Null-polygonal minimal surfaces in AdS_4 from perturbed W minimal models

We study the null-polygonal minimal surfaces in AdS_4, which correspond to the gluon scattering amplitudes/Wilson loops in N=4 super Yang-Mills theory at strong coupling. The area of the minimal surfaces with n cusps is characterized by the thermodynamic Bethe ansatz (TBA) integral equations or the Y-system of the homogeneous sine-Gordon model, which is regarded as the SU(n-4)_4/U(1)^{n-5} generalized parafermion theory perturbed by the weight-zero adjoint operators. Based on the relation to the TBA systems of the perturbed W minimal models, we solve the TBA equations by using the conformal perturbation theory, and obtain the analytic expansion of the remainder function around the UV/regular-polygonal limit for n=6 and 7. We compare the rescaled remainder function for n=6 with the two-loop one, to observe that they are close to each other similarly to the AdS_3 case.


Introduction
The AdS/CFT correspondence shows that minimal surfaces in AdS space-time are dual to the Wilson loops along their boundary [1,2], where the area corresponds to the expectation value of the Wilson loops at strong coupling. When the boundary is null-polygonal/lightlike, the minimal surfaces also give the gluon scattering amplitudes of N = 4 super Yang-Mills theory [3], implying the duality between the amplitudes and the Wilson loops [3][4][5] and hence the dual conformal symmetry [3,6,7] . This dual conformal symmetry completely fixes the n-point amplitudes/Wilson loops with n cusps up to n = 5. For n ≥ 6, however, it allows deviation from the Bern-Dixon-Smirnov (BDS) formula [8] by the remainder function [9][10][11], which is a function of the cross-ratios of the cusp coordinates on the boundary.
At strong coupling, the corresponding area of the minimal surfaces is evaluated with the help of integrability [12]. More concretely, one first solves a set of integral equations of the thermodynamic Bethe ansatz (TBA) form, or an associated Y-/T-system [13][14][15]. The cross-ratios are then expressed by its solution, i.e., the Y-or T-functions, and consequently the main part of the remainder function is given by these Y-/T-functions as well as the free energy associated with the TBA system.
In a previous paper [15], Sakai and the present authors pointed out that the TBA equations for the minimal surfaces with 2ñ cusps in AdS 3 coincide with those of the SU(ñ − 2) 2 /U(1)ñ −3 homogeneous sine-Gordon (HSG) model [16] with purely imaginary resonance parameters. Similarly, it was inferred there that the TBA equations for the minimal surfaces with n cusps in AdS 4 are those of the HSG model associated with SU(n − 4) 4 /U(1) n−5 .
These observations allow us to solve the TBA systems around the UV/high-temperature limit, where the two-dimensional integrable (HSG) model reduces to a conformal field theory (CFT). The deviation from the UV limit then corresponds to an integrable relevant/mass perturbation of the CFT. The corrections to observables are also regarded as finite size effects of the two-dimensional system, which can be computed by using the conformal perturbation theory (CPT). By the standard procedure [17], one can indeed derive an analytic expansion of the free energy around the UV limit. The Y-/T-functions are also expanded by the CPT with boundaries [18,19], based on the relation to the g-function (boundary entropy) [20].
Since the Wilson loops become regular polygonal in the UV limit, those expansions give an analytic expansion of the remainder function around this regular-polygonal limit. For the analysis in the opposite IR/large-mass regime, see [12,13,[21][22][23][24][25].
We have carried out the above program for the minimal surfaces embedded in AdS 3 [26,27]. In this case, the relevant CFT is the SU(ñ − 2) 2 /U(1)ñ −3 generalized parafermion theory [28] and, by turning off some mass parameters so as to leave only one mass scale (single-mass case), the TBA system is reduced to simpler ones for the perturbed SU (2) diagonal coset and minimal models. This is a key step which enables us to find precise values of the expansion coefficients in terms of the mass parameters in the TBA system, through the relation to the coupling of the relevant perturbation (mass-coupling relation) and the correlation functions. We then derived the expansion of the 8-and 10-point remainder functions in [26], and that of the general 2ñ-point remainder function in [27]. We observed that the appropriately rescaled remainder functions are close to those evaluated at two loops [29][30][31].
The purpose of the present work is to study the analytic expression of the regularized area of the null-polygonal minimal surfaces in AdS 4 by extending the above results in the AdS 3 case. In particular, we derive the analytic expansion of the remainder function around the UV limit by using the underlying integrable models and the CPT. In this case, the corresponding TBA or Y-system is obtained by a projection from that for the minimal surfaces in AdS 5 [14]. The relevant CFT in the UV limit for the n-cusp surfaces is now the SU(n − 4) 4 /U(1) n−5 generalized parafermion theory. The TBA systems with only one mass parameter are given by those for the perturbed unitary SU(4) diagonal coset models and W minimal models. We also argue that a similar correspondence to the perturbed nonunitary diagonal coset and W minimal models holds for the systems with a pair of equal mass parameters. These generalize the reduction in the AdS 3 case, and are used to find the precise expansion coefficients. Explicitly, we work out the leading-order expansion for n = 6 and 7. In these cases, the input from the perturbed W minimal models completely determines the leading-order expansion. For n = 6, we also compare the rescaled remainder function with the two-loop one which is read off from [32][33][34][35], to observe that they are close to each other. This paper is organized as follows: In section 2, we review the remainder function corresponding to the minimal surfaces in AdS 4 , and the associated TBA system. We explicitly check that the TBA equations for the n-cusp minimal surfaces are obtained from the SU(n − 4) 4 /U(1) n−5 homogeneous sine-Gordon model. In section 3, we discuss the TBA systems in the single-mass cases in relation to the perturbed SU(4) diagonal coset and W minimal models. In section 4, we discuss the expansion of the free energy, and derive the leading-order expansion for n = 6 and 7. In section 5, we extend the formalism of the expansion of the T-/Y-functions to the AdS 4 case, and derive the leading-order expansion for n = 6 and 7. Combining those results, we derive the analytic expansion of the remainder function for n = 6 and 7 in section 6. We also compare the rescaled remainder function for n = 6 with the two-loop one. In the appendix, we summarize a computation of a three-point function in a non-unitary W minimal model.

TBA equations for minimal surfaces in AdS 4
In this section, we review the computation of the regularized area of the minimal surfaces in the AdS space with a null polygonal boundary using integrability. As studied in [13][14][15], such an area is governed by a set of non-linear integral equations of the TBA form or the associated T-/Y-systems. Those equations coincide with the TBA equations of the homogeneous sine-Gordon model.

Functional relations and TBA equations
The basic idea to compute the area of the minimal surfaces is as follows. We start with the non-linear sigma model that describes the classical strings in AdS 5 . After the Pohlmeyer reduction, the equations of motion for classical strings are mapped to a linear system of differential equations. Due to the integrability of the linear system, one can introduce a spectral parameter θ. Using the bispinor representation, this system is brought to the SU(4) Hitchin system with a Z 4 -symmetry. Solutions of this linear problem show the Stokes phenomena [12]. The smallest solution is uniquely determined in each Stokes sector. Their Wronskians evaluated at special values of the spectral parameter form the cross-ratios of the cusp coordinates. From the Plücker relations, these Wronskians satisfy certain functional relations called the T-system. For AdS 5 , it reads as the following relations among the T-functions T a,s (θ), where a = 1, 2, 3; s = 1, 2, · · · , n − 5 for the n-cusp minimal surfaces and f ± (θ) := f (θ ± iπ 4 ). The boundary conditions for the T-functions are T a,0 = 1 (a = 1, 2, 3), T 0,s = T 4,s = 1 (s ∈ Z). (2.2) At the boundary s = n − 4, we have also to impose the boundary condition related to the formal monodromy [14]. For n ∈ 4Z, the condition is simply given by They satisfy a set of functional relations called the Y-system: The boundary conditions for the Y-functions are Y a,0 = Y a,n−4 = 0 (a = 1, 2, 3) and Y 0,s = Y 4,s = ∞ (s = 1, · · · , n − 5). The Y-system has many solutions in general. To determine a solution of the Y-system, we need to know the analytic structure of the Y-functions including their asymptotics for large |θ|, which has been studied in [14]. The asymptotics is specified by auxiliary complex (mass) parameters m s and constants C s , D s . For real m s , it is given as θ → ±∞. One can show that the Y-system can be rewritten into a set of integral equations of the TBA form. Those equations for the minimal surfaces in the AdS 5 space are given by where * stands for the convolution, f * g := ∞ −∞ dθ f (θ − θ ′ )g(θ ′ ). The functions α s , β s and γ s are defined by , (2.8) and the kernels are by The constants D s are obtained from γ s by D s = i π dθ γ s (θ), whereas the constant µ in In this paper, we particularly focus on the minimal surfaces in the AdS 4 subspace, which correspond to the amplitudes with the four-momenta of the external particles lying in a three-dimensional subspace. In this case, the above integral equations are simplified to the TBA equations of a known integrable system. To reduce the problem into AdS 4 , we need a projection of the original system for the AdS 5 space. This projection relates the smallest solutions of the linear problem to those for the inverse problem via a gauge transformation.
This relation results in the following conditions in the TBA system, where the latter relation leads to µ 2 = 1. In this paper, we particularly consider the case with µ = 1 and C s = 0, so that one can analyze the area for small m s by the underlying integrable models and conformal field theories. Then, we obtain the simplified integral equations, where α s and β s reduce to .
So far, we have focused on the real mass (m s ) case. One can generalize these results to the complex-mass case as in [14]. If the masses in the TBA equations are complex, m s = |m s |e iϕs , the driving terms of the TBA equations are modified as −m a,s cosh θ → − 1 2 (m a,s e θ + m a,s e −θ ) = − |m a,s | 2 (e θ−iϕs + e −(θ−iϕs) ), (2.13) where m 1,s = m 2,s / √ 2 = m s . Thus, definingỸ a,s (θ) = Y a,s (θ + iϕ s ), the TBA equations become Note that this integral equations are valid only when |ϕ s − ϕ s ′ | < π/4 for all s, s ′ . If at least one of |ϕ s − ϕ s ′ | is greater than π/4, we need to modify the TBA equations due to the poles in the kernels. The complex masses m s provide 2(n − 5) independent real parameters for the TBA system, which match the number of the independent cross-ratios formed by the cusp coordinates of the AdS 4 minimal surfaces. Here, we denote the SU(N) affine Lie algebra at level k by SU(N) k . This SU(N) k /U(1) N −1 coset CFT has the central charge, and its primary field Φ Λ λ (z) with weight λ in the highest-weight representation labeled by Λ has the conformal dimension, is half the sum of the positive roots (the Weyl vector) of the Lie algebra su(N). The action of the HSG model then takes the form, where Φ is a combination of the weight-zero adjoint operators Φ , which is parametrized by the N − 1 real mass parameters M s (s = 1, · · · , N − 1) and the real resonance parameters σ s . This perturbing operator Φ has the dimension, On dimensional grounds, the coupling of the integrable relevant/mass perturbation is expressed by the dimensionless coupling κ and the mass scale M as (2.20) We note that the above action describes a multi-parameter integrable perturbation, which is a notable feature of the HSG model.
The particles in this model are labeled by two quantum numbers (a, s) and have masses where a = 1, ..., k − 1. The S-matrix of the diagonal scattering between the particles (a, r) and (b, s) is then given by [37] S rs ab (θ; σ rs ) = S min ab (θ) where σ rs := σ r − σ s , and is the S-matrix of the A k−1 minimal affine Toda field theory (ATFT) [38,39]. In the second factor, I rs = δ r,s+1 + δ r,s−1 is the incidence matrix of the Lie algebra su(N), η r,s (= η −1 s,r ) are arbitrary k th roots of −1, and S F ab is given by The parity-invariance of S rs ab is broken due to η r,s and σ rs . Next, we recall that, for a diagonal scattering theory with the S-matrix S AB , the TBA equations in the fermionic case take the form (see for example [40]), where K AB (θ) are the kernels defined by On the right-hand side, m A = M A L is the dimensionless combination of the mass parameter M A and the length scale/inverse temperature L. We have also assumed above that the kernels are symmetric: K AB (θ) = K BA (θ). Once the resonance parameters are set to be vanishing, σ rs = 0, the kernels for the HSG model are indeed symmetric and one can apply this formula.
for the kernels K AB = K rs ab . When the mass parameters are complex, m s = |m s |e iϕs , the phases correspond to the purely imaginary resonance parameters σ s = iϕ s . One finds that the TBA equations in this case are given by (2.14).

Remainder function
In the previous two subsections, we have seen that the null-polygonal minimal surfaces with n cusps in AdS 4 are characterized by the TBA equations for the SU(n − 4) 4 /U(1) n−5 HSG model. We would like to know the area of such minimal surfaces. Here we see that the area can be expressed in terms of the T-/Y-functions, the free energy and the mass parameters associated with the TBA system.
The area shows divergence, since the surfaces extend to the boundary of AdS at infinity and have the cusp points there. Introducing the radial-cutoff, the regularized area is given by the Stokes data of the linear problem. For the n-cusp minimal surfaces, it takes the form, where A div is the divergent part, A periods is the period part which depends on the mass parameters governing the asymptotics of the Y-functions. A BDS−like is given by distances among the cusp points, which is similar to the BDS expression but different. A free is the free energy associated with the TBA system.
The remainder function is now defined by the difference between the regularized area and the BDS formula, The explicit form of the remainder function at strong coupling is then given by The first term ∆A BDS := A BDS-like − A BDS is expressed in terms of the cross-ratios, and its general expression for n / ∈ 4Z is found in [13]. Here, we list the expressions for n = 6 and 7, which are used in the following sections: , (2.32) for n = 6 and ∆A (n=7) for n = 7 (see also [23]). The cross-ratios u i,j above are defined by with f [r] := f (θ = iπr/4) as follows [14], Other cross-ratios are generated by the Z n -symmetry, x µ i → x µ i+1 , which corresponds to the shift of the argument In addition, other parts A periods and A free are given by The explicit forms of K ss ′ are found in [14] for n ∈ 4Z, and are conjectured in [21] for n ∈ 4Z. Here we list the results for n = 6, 7 only: Although ∆A BDS is given by the cross-ratios directly, A periods and A free are related to the cross-ratios indirectly through the Y-/T-functions and the mass parameters. Indeed, the Y-functions are uniquely determined by solving the TBA equations for given masses and, once Y a,s are obtained in terms of m s , the mass parameters and hence the Y-/T-functions are related to the cross-rations through (2.35). As a result, the remainder function at strong coupling is expressed as a function of the cross-ratios.

UV expansion
In the following sections, we discuss an analytic expansion of the remainder function around the high-temperature/UV limit, where the mass parameters m s become vanishing and the corresponding Wilson loops become regular-polygonal. This is achieved by several steps: First, we note that around this limit the deformation term in (2.18) is treated as a smallmass perturbation for the coset/generalized parafermion CFT [28]. Then, the free energy of the TBA system, which is given by the ground-state energy in the mirror channel, is obtained analytically by the conformal perturbation theory [17]. It is expanded in terms of the correlation functions of the deformation operator Φ. Next, we use the relation between the Y-/T-function and the g-function [20]. The g-function is regarded as a boundary contribution to the free energy, and analytically expanded by the CPT with boundaries [18,19].
In the course of the discussion, we first set the mass parameters to be real to keep the boundary integrability. Their phases are recovered after the expansion is obtained, so that the Z n -symmetry is maintained.
These expansions are first given in terms of the coupling λ. To find the expansion in terms of the mass parameters, we need the precise form of Φ and the relation between λΦ and m s . Once this mass-coupling relation is found, one can obtain the expansion in terms of the cross-ratios through (2.35) as discussed in the previous subsection.
Since there are multiple deformation operators in our case, it is a rather difficult problem to find the exact mass-coupling relation due to operator mixing. However, when some mass parameters are turned off so as to leave only one mass scale (single-mass case), the TBA system reduces to simpler ones and the problem becomes tractable.
In the next section, we begin our discussion of the UV expansion by considering the perturbation with single mass scale for the AdS 4 minimal surfaces. We see that the TBA systems in such cases reduce to those of the perturbed SU(4) diagonal coset models or W minimal models. For the 6-and 7-cusp cases (n = 6, 7), it turns out that the input from the W minimal models is enough to completely determine the leading-order expansion.
3 Perturbation with single mass scale and W minimal models  [41]. In particular, when r = 1, they become those of the unitary minimal model Mñ −1,ñ perturbed by the φ (1,3) operator [42,43]. are enough to completely determine the leading-order analytic expansion around the UV limit [26].
As discussed in the previous section, the n-cusp minimal surfaces in

Perturbed unitary diagonal coset/W minimal models
When only one mass parameter is turned on, M s = δ s,r M, one expects from the AdS 3 case that the TBA system of the HSG model reduces to that for the unitary In particular, when r = 1, the above model becomes equivalent to the perturbed unitary W minimal model, Here, we have used the relations (3.11) and (3.12).
This expectation is also supported by an observation that the TBA system of the SU(N) given by the TBA system of the perturbed SU(N) 1 × SU(N) m /SU(N) 1+m diagonal coset model [45]. Indeed, one can explicitly check that the above correspondences of the TBA systems are correct by comparing the TBA equations of the HSG model and those of the

Perturbed non-unitary diagonal coset/W minimal models
When a pair of the mass parameters are turned on, M s = (δ s,r + δ s,n−4−r )M, with n odd, the TBA system is characterized by the diagram (A 3 × T (n−5)/2 ) r , namely, the A 3 × T (n−5)/2 diagram with a mass parameter only for the r th column. Taking into account the above and AdS 3 cases, one then expects that the TBA system in this case reduces to that for the non-unitary The relation 1−h (1,1,adj) = 2(1−∆) is consistent with the UV expansion. In particular, when r = 1, this perturbed diagonal coset model becomes equivalent to the perturbed non-unitary These are particular examples of the correspondence between the TBA system characterized by the diagram (G × T l ) r and the non-unitary diagonal coset model for G, which has been suggested in [47]. In the following sections, assuming, in particular, the correspondence for the 7-cusp (n = 7) case, we derive the analytic expansion of the remainder function around the UV limit, to find a good agreement with the results from the numerical computation. We regard this also as a non-trivial check of the above correspondence.

W minimal models
In the previous subsections, we observed/argued that the TBA systems for the AdS 4 minimal surfaces in the single-mass cases reduce to those for the diagonal coset/W minimal models.
As mentioned, we consider the cases corresponding to the W minimal models to determine the UV expansion of the remainder function for n = 6 and 7. For later use, we thus summarize the W minimal model below.
In the following, we focus on the W A (p,q) k−1 minimal model [48], where p, q (p < q) are positive and relatively prime integers. The central charge of the model is The primary fields Φ l,l ′ have the dimensions Here, l = (l 1 , · · · , l k−1 ) and l ′ = (l ′ 1 , · · · , l ′ k−1 ) are vectors of positive integers satisfying Λ l,l ′ is given by where ω i (i = 1, · · · , k − 1) are the fundamental weights of A k−1 normalized as We also define the effective central charge by where h 0 denotes the lowest conformal weight. For the unitary model with q = p + 1, the lowest weight is 0, but otherwise it is evaluated as [49] k−1 minimal model is represented by the coset model as [50] W A We note that m is not generally a non-negative integer corresponding to an integrable representation. Instead, the general m corresponds to an admissible representation. Let (µ 1 , µ m , µ m+1 ) be the weighs of su(k) for SU(k) 1 , SU(k) m and SU(k) m+1 , respectively. Since µ 1 is determined by other two weights [49][50][51], one can label the fields in the coset model by Then, the dimension of the field is given by where ρ is the Weyl vector of su(k), i.e., Since ρ 2 = k(k 2 − 1)/12, comparing (3.14) and (3.6) gives (3.16) up to field identifications.
For example, the perturbing operator φ (1,1,adj) for the single-mass cases is labeled by and has the dimension In addition, for the non-unitary model W A with n odd, which is used later, the vacuum or ground-state operator φ 0 is labeled by and has the dimension .

Level-rank duality and decomposition of coset models
In subsection 3.1, we discussed the relation between the TBA systems of the HSG model in the single-mass cases and those of the perturbed unitary W minimal models. This relation is directly found by using a decomposition of the generalized parafermion model into a product of the diagonal coset models based on the level-rank duality [52,53].
The above equivalence between the perturbed parafermion and diagonal coset/W minimal models can be generalized to the case of SU(N) k /U(1) N −1 . We see that The first equation is due to [57], and the second expression is due to [58]. The matching of the central charges follows from In the decomposition ( One finds a similar "decomposition" also for the TBA system characterized by the A k−1 × To see this, we first note the relation among the central charges, and then denote the relation after a successive use of it by In parallel with the decomposition (3.21), we find that the rightmost factor on the r.h.s., We note that, in the rank 2 cases with N = 3, there is only one factor of (W A This was used to determine the UV expansion of the remainder function for the 10-cusp minimal surfaces [26]. In the next section, we use the relation for the AdS 4 case with n = N + 4 = 7, 25) to determine the UV expansion for the 7-cusp minimal surfaces.
The "decomposition" (3.24) based on the counting of the central charges tells us which W minimal model appears in the single-mass case. It would be of interest to substantiate this relation at a more fundamental level. A free = π 6 c n + f bulk

UV expansion of free energy
where c n is the central charge and f bulk n is the bulk contribution. In the case of our interest, SU(n − 4) 4 /U(1) n−5 , the central charge is given by The general form of the bulk term is not known. Here we assume, as in the AdS 3 case [26,27], that this term just cancels the period term A periods around the UV limit, i.e., This is equivalent to requiring that the remainder function is expanded by l 4p/n for n / ∈ 4Z, as is the case for the T-/Y-functions discussed in the next section. For n = 6, this is indeed the case [22], and we argue below that this holds also for n = 7. We expect it to be true for any n ∈ 4Z.
The expansion coefficients f (p) n are obtained from the connected n-point correlation functions of the perturbing operator Φ at the CFT point. In particular, f (2) n is given by and C (2) n is given by with γ(x) = Γ(x)/Γ(1 − x). We still need to determine the function G(M s ). As discussed below, it is trivial for n = 6, whereas for n = 7 it is determined by using the relation between the TBA system and the W minimal models in the previous section.

Case of six-cusp minimal surfaces (n = 6)
In this case, there is only one mass scale. Thus the above function is trivially given by G(M 1 ) =M 1 , which is equal to 1 for real m 1 . As discussed in the previous section, the HSG model for n = 6 is equivalent to a perturbed Z 4 -parafermion or SU(4) 1 × SU(4) 1 /SU(4) 2 = W A (5,6) 3 model. The constant κ 6 is thus read from the exact mass-coupling relation in [61] .

(4.9)
This is indeed obtained by setting µ = 1 in the results in [26]. As discussed in [26], this expression is continued to the complex-mass case as G 2 (M s ) → |G(M s e iϕs )| 2 so as to maintain the Z n -symmetry. The continuation in this case is, however, trivial, to give |G| 2 = 1.

Case of seven-cusp minimal surfaces (n = 7)
In this case, there are two mass parameters (m 1 , m 2 ), which we first set to be real. To fix the function G(M 1 ,M 2 ), we use the strategy explained in the previous section (see also [26]).
From the symmetry and the dimensional analysis, we see that this function takes the form where F 11 = F 22 and F 12 = F 21 . We would like to fix such coefficients. For this purpose, we consider the following two cases.
Let us first consider the case where (m 1 , m 2 ) → (l, 0). In this case, the TBA equations reduce to those for an integrable perturbation of the W minimal model, The perturbing operator is the relevant operator Φ with dimension ∆ =∆ = 3/7. The bulk term in this TBA system is [62] f bulk = − 1 2 l 2 . (4.12) We can also read off the mass-coupling relation for this perturbed model from [61]: .  . (4.17) The free energy is expanded around the UV fixed point aŝ where the bulk term is given by [62] f 19) and the coefficientsf (p) are expressed aŝ The coefficientsĈ (p) are given by the correlation functions of the vacuum and the perturbing operators, and the integral forms ofĈ (p) are found in [17,63]. Here, we are interested in the first correction given bŷ where CΦ 0ΦΦ0 is the three-point structure constant. This structure constant is computed in Appendix A, and given by (A.13). Thus the first correction is finally given bŷ 16 7 .

(4.22)
From this result, we can fix F 12 . For m 1 = m 2 (M 1 =M 2 = 1), we find from (4.10) that (4.23) This correction must be twicef (1) . Using (4.7), (4.13) and (4.22), we thus obtain . (4.24) In summary, for n = 7, the function G(M 1 ,M 2 ) has the following form, We also find that the bulk terms in the above two cases, (4.12) and (4.19), fix the form in the general case to be f bulk (l) = −A periods , as expected. Thus, from the connections to the W minimal models, we indeed find this relation for n = 7.
So far, we have considered the case of real m s . Let us now consider the UV expansion of the free energy when the masses are complex. By the relation to A periods , or the argument in [26] to maintain the Z n -symmetry, the bulk term is given by where |m s | =M s l. Similarly, following [26], we find that the function G is continued as We have then fitted the free energy by the function, and found the best values of the fitting for each value of ϕ. Note thatỸ a,s and hence A free depend on the phases only through ϕ in this case. In Fig. 2, we plot the ϕ-dependence of the coefficients b and f 7 . The solid lines represent our analytic prediction while the dots show the numerical data from the TBA equations. Our analytic expressions show a good agreement with the numerical data, which strongly supports the correspondence to the non-unitary W minimal models proposed in subsection 3.2, as well as the continuation to the complex masses.

UV expansion of T-functions
To derive the UV expansion of the remainder function, we need to expand the Y-/Tfunctions, as well as the free energy part discussed in the previous section. This is achieved by using an interesting relation between the T-function and the g-function (boundary entropy) [18,65]. Here, we extend the discussion for the minimal surfaces in AdS 3 [26,27] to the AdS 4 case. We concentrate on the case with n / ∈ 4Z.

T-functions for SU(N ) 4 /U(1) N −1 HSG model
The first step to derive the expansion of T a,s is to compare the integral equations for the Tand g-functions of the SU(N) k /U(1) N −1 HSG model with level k = 4. In this subsection, we consider those for the T-functions, which are obtained by a procedure similar to the one from Y-systems to TBA equations. The extension to general k may be straightforward. For the reason explained in the next subsection, we also set m s to be real.
Let us start our discussion by considering the asymptotic behavior of T a,s for large |θ|.
To see this, we note that, when N ∈ 2Z + 1 with the boundary conditions (2.2), (2.3), one can invert the relation between the Y-and T-functions (2.4) for AdS 5 , to express T a,s by Y a,s . This is also possible for N ∈ 4Z + 2 after imposing the AdS 4 condition Y 1,s = Y 3,s and T 1,s = T 3,s . In such cases, the asymptotic behavior of Y a,s (2.6) implies that of T a,s , log T a,s → −ν a,s cosh θ , This relation is inverted as and (J N −1 ) rs := δ rs /2(cos πr N − cos π k ). We have also used V −1 Given the asymptotics (5.1) , we next subtract the linear terms in l = ML from log T a,s and define U a,s := log(T a,s e νa,s cosh θ ) , (5.8) so that U a,s → 0 for large |θ|. From (5.3) as well as the T-system (2.1) and the relation between T a,s and Y a,s (2.4) with T k−a,s = T a,s , we find that Note that terms with ν a,s cancel each other due to (5.3), and that the above relation involves the same s only. Assuming that U a,s are analytic in the strip −π/k < Im θ < π/k and vanishing rapidly enough for large |θ|, which is expected from the relation to Y a,s , one can Fourier-transform the above equations. Further taking into account the boundary conditions on T a,s and using again (5.6) and (5.7) with N being replaced by k, we obtaiñ where tildes stand for the Fourier transform,f (ω) = dθ e iωθ f (θ), and J ′ k−1 is given by (J ′ k−1 ) ab := δ ab /2(cos πa k − cosh πω k ). Taking into account U 1,s = U 3,s for k = 4, and Fouriertransforming back (5.10), we find the integral equations of U a,s for k = 4: 12) and K 1 , K 2 are given in (2.9). The reflection factors are constrained by the conditions from the unitarity, crossingunitarity and boundary bootstrap [66,67]. In our case, they read as R a,s (θ)R a,s (−θ) = 1 , R a,s (θ)Rā ,s (θ − iπ) = S ss aa (2θ) , (5.13)

g-functions
Here,ū = π − u and we have used (a, s) = (ā, s) = (k − a, s). The location of the poles specified by u a bc is the same as that for the A k−1 minimal ATFT. Note that the boundary bootstrap equations involve the same label s only. Given a set of the reflections factors R a,s , one can deform it as R ′ a,s = R a,s /Z a,s [68], where the deforming factors Z a,s need to satisfy Z a,s (θ)Z a,s (−θ) = 1 , Z a,s (θ) = Zā ,s (iπ − θ) , (5.14) in order to maintain the conditions (5.13). are non-trivial only in the case s = r, where they reduce to those for the minimal ATFT [19]. This assures that they indeed satisfy the conditions (5.14). We also note that when k = 2, the indices a, b take only 1, and the deforming factors of the form (5.16) reduce to Z |1,r;C 1,s which were used to analyze the T-functions for the minimal surfaces in AdS 3 [26,27].
Given a pair of sets of the reflection factors, the g-functions associated with the corresponding boundaries satisfy [19,69] log where c |α are certain constants related to the vacuum degeneracy, and When we choose R |b,r;C a,s and R |1 a,s for the pair, the right-hand side of (5.18) is determined only through the deforming factors Z |b,r;C a,s . By further using the relations Y a,s (θ) = Y a,s (−θ) and Y 1,s = Y 3,s for k = 4, we find that G a,s (C) := log g |a,s;C (l) By comparing (5.11) and (5.20), we see that G a,s (C) = U a,s πi k C . Moreover, assuming that c |1 = c |a,s;C as in the case of AdS 3 , and subtracting the linear terms in l ∝ ν a,s from both sides, we arrive at the relation, The ratios of G (0) |α := log g |α − f |α l on the left-hand side, with f |α being a constant, are the quantities which are directly computed around the UV limit by the conformal perturbation theory with boundaries [18,19].

Expansion of T a,s
Another input for the expansion of the T-functions is their periodicity. To see this, we first note that, from the Y-system (2.5) with Y 1,s = Y 3,s and the boundary conditions given in section 2.1, the Y-functions have the quasi-periodicity, where n = N + k and k = 4. Since our T-functions are expressed by the Y-functions for N / ∈ 4Z, they inherit the same quasi-periodicity, Taking also into account the structure of the CPT, we find that T a,s are expanded as a,s . For lower orders, one can check from the T-system (2.1) that the terms t p,q ′ a,s with q ′ odd are indeed absent, and that the first two non-trivial coefficients are t To compute these coefficients using the relation to the g-functions (5.21), we still need to find which boundary the reflection factors R |b,r;C a,s correspond to. For this purpose, we recall that similar reflection factors for the SU(ñ − 2) 2 /U(1)ñ −3 HGS model in the AdS 3 case corresponded to a boundary labeled by a fundamental representation of su(ñ − 2). It is thus expected that the reflection factors in the present case also correspond to a boundary labeled by a definite representation. Expressing the weight vector by the Dynkin label as λ = [λ 1 , λ 2 , · · · ], we infer the following correspondence, |a, s; C ←→ λ (a,s) with (λ (a,s) ) j = aδ j s .
The result of the CPT and (5.21) then give [18,19] λµ is the modular S-matrix for SU(N) k given by the formula [70], , From the fact that the HSG model is obtained from an integrable deformation of the coset model by weight-zero adjoint operators, the CPT also gives [18,19] t where the coefficients are argued to be continued from the real-mass case as t a,s (|m s |e iϕs ). We confirm below that the expansions obtained in this way indeed agree with numerical results.

Case of N = 2 (n = 6)
In the next section, we discuss the UV expansion of the remainder functions for the 6-and 7-cusp minimal surfaces, which correspond to N = 2 and N = 3, respectively. Here, we list the relevant data for the expansion of T a,s . First, when N = 2, the coefficients t (p,2q) a,s with p odd vanish due to (5.23). At the lowest order, (5.26) and (5.27) give 1 where we have omitted the index s since it takes 1 only in this case. Denoting λ a,s by a and ρ adj = λ 2,s by 2, the ratios of the modular S-matrix elements appearing in t Collecting these results, we find where κ 6 G is given by (4.8) with G =M 1 = 1. In addition, substituting the expansion (5.24) into the T-system (2.1) with T 1,1 = T 3,1 , we also find that t The ratio of t In [22], the expansion of the Y-functions for N = 2 was numerically determined up to and including O(l 4/3 ). We can compare this with the above results. To this end, we note that the relation between the Y-and T-functions in this case reads as Y 1 = 1/T 2 , Y 2 = 1/(T 1 ) 2 , and that the Y-functions in [13,22] and those in this paper are inverse to each (5.35) The ratios of the modular S-matrix elements appearing in t where κ 7 G is given in (4.25). In addition, substituting the expansion (5.24) into the Tsystem (2.1) with T 1,s = T 3,s , we find that t  1,1 ) 2 ≈ 0.554958 is the exact value in the UV limit. The points in Fig. 3 (b) show the fitted values ofỹ (2) 2,1 for each ϕ. We find a good agreement with our analytic expression (solid line) again.

UV expansion of remainder function
Based on the results so far, we derive the UV expansion of the remainder function in this section.

Remainder function for six-cusp minimal surfaces
In the case of n = 6 (N = 2), the relevant cross-ratios for ∆A BDS in (2.32) are These are also rewritten by using T 2,1 . In the UV limit, the cross-ratios become u r,r+3 = 1/4 and equal to each other. Form (6.1) and the expansion of the T-functions, one finds that ∆A BDS is expanded in terms of t Since the period term and the bulk term in the free energy part cancel each other, we arrive at the expansion of the remainder function, where we have introducedt These results agree with those in [22].

Remainder function for seven-cusp minimal surfaces
In the case of n = 7 (N = 3), the relevant cross-ratios for ∆A BDS in (2.33) are These are also rewritten by using T Due to the cancelation between the period and bulk terms, we arrive at the expansion of the remainder function, 7 + D In Fig. 4, we show plots of the 7-point (7-cusp) remainder function for m 1 = e − π 40 i l, m 2 = e − π 20 i l from numerics (points) and from our analytic expansion (solid line). They are in good agreement around the UV limit.

Rescaled remainder function
In [29], it was observed numerically for the 8-cusp minimal surfaces in AdS 3 that the remainder functions at strong coupling and at two loops are close to each other, but different, if they are appropriately shifted and rescaled. In [26,27], this was analytically demonstrated around the UV limit for the general null-polygonal minimal surfaces in AdS 3 .
Similarly, one can define the rescaled remainder function for the AdS 4 case bȳ R n := R n − R n,UV R n,UV − R n,IR . (6.9) Here, R n,UV is the n-point remainder function in the UV limit, which is read off from (6.3) and (6.8). R n,IR is the n-point remainder function in the IR limit where |m s | → ∞. To find this constant, we note the asymptotics of Y a,s (2.6) valid for real m s and | Im θ | < π/2, and successively use the Y-system (2.5), to express Y a,s , e.g., by Y [0] a,s and Y a,s in the IR limit.
On the weak-coupling side, the remainder function at two loops for n = 6 in the AdS 4 case is read off from the results in the AdS 5 case [32][33][34][35]. In particular, one can find the UV expansion of the remainder function from a very concise expression in [35] and the expansion of the T-functions in the previous section: The value in the UV limit l → 0 has been given in [32,34]. The rescaled remainder function is defined similarly to (6.9). Since the two-loop remainder function vanishes in the IR limit, the rescaled remainder function at two loops is expanded as The ratio of the rescaled remainder functions at strong coupling and at two loops is then which is close to 1. By numerics, we also find that the two 6-point rescaled remainder functions are close to each other for all the scales as shown in Fig. 5 (a). We also show the 7-point rescaled remainder function from the numerics in Fig. 5 (b). For both the 6and 7-point cases, we find a good agreement with our analytic expansions around the UV limit. It would be of interest to compare the 7-point rescaled remainder function at strong coupling with the one at weak coupling, which is yet to be computed.

Cross-ratios and mass parameters
We have expanded the remainder function by the mass parameters. In order to express it by the cross-ratios, one needs to invert the relation between the former and the latter.
For n = 6, it follows from (6.1) and the expansion of the T-functions in the previous section that Inverting this relation, one can express the mass parameter m 1 = e iϕ l by the cross-ratios [22].
In the notation in this paper, the result reads as By inverting this relation, one can express the mass parameters by the cross-ratios. For example, when ϕ 1 = ϕ 2 , the inversion is simple, but generically it is not.

Conclusions and discussion
In this paper we have evaluated the regularized area of the null-polygonal minimal surfaces in AdS 4 , and the remainder function for the corresponding Wilson loops/amplitudes at strong coupling. They are described by the TBA integral equations or the associated T-/Ysystem of the HSG model, which is regarded as the integrable perturbation of the generalized parafermion CFT by the weight-zero adjoint fields. The connection to the HSG model as well as to the corresponding CFT allows us to derive the analytic expansion of the remainder function around the UV/regular-polygonal limit by using the conformal perturbation theory.
Generalizing the results in the AdS 3 case, we have found or argued that the TBA systems in the single-mass cases are given by those for the perturbed SU(4) diagonal coset models and W minimal models. This is used to find the precise expansion coefficients through their mass-coupling relations and correlation functions. We have derived the leading-order expansion explicitly for n = 6 and 7. For the 6-point case, we have also compared the rescaled remainder function with the two-loop one. They are close to each other, but different, similarly to the AdS 3 case. Although we have focused on the n / ∈ 4Z case in this paper, it would be an interesting problem to generalize our analysis to the minimal surfaces with general n, and to compare their remainder functions with those at weak coupling.
As noted in section 4, the TBA equations for the AdS 4 minimal surfaces generally exhibit a numerical instability around the UV limit. In spite of that, our analytic expansion works well, which proves our formalism to be useful in this respect as well. It would also be desirable to establish the proposed connection to the TBA systems of the non-unitary diagonal coset/W minimal models, and to substantiate the "decomposition" discussed in A Three-point function in W minimal models In this appendix we review the free field representation of the W A (p,q) k−1 minimal model and compute the three-point function of the ground-state and perturbing operators for k = 4 and p = 5, q = 7, that is used in section 4 to analyze the 7-point remainder function. To lighten the notation, we refrain from using boldface letters for the weight vectors.

A.1 Free field representation
The W A k−1 minimal model [48] is realized by the scalar fields ϕ = (ϕ 1 , ..., ϕ k−1 ) in the sl(k) conformal Toda field theory with the Lagrangian, Here, e j are the simple roots of sl(k), ( , ) denotes the inner-product,μ is the scale parameter and b is the dimensionless coupling. The system has the background charge, where ρ = j ω j is the Weyl vector of sl(k) and ω j are the fundamental weights satisfying (e i , ω j ) = δ ij . The energy momentum tensor is The central charge is given by For the W A