Holographic entanglement entropy and complexity of microstate geometries

We study holographic entanglement entropy and holographic complexity in a two-charge, $\frac{1}{4}$-BPS family of solutions of type IIB supergravity, controlled by one dimensionless parameter. All the geometries in this family are asymptotically AdS$_3 \times \mathbb{S}^3 \times \mathbb{T}^4$ and, varying the parameter that controls them, they interpolates between the global AdS$_3 \times \mathbb{S}^3 \times \mathbb{T}^4$ and the massless BTZ$_3 \times \mathbb{S}^3 \times \mathbb{T}^4$ geometry. Due to AdS/CFT duality, these geometries are dual to pure CFT heavy states. We find that there is no emergence of entanglement shadow for all the values of the parameter and we discuss the relation with the massless BTZ result, underlying the relevance of the nature of the dual states. We also compute the holographic complexity of formation of these geometries, finding a nice monotonic function that interpolates between the pure AdS$_3$ result and the massless BTZ one.

problem. With this simplification, in sec. 4 we describe the computations of the holographic entanglement entropy, pointing out that no entanglement shadow emerges and briefly discussing the relation with known results for the BTZ black hole. Finally, in sec. 5, we compute the complexity for these states. We close the paper with a discussion in sec. 6.

Microstate Geometries
In the context of type IIB string theory on M 4,1 × S 1 × T 4 a wide set of microstate geometries has been constructed, both of 1 4 -BPS and 1 8 -BPS nature . These are smooth, horizonless and asymptotically flat geometries, whose conserved charges are the same as the "naïve" D1D5P black hole and that, in the decoupling region, behaves approximately as AdS 3 × S 3 × T 4 . These solutions can be described by some scalar "profile" functions Z I (as well as by a set of 1-and 2-forms, that we will briefly describe later), similarly of what happens for the D1D5P black hole. The core difference is that for microstate geometries there are no singularities in the geometry nor poles in the profiles; moreover, due to supersymmetry [38], the Asymptotically flat region is attached to the AdS 3 × S 3 × T 4 region simply by "adding back the 1" in the profile functions Z 1 , Z 2 , that encodes the D1 and D5 charges [41,42,45]. Finally, we will always be able to work in the decoupling limit whose asymptotics is AdS 3 × S 3 × T 4 , that is well suited for the study of Holographic Entanglement Entropy [51][52][53].
In this paper, we will focus on recently build superstrata (k, m, n) [38,[41][42][43]45]: these solutions are controlled by three integers and they are invariant under rotation of the compact T 4 . The generic ansatz takes the factorised form R 1,1 × B 4 × T 4 ds 2 10 = where everything is z i -independent. We have marked with an over-bar the forms which have legs only in the six-dimensional non-compact space, while we have explicitly written down the T 4 directions. These geometries can either be 1 4 -or 1 8 -BPS. Restricting to the v−independent base B 4 = R 4 , eq. (2.1) becomes 2 4 , ds 2 4 = Σ dr 2 r 2 + a 2 + dθ 2 + (r 2 + a 2 ) sin 2 θ dφ 2 + r 2 cos 2 θ dψ 2 , When F = 0 the geometry has two charges and it is 1 4 -BPS, while it has three charges and it is 1 8 -BPS otherwise. Notice that ds 2 6 is intended in the Einstein frame. It is worthy introducing some additional objects, that are gauge invariant under the remaining gauge freedom B 2 → B 2 + dλ 1 , where λ 1 is an u , v−independent 1-form and have legs only on the base space R 4 [44,45]: The functions introduced before should fulfil some constraints encoded in a set of differential equations, often informally dubbed "layers". The first layer is: while the second one is The crucial point to be stressed is that these geometries have no causal-disconnecting horizon nor curvature singularities, they are geodesically complete and present the same conserved charges as the D1D5P black hole. Interpreted from the AdS 3 /CFT 2 view point, they are dual to heavy states in the CFT, which are pure states, in contrast with the putative dual state of the naïve black hole geometry, which instead is a thermal one.

Factorisable geometries
As noted in [43,44,46,54], there exists a class of superstrata, described in the previous section, that have a factorisable form. Using the notation of [42], we will focus on the case with (k, m, n) = (1, 0, n), whose metric is we can write When n = 0 these are three-charge geometries, as can be easily seen from F n = 0, which signals a non-vanishing momentum along the S 1 .
We can write the full six-dimensional metric in the Einstein frame in the factorised form as a three-dimensional Asymptotically AdS 3 fibered along the S 3 , where we have split the coordinates as x M = (x µ , θ a ) with x µ = (τ, σ, r) and θ a = (θ, φ, ψ), G ab is a deformation of the S 3 written in Hopf coordinates. We have defined the three dimensional Einstein metric as ds 2 =g µν dx µ dx ν , endowed with the regularity condition The physical interpretation of these parameters will be clear in sec. 2.3, once we have introduced the dual CFT description.

The Two charge solution
A useful subset of geometries, the one we will mainly use, is the (k, 0, 0) type [20]. They are two-charge solutions, whose metric is generally non-factorisable, except when k = 1. In the factorisable (k, m, n) = (1, 0, 0) case some holographic studies have already been performed, and we will discuss them in the following section. For this particular configuration, the metric takes the simple form: (2.15) We are mainly interested in this one-parameter family of solutions, controlled by the parameter η ∈ (0, 1), because it shows some important and peculiar features. In particular, we can identify two interesting regimes in the η-parameter space: • η → 0, or a b: in this region the geometry approaches the massless limit of the BTZ black hole [19,20]; • η → 1, or a b: the geometry reduces to vacuum AdS in global coordinates.
It is known that the former geometry presents Entanglement Shadows [21][22][23], while the latter presents none. The main goal of this paper is to study if these Shadows could actually appear in the intermediate region where our solutions live. We will introduce this concept more deeply in sec. 3.

Holographic interpretation
All the geometries described above are supergravity solutions with an AdS 3 × S 3 × T 4 decoupling region, hence they are dual to a CFT, often dubbed D1D5 CFT [16-18, 52, 53]. In the moduli space there exists a special point, called "free-orbifold point", where this theory can be described as a (1+1)-dimensional supersymmetric non-linear sigma model whose target space is the symmetric orbifold of (T 4 ) N /S N . It has an SO(4) S 3 SU (2) L ×SU (2) R affine algebra of currents, dual to the isometries of S 3 , as well as a global "custodial" symmetry SO(4) T 4 SU (2) 1 × SU (2) 2 , dual to the isometries of T 4 . We will denote spinorial indexes α,α for SU (2) L × SU (2) R and A,Ȧ for SU (2) 1 × SU (2) 2 to label the CFT operators. The elementary field content is made by Left-and Right-Moving bosons and fermions In the theory, it exists a set of twist operators that joins together k-copies of elementary strings -or strands -to create a single strand of winding k; we will have then an untwisted sector and a twisted one (for more details, we refer to [18,53]). From the elementary fields it is possible to build the generators of the superconformal algebra : (2.17) Expanding them in modes, it is possible to select the generator of the global subalgebra. The theory splits into two sectors: the Ramond and the Neveu-Schwarz sector; in the latter there exists a single vacuum |0 NS , while the first one has 16 R-and custodial-charged vacua, both in the twisted and untwisted sector, that are The more relevant for us will be the highest-weight vacuum state |++ k and the custodialsinglet |00 k = ε AB |AB k . With these two vacua at hand, we can build the heavy states dual to the (k, m, n) geometries of sec. 2 by means of the action of the generators of the global subalgebra; in fact, we have that the dual CFT state of the supergravity solution is [41,42]  and whose pictorial representation is furnished in fig. 1. For example, the simplest state whose supergravity dual has the two-charge geometry (2.14) is with the constraint N (++) + N (00) = N . The duality relates N (00) , N (++) with the supergravity parameters a, b controlling the geometry through the relations: and precisely where we have introduced the two adimensional parameters η = 1 − ε that are sometimes used in the literature. As said on the supergravity side, when a → a 0 (η → 1) the geometry approaches the vacuum AdS 3 × S 3 × T 4 ; this is easy to see in the CFT side since the state |+ + N 1 goes under spectral flow to the NS-vacuum |0 N 1 . On the contrary, when a → 0 (η → 0), on the supergravity side it approaches the massless BTZ geometry, while the state in the CFT approaches the pure state |00 N 1 ; this simply means that the geometry dual to the pure state |00 N 1 differs from the one of the thermal state dual to the naïve black hole geometry by stringy corrections that are not captured in the supergravity regime under scrutiny. This fact will be relevant in the following discussion when we will stress the difference between the holographic results on pure (micro)states and on the naïve massless BTZ geometry dual to a thermal state.

The late behaviour and emergence of a "quantum" scale
As briefly mentioned in the previous section, some investigation on the holography of the microstate geometries have been conducted; for example, a set of nice computations of 4-point functions involving the two heavy operators dual to the geometry and two light operators dual to supergravity mode on top on that geometry were computed [44,52,[55][56][57][58][59]. We will now focus on the results of [20], since there the computation on the geometry (2.14) was put forward. There the authors computed the HHLL correlator involving two light operators made with the two elementary bosonic fields (2.27) In the a → 0 limit it is easy to see that this correlator approaches that has to be contrasted with the result on the naïve massless BTZ black hole (2.29) 2 We recall that the plane/cylinder map is C micro (⌧ ) Figure 2: A pictorial representation of the HHLL correlator computed in both the naïve massless BTZ geometry (in dash-dotted red) and in the pure microstate geometry (in violet). Up to a certain time τ ∼ a 2 0 /a 2 , the two correlators present the same decaying behaviour; after that, the two starts to differ: the BTZ one maintains its decaying behaviour, while the microstate one starts oscillating, as imposed by unitarity.
It is easy to see that the two results agree up to a time scale τ ∼ a 2 0 /a 2 , that is when there is no time for the perturbation to probe the different geometric structure of the two geometries; after that, the correlator computed on the naïve geometry decays exponentially while the other one start oscillating indefinitely, as prescribed by unitarity. We report a pictorial representation of that in fig. 2.
We have thus seen appearing a dramatic difference of an interesting observable in the two geometries, that signals a different behaviour of observables between pure and thermal states. We want to study other interesting observables on microstates in order to understand better the (holographic) properties of these geometries. In the following sections, we will thus focus on the study of Holographic Entanglement Entropy and Holographic Complexity.

Holographic Entanglement Entropy in separable six-dimensional geometries
One of the most relevant observables that can be computed holographically is the Holographic Entanglement Entropy (HEE) via the so-called Ryu-Takayanagi (RT) formula [5,7,14,60,61]. The main idea is that one can associate the Entanglement Entropy of a subregion A on the AdS-boundary, where the CFT lives, to the Area A of the minimal area surface in the bulk, whose boundary is the same as the one of A, i.e.
One of the issue that may arise in the study is the emergence of an Entanglement Plateux [21] or entanglement shadow [22,23]; it consists in a region unexplored by any minimal surface attached to any possible region on the boundary and that could lead to a limit to the possibility of a complete bulk reconstruction from CFT data. It is known that this phenomenon appears on black hole solutions, as well as in some higher-dimensional bubbling geometries, e.g. on the so-called Lin-Lunin-Maldacena (LLM) geometries [62]. We thus would like to address the computation of holographic entanglement entropy in microstate geometries 3 for all the possible intervals on the boundary CFT, focusing on the study of possible entanglement shadows in the two-charge geometry (2.14). As we have already said, this type IIB solution is a one-parameter family of geometries that is factorisable as AAdS 3 × S 3 × T 4 and that somehow interpolates between the pure global vacuum AdS 3 × S 3 × T 4 (when a → a 0 ) and the massless BTZ 3 × S 3 × T 4 geometry (when a → 0). Since the former shows no entanglement shadow while the latter does [22] 4 , we are interested in addressing this issue for generic values of the ratio a/a 0 . In order to compute it we will need some ingredients: • A slight modification of the RT proposal for geometries that have only an AdS factor, a proposal that was put forward in [51,52]; • A proof that this proposal reduces to compute the HEE via standard RT on AdS 3 for factorizable (or separable) geometries, allowing us to reduce consistently to three dimensions. This proof will be stated in sec. 3.1.
• A numerical procedure for extracting the shadow radius and then for computing the entropy.
After all these steps, we will show that for any value of the ratio a/a 0 , the geometry (2.14) shows no entanglement shadow whatsoever, even for a approaching zero. One thus may wonder how the non-vanishing shadow result of [22] is obtained in the limit a → 0. It is due to a somewhat non-trivial mechanism: what happens is that, at a certain value of the ratio a/a 0 , namely a/a 0 ∼ 0.55 corresponding to approximately to a dual CFT pure state that has as many |00 1 strands as |+ + 1 strands, the non-linear equation that fixes the shadow radius r * in terms of the size of the boundary interval α ∈ (0, π) starts to admit two possible solutions, an r * = 0 solution (valid ∀a) and an r * = 0 one, that instead exists only for a ∈ (0, 0.55)a 0 . This means that for some values of the ratio a/a 0 there exist two extremal surfaces that are connected with the interval; the key point is that the one with r * = 0 is maximal, while the other with r * = 0 in minimal (and thus its area computes the HEE). While this seems to hold for all possible values of the ratio a/a 0 , the specific case a = 0 should be treated with extreme care, since the r = 0 point is the location of the point-size horizon and this it is not part of the spacetime. This means that we are not allowed to keep the r * = 0 geodesic since it becomes unphysical, leaving us with the r * = 0 geodesic as the only extremal geodesic whose boundary is the same as the boundary of the interval, reproducing the result known in the literature [22] 5 .

Ryu-Takayanagi formula in separable geometries and reducing formula
One of the issues that arises in the study of entanglement entropy in microstate geometries is that these have a truly six-dimensional nature and they reduce to an AdS 3 ×S 3 structure only asymptotically. It is thus necessary to extend the definition of the Ryu-Takayanagi formula suitably. A proposal to extend the prescription for calculating the HEE was presented in [51]; it applies to stationary geometry that have AdS×S asymptotics, and generalises the covariant formula put forward in [5,7]. According to [51], given a 1D spacial region A = [0, l], the HEE should be defined as: where A is a co-dimensional 2 extremal surface in the six dimension spacetime such that at the boundary it reduces to ∂A × S 3 , and where G N is the six-dimensional Newton constant.
Here we prove that for a special class of geometries it is possible to reduce the sixdimensional problem to a three-dimensional one, to whom the original RT prescription applies.
We have shown in sec. 2 that it is always possible to rewrite our six-dimensional metric with an almost product structure as where we have split the six dimensional coordinate system as x M = (x µ , θ a ). Now we defineg which is the reduced three-dimensional Einstein metric, where det G = det G αβ and G 0 is the round metric of S 3 . For some 3-charge solutions, in particular for the superstrata (1, 0, n) in eq. (2.20), g E µν does not depend on the coordinates on S 3 . For a metric that has this property, it is possible to prove that the extremal surface in six dimensions is equivalent to x µ × S 3 , where x µ (λ) is a geodesic ong E µν . This greatly simplifies the computation of the HEE since it allows to restrict our attention to the pure three-dimensional part of the metric, which is asymptotically AdS and whose HEE can be computed via the usual RT prescription: where L γ is the lentgh of the space-like geodesic with minimal area on the AdS 3 part of the metric. In addition we have used the fact that for microstate geometries we have where c is the central charge of the dual CFT, where n 1 , n 5 are the numbers of D1, D5 branes and R AdS the AdS radius. According to eq. (3.2), in the full six-dimensional metric we need to find an extremal four-dimensional surface in order to compute the HEE, which we parametrize as x I (λ, θ a ), I = {µ, a} 6 . The induced metric on the submanifold is then where we have defined and where we have denotedẋ µ ≡ ∂x µ ∂λ . Following the idea of Ryu and Takayanagi, x I should extremize the area functional (3.7) We want to find under which assumptions the minimization problem in higher dimensions is equivalent to the one in three dimensions. In terms of extremal surfaces, we want to show which conditions the original metric must satisfy in order forx I = (x µ (λ), θ a ) (where x µ (λ) is a geodesic ofg E ) to be a solution of the minimization problem (3.7). Extremizing the functional I is equivalent to solve Euler-Lagrange equations: In order to prove that the three-dimensional geodesic is a solution, we will compute eq. (3.8) inx or equivalently considering ∂ a x µ = 0. Let us look in detail at the induced metric (3.6) 7 : Looking at the components of g * IJ we immediately realise that, when considering derivatives w.r.t. x ρ andẋ ρ in (3.8), there is no difference in differentiating the full induced 7 For later use, we report the induced metric evaluated at the solution: and its inverse:ḡ where we have denoted g λλ the inverse of: Moreover we have detḡ = g µνẋ µẋν det (G αβ ) =g E µνẋ µẋν det (G 0 ) . (3.11) metric or directlyḡ, since terms proportional to ∂ a x µ are not involved in differentiation and can be simply put to zero. It then implies that The only non trivial term in the Euler-Lagrange equations is the last one, but we can see that (3.14) Firstly, we have where the only components that contribute are (I, J) = (λ, a) or (a, b), and computed in the solution they give We can finally perform the last derivative, substituting g µν withg E µν and defining finding that where ∇ 0 a is the covariant derivative w.r.t. the round metric of the 3-sphere. The three-dimensional geodesic x µ is thus a solution if and only if this term vanishes. From eq.(3.18) we see that this happens if two conditions are satisfied: 1. ∂ a g E µν = 0; in this way the first and third term in the second line of eq. (3.18) are zero. But this requirement is exactly the definition of a factorizable metric, i.e. a metric whose three-dimensional Einstein metric of (3.3) does not depend on angular coordinates; 2. ∇ 0 a A a ρ = 0; this is a gauge choice that can always be employed. It is called the de Donder gauge.
Summarising our results, we have proven that for a factorisable six-dimensional metric in the de Donder gauge, the full HEE defined in eq. (3.8) reduces to: which is the same minimization problem we have to solve to find the geodesics ofg E µν . Property 1 is non-trivial to be realised and might depend on a clever coordinate choice in the original six-dimensional metric. As expressed in the previous section, we will study special microstate geometries whose six-dimensional metric are indeed factorised, hence we will make extensive use of eq. (3.19) in order to find their HEE.

Computing the Holographic Entanglement Entropy
Since in sec. 3.1 we have proven that for factorisable geometries in the de Donder gauge the six-dimensional problem reduces to the three-dimensional one, we can start directly with the stationary three dimensional metric ds 2 = g tt dt 2 + g rr dr 2 + g yy dy 2 + 2g ty dt dy , where we have assumedg E ry = 0 =g E rt = 0, which is always the case for our geometries. It is convenient to parametrize the geodesic x µ (λ) in terms of proper time, i.e. whereẋ µ ≡ dx µ dλ . We recall that geodesics are defined by the minimization of the functional: Since we are assuming that the metric does not depend explicitly on t and y, Euler-Lagrange equations with the respect to these coordinates gives two constants of motion, namely: where C 1 and C 2 are two constants of motion to be fixed later. Solving (4.2),(4.3) taking into account the condition (4.1), one finds: (4.5) (4.6) The two integration constants C 1 and C 2 are determined by the choice of boundary conditions. We are interested in a spatial region A at fixed time (t =t), made of an interval of length l. The endpoints of the geodesic have to lie at the boundary of AdS, but for r → ∞ the area diverges, as expected for the HEE in the dual CFT. So, as usual, we introduce an IR cut-off r 0 , which we will consider as the AdS boundary for computations, and at the end we will take the results for large r 0 . Thus boundary conditions reads: where r * is the geodesic turning point, such thatṙ| r=r * = 0. In the end we compute the HEE via eq. (3.5).
Up to now, we can apply our result to the generic stationary factorisable (1, 0, n) family of solutions; in order to give a concise and consistent explicit result, in the following we will restrict ourself to the static (k, m, n) = (1, 0, 0) case.

The static k = 1, n = 0 geometry as an explicit example
We study now the easiest, static case: the two charge (k, m, n) = (1, 0, 0) geometry in (2.14). When the metric is static we can take a submanifold at constant t and the only relevant components of the metric remain g yy (r) and g rr (r). eqs. (4.4)-(4.6) simply reduce to: The turning point is now defined through g yy (r * ) − C 2 = 0 (4.10) Boundary conditions are the same as the stationary case, apart the fact that condition (4.7) is automatically satisfied and (4.7b) and (4.7c) simplify to (4.12) Let us start from determining the turning point r * , that is given by the solution of where r * is determined by the constraint 8 : For convenience we will indicate with α the ratio l R , α ∈ [0, 2π], and we define f a/a 0 (r) = α−arccos  , a) of f a/a 0 (r * ) in eq. (4.16) for α = π, that is the maximal-size interval on the boundary. While r * = 0 is always a solution, for a (0.55 ± 0.01)a 0 there exists two possible r * for any a. We thus need to compute the Area for both of those solutions.
The Entanglement Entropy of a region A, made of an interval of length l is thus Unfortunately we were not able to invert eq. (4.14) analytically and thus express S A as a function of l. We will discuss it in the two interesting regimes a b and b a to give an analytical intuition about the results, while we resort to a numerical computation in the generic regime.

Numerical analysis
We resort to numerical computations to find the roots of f a/a 0 (r * ), and we employ the NumPy library of Python for the numerical procedure. As reported in fig. 3, while r * = 0 is a solution for all values of a, there exists a range of values for a where the second branch of possible roots exists. This range is approximately a ∈ (0, 0.55)a 0 for the maximal-size boundary interval α = π and on that branch r * > 0, signaling a possible existence of an entanglement shadow on these geometries. This means that, for a 0.55 a 0 , there are two extremal surfaces that are attached to the boundary. In order to understand which one of these two extremal surfaces is minimal, we need to (numerically) compute the entanglement entropy. Surprisingly, the minimal surface is always the curve with vanishing entanglement shadow radius, for which we have (4.18) implying that, on pure microstate geometries, there is no entanglement shadow whatsoever. This is somehow expected, since these geometries are dual to pure CFT states, and it seems unnatural to find an obstruction to bulk reconstruction on the supergravity side for these states. Since the HEE seems to be well defined up to a = 0, one may wonder how it is possible to recover the standard massless BTZ result, that has a non-vanishing shadow. As explained above, this is due to the fact that, in the strict a = 0 case, for the naïve massless BTZ geometry, the r = 0 point of the space-time is the location of the (string-sized) black hole horizon; this means that the geodesic that passes through r = 0 has no physical meaning, and has to be discarded 9 . This means that only the HEE for the curve with non-vanishing r * is physically viable and thus we reproduce the results of [22]. We will now move to compute analytically the entanglement entropy in the asymptotic regions a b and a b, to have at least some approximate analytical expressions for the entanglement entropy in these two regions, that represent perturbative corrections to the vacuum AdS 3 and BTZ 3 case, respectively.

The b a limit
In the a/a 0 → 1 or b a case, the problem is exactly the same as pure AdS 3 10 up to the order b 2 , so we have (4.19) and then the HEE is the same as the pure AdS 3 case This is obvious since we have seen that, up to a certain value of a, r * = 0 is the only solution to the equation f a/a 0 (r * ) = 0.

The b a limit
In the a/a 0 1 or b a regime the geometry approaches the naïve massless BTZ geometry. The eq. (4.14) for r * > 0 simplifies a lot and allows us to invert the relation , (4.21) so that the entanglement entropy becomes where we recall that r 0 is the IR cut-off. We have then reproduced the usual BTZ result plus corrections that reduces the size of the entangling shadows. This result is the approximate expansion near a = 0 of the blue curve of fig. 4, that turns out not to be the minimal surface. It is still worth to compute the correction to it, since one may wonder what happens in geometries with small corrections w.r.t. the naïve black hole geometry. 10 In this approximation up to O(b 2 ) the metric is: there is no difference between the computation in this metric and that one for pure AdS since at order b 2 the only term which receive corrections is g tt , which never enters the calculations.

The Holographic Complexity on microstate geometries
In the last years, one new observable gained attention in the black hole community, as well as in the quantum information one: the (holographic) complexity [2, 3, 8-10, 12, 64-75]. The two proposals for computing holographically the complexity are the "complexity = action" or "C=A" proposal, where the complexity is proposed to be dual to the on-shell action on the Wheeler-de Witt patch, while the second one is the "complexity = volume" or "C=V" proposal, where the complexity is proposed to be dual to the volume of the maximal spacelike co-dimension 1 surface in the bulk. We will study the latter for its simplicity in the case at hand. The first thing to notice is that, for factorisable metric in the de Donder Gauge, one can easily generalise the computation of sec. 3.1 and prove that one may reduce to compute the volume of the maximal spacelike slice in the Euclidean three dimensional metric.

The Complexity = Volume of the two-charge geometry
If we assume that the (holographic) Complexity of the state on a boundary time-slice Ω is given by the volume of the extremal time-slice B (such that Ω = ∂B) via the formula we have that, in the (1, 0, 0) geometry, the induced 2-dimensional geometry is where we recall that η = a 2 /a 2 0 , so that it is straightforward to compute the holographic complexity as where we set G 3 = 1 and we recall that = a 0 . As usual when computing holographic quantities in AdS, the value of the complexity turns out to be divergent; this is why we introduced a regularising scale r 0 . We can then compute the regularised Complexity (or "complexity of formation" [12]) by subtracting the "vacuum" contribution, i.e. C vac V ≡ C V (η = 1) 11 : This has the correct behaviour; when η → 1, i.e. when we are approaching the most "simple" state, C reg V (η) → 0; also, when η → 0, i.e. when we are approaching the most "complex" state, C reg V (η) → 2πa 0 . In other words: • when η → 1 so that the 1 4 -BPS state (2.22) approaches ψ {N (++) ,N (00) } ∼ |++ N 1 , we have that C reg V (η) → 0; • when η → 0 so that the 1 4 -BPS state approaches ψ {N (++) ,N (00) } ∼ |00 N 1 , we have that C reg V (η) → 2πa 0 ; we can also see from fig. 5 that the behaviour of C reg V (η) is indeed monotonic.

Discussion
In this paper, we have discussed the holographic entanglement entropy for a one-parameter family of microstate geometries that are asymptotically AdS 3 × S 3 × T 4 . This family of geometries is controlled by the dimensionless parameter η ∈ (0, 1) and has the peculiarity that, for η → 0, it approaches the massless BTZ geometry, while for η → 1 it approaches the vacuum global AdS 3 geometry.
We have first shown that, for factorisable six-dimensional geometries in the de Donder gauge, the problem of computing the HEE reduces to a pure three-dimensional one; after that, we have resort to a numerical procedure to compute the minimal-area geodesic, finding that for all solutions of the one-parameter family we have considered there is no entanglement shadow. The key point is that the non-linear equation, relating the size of the boundary interval l = αR to the turning point radius r * (and thus to the entanglement shadow), admits two solutions in the η ∈ (0, 0.55) region: one is r * = 0 while the other has r * > 0. When computing the length of the associated geodesics, it turns out that the latter is always bigger than the former, thus indicating that this family of geometries never shows any entanglement shadows.
It seems to be in contrast with the result for the massless BTZ case, which does show entanglement shadow [22]; what happens is that the result is recovered by a somewhat non-trivial mechanism. In fact, for a = 0, the geodesic with r * = 0 does not exist since the r = 0 is the location of the (degenerate) black hole horizon; this means that in this case there exists only one geodesic, namely the one with r * > 0, and we thus reproduce the known result for the massless BTZ black hole. This suggests that the naïve BTZ geometry (dual to a thermal state) is strikingly different from the microstate one (dual to the pure state |00 N 1 ), even if the difference is not captured by the supergravity regime. We have also computed the holographic complexity via the "complexity = volume" duality, since the results of sec. 3.1 can be easily generalised to the computations of the volume of the extremal slice for this family, finding a monotonic function in the parameter η, that interpolates smoothly the known results for vacuum global AdS 3 with the one for massless BTZ case [12].
One may wonder to extend the results of this paper among many directions; one may try to use the results of sec. 3.1 to stationary but not static geometries, as the 1 8 -BPS factorisable geometries of the (1, 0, n) superstrata, where one needs to refine the numerical procedure, as well as trying to compute the complexity via the "complexity = action", trying to understand if it is possible to reduce the problem from a six-dimensional to a three-dimensional one even in this case. In the case of non-stationary geometries that are separable, e.g. the (1, 0, n) superstrata, the holographic entanglement entropy computations become more involved, and different numerical procedure may be needed, but the basic ingredients that are necessary to achieve that are the same employed in this paper, and we expect similar results to hold, i.e. that any entanglement shadow emerges, even for finite n.
that immediately tells us that, taking the largest possible interval on the boundary α = π, we have a non-vanishing shadow matching the results of [22]; also, in the massless limit r h → 0, we have matching the results of [76]. Again, in the massless limit

A.2 The Complexity = Volume
Since the metric is static, the maximal spacelike slice is obtained by setting t =const. and then V BTZ = 2π We can now easily compute the complexity of formation, that is that is a constant independent of the mass of the BTZ black hole [12].