A one-loop test for construction of 4D N=4 SYM from 2D SYM via fuzzy sphere geometry

As a perturbative check of the construction of four-dimensional (4D) ${\cal N}=4$ supersymmetric Yang-Mills theory (SYM) from mass deformed ${\cal N}=(8,8)$ SYM on the two-dimensional (2D) lattice, the one-loop effective action for scalar kinetic terms is computed in ${\cal N}=4$ $U(k)$ SYM on ${\mathbb R}^2 \times$ (fuzzy $S^2$), which is obtained by expanding 2D ${\cal N}=(8,8)$ $U(N)$ SYM with mass deformation around its fuzzy sphere classical solution. The radius of the fuzzy sphere is proportional to the inverse of the mass. We consider two successive limits: (1) decompactify the fuzzy sphere to a noncommutative (Moyal) plane and (2) turn off the noncommutativity of the Moyal plane. It is straightforward at the classical level to obtain the ordinary ${\cal N}=4$ SYM on ${\mathbb R}^4$ in the limits, while it is nontrivial at the quantum level. The one-loop effective action for $SU(k)$ sector of the gauge group $U(k)$ coincides with that of the ordinary 4D ${\cal N}=4$ SYM in the above limits. Although"noncommutative anomaly"appears in the overall $U(1)$ sector of the $U(k)$ gauge group, this can be expected to be a gauge artifact not affecting gauge invariant observables.


Introduction
The correspondence between four-dimensional (4D) N = 4 U(N) supersymmetric Yang-Mills theory (SYM) and type IIB superstring theory on AdS 5 ×S 5 is one of the most typical examples of the AdS/CFT duality conjecture [1,2,3]. The correspondence between the 4D N = 4 U(N) SYM in the large N and large 't Hooft coupling limit and the classical gravity limit of the superstring has been supported by numerous pieces of evidence and is almost established. On the other hand, the strong claim of this correspondence between the gauge theory with finite N and quantum superstring theory has been poorly explored and still remains conjectural. This is partly because of a lack of numerical as well as analytical tools beyond perturbative treatment on the gauge theory side. A possible way to overcome this situation is to construct a lattice formulation of 4D N = 4 SYM as its nonperturbative framework. Indeed, in the case of a lower-dimensional version of the duality in the D0brane system [4], numerical simulations [5,6,7,8,9,10,11] have been developed to provide quantitative nonperturbative tests of the duality beyond the supergravity approximation and bring new aspects into black hole physics. A similar numerical study for the 4D theory will enable an explicit check of this conjecture. Furthermore, if the strong duality conjecture is true, the discretized formulation will provide a nonperturbative description of type IIB superstrings.
Constructing lattice formulations for 4D supersymmetric gauge theories is, however, not straightforward. As an obstacle, it seems impossible to maintain all the supersymmetry on the lattice because of the breakdown of the Leibniz rule [12,13,14]. Indeed, a no-go theorem has been proved for constructing a lattice theory with keeping translational invariance, locality, and the Leibniz rule [15]. In order to circumvent this problem, several lattice formulations that keep nilpotent supersymmetries (up to gauge transformations), which do not generate space-time translations, are constructed by applying the so-called orbifolding procedure [16,17,18,19,20,21,22,23] or topological twists [24,25,26,27,28,29,30,31] to the discretization. For one-and two-dimensional theories, we can see by a perturbative argument that the continuum limit gives the target theories without any fine-tuning as a result of the exact supersymmetries on the lattice. This has been nonperturbatively shown by numerical simulations as well [32,33,34,35,36,37,38]. However, for 4D supersymmetric gauge theories including N = 4 SYM, the nilpotent supersymmetries are not sufficient to forbid all the relevant operators that prevent the full 4D supersymmetry from restoring in the continuum limit. Therefore, we normally need to tune a number of parameters in taking the continuum limit, which makes it almost impossible to carry out a numerical simulation 1 . For some progress in 4D N = 4 SYM from 4D lattice, see [41,42,43]. Regarding the planar part of the 4D N = 4 SYM, its nonperturbative construction has been given by the plane-wave matrix model [44,45].
In [46,47,48], a new approach to circumvent this issue is proposed for 4D N = 2, 4 SYM theories, where two different discretizations by lattice and matrix [49,50,51] are combined. The strategy is as follows. We first construct lattice formulations reminiscent of the "plane-wave matrix string" [50,52], which are mass deformations of the lattice theories of two-dimensional (2D) N = (4,4) and N = (8,8) SYM with U(N) gauge group given in [25,26,27]. As a result of this deformation, fuzzy sphere configurations realize as classical solutions of the theories. In particular, if we expand field variables around a solution representing k-coincident fuzzy spheres, fluctuations can be regarded as fields of the supersymmetric U(k) SYM on the direct product space of the 2D lattice and the fuzzy sphere. The degrees of freedom of the fuzzy sphere are n 2 ≡ N 2 /k 2 . As mentioned above, the continuum limit of the 2D lattice can be taken without any fine-tuning. In addition, by taking a large-N limit of the original 2D lattice theories with scaling the deformation parameter appropriately, the fuzzy sphere becomes the 2D noncommutative (Moyal) plane R 2 Θ . In the rest of this paper, we call this limit the "Moyal limit". Therefore, if we first take the continuum limit of the lattice theory followed by the large-N limit, we obtain the 4D SYM theories on R 2 × R 2 Θ . In particular, it is argued [53] and shown in the light-cone gauge [54] that the commutative limit (R 2 Θ → R 2 ) of the noncommutative N = 4 SYM theory is continuous. Therefore, numerical simulation of the ordinary 4D N = 4 SYM theory is expected to be possible by this hybrid formulation.
In [46,47], it is discussed that there appears no radiative correction preventing restoration of the full supersymmetry of the 4D N = 4 theory based on power counting. This argument relies on an assumption that the mass deformation of the 2D theory is soft in 4D theory as well as 2D theory. The deformation parameter M has a positive mass dimension and the deformation is indeed soft in 2D theory, which does not ruin the feature of the 2D lattice theory that no fine-tuning is needed in taking the continuum limit. On the other hand, the situation is more subtle from the viewpoint of the 4D theory, since the same parameter M comes in both the radius of the fuzzy sphere and the noncommutative parameter. Furthermore, it is related to the UV cutoff in the fuzzy sphere directions. Thus, we cannot say that M is an IR deformation in the usual sense. There is still the possibility of unexpected divergences radiatively generated by the so-called UV/IR mixing 1 The exceptions are N = 1 pure SYM theories in three and four dimensions. The exact parity or chiral symmetry rather than supersymmetry on the lattice plays a key role in restoring the supersymmetry and all the other symmetries in the continuum limit [39,40].
due to the noncommutativity [55] or via the relation to the UV cutoff of the fuzzy sphere, which may spoil some symmetries to be restored in the 4D continuum theory.
The purpose of this paper is to check if there appear such unexpected divergences by perturbative calculation in Feynman-type gauge fixing at the one-loop order. As mentioned above, the lattice continuum limit can be safely taken even after the deformation, which allows us to start with the deformed theory on the continuum 2D space-time; namely, the deformed 2D N = (8,8) SYM theory on R 2 . We explicitly calculate the oneloop radiative corrections to the one-and two-point functions of bosonic fields, which have larger superficial degrees of UV divergences and are more nontrivial compared to higherpoint functions. For the SU(k) part, we will see that there is no unexpected divergence and the 4D rotational (SO(4)) symmetry is restored at this order. On the other hand, for the overall U(1) part, there appears a non-trivial correction regarded as the so-called "noncommutative anomaly" [56], which breaks the SO(4) symmetry. However, according to the results for theories on R 2 × R 2 Θ as well as R 4 in the light-cone gauge [54,57], it is considered that this anomaly arises only accompanied with wave function renormalizations. This implies that the anomaly is a gauge artifact and does not affect correlation functions among gauge invariant observables, in which the SO(4) symmetry is expected to be restored.
The organization of this paper is as follows. We present a brief review of the continuum deformed 2D N = (8,8) SYM theory in the next section, and expand fields around a classical solution of k-coincident fuzzy spheres by using the so-called fuzzy spherical harmonics in section 3. In section 4, the successive limits leading to the 4D target theory are presented, and propagators are explicitly given for perturbative calculations. In section 5, the one-point functions of bosonic fields are computed at the one-loop order. As a warm up exercise, calculations are presented in some detail. In section 6, we calculate the radiative corrections to the scalar two-point functions at the one-loop level. Since we are interested in the Moyal limit, each Feynman diagram is evaluated in this limit. The use of three theorems proved in appendix D considerably simplifies the computations. In section 7, the one-loop effective action for the scalar kinetic terms is obtained in the successive limits. Section 8 is devoted to a summary of the results obtained so far and discussions of future subjects. In appendix A we give the explicit form of the deformed action in the balanced topological field theory (BTFT) description. In order for this paper to be as self-contained as possible, we derive various useful properties of the fuzzy spherical harmonics in appendix B. Computational details of the one-point functions are collected in appendix C. Appendix D is devoted to proofs of the theorems given in section 6. We give precise relations between the fuzzy sphere and the Moyal plane by taking the Moyal limit of the fuzzy spherical harmonics in appendix E. To examine the nonperturbative stability of the k-coincident fuzzy sphere solution, tunneling amplitudes to some other vacua are evaluated in appendix F.
(α is an index for the gauge group generators), which satisfy The eigenvalues of J 0 are ±1 for the fermions with the index ±, ±2 for φ ± , and zero for the other bosonic fields. As mentioned above, φ ± and C form an SU(2) R triplet and each pair of (ψ +µ , ψ −µ ), (ρ +i , ρ −i ), (χ +A , χ −A ), (η + , −η − ), and (Q + , Q − ) forms a doublet. The invariance of the action under Q ± -transformations is most easily seen by the fact that the deformed action (2.4) is recast as In fact, the invariance is shown by (2.10) and the commutation relation between (J ±± , J 0 ) and Q ± : In appendix A we give the explicit form of the action in the BTFT description.

Fuzzy sphere solution and mode expansion
In this section, we expand the action around a particular supersymmetry preserving solution (a k-coincident fuzzy sphere solution) and explicitly give its mode expansion, which is convenient to carry out perturbative calculations in the subsequent sections.

Action around fuzzy sphere solution
As a result of the deformation by (2.5), the theory has fuzzy sphere solutions as minima of the action (S = 0) preserving Q ± supersymmetries: where L a belong to an N-dimensional (not necessary irreducible) representation of the SU(2)-algebra satisfying Among a lot of possible solutions, we consider k-coincident fuzzy S 2 with the size n described by are the n-dimensional irreducible representation of su (2) with N = nk. The fields X a are expanded around the solution (3.3) as Introducing the "field strength" (3.5) and the "covariant derivatives" recasts the bosonic and fermionic parts of the action (2.4) : S = S b + S f as Upon perturbative calculations, we fix the gauge to a Feynman-type gauge: Under the gauge transformation (the latter is obtained from δX a = −i[c, X a ]), F (A, X) changes as Thus, the gauge fixing terms to the Feynman-type gauge and the associated Faddeev-Popov ghost terms are given by where c andc are ghost and anti-ghost fields, respectively. The action after the gauge fixing is given by the summation of (3.7), (3.8), and (3.12):

Fuzzy spherical harmonics and mode expansion
As discussed in [46], the derivatives and the gauge fields along directions of the fuzzy S 2 with the radius R = 3 M are given by two linearly independent combinations of i M 3 [L a , · ] (a = 8,9,10) and by the two corresponding combinations of X a , respectively. Integration over the fuzzy sphere corresponds to taking a partial trace over the n dimensions in the total trace "Tr". This means that the action (3.7) and (3.8) can be regarded as the action of mass-deformed 4D N = 4 SYM theory on R 2 ×(fuzzy S 2 ). In doing perturbative calculations, it is convenient to expand all the fields in the action (3.13) by the momentum basis on R 2 and by the basis of fuzzy spherical harmonics on the fuzzy S 2 . The fuzzy spherical harmonics are n × n matrices, and their definitions and relevant properties are presented in appendix B.
The fields A µ (x), X i (x), c(x), andc(x) are expanded by using the scalar fuzzy spherical harmonicsŶ where n = 2j + 1, and a µ, J m (p), Substituting (3.14), (3.15), and (3.17) into the action (3.13) and using these vertices, we obtain the action with respect to the modes both for the momentum in the 2D flat directions and the angular momentum in the fuzzy S 2 directions. We write it as where S 2,B and S 2,F are bosonic and fermionic kinetic terms, S 3,B and S 3,F denote bosonic and fermionic 3-point interaction terms, and S 4 represents (bosonic) 4-point interaction terms. The interaction terms are further decomposed as Here "tr k " denotes the k-dimensional trace acting on the modes. The 3-point interaction terms are expressed as (3.38) For later convenience, we present another expression for S CD 4 in terms of only the D coefficients rather than the C and D coefficients: Step 1 (Moyal limit): Take n = 2j + 1 → ∞ with the fuzziness Θ ≡ 18 M 2 n and k fixed. Then, the IR and UV cutoffs on S 2 become M ∝ n −1/2 → 0 and Λ j ≡ M 3 · 2j ∝ n 1/2 → ∞, respectively. Namely, the fuzzy S 2 is decompactified to the noncommutative (Moyal) plane R 2 Θ .
At the level of the classical action or tree level amplitudes at least, the theory coincides with the ordinary N = 4 U(k) SYM on R 4 after these steps. One might further expect that Step 1 would be safe even quantum mechanically since 4D N = 4 SYM is UV finite and the deformation by the mass M would be soft. However, the situation is not so simple. Although the deformation parameter M giving masses naively seems soft, the softness is not clear in the sense that M gives not only the IR cutoff but also the UV cutoff. In addition, the obtained theory at Step 1 is a noncommutative field theory; i.e, there might appear nontrivial divergence through the so-called UV/IR mixing in nonplanar diagrams [55]. In fact, we have to consider non-planar contributions as well as planar ones and take care of both the UV and IR divergences. In the following, we first compute the one-point functions at the one-loop order in the next section (section 5), and then explicitly calculate the two-point functions of the scalar fields X i in section 6. This will give a check of whether we can take the limits safely even in the quantum mechanical sense. Upon the perturbative calculation, we rescale all the fields as so that the kinetic terms take the canonical form. The kinetic terms of gauge fields a µ and a scalar x 7 are written as where K, K ′ = 1, 2, 3, and the kinetic kernel is a 3 × 3 matrix: Then, we obtain the propagator Here, s, t, s ′ , t ′ (= 1, · · · , k) denote the color indices. The inverse of the kinetic kernel is given bỹ The propagators for the other bosons and ghosts can be easily read off. For scalars, with i, i ′ = 3, 4, 5, 6. For ghosts and y-fields, (4.10) Finally, the fermion propagators are obtained as where r, r ′ (= 1, · · · , 8) label the spinor components of (3.27), and the inverse of the kernel is given bŷ (4.13)

One-point functions at one-loop
Due to the Q ± supersymmetries, gauge invariant one-point functions should not be induced radiatively for any n = 2j + 1 and M. As a warm-up exercise, let us check this at the one-loop level by computing tadpole diagrams. We also see that the one-point functions do not contribute to the one-loop effective action in the Moyal limit. In this section,p and (J,m) denote external momentum and angular momentum, respectively.

One-point functions of x i and a µ
The one-loop contribution to 1 k tr k x i,Jm (p) (i = 3, 4, 5, 6) comes from the Wick contractions among 1 k tr k x i,Jm (p) and −S F 3,F . We obtain at the one-loop order. We introduced a cutoff Λ p for the loop momentum integration. "tr 8 " stands for trace over spinor indices. It is easy to see that tr 8 γ iDJ, κ (p) −1 vanishes for each i and κ. Hence, The one-loop contribution to 1 k tr k x 7,Jm (p) can be written as where the subscript "1-loop" means the Wick contractions generating one-loop diagrams. We see that each of the three contributions arising from the contractions with −S C 3,B , −S D 3,B and −S F 3,F vanishes separately, because the integrands are odd functions of loop momenta or vanish by themselves due to

One-point function of y
The one-point function of y can be written as The contractions with −S E 3,B and −S D 3,B lead to diagrams with loops of bosons or ghosts, and the contractions with −S G 3,F to loop diagrams of fermions. This time each diagram does not vanish separately. Let us compute these three contributions explicitly.
First contribution The first contribution is tadpoles of y-loops: The sum ofÊ is calculated in appendix C.1. Plugging the result (C.7) into (5.9) leads to Second contribution The second contribution is tadpoles of boson (a µ , x i ) loops and ghost loops: .

(5.12)
Third contribution The third contribution is tadpoles of fermion loops: Here, the spinor trace reads (5.14) and The sum ofĜ is computed in (C.23). Plugging these into (5.13), we obtain Total contribution Gathering the three contributions (5.10), (5.12), and (5.16), the one-point function becomes Here, we see that the O(1/p 2 ) and O(1/p 4 ) terms of the large-|p| expansion of the integrand vanish and the integral converges owing to the Q ± supersymmetries. In fact, after the p-integrals and the summation of J, we end up with This indicates that tr k y 0 0, ρ=−1 is generated by a one-loop effect. However, we should note that tr k y 0 0, ρ=−1 is not gauge invariant, as discussed in appendix C.4. Let us consider the gauge invariant combination from (C.25): Thus, we conclude that the expectation value of the gauge invariant part Tr Ỹ (p) vanishes at the one-loop level.

Summary of the one-point functions
The results obtained in this section give an explicit check at the one-loop order for the statement that any gauge invariant one-point operators are not radiatively induced for arbitrary n and M. Equation (5.18) multiplied by M 2 /9, which is the one-point function with the external line truncated, is seen to vanish in the Moyal limit (Step 1 in the successive limits). Therefore, the one-point functions give no contribution in the one-loop effective action in the successive limits.

Two-point functions of scalar fields at one-loop
We express the two-point functions of scalar fields X i as Here, A i,ī (p; J m;Jm) and B i,ī (p; J m;Jm) are contributions from planar and nonplanar diagrams, respectively, whose external legs are removed.

List of one-loop graphs
Let us write each of A i,ī and B i,ī as a summation of six contributions: The superscript 4XY means contributions from diagrams containing a single 4-point vertex that has the vertex coefficientsX andŶ , while 3XY stands for contributions from diagrams consisting of two 3-point vertices one of which includes the vertex coefficientX and the other hasŶ (X, Y = C, D, E, F , G). In (6.2), we have further divided each of into two parts with the symbols "(1)" and "(2)", in which the latter reflects that the interaction terms S C 3, B including x 3, J m (p), x 4, J m (p), and 3,4 , and B 3CC 7,7 . In addition, due to the propagator connecting x 7 and a µ (4.5), the expressions of A i,ī and B i,ī are different depending on whether the external line includes x 7,Jm (p) or not. So we separately treat A i,ī , B i,ī with (i,ī) ∈ {3, · · · , 6} and A 7,7 , B 7,7 . Note that A i,7 and B i,7 are identically zero.
In the following, we explicitly present all the contributions that do not trivially vanish. In order to simplify notations, the following symbols are employed: with Q = J + δ ρ,1 and U = J + 1 2 δ κ,1 . In expressing index structures of planar and non-planar contributions, the relations (B.24), (B.99), and (B.118) are used.

A 7,7 and B 7,7
The planar contributions are By using (3.39), A 4CD 7,7 can also be expressed as The non-planar contributions are (6.31)

Strategy to evaluate the one-loop diagrams
In the following, we compute the one-loop diagrams listed in (6.4)-(6.31) in the Moyal limit of Step 1: : fixed. (6.32) As in the computation of the one-point functions, UV divergences from the momentum integrations are regularized by the UV cutoff Λ p 4 . Even after setting the cutoff, we often encounter divergences in the sums over J ′ and J ′′ from the region of J ′ = 0 or J ′′ = 0. Since they are angular momenta in the fuzzy S 2 , they can be regarded as "IR" divergences on the S 2 . We consider the case with the external angular momentum J nonzero such that the Moyal limit of the momentum u ≡ M 3 · J is kept finite as M → 0. The external momentum |p| in the original R 2 is assumed to be of the same order as u. From the triangular inequality the IR divergences may arise when one of J ′ and J ′′ (say J ′′ ) is equal to zero. We first remove the region of J ′′ = 0 from the summation in that case, and call the remaining part the "UV part". The J ′′ = 0 part is treated separately by introducing the IR cutoff δ, which is removed after taking the Moyal limit (6.32). By using the expression of the vertex coefficients (B.23), (B.95), and (B.112), we can carry out the sum over variables other than J ′ and J ′′ in the UV part of the planar contributions (6.4)-(6.11) and (6.19)-(6.25). The result of each contribution is expressed as a sum of the following building block: Similarly, for the non-planar contribution in (6.12)-(6.18) and (6.26)-(6.31), the corresponding building block takes the form 5 We have rescaled the external and internal momenta p µ and q µ as p µ = M 3 p µ and q µ = M 3 q µ . Correspondingly, we have rescaled the UV cutoff as Λ p = M 3 Λ p and the momentum integral Λ p denotes the integration with the rescaled UV cutoff. f (J ′ , J ′′ ; J) regularization corresponds to the dimensional reduction regularization (d = 2 − 2ǫ) which respects the gauge symmetry, by ln Λ 2 p ⇔ 1 ǫ − γ + ln(4π) with γ being the Euler constant.
is a function of the form where C is an O(1) constant, and N J , N ∆ , N 1 and N 2 are integers. g( p, q) is a homogeneous polynomial of p and q consisting of In the calculation, the function is used, and the polynomials appear as P i (J) and Q k (J , the expressions (6.34) and (6.35) hold respectively up to additive terms that are irrelevant in the limit (6.32). It should be noted that the summand of (6.35) is different from that of (6.34) just by the sign factor −(−1) J ′ +J ′′ +J . Note that the 6j symbol J ′ J ′′ J j j j vanishes outside the region of J ≤ J ′ + J ′′ ≤ 4j and |J ′ − J ′′ | ≤ J. In order to estimate the expressions (6.34) and (6.35) in the limit (6.32), we separate the range of the summation of angular momenta J ′ and J ′′ , into two regions satisfying    Region I : Region II : The regions are depicted in Fig. 1. In Region I, we can evaluate the summations of J ′ and J ′′ by integrations in the Moyal limit, but this it is not justified in Region II. Since J is typically of the order of O(M −1 ), J ≪ J B ≪ j in the limit M → 0. Correspondingly, we divide A UV into two parts: we do the same for B UV .
Here the amount of computation is considerably reduced by looking at the asymptotic behavior of the integrands. We assume that the homogeneous polynomial g( p, q) behaves asymptotically as for |q| ≫ 1. Then we can claim the following theorem with respect to the summation in Region II: Then A UV II vanishes in the Moyal limit (6.32) if one of the following conditions is satisfied; In Region I, on the other hand, the summation over J ′ and J ′′ can be approximated by an integration over the variables u ′ ≡ M 3 · J ′ and u ′′ ≡ M 3 · J ′′ when M is sufficiently small. Under the assumption that A UV II vanishes in the Moyal limit, the IR property of the function 6 g(p, q − tp) has a key role, as discussed in Appendix D.2. In the expansion which is a polynomial of t, is assumed to behave as up to multiplicative O(1) factors. Then we can prove the theorem with respect to the summation over Region I:

Theorem 2 (in Region I)
Suppose that A UV II vanishes in the Moyal limit by satisfying the condition of Theorem 1. In the case of a, b ≥ 1, A UV I vanishes in the same limit if both of and are satisfied. When one of a and b is zero (say b = 0), the condition for A UV where ℓ 0 is the smallest ℓ such that α ℓ (0) = 0.
Proofs of these theorems are given in Appendix D. As we mentioned above, when the planar contribution has the form (6.34), the corresponding non-planar contribution takes the form (6.35). In Region I, where J, J ′ , J ′′ ≪ j is satisfied, the Wigner 6j symbol can be approximated as (D.21). Since it is negligible for J ′ + J ′′ + J odd, the phase factor (−1) J ′ +J ′′ +J in (6.35) is irrelevant in Region I. In addition, looking at the proof in Appendix D.1, we can evaluate the non-planar contribution in exactly the same way as the planar contribution because the difference between them is only the factor (−1) J ′ +J ′′ +J . Namely, |B UV II | can also be bounded from the above by the r.h.s. of (D.17). Therefore, we can claim These three theorems are useful in evaluating (6.4)-(6.31).

Contributions from one-loop diagrams
We explicitly calculate the diagrams listed in section 6.1. Although there are a number of diagrams, applying the theorems avoids carrying out brute-force computation for all of them. We first look at the diagrams that concern to the two-point function x i x i (i = 3, . . . , 6), where the computations of (6.4), (6.12), (6.8), and (6.16) are presented as typical examples. For the other diagrams, we simply show the results. After that, it is shown that the diagrams for x 7 x 7 give identical results with those for x i x i (i = 3, . . . , 6) in the Moyal limit.

Diagrams from 4-point vertices:
Let us first look at the planar contribution (6.4). By using the UV part can be written as it turns out that the last two terms do not contribute in the Moyal limit, and we have (6.56) Here and in what follows, the ellipsis (· · · ) expresses terms that vanish in the Moyal limit 7 . For the IR part of (6.4) (contribution from J ′′ = 0), introducing the IR cutoff J ′′ = δ (0 < δ ≪ 1) regularizes it as and used (6.53). We next evaluate the non-planar contribution (6.12). Theorem 3 leads to its UV part as Similarly to (6.57), the IR part of (6.12) becomes (6.62)

Diagrams from 3-point vertices:
The UV part of the planar contribution (6.8) can be expressed as It is easy to see that the second term in the integrand does not contribute in the Moyal limit due to Theorems 1 and 2. For the first term, we rewrite the numerator as and apply the theorems, leading to where the function L( A, D; p) is given by (6.39). The IR part of (6.8) from J ′′ = 0 is regularized by the IR cutoff δ and becomes (6.66) For the non-planar contribution (6.16), Theorem 3 gives its UV part as We have the IR part of (6.16): Here we present the results of other one-loop diagrams for x i x i (i = 3, ..., 6), except for A 3CC(2) and B 2CC (2) . The results of the UV parts of the planar diagrams are expressed as The IR parts of the planar contributions are For the non-planar diagrams, the results of the UV parts are obtained as The IR parts are given by We evaluate the diagrams (6.19)-(6.24) and (6.26)-(6.31) by taking the differences between (6.4)-(6.8) and (6.12)-(6.16) with help of the theorems. The difference between (6.19) and (6.4) reads , (6.81) whose UV part vanishes in the Moyal limit from the same reason why the last term in (6.54) vanishes. From Theorem 3, the UV part of the non-planar counterpart does not contribute. It is easy to see that the IR parts of (6.81) and the corresponding expression for the non-planar diagrams (the contribution from J ′′ = 0) both vanish in the limit. Therefore, (6.19) and (6.26) coincide to (6.4) and (6.12) in the Moyal limit, respectively. Let us next evaluate the difference between (6.21) and (6.6): We read off the parameters in the theorems for each of the terms that satisfy w ≤ 0, W − ≥ 2, D 0 ≥ 1, and D 1 ≥ 2, meaning that the UV parts of A 3DD 7,7 and B 3DD 7,7 coincide with those of A 3DD i,i and B 3DD i,i in the Moyal limit, respectively. In addition, the IR parts of (6.82) and the corresponding non-planar contributions both behave as (M/3) 2 × O(δ 0 ), which is irrelevant. Therefore, we can say that A 3DD 7,7 and B 3DD 7,7 has the same contribution as A 3DD i,i and B 3DD i,i in the limit, respectively. By repeating the same manipulation, the quantities (6.19)-(6.24) and (6.26)-(6.31) coincide with (6.4)-(6.8) and (6.12)-(6.16), respectively. Furthermore, we can show that the residual diagrams (6.9), (6.10), (6.17) and (6.18) for x 3,4 x 3,4 vanish in the same way.
Combining the results obtained above, we see that the amplitudes A i,ī (p; Jm;Jm) and B i,ī (p; Jm;Jm) (i,ī = 1, · · · , 7) in (6.1) vanish for i =ī and those for i =ī become independent of the value of i in the Moyal limit.

Amplitudes and effective action
In this section, we sum up the contributions from the various diagrams obtained in the previous section, and evaluate the summations with respect to J ′ and J ′′ to obtain the one-loop effective action for the scalar kinetic terms in the successive limits (the Moyal limit and the commutative limit).

Planar diagrams
The UV part of the planar contributions is given by the summation of (6.56), (6.65), (6.69)-(6.71): and the IR part is given by the summation of (6.57), (6.66), (6.72)-(6.74): Note that the UV divergences containing ln Λ p are canceled in the sums and do not appear in either (7.3) or (7.4), which is expected from the supersymmetry and supports the softness of the mass M. We easily see that (7.2) and (7.4) vanish in the Moyal limit (6.32) (followed by δ → 0 for the IR part) : 5) because (7.2) behaves as O 1 n ln n → 0 as n = 2j + 1 → ∞. Therefore only (7.3) has a sensible contribution in the Moyal limit. In the following, we separately evaluate the contributions of (7.3) from Region I and Region II: In terms of the rescaled variables the 6j symbol can be approximated as (D.23). Also, and all the L functions appearing in (7.3) have the same leading-order behavior: Recall that the external momentum |p| is assumed to be the same order as u. In this region, the summation can be approximated by the integral and the prefactor 1 2 reflects the fact that only the cases of J ′ +J ′′ +J being even contribute to the summation.
Then the double sum part in Region I can be expressed as where the integration variables have been changed from (u ′ , u ′′ ) to (z, w) by and we have defined The leading contribution of the integration (7.12) comes from the region z ∼ ∞ where the integrand behaves as which gives a singular behavior upon the integration: For the quantity subtracted by the singular part, we have analytically computed both I(0) and lim a→1−0 I(a) to provide the identical result π (− ln 2 + 1). Furthermore, numerical computations for general 0 < a < 1 (from, e.g., Mathematica) strongly suggest that I(a) is indeed a constant independent of a: I(a) = π (− ln 2 + 1) for 0 ≤ a < 1. We proceed assuming that this is correct.
Combining the above results, we eventually have ln p 2 + u 2 − ln 2 + 1 (7.20) in the Moyal limit.

A UV, double, II
i,i In Region II, J ′ and J ′′ satisfy Recalling that J B = O (j α ) and J = O j 1/2 , we see In this region, the summation of J ′ and J ′′ cannot be evaluated by integrals as we have done in Region I. As seen in appendix D.1, the 6j symbol J ′ J ′′ J j j j in this region can be well approximated by using Edmonds' formula (D.6), which leads to Together with the above approximations, it turns out that (7.3) in Region II takes a simple form: (7.25) Note that no ∆-dependence remains in the leading term except for the d-function.
The variable J + in the sum runs by two steps as signified, because J + must take even (odd) integers for a fixed ∆ being even (odd). Thus the summation separates into those over even integers for both ∆ and J + and odd integers for both ∆ and J + . Let us consider replacing the latter summation over odd ∆ and odd J + with the summation over odd ∆ and even J + by increasing or reducing the value of J + by one. Since the error induced by this replacement is of the order O J −2 + , we can rewrite the summation in (7.25) as where R(J) ≡ (J + : even) Thus we end up with a simple sum to be evaluated as (J + : even) with Λ j ≡ M 3 · 2j, and the amplitude becomes

Total contribution from the planar diagrams in the Moyal limit
Combining the results (7.5), (7.20), and (7.30), we obtain the Moyal limit of the total contribution from the planar diagrams: The dependence on u B cancels between the contributions from Region I and Region II as it should. The amplitude depends on the external momenta in the 2D plane p and in the decompactified fuzzy sphere u only through the combination p 2 + u 2 , which suggests the restoration of 4D rotational symmetry from R 2 × (fuzzy S 2 ) in the Moyal limit.
Similarly to the planar case, (7.33) and (7.35) vanish in the Moyal limit, is the existence of the sign factor −(−1) J ′ +J ′′ +J in the summation.
In Region I, the 6j symbol is negligible unless (−1) J ′ +J ′′ +J = 1 from (D.22). Hence we can repeat the same calculation as in the planar case (section 7.1.1) to evaluate B UV, double, I i,i , which leads to In Region II, the expression reduces to where Y ≡ cos(2β) and X ≡ cos β. On the other hand, the summation can be converted into an integral as (J + : even) Then we can evaluate the summation in (7.39) as − (−1) J (J + : even) where H J denotes the harmonic number and γ is the Euler constant. In particular, H J is evaluated as for small M. Plugging these results into (7.39), we obtain the non-planar contribution in Region II: Combining (7.36), (7.38) and (7.47), the Moyal limit of the total contribution from the non-planar diagrams becomes The last two terms − ln 3 M u − γ in the non-planar amplitude have no counterpart in the planar contribution (7.31). They arise from the asymptotic behavior of the harmonic number (7.46) that has been recognized as a "noncommutative anomaly" in scalar field theory on fuzzy S 2 [56]. Although this anomaly is finite in the theory on the fuzzy S 2 since the external angular momentum J is finite, it becomes singular in the Moyal limit; i.e., the large radius limit of the fuzzy sphere to the Moyal plane R 2 Θ . Actually, in (7.48), 3/M is nothing but the radius of the fuzzy S 2 which diverges in sending M → 0 with fixed u. Note that the terms are expressed as − ln √ Θ u − 1 2 ln n 2 − γ, in which the first term signifies the UV/IR mixing phenomenon [55]. Due to their u-dependence, the noncommutative anomaly in the non-planar amplitude prevents restoration of the 4D rotational symmetry, which makes a contrast to the planar case.

Moyal limit of the modes
In order to obtain the one-loop effective action in the Moyal limit, we have to know the concrete form of mapping from the modes x i, J m (p) in the expansion by the fuzzy spherical harmonics (3.14) to the modes x i (p, p ′ ) expanded by plane waves on the Moyal plane, and p ′ ·x = p ′ 1ξ + p ′ 2η . When n(= 2j + 1) is large, x i, J m (p) can be expressed by x i (p, p ′ ): Since we are eventually interested in the commutative limit Θ → 0, let us first consider the Moyal limit with Θ being small and j ≫ 1. According to (E.29) and (E.84) in appendix E, the fuzzy spherical harmonicsŶ (jj) J m can be approximated aŝ with J ε being an integer of the order of O(M −ε ) (0 < ε ≪ 1). ϕ p ′ is a phase of the complex combination of the momentum: p ′ 1 + ip ′ 2 = |p ′ |e iϕ p ′ . Plugging (7.52) into (7.51) leads to where we have used tr n e ip·x e iq·x = 2π Θ δ 2 (p + q) for n ∼ ∞, (7.54) and the second argument of x i (p, ue iϕ p ′ ) specifies the momentum in R 2 Θ in the form of the complex combination. In addition, we divide the summation over J into two parts:

Scalar kinetic terms in the one-loop effective action
Let us first consider rewriting the tree-level kinetic terms of the scalar fields in terms of the modes x i (p, p ′ ): In the Moyal limit, we replace the modes with (7.53) and take the limit of M → 0 and n → ∞ with fixing Θ = 18 M 2 n . It turns out that the contribution from 0 ≤ J ≤ J ε disappears, and the result reads with the 4D coupling g 2 4d ≡ 2πΘg 2 2d and four-momentum p ≡ (p, p ′ ). We have also used (7.59) This reproduces the tree-level kinetic terms of 4D scalar fields. We repeat the same procedure for the one-loop part of the effective action for the operators tr k (x i, J m (−p)x i, J −m (p)) and tr k (x i, J m (−p)) tr k (x i, J −m (p)) which is nothing but the negative of the scalar two-point function (6.1) with (7.31) and (7.48). Since all the fields were rescaled as (4.1) in the perturbative calculation, we rescale them back to the original expression. Then, the result in the Moyal limit becomes We decompose the modes to the SU(k) part and the overall U(1) part: and express the effective action up to the one-loop order ((7.58) + (7.60)): The wave function renormalization: (µ R is the renormalization point) absorbs all the divergences arising in the Moyal limit and recasts the effective action as At this stage, the limit of Step 2 (commutative limit Θ → 0) can be trivially taken to give the final result (7.65) and (7.66). The U(1) part is not SO(4) invariant due to the noncommutative anomaly in ∆Γ (recall that u is the momentum in the plane obtained from the fuzzy S 2 ), while the anomaly is harmless in the SU(k) part at the one-loop level.
Beyond the one-loop order, however, such SO(4) breaking could also affect the SU(k) sector in the kinetic terms. Here, the U(1) part does not couple with the SU(k) part in the quadratic terms of the effective action, which is the case for any quadratic term to all the orders for a group theoretical reason. We also expect that the interaction terms receive no radiative corrections except those absorbed by the wave function renormalization which is the same as the situation in the ordinary N = 4 SYM on R 4 [57,59,60]. For n-point amplitudes with n ≥ 3, the UV divergence will be at most logarithmic from the power counting and parity invariance in the Moyal limit. The leading divergence would be canceled by supersymmetry as seen in the two-point amplitudes, and the result would be UV finite. In such UV-finite amplitudes, there is no obstruction in the commutative limit on the convergence to the corresponding results in the ordinary N = 4 SYM. As in the ordinary N = 4 SYM [57], radiative corrections to the quadratic terms in the effective action would be gauge-dependent, and thus the noncommutative anomaly that appears accompanied by the wave function renormalization would be a gauge artifact not affecting gauge invariant observables. This is supported by the analysis of the 4D N = 4 SYM theory in R 2 × R 2 Θ in the light-cone gauge [54], which shows that the limit Θ → 0 is continuous to the ordinary theory defined on R 4 to all the orders in perturbation theory; namely, the noncommutative anomaly does not appear.

Conclusion and discussion
Starting with the mass deformation of 2D N = (8, 8) U(N) SYM ,which preserves two supercharges, we have obtained 4D N = 4 U(k) SYM on R 2 × (fuzzy S 2 ) around the fuzzy sphere classical solution of the 2D theory. The radius of the fuzzy S 2 is proportional to the inverse of the mass M and the noncommutativity Θ is proportional to the inverse of the product of n = N/k and the mass squared M 2 . It is clear at the classical level that the two successive procedures, (1) decompactify the fuzzy S 2 to a noncommutative plane and (2) turn off the noncommutativity of the plane, derive the ordinary N = 4 SYM on R 4 . As a nontrivial check at the quantum level, we have computed the one-loop effective action with respect to the kinetic terms for scalar fields X i (i = 3, · · · , 7), where the gauge is fixed to a Feynman-type gauge. The IR singularities turn out to be harmless in the above limits by introducing the IR cutoff δ at the intermediate step in the computation.
For the SU(k) sector in the gauge group U(k) of the 4D theory, which contains only the contribution of planar diagrams, the result coincides with the ordinary 4D SYM on R 4 after the wave function renormalization. In particular, the SO(4) rotational symmetry in R 4 is not ruined by the quantum correction. On the other hand, the overall U(1) sector including the contribution of non-planar diagrams has shown a "noncommutative anomaly", which has no counterpart in the ordinary SYM. Due to this anomaly, the SO(4) symmetry does not appear to be restored. Also, such an anomaly may affect the SU(k) sector beyond the one-loop order. However, we expect that it arises only accompanied by the wave function renormalization. Since the wave function renormalization is gauge dependent as in the ordinary 4D N = 4 SYM, the anomaly is expected to be also a gauge artifact and will not arise in computing gauge invariant observables as far as the gauge symmetry is respected. Of course, it is desirable to calculate other kinetic terms and interaction terms of the one-loop effective action as well as higher-loop corrections in order to make the expectation firmer. Due to the technical complexity, we will leave this for future work. It will be important to confirm the harmlessness of the noncommutative anomaly by numerical simulations.
Rigorously speaking, the gauge invariant observables should be invariant under the gauge transformation of the theory on R 2 × (fuzzy S 2 ) before taking the Moyal limit. They should be nonlocal in the fuzzy S 2 directions and have the angular momentum J = 0. Interestingly, however, for field variables in the effective action with J ≪ j, we can consider local observables with nonzero J in the following reason. The scalar field X i (x), whose mode expansion is given by (3.14), transforms under the gauge transformation with the parameter Ω(x) = d 2 p (2π) 2Ω (p) and (C.27) as where we have used (B.18) and (B.24) with the notation (6.3). The expression does not vanish by taking the partial trace tr k , but does under the total trace Tr = tr n tr k . The total trace Tr X i (x) yields the observables tr k x i, 0 0 (p). For the case of external angular momenta sufficiently smaller than the cutoff, i.e. J 1 , J 2 ≪ j and thus J ≪ j, the contribution ofĈ J m (jj) J 1 m 1 (jj) J 2 m 2 (jj) , which includes the 6j symbol as in (B.23), is greatly suppressed when J 1 + J 2 + J is odd, according to the formula (D.21). This allows us to replace the sign factor (−1) J 1 +J 2 +J with unity. Hence, we can effectively consider the partial trace tr k X i (x) or equivalently tr k x i, J m (p) as gauge invariant observables 10 . By repeating a similar argument, tr k X i (x) ℓ (ℓ = 1, 2, · · · ) can be regarded as gauge invariant observables.
Since the overall U(1) part is uninteresting in the target theory (the ordinary N = 4 SYM on R 4 ), it is better to consider the observables subtracted by that part. For example, 10 Nevertheless, the two-point function of tr k x i, J m (p) yields the SO(4) breaking term ∆Γ in (7.65).
There we used the ordinary renormalization prescription of local field theory, i.e., subtraction by local counter terms, which would not respect the full gauge invariance in noncommutative gauge theory. In fact, under the gauge transformation (8.1), the local counter term corresponding to (7.64) varies by the amount of the product of the divergent factor (ln M ) and the suppression for J 1 + J 2 + J odd, which could be nonvanishing. We expect that finite quantities free from the renormalization factors such as (8.4) will avoid the issue and show restoration of the SO(4) symmetry.
with use of the field variables it would be worth seeing the restoration of the SO(4) symmetry at the nonperturbative level by numerical simulation of the quantities (L = 2, 3, · · · ): The subscript "conn." in the numerator means that the connected part of the L-point correlation function is taken. The denominator is introduced as having finite quantities independent of possible wave function renormalizations. The point x and the index i ∈ {3, · · · , 7} in the denominator can be freely chosen as a reference. Finally, we comment on the nonperturbative stability of the k-coincident fuzzy S 2 solution (3.1) with (3.3), which will be relevant in numerical simulation. As an illustration, tunneling amplitudes from the fuzzy S 2 solution 1. to the trivial vacuum (X a = 0) 2. to the (k − 1)-coincident fuzzy S 2 solution are evaluated in appendix F. Although the tunneling amplitudes are expected to be suppressed due to the infinite volume of the space-time R 2 [46], we will see this in more detail. Indeed, by scaling the length of the spatial direction in the two-dimensional space-time faster than 1/M, all the results are shown to become zero in the successive limits (Step 1 and Step 2 in section 4). This supports the nonperturbative stability of the solution (3.1) with (3.3) in taking the successive limits.

A Deformed action in BTFT form
The deformed action in the BTFT description S = S b + S f (S b and S f denote its bosonic and fermionic parts, respectively) is explicitly given by

B Fuzzy spherical harmonics
In this appendix, we give definitions and properties of various fuzzy spherical harmonics [61,44] that are relevant in the text. Let |j r (r = −j, −j + 1, · · · , j) an orthonormal basis of n(= 2j + 1)-dimensional space of a spin-j representation of SU(2) normalized by Here j is assumed to take a non-negative integer or half-integer value. The SU(2) generators L a (a = 1, 2, 3) satisfying [L a , L b ] = iǫ abc L c act on the basis as L 3 |j r = r|j r .
By expressing {|j r } as n-dimensional unit vectors as any n × n matrix M can be written as The adjoint action of L a to M is defined as Then, it is easy to see that B.1 (Scalar) fuzzy spherical harmonics (Scalar) fuzzy spherical harmonics is defined bŷ where C J m j r j −r ′ ≡ j j r − r ′ |J m is a Clebsch-Gordan (C-G) coefficient vanishing unless m = r − r ′ . In the basis (B.3),Ŷ (jj) J m is an n × n matrix whose (r, r ′ ) component is given by √ n (−1) −j+r ′ C J m j r j −r ′ . Note that J and m = r − r ′ take integer values as seen from the C-G coefficient.
From the definition (B.5), C J m j r j −r ′ is real, and the relation Then, the hermitian conjugate ofŶ For the n-dimensional trace "tr n ", the orthonormality follows from the orthogonality of the C-G coefficients Next, let us compute the trace of the product of three fuzzy spherical harmonics given by (3.18) in the text, which is equivalent tô From the definition (B.7) and the identity 12 we haveĈ (See, e.g., eq. (12) in Chapter 8.7.3 in [58]) and the property of the 6j symbol, The first equality of (B.13) leads tô J m−n ′ stands for a harmonics of the orbital angular momentum (J, m − n ′ ) on the fuzzy S 2 . Combined with a wave function with spin S, χ S n ′ , which satisfies Y S n ′ J m,J (jj) χ S n ′ represents the irreducible representation of the total angular momentum (J, m) obtained from the tensor product (J, m − n ′ ) ⊗ (S, n ′ ).
In the text, S ± , S 3 are related to the SU(2) R generators (2.11) as χ S n ′ comes from the wave functions of the doublets (ψ +µ , ψ −µ ), (ρ +i , ρ −i ), (χ +A , χ −A ), (η + , −η − ) for the S = 1 2 case, and from the wave functions of the triplet ( 1 We can see that J, m ∈ Z in Ŷ ρ J m Taking the sum over n ′ leads to where we have used 13 For the vector fuzzy spherical harmonics (B.31), the identity and (B.46) imply For cases including (B.32) and (B.33), similar formulas are obtained, and we conclude that For the spinor fuzzy spherical harmonics (B.37) and (B.38), holds.

B.5 Some C-G coefficients
The C-G coefficient is related to the 3j symbol as Here we present the explicit form of some C-G coefficients that will be used later. In this subsection, we will show that the following four relations hold: σ are the Pauli matrices, and Proof of (B.64) Note that Using (B.61), we can see Similarly, we obtain   For ρ = 1, let us consider, e.g., the i = 3 component: Proof of (B.67) For κ = 1, let us consider, e.g., the α = 1 2 component:

B.7 Vertex coefficients
We start by showing the formula for the trace of the product of three kinds of spin-S fuzzy spherical harmonics: Applying the identity 14 with Q ≡ J + δ ρ, 1 ,Q ≡ J + δ ρ, −1 , and In the remaining part of this section, we compute the vertex coefficients defined by (3.19)- (3.22) in the text.
so that theÊ coefficient is expressed aŝ Here, it can be seen from the formulas in section B.5 that holds. This and (B.82) lead tô Finally, we use which follows from (B.52), and note m 1 ∈ Z, and (B.92), to arrive at the formulâ Equivalently, we can rewrite (3.20) as

C Computational details of one-point functions
In this appendix, the details of the computation of (5.9), (5.11), and (5.13) are presented. The gauge transformation property of the one-point function y J m, ρ (p) is also discussed.
From the definition of vector and scalar fuzzy spherical harmonics, p µ = 3 M p µ and J = 3 M u typically of the order of O(M −1 ). In Region II defined by (6.41) and (6.42), it is convenient to change the summation variables from J ′ and J ′′ to Then, J + ≫ ∆ for a sufficiently small M. By noting that both J + and ∆ take even integers or odd integers since J ′ = 1 2 (J + + ∆) and J ′′ = 1 2 (J + − ∆) should be integers, the summation can be rewritten as (D.4) From |∆| ≤ J, it can be seen that there is a constant C 1 such that which means that there is a constant C 2 such that Next, let us evaluate the integral I a,b in (D.2). Recall that the polynomials P i (J) and Q k (J) in the denominator of the integrand are actually of the form (6.40). Because of | p| ≪ J ′′ and |∆| ≪ J + in Region II, we see that 1 q 2 + P k (J ′ ) and 1 ( q+ p) 2 + Q k (J ′′ ) are bounded from the above as 1 where c ′ and c ′′ are some constants of the order of O(1). Therefore, with use of (6.45), I a,b is bounded as where w ≡ N 1 + N 2 − 2(a + b) + n q + 2, W + ≡ −N J − N ∆ − n p + 2 − 2w and W − ≡ −N J − N ∆ − n p + 2 − w. C + , C 0 and C − are M-independent constants. That immediately proves the theorem.

D.2 Theorem 2
In Region I, we express A UV where we have used the Feynman integral formula, and Combining (D.20), (D.23), and (D.24), we have where the integration region D u stands for Similarly to the case of Theorem 1, let us consider the situation that the q-integrals in I a,b converge in the UV region, because the assumption of Theorem 2, that A UV II vanishes, implies that there is no UV divergence in A UV I . The only possible divergence in the integration in (D.27) is from the IR region: u ′ ∼ 0 or u ′′ ∼ 0.
Note that g(p, q − tp) can be expanded as because parity-odd terms with respect to q µ → −q µ trivially vanish in the integration. The coefficient α ℓ (t) that depends on p 2 is a polynomial of t. After the rescaling q µ = X(t) 1/2q µ , we obtain As mentioned above, we are considering the situation that theq-integrals UV converge, i.e.
for each ℓ, and a, b ≥ 1 in the computation here.

E Flat limit of fuzzy spherical harmonics
In this appendix, we express the fuzzy spherical harmonics in terms of plane waves on the Moyal plane in order to prepare to obtain the one-loop effective action in the Moyal limit (the limit at Step 1 in section 4). Once the expression is obtained, taking the commutative limit (at Step 2) is straightforward.

F.4 Tunneling amplitude to the solutions (8.7)
We can evaluate a similar bound for the process in which one of the k L (n) a in the fuzzy sphere solution splits into L (ℓ) a and L (n−ℓ) a with ℓ = 1, 2, · · · , n − 1. Then, the upper sign is taken in (F.10). The result reads (F.14) The same procedure of sendingL as in appendix F.3 maintains the stability.