shall be determined later when we choose λ to be the affine parameter.
We further define W = W + ∩ W − and J ± = E ± ∩ W ± . W is on the cutoff surface δ which becomes A in the δ → 0 limit. These geometrical objects can be visually seen in Fig. 1

Subtleties in the complexity-action calculation
Before we plunge into the calculation of the action in V, we want to point out a few issues that are worth careful consideration. First, the null hypersurface, E, entering the definition of V can present caustics. We show that for the kink singularity we consider here, the caustics are outside the region of interest and therefore pose no problem. Second, since the subregion we consider is not closed we need a IR cutoff. We show that a naive choice for this IR cutoff leads to inconsistencies and present a consistent choice. Finally, we comment on the limitations of the available techniques for computing the action in a bulk region with boundaries.

Caustics
According to the focusing theorem, lightsheets end on caustics in finite amount of affine time. If the lightsheet ends before it intersects with W ± , the caustics should also be included as part of the hypersurface that delimits the region.
The expansion rate Θ of the lightsheet congruence is where g ±,αβ (λ) is the induced metric on equal λ slice of the light sheet and λ is the affine parameter of the geodesic X µ ± (λ). We solve the geodesic equation to obtain X µ ± (λ), determine the induced metric nd obtain Θ ± Caustics occur when Θ ± diverges, that is, at Note that this result does not depend on convexity of the kink.In order to see if we encounter caustics before reaching the intersection J ± , we solve for the λ on J ± by combining (4.1) and (4.2), where, It is easy to see that λ * < λ c for any ρ and θ, and for both convex and concave kinks. Therefore, the caustics are always outside the region V and we don't need to worry about them.

IR cutoff
Another tricky issue is how to choose the IR cutoff of the region. It is tempting to naively use the same cutoff R for the radius coordinate on both A and E. However, with this choice of IR cutoff the hypersurfaces E ± and W ± do not match exactly to enclose the region. Thus, we can only set a constant cutoff R for either of A or E. And we determine the cutoff on the other hypersurface by following appropriate null rays. This is illustrated by the dashed arrows in Fig.2. As we have already parameterized E ± in terms of coordinates (ρ, θ) on E, it is more convenient to work with a constant IR cutoff at ρ = R where ρ is defined to be the projected radial coordinate on E. We want to use coordintes (ρ, θ) to parameterize the whole region V. So far we have used a parametrization along E up to the surface J as X µ (λ), 0 < λ < λ * . We continue along W towards the boundary surface W by following the null rays that generate W, with integral curve The starting point X µ ± (λ * ) is at the joint surface J ± , while the ending at W is solved through X z ± (η * ) = δ as, 12) Figure 2: Region V with both spatial directions shown explicitly -the time direction is suppressed. The yellow plane is the boundary space, with the red curve denoting the kink. The blue surface is the HRT surface E and the purple one is J. The null hypersurfaces E and W are the volume between E and J and the volume between the boundary and J, while the red and green dashed arrows are typical null rays on them. It is clear that if we choose ρ = R constant as IR cutoff on E as shown here, the red null ray matches a corresponding green null ray only when the IR cutoff on the boundary is given by the purple dashed curve instead of the naive orange circle. where is a shorthand notation used throughout the paper. The surface W is parameterized as X µ (η * , ρ, θ) where ρ < R sets the cutoff boundary, as shown in Fig. 2 by the purple dashed curve. Now that we have parameterized all the boundary hypersurfaces by (ρ, θ), we can naturally extend the parameterization to the whole region V together with ζ = f (λ) and η: with range of parameters The last condition is to restrict the parameterization V ± to ±t > 0 region, and hence V = V + V − . Due to time reflection symmetry, the action should be twice the action in either of V ± .
For convenience, we also write the bulk reparameterization explicitly in the form of a coordinate transformation (for convex kink): (4.16)

Higher codimension manifolds on the boundary
We can write down the most general form of gravitational action as that are relevant for the spacetime region in which we compute action. In particular, Σ Note that for Lorentzian spacetime there can be null manifolds on the boundary, which has degenerate metric. Contributions from null Σ (D−1) i has been studied [24], which shows that the volume form should be modified and the integrand should be the surface gravity: where λ is the null parameter and κ is the surface gravity associated with null vector field ∂/∂λ. Inspired by the CA conjecture, further studies have been done for contributions from higher codimension manifolds. The codimension 2 non-null manifolds, called joints, were studied in [25] which provides the integrand as where the joint is specified as intersection of two codimension 1 manifolds Σ . We only present here the case when both Σ are null, as it is the only relevant case for our computation. k i,j = ∂/∂λ i,j are null generators of the codimension 1 manifolds, and the sign depends on the orientation of the intersection.
The action is, (4.20) Note that, in principle, higher co-dimension singular can also be present. It was argued in [26] that a high codimension conical singularity can be regulated to a geometry with only lower dimensional singular surface, and through the regulation it was shown that the conical singularity does not contribute to the action. However, there is no staightforward way to generalize the regulation method to deal with general singular features, like polyhedral singularity (the intersection of several hypersurfaces). Polyhedral singularity naturally appear in any subregion CA computation, the common one being the codimension 3 manifold ∂W sitting at the intersection of all of the four E ± and W ± hypersurfaces. In our case when there is additional singular feature on the surface of the boundary subregion, even higher codimension singularities are present on ∂W . Understanding if polyhedral singularities can be reduced via some type of regularization and what excatly their contribution is is an issue that deserves further study.
After having discussed the subtleties present in the calculation of subregion complexity-CA we outline the calculation in the next two subsections, the technical details are left for the appendices.

Bulk contributions
Let us first consider the bulk action, It is convenient to write the metric in the new set of coordinates ξ a = {ζ, η, ρ, θ}, where g 0 is the original Cartesian coordinates of AdS 4 . Its determinant has the simple form We first integrate along ζ and η directions. Using the range of parameters specified in eq(4.15), we get (4.25) Explicitely, where A δ denotes the range of (ρ, θ) over which we integrate and the subscript δ reminds us that the bulk region is cutoff by z = δ. The A δ integration can be written as The ρ integration can be carried out analytically. Using the technique developed in the Appendix A, we evaluate the integration in A δ to get where the integrands for the numerical integrations are and H ± , K, and ω(h) are defined in (4.13),(2.5) and (2.6) respectively. Recall that h(θ) is the function that defines the HRT surface (2.3).

Boundary contributions
The boundary of the spacetime region we are interested in is a null codimension-1 hypersurface. For the null hypersurfaces E ± and W ± , we can choose an affine parametrization and make the YGH term vanish. Thus, the only term left is the I joint , which we write as, where I J ≡ I J + = I J − due to symmetry. In terms of affine parameters, the null generators of the four hypersurfaces are where α, β are positive normalization factors, and the signs are chosen so that these are outward pointing one forms. The integrands of these joint terms are respectively The joint contribution can then be expressed as where g  is given in Appendix B and the integration is carried out in Appendix A.

Final result and discussions
Adding the contributions from bulk and boundary, the total action is obtained in (A.23), where the coefficients arē We can see explicitly the volume law term that is proportional to the volume V A 2 at divergence order δ −2 and δ −2 log δ, and the area law term that is proportional to the area V ∂A = 2R at divergence order δ −1 and δ −1 log δ. In addition, there are divergences of order log δ and order log 2 δ.

Geometric origin and cutoff dependence
In light of the discussions in [12], the divergence structure of the holographic complexity of subregion A can be expressed in terms of volume integration in A and surface integration on ∂A In the above equation, we have R denoting the spacetime curvature, K denoting the exterior curvature of the time slice,K denoting the exterior curvature of ∂A, and s, t denoting the spacelike and timelike normal vectors of the ∂A. We added dependence of ξ denoting the collection of CFT parameters, which are dimensionless and should determine the coefficients of various combinations of curvatures in the integrands. This expression is only for subregions with smooth surface, similar to gravitational action only with YGH term. In general, there could be higher codimension defects on the surface, like the cube example we mentioned in Sec. 4.2.3, and these defects could contribute independently to the complexity. Thus the most general form would be an expression similar to (4.17). In this paper, we only deal with a kink shape in two spatial dimensions, where the only singularity on the surface is the point-like kink tip. Thus the only extra term we expect is a local contribution that does not involve any integrations. For higher dimensional subregions with singular surface, integration might be needed for contributions from non-point-like creases.
The main point of this notation is that these integrands are local functions in A or on ∂A. In our kink case, all curvatures involved in the volume and area integrations vanish: R = 0 because we are in Poincare patch, K = 0 as we are on a trivial flat time slice, andK = 0 for the straight sides of the kink. Hence, we expect the integrands to be constants, and the integrations simply give the volume and surface area of A. These are the volume law terms and area law terms shown above.
The locality of the integrands also imply that the coefficients of the volume and area terms are independent of the kink angle Ω or h 0 . As expected, the volume law coefficients v 0 = −1/2 andv α 0 = 1 are constants. However, we find that our area law coefficients depend on the opening angle, as seen from (A.24): Afterwards, it increases monotonically towards infinity. When the openning angle crosses the flat angle, hence the smooth limit case, the function has value 1 (in unit of L 2 /4πG).
As a comparison, we can look at a smooth surface case, like a spherical region in Poincare patch [12]. For example, a disk region in CFT 3 has "CA" complexity (5.5) The volume law coefficients are exactly the same, while the area law coefficients differ by a h 0 dependent shift ∆(h 0 ). This function has been plotted in Fig. 3, from which it is clear that it vanishes for zero opening angle, and approaches 1 (in unit of L 2 4πG ) in the h 0 → ∞ smooth limit. It has a small dip into the negative values when the openning angle was very small, and then increase monotonically towards infinity as the openning angle continue increasing to the Ω → π limit, which corresponds to moving to right infinity along the "convex" branch and coming back along the "concave" branch to h 0 → 0. This peculiar area law contribution has the origin as the subleading term in the volume of W , as shown in (A.25). One can easily show that for ordinary constant IR cutoff on the boundary, no subleading term at this order would appear. After all, coefficients at this order are still cutoff dependent, so this extra contribution may not have important physical meaning.
It is also interesting to observe the sign difference between the volume and area terms. If we assume that the terms with log L αδ dominates (more details discussed in Sec. 5.3) over the terms without extra log δ, we end up with positive volume law term and negative area law term. The positive volume law is easy to understand as complexity naturally grow with the size of subregion. If we relate the area law terms with entanglement with outside the subregion, the negative sign can be understood as loss of detailed information of the entanglement when the outside is traced out. This loss of information also occurs for thermal state, when the complexity growth that is dual to the wormhole growth is completely hidden for the one-sided thermal state.

Cutoff Independent Terms
Next we investigate the cutoff independent coefficients, which naturally groups into the ones for log terms, denoted by k 1 , and for log 2 terms, denoted by k 2 , which have superscripts 0, α, β labelling the exact form of the logs.
We start by looking at the k 1 functions. These functions are shown in Fig. 4, where convex and concave cases are plotted separately. Note that besides the individual contributions, we also present the sum of them as the black curves. The sum represents the total coefficient of log δ divergence if no extra δ dependence is introduced (for instance from the constants α, β, which will be discussed later).  One observation is that for convex openning angles, the sum is mostly contributed by k 0 1 (h 0 ) for large angles, while k α,β 1 (h 0 ) are nealy zero. k α,β 1 are much more important in small angle region. On the other hand, for concave kinks, k β 1 remains small, while k α 1 and k 0 1 are roughly opposite of each other. In terms of additivity, the sum is mostly close to k β 1 as far as the plot shows.
Finally we look at the log 2 terms k 2 (h 0 ). From (5.2) we obtain the identity 3 This indicates a combination of the log 2 level divergence structure where the second term is finite. This means that the appearance of log 2 in three different terms is only an artifact from the computational technique. The true source of log 2 terms are only the two log terms inside brackets in the second and third lines of (5.1). The contents in these two brackets have a more direct geometrical meaning. As shown in (4.32) and (4.33), log(L/αδ) and log(L/βδ) are constant pieces in a  which can be taken out of the integral, and what is left just gives the volumes of the joint surfaces. Thus, we can show that This reminds us of the generalized Bousso bound [27]. Note that from (4.6), the expansion rates on E vanishes on E, which implies that all four null normal hypersurfaces bounded by E are light-sheets. In particular, E for both convex and concave kinks are light-sheets. On the other hand, the expansion rate on W of congruence towards the bulk is always negative, so W is also a light-sheet. functions. The sum of them is also presented as black curve. Note that the true total log 2 coefficient should be half the sum.
The generalize Bousso bound states that the area differences of two sections on the lightsheets bound the matter entropy on the part of the light-sheets between the two sections. This powerful statement and was proven with weak assumptions recently [28,29]. In particular, Therefore, the sum of the two bounds the entropy inside the region V, because W ∩ E is a Cauchy surface for V 4 . The vanishing expansion rates on the HRT surface is guaranteed by the extremality condition of its area, thus E is always a light-sheet. However, there might be a problem if the light-sheet ends before it reaches the joint J. In the present case we don't hve to worry about this complication since the caustic occurs after the intersection at J, as discussed in Sec. 4.2.1. Therefore, it is not clear if this relation with the Bousso bound holds for the complexity of any subregion but it seems to be valid for a class of shapes with the "good" property that caustics do not occur within the region V.
The above arguments tend to imply that these log 2 terms come from the log piece in the generalized Bousso bound. We have the coefficients plotted in Fig. 5. We further notice that these coefficients do not depend on which of the convex and concave kinks is being computed, which means that it is purely determined by the entangling surface ∂A alone. This strongly suggests that this contribution originats from the entanglement entropy on the boundary.
As a summary, let us list the asymptotic behaviors of all the coefficient functions at various limits in Table 1. The γ appearing in them is the one defined in (2.8). The middle column of the Table 1 shows that all of these contributions vanish in the smooth limit. Thus, we conjecture that they all come from the kink singularity.

Determine α and β from limiting behavior
It was argued in [12] that the free parameters α, β should be chosen as where α,β are some scales in the setup. The reason is that as a boundary quantity, the holographic complexity should depend on L explicitly only through the combination L 2 /G, thus L/α and L/β cannot have explicit L dependence. The natural ways to choose the scales α,β are left as only δ or R.
It was also pointed out that by setting α = R, the typical size of the subregion, the complexity would be superextensive in the sense that it grows faster than the volume. We argue that the super-extensivity is expected from long-range entanglement in the boundary theory. Naive extensivity holds if we only consider complexity of the UV structure of the state, but there can in general be long-range entanglement, whose complexity "density" accumulates logarithmically with the size, similar to the depth of MERA network for CFT states 5 .
A further indication that we should take α = R is that only this choice yields a complexity that is differentiable with respect to the angle Ω in the smooth limit. Note that taking α = R we can combine k 0 1 and k α 1 by simple addition. From Table 1 we see that the |π/2 − Ω| dependence drops off in the smooth limit. In terms of = π/2 − Ω and using (2.7) 6 , we can write Hence simply adding k 0 1 and k α 1 eliminates the | | dependence at least at leading order and makes the complexity a differentiable function of angle Ω at the smooth limit. While the importance of this differentiability remains unclear, this cancellation suggests taking α = R.
As a side comment, we also like to point out that the cancellation of | | contributions leads to the following behavior (5.12) Note that subregion-CA does not satisfy an identity similar to the one for subregion-CV proposed in [12] However, it is natural to expect that a quantity defined as could encode some physical information like the complexity of entanglement between the two regions A and B. In this case, C E ∼ S EE (A) the entanglement entropy when B =Ā. For the general case B =Ā, one may expect that C E ∼ I(A : B) the mutual information, or other measure of bipartite entanglement like the entanglement of purification [31].
Note that there is an extra log contribution in (2.9) to the entanglement entropy of kink, which goes like 2 at the smooth limit according to (2.10). Thus the behavior in (5.12) can possibly be explained as the kink contribution to the entanglement entropy. 5 One may still get extensivity by simply counting the number of gates in the MERA network as complexity.
But we claim that gates in MERA at different scale level could in principle have different weights for complexity counting. To operate at larger scale will of course cost more "simple" operations based on a sense of locality for quantum gates. 6 Note that in (2.7), Ω is defined to be convex. For general Ω, we have Ω = π/2 ± π 2h 0 where + sign is chosen for concave angle and − for convex angle. Thus the term with ± sign is actually ∼ , while the term without ± is ∼ |π/2 − Ω|. 7 This identity only holds in time-symmetric bulk geometry when the HRT surface lies on the maximal time slice in the bulk. In more general cases, CV (A) + CV (Ā) < CV (A ∪Ā) due to the maximality of bulk time slice volume on the right side.
There is another interesting property of the coefficients in the Table. 1. If we simply add all the k 1 functions, at small angle behavior we have (5.14) Note that all the log(h 0 )/h 0 behavior exactly cancels. We examined this cancellation analytically, it is not a numerical artifact. This suggests a simple addition of all the three terms, and hence the choice of β = R. Summarizing, with the choice α = β = R the complexity is, We can identify the constants appearing in the aymptotic behavior of k 1 (h 0 ) as "corner charges": similar to the studies of entanglement entropy [19]. Note that there are two κ charges due to the lack of symmetry between A andĀ in complexity. The corner charge for smooth limit is given by (5.11) as lim Ω→π/2 where we chose the linear coefficient with respect to the deviation while in the entanglement entropy case it was the quadratic coefficient. There are no obvious relations between the corner charges for complexity due to lack of analyticity in the computation, and the universality for them to describe the underlying CFT is also unknown before more examples are investigated. Finally, it was suggested in [6] that α should be interpretted as choice of reference state in the definition of field theory complexity. In light of this, we suggest that where the parameterσ characterizes the change of the normalizations due to change of the reference states. This scale factor does not affect the analysis above. The only change is that, according to (5.9), besides the results we have in (5.15) we have additional term as where S B is the generalized Bousso bound on entropy within the causal diamond V. This indicates that the complexity depends on reference state as linearly inσ in unit of the Bousso entropy bound.

Conclusions and future directions
We studied complexity-action for a particular singular subregion i.e. the 3D kink. The calculation is quite involved and great care was taken to develop appropriate techniques that can be generalized to other singular and higher dimensional regions. The result we obtain for subregion-CA consists of a volume term, an area term, and cutoff independent divergent terms of order log δ and log 2 δ. The kink computation presented here provides some insight on the choice of normalization parameters α, β. The scales present in the problem are R and δ. Either of them could in principle be a good choice for α,β . However, if the differentiability of complexity with respect to the angle is a desirable property we arrive to α,β ∼ R as a natural choice. Incidentally, this choice also leads to the cancellation of certain terms in the log coefficients when they are added up. Allowing extra scaling freedom to account for the reference state we obtain (5.19).
Let us make couple of observations regarding the possible contribution from entanglement to complexity-action. We find a negative area contribution to subregion complexity. If we relate this area term with entanglement this negative contribution can intuitively be understood as the information lost when we trace out the outside region. However, we cannot verify this quantitatively, because the contribution from entanglement might not be proportional to the entanglement entropy 8 . In general entanglement might be weighted differently in entanglement entropy than in complexity. Nevertheless, from this hypothesis, we can further make two useful comments: • In subregion-CV, an area law term is prohibited in a time-symmetric configuration [12].
The contribution discussed above, however, should exist universally, including the timesymmetric case. This indicates that this feature of complexity could only be reproduced by CA not CV.
• We know that in addition to the area term, the entanglement entropy of the kink subregion has a contribution from singular kink tip, which goes like log δ. We also know that for CA, the leading contributions come with an extra log δ. Hence we expect the singular contribution in the entanglement entropy of kink should induce a log 2 contribution in the subregion complexity. In fact, the log 2 contribution we got actually has the same symmetry Ω → π −Ω as the entanglement entropy, implying that it could be determined solely by the entangling surface. We thus conjecture that the log 2 contribution in CA are totally related with the entanglement of subregion with its complementary.
We also point out a relation between subregion complexity and the Bousso entropy bound. The Bousso bound for V seems to set the rate of change of complexity with respect to the reference state labelled byσ defined in (5.19).
We discuss super-extensivity of subregion complexity, due to the extra leading log in the volume and area law terms. A possible interpretation is that as the size of subregion grows, more and more long range entanglements, like more layers in MERA network, should be counted whose complexity density may be similar to the short range entanglement layers. The number of layers would be log of the size of the subregion, thus the overall leading contribution should go like V log V instead of V for simple extensivity. The same argument goes for area law terms.
To conclude the discussion, we list some future directions related to our work: • In the case of a kink singularity studied here , the caustics are always outside region V and thus, played no role in our calculation (Section ??). However, for more general singular regions this need not be the case, caustics might occur on the relevant part of E. It is not known if the caustic itself provides additional contributions to the complexity. Due to the vanishing of the induced metric at the caustic, it would be either a higher co-dimensional singular feature, or a null joint. Action contributions for both cases are unknown and worth studying.
• Methods of computing action contributions from higher co-dimensional surfaces are also necessary for existing polyhedral corners on ∂V.
• The connection found between complexity contribution and entanglement is worth exploring. More detailed studies on such contributions in holographic complexity and the implication from certain field theory definition of complexity are interesting directions to pursue.
• We can modify the boundary state or the theory to see how the cutoff independent terms change. In the study of entanglement entropy, certain universality of the coefficient functions was found, namely the "corner charges". It would be nice if subregion complexity also has some universality that one can use to reveal information of underlying CFT.
• Similar computation can be extended to higher dimensional singular surfaces, including smooth cones, polyhedral cones and their product with flat dimensions (creases). It will be interesting to see the dependence of the "corner charges" on dimensionality. New divergence structures are also expected.
and PHY-1820712. E.C. also thanks the Aspen Center for Physics, supported by National Science Foundation grant PHY-1607611, where part of this work was performed.

A Details of the action computation
We adopt the following procedure to do ρ and θ integrations over the range defined by A δ . Using eq(4.27), we obtain The ρ integration is usually carried out analytically, so we can define Then we series expand f 1 (h, δ) in powers of h, and extract the part that is divergent at h = 0 As f div is in power series of h, its integration can be carried out explicitly while we do integration of f 2 (h, δ) for each divergent powers of δ numerically. Namely, for the expansion we get for ith order divergence. Note that although the regularity at h = 0 guarantees that there's no new divergence from integration at h = 0, it can still produce weakend divergence from the second term above. For instantce 2 (0) = 0. In sum, the UV divergent part of the integral F is There is a subtlety in this procedure. Although f 2 (h, δ) is regular at h = 0 by design, after expansion in δ, it may not be regular order by order. We are lucky that in our case the coefficients f Now we apply this procedure to the real calculations. In what follows, the functions ω(h), H ± (h) and the integration constant K appear frequently in our calculations. They are defined in equations (2.6) , (4.13) and (2.5) respectively. For sake of completness we list them again here:

A.1 Bulk contributions
First look at the bulk contribution eq(4.25): where the ± are for convex and concave kinks respectively. The ρ integration can be done analytically to get f 1 (h, δ), which decomposes as 2δ Rh , Note that the non-zero limit of B 2 at h = 0 will contribute to the δ −1 divergence after integrated against the UV cutoff region, as indicated by the second term in eq(A.6). Overall, the UV divergences are: where the integrands for the numerical integrations are (A.14)

A.2 Joint contributions
Next look at the joint contribution eq(4.33). After the ρ integration, we have The result is thus where the h = δ/R contributions give Note that the second lines in eq(A.20) and eq(A.21) are exactly opposite, indicating no dependence on β. Roughly speaking, these contributions are from ∂A. It is interesting to see that although the β dependence does not vanish in general, it might only come from the interior of the boundary region A. We list the integrands of the numerical integrations as follows The naming rules are 1) J stands for joint contributions instead of bulk ones denoted by B; 2) the subscripts denote the order of divergence, i.e. 2 for δ −2 , 1 for δ −1 , and each 0 for one log δ; 3) the superscripts show which joint surface source this term; 4) the tildes indicate that one of the log is of the form log(L/δ) instead of log(R/δ), which may or may not be important for future study.

A.3 Total Results
Now we can wrap up all the contributions and summarize as the following where the coefficients arē

B Induced Geometry on the Null Hypersurfaces
In this appendix, some detailed calculations about the null hypersurface geometry will be carried out. These results are involved in several discussions like Sec(4.2) and the joint contribution computations. Because we are always using w α = (ρ, θ) defined on the HRT surface as the parameters, the light sheet geometry is mostly convenient to be studied as induced geometry following the flow of V µ ± and U µ ± . Starting from the HRT surface, from eq(4.2) we get the induced metric on E ± : As λ is chosen to be affine parameter, the expansion rate of the congruence V µ ± can be computed
(B.4) and we drop the ± superscript due to time reflection symmetry. We also restore the > and < superscripts for concave and convex kinks respectively. We recognize the induced metric on the joint surfaces as g andg at different ends of the affine parameter ranges, and obtain the relations: (B.5)