On the existence of the NS5-brane limit of the plane wave matrix model

We consider a double scaling limit of the plane wave matrix model (PWMM), in which the gravity dual geometry of PWMM reduces to a class of spherical NS5-brane solutions. We identify the form of the scaling limit for the dual geometry of PWMM around a general vacuum and then translate the limit into the field theoretic language. We also show that the limit indeed exists at least in a certain planar 1/4-BPS sector of PWMM by using the localization computation analytically. In addition, we employ the hybrid Monte Carlo method to compute the matrix integral obtained by the localization method, near the parameter region where the supergravity approximation is valid. Our numerical results, which are considered to be the first computation of quantum loop correction to the Lin-Maldacena geometry, suggest that the double scaling limit exists beyond the planar sector.


Introduction
The gauge/gravity correspondence claims the equivalence between a certain class of gauge theories and string theories [1][2][3]. The dual description provided by this relation often gives a useful method of analyzing the gauge theories and string theories. This is also applicable to the little string theory (LST), which is considered to be defined on NS5-branes in the decoupling limit [4]. Although any direct definition of LST using a Lagrangian has not been established yet, the gravity description enables us to extract information of LST.
Based on the gauge/gravity duality, it was conjectured that the type IIA LST on R×S 5 is described by the plane wave matrix model (PWMM) [5] in a double scaling limit [6,7], which we call "the NS5-brane limit" in this paper. The gravity dual of PWMM possesses a scaling limit where the geometry reduces to an NS5-brane solution. If the gauge/gravity correspondence is true, it implies that the limit also exists on the (gauge-)field theory side. Thus, the IIA LST is expected to be described as this limit of PWMM. Since PWMM is a well-defined matrix quantum mechanics, this relation is expected to give a direct definition of the LST. In this paper, we investigate this relation.
The gauge/gravity duality in this context is for theories with SU (2|4) symmetry. The field theory side in this duality consists of PWMM, N = 8 SYM on R × S 2 , N = 4 SYM on R × S 3 /Z k and the IIA LST on R × S 5 [6] (see also [8][9][10]). These theories have many discrete vacua, which preserve the SU (2|4) symmetry. The vacua in PWMM are given by fuzzy spheres and classified by the representation of the SU (2) Lie algebra. For SYM on R × S 2 and SYM on R × S 3 /Z k , the vacua are labeled by monopole charges and holonomy, respectively. The description of the vacua of the IIA LST is not clear due to the lack of the direct definition of the theory. However, from the gravity description, one can find that each non-trivial vacuum of the LST carries not only the NS5-brane charges but also some D2-brane charges 1 .
For each vacuum of these theories, the SU (2|4) symmetric dual geometry was constructed [6]. This class of geometries has R × SO(3) × SO(6) isometry, and they locally contain R × S 2 × S 5 . A remarkable feature is that the equations that determine those geometries can be rewritten into the form of the Laplace equations in certain axially symmetric electrostatic systems. The system consists of background potential and a set of conducting disks, which specify the corresponding field theory and a vacuum of the theory, respectively. The geometry is completely fixed by the electrostatic potential of the system.
The geometry dual to the IIA LST is a gravity solution for spherical NS5-branes. It was pointed out that this NS5-brane geometry can be obtained by taking a limit of the dual geometry of PWMM such that one of the conducting disks becomes infinitely large [6,7]. Then, this limit is translated into the language on the field theory side, and found to be a certain double scaling limit in PWMM, in which the 't Hooft coupling also becomes large as the matrix size approaches infinity. Through the gauge/gravity correspondence, this relation suggests that the IIA LST is realized as the double scaling limit of PWMM. The same argument can also be made for N = 8 SYM on R × S 2 and N = 4 SYM on R × S 3 /Z k [11].
The form of the double scaling limit was found only for the case of a simple vacuum of PWMM [7]. In this case, the resulting IIA LST is the theory around the trivial vacuum of the LST, which has only one stack of fivebranes. In this paper, we generalize this argument to the situations where the IIA LST around a general non-trivial vacuum are realized, which also carries some D2-brane charges. We find the form of the double scaling limit in general gravity solutions and then rewrite it using the field theoretic parameters.
For this purpose, we use the remarkable result found in [12,13] based on the localization computation 2 which states that the eigenvalue distribution of an adjoint scalar field on the gauge theory side is identified with the charge densities on the conducting disks. Through this relation, we can identify the parameters on both sides and express the double scaling limit in terms of the parameters of PWMM. Furthermore, the direct identification between the eigenvalue densities and the charge densities also makes it possible to see that the double scaling limit exists at least in the planar quarter-BPS sector of PWMM which is considered in the localization computation.
In addition to the analytic computation, we numerically compute quantities of the BPS sector in PWMM with finite matrix size N by applying the hybrid Monte Carlo method to the matrix integral obtained in the localization computation. Such numerical computation allows us to discuss the double scaling limit beyond the planar sector because it can provide information of 1/N corrections. As a result of the numerical computation, we find evidence that the double scaling limit seems to exist at the non-planar level.
This paper is organized as follows. In section 2, we review how the dual geometry is related to the electrostatic system. In section 3, we consider the double scaling limit, in which the dual geometry of PWMM is reduced to an NS5-brane solution. We first review the simplest case obtained in [7] and then generalize it to the cases for non-trivial vacua. In section 4, after we review the correspondence between the scalar eigenvalue densities and the charge densities, we see how the double scaling limit is realized on the gauge theory side. In section 5, we present numerical results that support the existence of the double scaling limit. Finally, we summarize the paper in section 6 with discussion.

Dual geometries of PWMM and LST
The supergravity solutions dual to the (gauge) theories with SU (2|4) symmetry were obtained in [6,8] by using an SU (2|4) symmetric ansatz. The geometry is written by a single function V (r, z), as 2 and the dots and primes indicate ∂ ∂ log r and ∂ ∂z , respectively. Note that the geometry contains S 2 × S 5 , which reflects the bosonic subgroup SO(3) × SO(6) of SU (2|4).
The supersymmetry requires V (r, z) to satisfy The positivity and regularity of the metric impose further conditions. The positivity restricts the asymptotic behavior of V (r, z) for large r and z. The regularity at the points where S 2 or S 5 shrinks to zero requires either one of the following conditions to be satisfied: (A) ∂ ∂r V (r, z) ∼ r at r = 0 or (B) z = constant if ∂ ∂r V = 0. One can then construct a 3cycle or 6-cycle that is a fiber bundle over a non-contractible curve connecting two points where S 2 or S 5 shrinks, with fiber given by the sphere. The integrals of fluxes H 3 and F 4 over those cycles yield NS5-brane charges and D2-brane charges, respectively [6].
The equation (2.2), which determines V (r, z), is just the Laplace equation in the axisymmetric three-dimensional space and the regularity condition on V (r, z) can be rephrased as the presence of some conducting disks. Thus, the geometry is described in terms of the axisymmetric electrostatic system with the conducting disks, and V (r, z) is interpreted as an electrostatic potential. The positivity of the metric requires existence of background potential in the electrostatic system, which we will introduce explicitly in the following subsections.
The electric charges and the z-coordinates of the conducting disks are parameters of this geometry. As we will describe in the next subsection, they are proportional to D2brane and NS5-brane charges, respectively, through the cycle integrals, and thus quantized. On the other hand, the radii of the disks are not free parameters of the solution, since the regularity of the solution requires that the charge density on any finite disk must vanish at the edge. This relates the radius of the disk to the charge.

Dual geometry of PWMM
In the following, we focus on the dual geometry of PWMM. In order for the solution to become the D0-brane geometry in the UV region, the presence of an infinite conducting plate (say at z = 0) is required 3 . Then only the space with z > 0 becomes relevant for this geometry. We also need background potential of the form where V 0 is a positive constant. This is determined from the positivity of the metric. There are also some finite conducting disks along the z-direction. We denote the z-coordinate and the total electric charge of the sth disk by d s and Q s , respectively (Fig. 1). Through the cycle integrals mentioned above, these quantities are related to the NS5-brane charge of the sth stack N 3 When the solution is lifted up to 11 dimensions, the infinite conducting plate is required in order for the asymptotic geometry to be the plane wave geometry.
The solution to the Laplace equation (2.2) generally takes the form [13,14], f s and R s are an electric charge density and a radius of the sth disk, respectively. The support of the function f s is the interval [−R s , R s ]. Let σ s (r) be the charge density along the radial direction on the sth disk. Then, it is related to f s by These relations imply that f s can be interpreted as the charge density projected onto a diameter of the disk. The regularity of the gravity solution requires that each charge density σ s (r) vanish at the tip of the disk or equivalently d du f s (u) be finite at the edges. In terms of these expressions, one can show that V s satisfies the following differential equation: (2.8) The total charge of the sth disk can be written as The Laplace equation (2.2) is rewritten in terms of the density f s (u) into the following integral equation, where C s is the constant value of the potential on the sth disk and K s is defined by (2.11) where δ is an arbitrary real number. See [12][13][14] for a derivation of the equation (2.10). For general disk configurations, no analytic solution of (2.10) is known. However, in some scaling limits considered in [7], one can evaluate the charge density.

Dual geometry of LST
The electrostatic system for a general vacuum of LST consists of two infinite conducting plates at z = 0 and z = d and some finite conducting disks placed between them. The positivity of the metric requires the presence of the background potential, where I 0 is the modified Bessel function of the 0th order, g 0 is a constant and 0 ≤ z ≤ d.
The positions and charges of the disks are related to the brane charges in the same way as (2.4) (see Fig. 2). The potential for this system can be written as where Λ is the number of the finite conducting disks and (2.14) f s represent the charge densities on the finite conducting disks. The infinite summation in the second term in (2.13) corresponds to contributions from infinitely many mirror images of the conducting disks. The charge densitiesf s satisfỹ whereC s is the value of the potential on the sth disk and K s (d) represents the integral operator defined in (2.11).
If we consider the simplest case where there is no conducting disk between the two infinite plates, the geometry at large r reduces to the geometry with linear dilaton and H flux, which is a typical feature of NS5-brane solutions [7]. This fact enables us to associate this class of solutions to the NS5-branes. Note that for a generic disk configuration, the geometry possesses not only NS5-brane charges but also D2-brane charges.

NS5-brane limit on the gravity side
In this section, we consider the NS5-brane limit, in which the dual geometry of PWMM reviewed in section 2.1 is reduced to the NS5-brane geometry in section 2.2 as a result of a double scaling limit.
As reviewed in the previous section, the dual geometries of both PWMM and LST can be described in terms of electrostatic potentials of axisymmetric electrostatic systems ( Fig. 1 and Fig. 2). Comparing two systems, one can find that the NS5-brane limit corresponds to the limit where the radius of the disk at the highest position in z in the electrostatic system for PWMM is sent to infinity. It turns out, however, that only taking this limit does not lead to the electrostatic system for LST. Indeed, in order to obtain the background potential of LST (2.12), one also needs to tune the magnitude of the background potential in a suitable manner. Thus, the NS5-brane limit turns out to be a double scaling limit.
The NS5-brane limit for the trivial vacuum of LST (i.e. the case without any finite conducting disks) was first investigated in [7]. Here, we extend this study to the case for a general vacuum.

NS5-brane limit for the trivial vacuum
Let us first review the NS5-brane limit for the trivial vacuum of LST [7]. The electrostatic system in this case consists of two infinite conducting plates and the background potential (2.12) (Fig. 2 with Λ = 0). Since there are no charged disks, the potential of this system is given only by the background potential (i.e. the first term in (2.13)). We will show that this electrostatic system can be obtained as a double scaling limit of the particular electrostatic system for PWMM which consists of a single finite conducting disk and the infinite plate ( Fig. 1 with Λ = 0). In the electrostatic system of PWMM, let Q, R and d be the total charge, the radius and the z-coordinate of the conducting disk, respectively. The potential of this system is given by (2.5) and (2.6) as The radius R is determined from the total charge Q by (2.9): is the charge density on the disk and satisfies where C (0) is the value of the potential (3.1) on the disk at z = d.
Let us now find the form of the NS5-brane limit. First, note that the boundary condition at z = d of (3.1) does not agree with that of (2.12): the value of the former is C (0) , while the latter vanishes. To resolve this discrepancy, we add −C (0) z/d to (3.1). This addition of the linear term in z is justified because it does not change the supergravity solution (2.1). In other words, V (r, z) − C (0) z/d satisfies all the requirements for the electrostatic potential for PWMM: it is an axisymmetric solution to the Laplace equation and vanishes at z = 0. Then, the shifted potential can be expanded in Fourier series as where c n are Fourier coefficients. Note that the potential (2.12) of the electrostatic system for LST appears as the lowest Fourier mode in the expansion (3.4). Hence, the NS5-brane limit is given by the limit under which only c 1 survives while the other c n 's vanish. From (3.4) with r = R, one can evaluate c n as The leading part of the integral in (3.5) for large R was evaluated in [7] and found to be proportional to V 0 Rd 3 . Then, by noting that one finds that c n behaves as c n ∼ V 0 (Rd) 3/2 e −nπR/d when R → ∞. Therefore, we finally obtain the NS5-brane limit as where the fixed quantity is proportional to g 0 in (2.12). For later use, let us express the relationship (3.2) between Q and R more explicitly. In the limit R → ∞, one can solve the integral equation (3.3) as The derivation is given in appendix A. Substituting this into (3.2), one obtains This relation shows that the electric charge Q goes to infinity in the NS5-brane limit.
Since Q is proportional to the D2-brane charge by (2.4), the D2-brane charge also goes to infinity in this limit.

NS5-brane limit for general vacua
Next, let us investigate the NS5-brane limit for a general vacuum of LST. The electrostatic system for the general vacuum of LST consists of Λ conducting disks between two infinite conducting plates in the presence of the background potential (2.12) (see Fig. 2). The net electrostatic potential is given by (2.13). We will show that this electrostatic system can be obtained as a double scaling limit of that for PWMM with Λ + 1 conducting disks ( Fig. 1). We will find that the limit is given by where R Λ+1 and d Λ+1 are the radius and the z-coordinate of the highest disk, respectively. This limit is a natural generalization of the NS5-brane limit (3.6) in the previous subsection. Hereafter, to simplify the notation, we denote the radius, the z-coordinate and the charge of the highest disk just by R, d and Q, respectively, instead of R Λ+1 , d Λ+1 and Q Λ+1 . Let us first focus on the (Λ + 1)th disk with charge Q, which should become infinitely large in the NS5-brane limit. The potential generated by this disk is given by where f Λ+1 (u) is the charge density of the highest disk, which is determined by (2.10). The radius R is related to the charge Q through (2.10) and What we expect in the case of general vacua is that the potential (3.10) can be divided into two parts: one of them contributes to the background part in (2.13) in the double scaling limit, and the other contributes to the rest. This separation should appear as a result of decomposition of the charge density f Λ+1 (u) as where f (0) (u) is the charge density in the case of Λ = 0, which satisfies the same equation as (3.3). We set the support of f (0) (u) to be the same as that of We assume the charge associated with f (0) (u) is of the same order as Q when Q is large. Meanwhile, g(u) corresponds to the fluctuation created by the presence of the other disks and has support on [−R, R]. Note that, since g(u) is vanishing as Q s → 0 for any s, the ratio g(u)/f (0) (u) should be on the order of Qs Q for large Q. With the decomposition (3.12), the argument in the previous subsection ensures that the potential generated by f (0) (u) plus the background part (3.1) produces the background potential for LST in (2.12), in the NS5-brane limit (3.9).
We then show that g(u) corresponds to the infinitely many mirror images in the potential (2.13) for LST. Note first that g(u) satisfies for |u| < R, which follows from (2.10) and (3.3). Using (3.12) and (3.13), we find that each term in the right-hand side of (3.10) can be rewritten as (3.14) The derivation is shown in appendix B. Then, from (3.10) and (3.14), we obtain Combined with the contributions from the other disks, V s (r, z)(= V s,0 (r, z)) (s = 1, · · · , Λ), and the background potential for PWMM, in the NS5-brane limit, the total electrostatic potential becomes In the limit (3.9), the first two terms in the right-hand side becomes the background potential for LST (2.12) with g 0 proportional to the fixed quantity in (3.9). Thus, (3.16) indeed coincides with the electrostatic potential for a general vacuum of LST (2.13). Since the potential of the electrostatic system for PWMM becomes completely equivalent to that of LST as a result of the limit (3.9), this double scaling limit is indeed the NS5-brane limit, as desired. We can also understand the NS5-brane limit (3.9) from the action of the electrostatic systems. The action of the electrostatic system for PWMM 4 is as follows, Here, the charge densities are given by σ s (r)(δ(z − d s ) − δ(z + d s )), where the second term comes from the mirror images. The coefficients C s in (3.17) play the role of the Lagrange multipliers for the condition that the total charge of the sth disk is Q s . The equation of motion for σ s gives the condition that the potential on the sth disk is constant and is equal to C s . In appendix C, we show that the action (3.17) reduces to the action for LST in the NS5-brane limit (3.9). This gives another ground of (3.9).

NS5-brane limit in PWMM
In this section, we consider the gauge theory side. We first review the emergent bubbling geometry from PWMM [12,13]. Then, we show that the NS5-brane limit, which we found from the gravity side in the last section, also exists at least in the planar part of a quarter-BPS sector of PWMM. In showing this, we express (3.9) in terms of the parameters in PWMM.

The plane wave matrix model
The action of PWMM in the Euclidean signature is given by We use the same notation as in [12,13].
∂ 1 is the derivative with respect to τ and X 1 is the 1-dimensional gauge field. Throughout this paper, the mass parameter m is set to 2, without loss of generality. PWMM has the SU (2|4) symmetry. The vacua of PWMM are given by fuzzy spheres, which preserve the entire SU (2|4) symmetry and are labeled by the SU (2) representations. The vacuum configuration is given by X m = 0 and for the D2-brane charges and D s for the position of the disks, which is related to the NS5-brane charges. This usage is justified since these variables can be indeed identified with each other through the gauge/gravity correspondence. The identification for N

Emergent geometries
If the gauge/gravity correspondence holds true, the dual geometry should be somehow realized in the strong coupling limit of PWMM. In order to see the emergence of the dual geometry, we apply the localization method, which makes exact calculations possible for some supersymmetric sectors [16].
We first choose an appropriate BPS sector. Note that the emergence of S 2 and S 5 in the geometry (2.1) is rather trivial because any vacua of PWMM preserve SO(3) × SO(6) symmetry. So the only non-trivial part is the emergence of the space parametrized by r and z. It was found that the vacuum expectation values (VEVs) of operators made of the complex scalar field φ(τ ) = −X 4 (τ ) + X 9 (τ ) sinh τ + iX 10 (τ ) cosh τ, (4.4) describe the r-and z-directions 6 [12,13]. This can be seen as follows. If we make a Wickrotation to go back to the Lorentzian signature, the real and imaginary parts of φ at each fixed time are given by an SO(3) scalar and an SO(6) scalar, respectively, up to an SO(2) (⊂ SO (6)) rotation. So this field corresponds to a point on S 2 × S 5 in the dual geometry and thus corresponds to the perpendicular directions, that is, the space of r and z. On the other hand, when X 10 direction is further Wick-rotated while the time is kept to be Euclidean, the field (4.4) is shown to preserve a quarter of the whole supersymmetry and the vacuum expectation value of any operators made only of this field can be computed by the localization method. The localization computation for the partition function and the VEV of such operators was done in [17]. Let us review this computation. One first needs to impose a boundary condition to define the theory as the time direction of PWMM is not compact. A natural condition for the theory around a fixed vacuum is that all fields approach the vacuum configuration as τ → ±∞. After imposing this condition so that the theory is defined appropriately, one adds a Q-exact term to the action, where Q stands for a supersymmetry transformation which keeps φ invariant. Then, from the standard argument of the localization computation [16], one-loop calculations around the saddle point of the Q-exact term give an exact result. At the saddle point, all fields except X 10 take the vacuum configuration while there are non-trivial moduli for X 10 . The saddle point configuration is written as X a (τ ) = −2L a (a = 2, 3, 4), where all the other fields are vanishing at the saddle point. M is a constant Hermitian matrix commuting with L a 's. When L a are decomposed as in the right-hand side of (4.3), M can be decomposed as Hermitian matrix. After calculating the one-loop determinant around the saddle point (4.5), one obtains the following form for the partition function [17]: Here J runs from |D s − D t |/2 to (D s + D t )/2 − 1. We denote by the product in which the second factor in the numerator with s = t, J = 0 and i = j is not included. Then, any operator in the BPS sector, which is made only of φ, can be computed by the partition function (4.6); namely, the VEV of a function of φ, say f (φ), is computed as where · · · M stands for the expectation value with respect to the partition function (4.6). The partition function (4.6) describes the quarter-BPS sector of the theory around the vacuum labeled by the representation R. Insertions of operators of φ correspond to the quarter-BPS fluctuations around the vacuum. On the supergravity side, they will correspond to the fluctuations of fields which preserve the same supersymmetry. On the other hand, if there is no insertion of the operators, the partition function (4.6) is dominated by the vacuum configuration, which should contain the information of the background geometry of the dual string theory. If we can solve the model (4.6) completely, we can check whether these correspondences are true or not. Though solving this model for arbitrary parameters is very difficult, the problem is simplified in the large-N limit, where the eigenvalue integral can be evaluated by the saddle point approximation. This is also the regime where the supergravity approximation becomes valid.
An appropriate parameter region to see the correspondence is given as follows [12,13]. In order to suppress the bulk string coupling correction, we take the 't Hooft limit Moreover, to suppress the α corrections, we consider the region where for arbitrary s. In this parameter region, the effective action of (4.6) is given by δ(x − m si ), and µ s is the Lagrange multiplier imposing the normalization for ρ (s) (x): (4.12) x (s) m is the extent of the support of ρ (s) (x). In the 't Hooft limit, the saddle point approximation for the variables ρ (s) becomes exact and the theory becomes basically a classical theory with the action (4.11).
Let us now show that this classical theory is equivalent to the axially symmetric electrostatic system considered on the gravity side. Below, we verify this equivalence 7 by comparing the electrostatic action (3.17) and the quarter-BPS effective action (4.11).
Back on the gravity side, we first rewrite the electrostatic action (3.17) in terms of the charge densities f s (u). By using (2.5) and (2.7), one can show that the potential V s 's satisfy the following relations 8 , This equivalence was originally argued by using the equations of motion in [12,13]. 8 It is convenient to use the following expression for V (r, z), V s (r, z) = 4π dz r dr G(r, z; r , z )σ s (r )(δ(z + d s ) − δ(z − d s )). Here, is the Green function of the Laplacian in the cylindrical coordinate, where J 0 is the Bessel function of the 0th order.
By using these relations, we can rewrite (3.17) as where we neglected total derivatives and terms that contain no dynamical variables.
Comparing the gauge-theory side (4.11) and the gravity side (4.16), we find the equivalence relation under the identifications Therefore, S es and S ef f are equivalent up to irrelevant constant terms and total derivatives, and thus give the same extremum. Some consistency checks of this identification can be found in [12,13]. It turns out from (4.19) that the sth eigenvalue distribution ρ (s) (x) in PWMM is equivalent to the charge density on the sth disk in the dual gravity. Note that the charge density fully determines the geometry once the gravity solution is in the form of (2.1) with the Laplace equation (2.2) and the positivity and regularity conditions. In this sense, one can say that the dual geometry is formed by the eigenvalue distribution of the scalar fields in PWMM, through the relation (4.19). Therefore, this provides a glimpse into a manifestation of geometry from PWMM.

The double scaling limit in PWMM
The equivalence (4.17) clearly shows that the planar part of the quarter-BPS sector of PWMM has the same NS5-brane limit as the gravity side discussed in section 3. In this subsection, we rewrite the NS5-brane limit (3.9) using the parameters on the gauge theory side: where ω (t) (z) are the resolvents for each eigenvalue density, i0)). Now, let us consider the saddle point equations for ρ (Λ+1) (x) in the NS5-brane limit (3.9). In terms of D s and x From the equations for s = Λ and Λ + 1, one obtains the relation . As shown in appendix A, integrating (4.24) over x yields the saddle point equation for a well-known quartic matrix model, which can be easily solved. As a result, the eigenvalue density for ω (Λ+1) (z) is obtained as . The normalization of ρ(x) (4.12) imposes a condition among parameters:μ Recall that there is one more condition for ρ(x). As mentioned in section 2, the finiteness of the corresponding gravity solution requires each charge density f s (u) to have a finite derivative at the edges. Imposing the same condition on ρ(x) together with (4.26), we obtain This relation can be also derived from purely field theoretic view point by application of the least action principle. The detail of this derivation is explained in appendix A. Then the eigenvalue density (4.25) becomes This is exactly the same form as that for Λ = 1 given by (A.9). Finally, by substituting the relations (2.4), (4.18) and (4.27) to (3.9), we find that the NS5-brane limit can be written as The fixed parameterg s corresponds to g 0 in (2.12) on the gravity side, and hence, it is argued in [7] thatg s is considered as an effective coupling of the IIA LST. At finite N 2 , there are possible non-planar corrections to the quantities (4.19)-(4.21), which could correct the fixed parameter in the expression (4.29). However, we claim that the double scaling limit (4.29) should be valid with the non-planar corrections taken into account. Obviously, the correction appears when the ratio of the correction δg s to the originalg s approaches a finite value or infinity in the double scaling limit (4.29). In the former case, each g-dependent coefficient in the non-planar corrections appears such that the coefficient combines with 1/N 2 to formg s in the limit. Thus, the corrected fixed parameterg s +δg s becomes a polynomial ing s , and one can still fixg s for the double scaling limit. The latter case should be impossible if the gauge/gravity duality holds because the perturbative expansion in terms of quantum correction breaks down otherwise. Therefore, the double scaling limit should remain unchanged by the non-planar corrections.

Numerical results
In this section, we show our numerical results for the quarter-BPS sector in PWMM and discuss the existence of the double scaling limit. We employ the hybrid Monte Carlo method to perform the integration in the partition function (4.6), which is the effective theory of the quarter-BPS sector. Since the numerical results provide information of N 2dependence, one can discuss not only the existence of the double scaling limit but also quantum loop correction, which is interpreted asg s correction in the LST as we will see below.

Expansion with respect to the LST coupling
The double scaling limit (4.29) is understood as the limit that captures all orders in the genus expansion even if N 2 goes to infinity [7,12]. Since PWMM in this BPS sector is expressed by the matrix integral (4.6), the VEV of a quarter-BPS operator can be expanded in powers of 1/N 2 as where λ = g 2 N 2 and O is a quantity equivalent to the quarter-BPS operator, which is thus made of M . As discussed in the previous section, PWMM realizes the IIA LST on S 5 in the NS5-brane limit (4.29), which naturally suggests that the VEV is expanded with respect to the effective LST couplingg s . Hence, the expansion (5.1) is rewritten as where C(λ) is an overall factor, which may be, in general, infinitely large in the limit. The coefficients c n are supposed to be independent of λ because any LST observable is considered as a finite function ofg s up to the overall factor. Therefore, if the NS5-brane limit (4.29) successfully realizes IIA LST, we will observe that the ratios d n+1 (λ)/d n (λ) have the following λ-dependence: for each n. This is a non-trivial prediction, which we examine in the next subsection.

Numerical analysis
We simulate the quarter-BPS PWMM system (4.6) with Λ = 1 by using the hybrid Monte Carlo method. In the viewpoint of the electrostatic system, this case corresponds to the system consisting of one conducting disk at z = d 1 between two larger conducting disks at z = 0 and d 2 in the background potential (2.3). While the conducting disk at z = 0 has an infinitely large radius, the other larger one at z = d 2 becomes infinitely large after the NS5-brane limit is taken.
The quarter-BPS operators we measure are constructed of the "moduli" matrix M , which has the following structure in this case: In this paper, we compute TrM 2 s , which are projected traces onto each block matrix for s = 1 and 2.
In the following, we discuss two ways to see the validity of the NS5-brane limit.

Finiteness in the double scaling limit of PWMM
One way is to compute the quantities for s = 1 at fixedg s and check whether they take finite values at large N 2 (= N 2 ). Since M 1 corresponds to the middle conducting disk in the electrostatic picture and therefore to a D2-brane flux in the supergravity picture, the quantities for s = 1 should describe a non-perturbative object in the IIA LST. Thus, if the double scaling limit exists and reproduces the IIA LST, such matrix-model quantities are expected to be finite in the limit in general. Therefore, C(λ) in (5.2) should approach a finite constant for s = 1. The plot of TrM 2 1 M is shown in Fig. 3 forg s = 100, 200, 300, 400 and 500 (see appendix D.1 for our choice of N (s) 2 and D s ), the values of which are chosen so that they satisfy λ D 4 in (4.10). One can see that they look convergent at eachg s as N 2 tends to infinity. This supports the validity of the NS5-brane limit.
Computation of c 1 /c 0 Another way is to compute quantities with the 't Hooft coupling λ fixed and then read off the coefficients d n (λ) in (5.1) by fitting data by a polynomial function. If the ratio d n+1 (λ)/d n (λ) behaves as in (5.3) at large λ, or equivalently, if the ratio c n+1 /c n approaches a constant as λ goes to infinity, finiteness of each term in the genus expansion is guaranteed even in the NS5-brane limit. It means that the quantities have non-trivial dependence ong s in the computed parameter region, and therefore that a system on NS5-branes is considered to be realized in the parameter region.
The ratio c 1 /c 0 at each λ is computed by fitting of the numerically obtained expectation values. The expectation values are computed for several N 2 , and hence, the large-N 2 extrapolation through the relation (5.3) provides the coefficients, d 0 (λ) and d 1 (λ). See appendix D.2 for our choice of N (s) 2 and D s in this computation and appendix D.3 for d 0 (λ) and d 1 (λ) obtained by the fitting. We plot c 1 /c 0 for TrM 2 1 at λ = 200, 300, · · · , 900 in Fig. 4.
We fit the obtained data of c 1 /c 0 by a polynomial function of λ − 1 4 in the left panel in Fig. 4. This is motivated by the gauge/gravity duality, which predicts that α -expansion corresponds to expansion in powers of λ − 1 4 since R 2 S 5 /α = 2π(8λ) 1 4 in the NS5-brane limit 9 [12]. The figure shows that the linear fits in λ − 1 4 for data with λ ≥ 500, 600 and 700 are consistent with each other and that the ratio approaches c 1 /c 0 ∼ −0.3. Note that one cannot exclude the possibility that it converges to 0.
To numerically support the polynomial fitting in λ − 1 4 , we fit the log-log plot of c 1 /c 0 against 1/λ by a linear function with slope 1/4 in the right panel in Fig. 4. To properly detect the slope, the ratio c 1 /c 0 is subtracted by an offset s = −0.34 (20), which is measured by the fitting s + tλ − 1 4 for 600 ≤ λ ≤ 900. The good agreement seen in the figure reinforces the exponent −1/4 of λ in the fitting polynomial and suggests that the correction at large λ corresponds to the α -correction to the NS5-brane system. 9 R S 5 is the S 5 radius at the spatial point corresponding to the edge of the highest conducting disk in the electrostatic picture. In order to verify that we simulate the system at large enough λ to reproduce the physics on the NS5-branes, we check the λ-dependence of TrM 2 2 M . It is shown in Ref. [12] that this quantity grows as λ 1 2 at large λ, i.e. C(λ) ∼ λ 1 2 . The next-to-leading order of the quantity is naturally assumed to be the order of λ 1 4 , as discussed for the fitting of c 1 /c 0 . We observe that numerical results of TrM 2 2 M agree with fits at large λ by λ 1/2 (α + βλ −1/4 ) with fitting parameters α and β, shown in Fig. 5. Our data is consistent with the theoretically expected behavior; hence this supports the validity of our parameter choice.
Even though the above arguments support the linear fitting in Fig. 4 to some extent, the deviation from the linear fit does not look like a quadratic correction. However, since the 5-brane physics is not well understood yet, the exact correction to the linear term, which should correspond to higher α -corrections, is simply not known. Thus, it is possible  that some higher-order terms are accidentally large or that the fitting function is not even a polynomial but in a more complicated form. In addition, it is also a possible scenario that there is a phase transition around λ = 450, which may be caused by the large-N 2 limit. (See Fig. 6.)

Summary and discussion
We investigated a novel realization of the IIA LST on R × S 5 by PWMM in a double scaling limit, which is called the NS5-brane limit throughout this paper. The IIA LST on R × S 5 is known to have many discrete vacua and the theory around each vacuum is conjectured to be realized by PWMM in the limit. For the trivial vacuum, the form of the NS5-brane limit was obtained in [7], where the limit was first obtained in the gravity side and translated into the field-theory language based on some reasonable arguments.
In this paper, we obtained the NS5-brane limit for general vacua. We first found the NS5-brane limit on the gravity side in terms of the electrostatic description. Then, by making use of the localization method, we showed the equivalence between the effective action of a quarter-BPS sector in PWMM and the action for the electrostatic system. This equivalence made it possible to directly identify the parameters on both sides. Using this relation, we finally expressed the form of the NS5-brane limit in terms of the parameters in PWMM.
The use of the localization computation also yields a byproduct that one can see the  direct equivalence between the quarter-BPS sector in PWMM and the electrostatic system on the gravity side. This equivalence assures that the NS5-brane limit, which we found in the gravity side, also exists in the quarter-BPS sector in PWMM. However, since we took the planar limit of PWMM in deriving this equivalence, our analytical proof on the presence of the NS5-brane limit in PWMM is also limited to this sector. Note that our numerical results in section 5 presented evidence of the NS5-brane limit in more general sector beyond the planar limit.
The IIA LST around a trivial vacuum can be also obtained by different double scaling limits of the other SU (2|4) symmetric theories, N = 8 SYM on R × S 2 and N = 4 SYM on R × S 3 /Z k [11]. By applying the procedure used in this paper, it would be possible to show the existence of the double scaling limit to obtain the IIA LST around a general vacuum from these gauge theories.
In the numerical computation, we obtained results consistent with the existence of the double scaling limit. We also found that the coefficients in theg s expansion of TrM 2 1 , which is associated with the electric charge density (or some sort of D2-brane charge density in the supergravity picture), satisfied c 1 /c 0 ∼ −0.3.
Our result is the first computation to obtaing s correction in LST using PWMM, which can be interpreted as quantum loop correction in the Lin-Maldacena geometry in the limit (4.9), to the best of our knowledge. However, the linear fitting of c 1 /c 0 was not quite satisfactory. This should be due to the lack of knowledge of the 5-brane physics. In this regard, it would be interesting to clarify whether there is a phase transition in the large-N limit, inferred in Fig. 6.
Since the error is large, the ratio c 1 /c 0 for TrM 2 1 was consistent with 0, which means TrM 2 1 would be independent ofg s . If this is the case, theg s -independence of the quantity would suggest that there is a non-renormalizable theorem, or it is equally possible that only the next-to-leading correction ing s is absent while higher-order corrections are present. Unfortunately, it is not feasible to determine the higher-order terms to high precision by the current numerical method, or at least, in the parameter region of our choice.
The relation of the obtained double scaling limit with the M-theoretic viewpoint is also interesting. In [18,19], it was revealed that, in the strong coupling limit of PWMM, the low energy modes of the SO(6) matrices form the S 5 geometry with the correct radius predicted in [15]. Even though its interpretation of the emerged geometries is slightly different from the gauge/gravity, they should be related in some way. Λ = 0: The saddle point equation of f (0) is given by (3.3). If we take the d/R → 0 limit in (3.3), we obtain and ω (0) (z) is the resolvent: If we integrate (A.2) once, we obtain an equation which is equivalent to the equation of motion of the one-matrix model with a quartic interaction. Thus, we find that the solution is given as the same form as the quartic matrix model: Furthermore the normalization (3.2) gives another relation as By eliminating C using this relation, we obtain Note that the semi-positivity of f (0) (u) imposes the condition that R ≤ (8Q/V 0 ) 1 4 . Then, we use the action principle. By substituting (A.6) into (A.1), we find that the R-dependence of the effective action is given by This has an extremum point at Under the condition that f (0) (u) is semi-positive everywhere, we can find that this point is also the minimum of the action. Now, by substituting (A.8), we obtain Thus, the solution with df (0) du (±R) = 0 is indeed obtained through the least action principle. The same procedure can be also applied to calculating (4.28). The procedure is as follows. Firstly, in the NS5-brane limit, the leading behavior of ρ(x) is governed only by self-interaction since the effect from the eigenvalues for s = 1, · · · , Λ is on the order of Qs Q and hence suppressed. Then, by plugging (4.25) and (4.26) into the effective action (4.11) and neglecting the interaction terms involving ρ (s) (x) for s = 1, · · · , Λ, one obtains the expression of the action written by x m in the leading of D/x m . Finally, one finds that its minimum point is given by (4.27).
We first show that the right-hand side of (3.13), C Λ+1 −C (0) , vanishes in the NS5-brane limit (3.9). For this purpose, we integrate (3.13) over u from −R to R: As noted below (3.12), g(u) and f s (u) are on the order of Q s and g(±R) = 0. Using these conditions, one can show that the right-hand side of (B.1) vanishes in the NS5-brane limit. Thus, in the NS5-brane limit, C Λ+1 − C (0) → 0. Next, we introduce a function, In terms of this function, the potential produced by g(u) is just given by G(d, r, z) − G(−d, r, z). Plugging (3.13) into (B.3) with δ = ±d and taking the NS5-brane limit (3.9), we obtain G(±d, r, z) = G(±3d, r, z) where we have used C Λ+1 − C (0) → 0 and the following relation that holds as R → ∞: Repeating this procedure ν times, we arrive at Note that one can show that G(δ, r, z) is finite even in the NS5-brane limit and that G(δ, r, z) is on the order of 1/δ for sufficiently large δ. Thus, the first term on the righthand side turns out to vanish as ν → ∞. From the ν → ∞ limit of this expression, we finally obtain relation (3.14).

C Electrostatic action in the NS5-brane limit
In this appendix, we derive the action of the electrostatic system for LST by starting from the action for PWMM (3.17) and then taking the NS5-brane limit. A tricky term in (3.17) involved in the derivation is the term of C Λ+1 σ Λ+1 (r). This can be rewritten as Then, this V Λ+1 can be divided into V (0) and V s,n by using the equation ( holds for |n| ≥ 1 and its integration over z vanishes. However, in order to express the action for LST by V s,n (r, z) in a neat way, we rewrite (C.1) as by utilizing the fact that the integration of (C.2) over z vanishes for arbitrary s when |n| ≥ 1. By substituting (2.8), (3.16) and (C.3) into (3.17) and gathering the terms relevant for V s,n (r, z) (or f s (u)), we now obtain, up to surface terms, where S es,Λ+1 is an irrelevant term: By putting C s =C s + C (0) ds d and taking the NS5-brane limit (3.9), we obtain up to irrelevant constants. One can see that this gives the action of the electrostatic potential for LST. In terms of the charge densities f s (u), it is rewritten as It is easily seen that the equation of motion for f s (u) coincides with the integral equation off s (u) in (2.15). Thus, in the limit (3.9), the action of the electrostatic system for PWMM indeed reduces to the action for LST.

D Details of numerical analysis
In this appendix, we show our choice of the parameters for the numerical computation in section 5.2.

D.1 Parameter region for the NS5-brane limit
The expectation values are computed with 2 = 4. These are chosen so that λ D 4 in (4.10), which is actually required by the NS5-brane limit (4.29) as λ scales as (D ln N 2 ) 4 and hence λ D 4 . Note, however, that D 1 and D 2 − D 1 are not large but 1 in this case. Thus we expect that the system reproduces the theory on two NS5-branes with a D2-brane flux, which should contain substantial string and quantum correction. This is not a problem because the purpose of the simulation is to confirm the double scaling limit.

D.2 Parameter region for the large-N limit
We compute the expectation values by setting the parameters N (1) 2 = 4. Again, λ needs to satisfy λ D 4 . Since it is a stringent condition for numerical simulations due to costly computation of such large-size matrices, we choose the region of λ to be λ > D 4 /8 = 162. and λ (= 200, 300, 400, and 500) which we choose in our simulations. The cell whose row and column are λ and N (2)

D.4 Comparison withg s
We perform the large-λ extrapolation varying the range ofg s and compare the dependence of the range.
Below λ = 500, the restriction forg s provides no condition. For λ = 600, the restriction gives the same condition, and hence, the same coefficients for the fit. The error tends bigger than the one without restriction due to the reduction of the data points for the fit.