Finite-$N$ corrections to the M-brane indices

We investigate finite-$N$ corrections to the superconformal indices of the theories realized on M2- and M5-branes. For three-dimensional theories realized on a stack of $N$ M2-branes we calculate the finite-$N$ corrections as the contribution of extended M5-branes in the dual geometry $AdS_4\times \boldsymbol{S}^7$. We take only M5-brane configurations with a single wrapping into account, and neglect multiple-wrapping configurations. We compare the results with the indices calculated from the ABJM theory, and find agreement up to expected errors due to the multiple wrapping. For six-dimensional theories on $N$ M5-branes we calculate the indices by analyzing extended M2-branes in $AdS_7\times \boldsymbol{S}^4$. Again, we include only configurations with single wrapping. We first compare the result for $N=1$ with the index of the free tensor multiplet to estimate the order of the error due to multiple wrapping. We calculate first few terms of the index of $A_{N-1}$ theories explicitly, and confirm that they can be expanded by superconformal representations. We also discuss multiple-wrapping contributions to the six-dimensional Schur-like index.


Introduction
In typical utilization of the AdS/CFT correspondence [1] we calculate quantities in the boundary theory by using the gravity or string theory in the bulk. For this to be possible it is necessary that the quantum gravitational effect is suppressed because we do not have enough knowledge to carry out quantitative analysis of quantum gravity. Due to this restriction the majority of works about the AdS/CFT correspondence assume the large-N limit.
However, there is a possibility that some physical quantities in supersymmetric theories are protected from the quantum gravity corrections and we can perform an analysis on the gravity side even if N is finite. An example of such a quantity is the BPS partition function of the four-dimensional N = 4 supersymmetric Yang-Mills theory. It was shown in [2] that by geometric quantization of BPS configurations of D3-branes expanded in S 5 we can reproduce the exact BPS partition function for finite N.
Based on the result of [2] two of the authors proposed a prescription to calculate the finite-N corrections to the superconformal index [3] of N = 4 SYM as the contribution of D3-branes wrapped on topologically trivial cycles in S 5 [4]. The method was also applied to S-fold theories [5] and the consistency with the supersymmetry enhancement [6] was confirmed. Later, it was extended to orbifold theories [7] and toric gauge theories [8], and it was found that the prescription works at least for single-brane configurations. The contribution of multiple-brane configurations was first calculated in [9] in the Schur limit, and the analytic result in [10] I N I ∞ Schur = 1 + ∞ n=1 c n q nN +n 2 , c n = (−1) n N + 2n n!
was successfully reproduced. For analysis of D3-brane configurations in AdS 5 ×S 5 there are three important parameters: the AdS 5 radius L, the S 5 radius r, and the D3-brane tension T D3 . Two dimensionless parameters defined with them are where V 3 = 2π 2 is the volume of the unit 3-sphere 1 . The ratio L/r is the unit of the energy of Kaluza-Klein gravitons in S 5 normalized by the AdS radius, and on the boundary theory point of view it is interpreted as the scale dimension of a free scalar field φ, which is 1 in the four-dimensional theory. The second equation shows that the energy of a D3-brane wrapped around a large S 3 ⊂ S 5 is as N times as that of a unit Kaluza-Klein mode, and we identify the wrapped brane with an operator like det φ. In the large-N limit such wrapped D3-branes decouple while for finite N they are expected to contribute to the superconformal index as finite-N corrections.
In the prescription proposed in [4,9] the complete superconformal index is given by (3) I KK is the index of the supergravity Kaluza-Klein modes, which reproduces the gauge theory index in the large-N limit [3]. C runs over a set of D3-brane configurations. Each configuration consists of D3-branes wrapped around large 1 The volume of the unit n-sphere V n is given for small n by V 2 = 4π, V 3 = 2π 2 , V 4 = 8π 2 3 , V 5 = π 3 , V 6 = 16π 3 15 , and V 7 = π 4 3 . gives the classical contribution from wrapped D3-branes without fluctuations. If C consists of n wrapped D3-branes it is proportional to q nN , where q is the fugacity associated with the energy. The other factor I excitations C is the index of the theory realized on the configuration C. If C consists of a single D3-brane the theory is free U(1) supersymmetric gauge theory and the index is given by where i D3 C is the single-particle index of the fluctuation modes on the brane. Pexp is the plethystic exponential defined as follows. Let f be a function of fugacities given as a formal power series where x n are products of fugacities and c n are integer coefficients. Then, the plethystic exponential of f is defined by Because a large S 3 is a topologically trivial cycle in S 5 and a D3-brane wrapped around such a cycle is shrinkable there exist tachyonic modes with negative energy on C consisting of such branes. The treatment of such tachyonic modes is a key point to calculate the finite-N corrections. A simple analysis shows that on a single wrapped D3-brane there is one tachyonic mode and its energy is −1 in the unit of L −1 . Correspondingly, the single-particle index i includes the term q −1 . (We set other fugacities to be 1.) With the definition (6) the plethystic exponential of this negative-power term is Interestingly, this factor increases the order of the index by 1 and changes the overall sign of the correction. Although these facts are against intuition the corrections calculated in this way agree with known results. An interpretation of the correction is as follows. In the large-N limit the complete index is reproduced by the Kaluza-Klein index I KK . If N is finite we should consider giant gravitons [11] instead of the supergravity Kaluza-Klein modes. An important difference from the supergravity index is the existence of the upper bound of the size of giant gravitons. Namely, the finite-N index is obtained by somehow subtracting contributions of high momentum modes that do not have corresponding giant gravitons. The negative correction including the factor (7) is interpreted as the absence of giant gravitons of large momenta.
If there are δ tachyonic modes they raise the order of the correction by δ. The interesting exponent nN + n 2 in (1) can be interpreted as the effect of δ = n 2 tachyonic modes on the configuration with n D3-branes [9]. In the following we call the shift δ of the order of a correction "the tachyonic shift." The purpose of this paper is to apply the same idea to the theories on M2branes and M5-branes. The BPS partition functions of these theories were calculated in [12], and were reproduced by the geometric quantization of M5-and M2-branes, respectively, in the same reference. We extend their analysis to the superconformal indices according to the prescription in [4].
This paper is organized as follows. In Section 2, we investigate the finite-N corrections of the M2-brane theories. We first derive the formula for the finite-N corrections induced by a single wrapped M5-brane. We compare the results obtained by the formula with the index of ABJM theory with k = 1 [13], and find nice agreement. We also consider Z k orbifolds corresponding to the ABJM theory with the Chern-Simons level k = 2 and 3. The comparison of the gravitational analysis and localization formula again find nice agreement.
In Section 3, we consider the finite-N corrections of the 6d N = (2, 0) theories. We first derive the formula for the finite-N corrections induced by a single wrapped M2-brane, and estimate the error due to the multiple wrapping by using N = 1 case. Then, we calculate the index of A N −1 theories by using the formula. As a consistency check we confirm that they are expanded by indices of superconformal irreducible representations. We also consider the Schur-like limit of the index and discuss multiple wrapping contributions.
In Section 4, we summarize the results and discuss some extensions. In Appendices we show some technical detailks and results that we do not show in the main text.

3d N = 8 superconformal theories
The three-dimensional N = 8 superconformal theory realized on a stack of N coincident M2-branes is described by the ABJM theory with Chern-Simons level k = 1 [13]. The gravity dual is the M-theory in the AdS 4 × S 7 background. The AdS 4 radiusL, the S 7 radiusr, and the M5-brane tension T M5 satisfy the relationsL (We use hats for distinction from similar symbols used in the next section, in which we will use checked symbols.) With these relations we can easily see that the energy of a maximum giant M5-brane has energy N/2. This fact suggests that such wrapped M5-branes correspond to baryonic type operators in the ABJM theory [13].
In this section we investigate the fluctuations on such a wrapped M5-brane and calculate the finite-N corrections to the superconformal index. We compare the results with the index obtained by using localization formula. We will focus on configurations consisting of a single wrapped M5-brane, and we will not consider configurations with multiple wrapped M5-branes.

Superconformal index
The 3d N = 8 superconformal algebra isÂ = osp(8|4), whose bosonic subalgebra is so(2, 3) × so(8) ⊂Â. There are six Cartan generatorŝ The HamiltonianĤ and the spinĴ 12 are Cartan generator of so(2, 3) and the other four are Cartan generators of the R-symmetry so (8). To define the superconformal index we choose one complex superchargeQ that carries specific Cartan charges. We take the one with the following quantum numbers: The subalgebra ofÂ that keeps the chosen superchargeQ intact iŝ whereB = osp(6|2) is the superalgebra whose bosonic subalgebra is sl(2, R) × so (6). The central factor u(1)∆ is generated bŷ The superconformal index associated with the BPS bound∆ ≥ 0 is defined as theB character by 2 Due to the Bose-Fermi degeneracy for∆ > 0 this does not depend onx.

Wrapped M5-branes
In the large-N limit the superconformal index is reproduced by the Kaluza-Klein modes in AdS 4 × S 7 . The Kaluza-Klein index I KK is given by I KK = Pexp i KK , where i KK is the single-particle index [14] i The corresponding boundary theory is the ABJM theory with the Chern-Simons level k = 1. The full index of the ABJM theory including the contribution of monopole operators was calculated in [15], and the agreement of the ABJM index in the large-N limit I ABJM N =∞ and this Kaluza-Klein index I KK was confirmed. Based on the idea in [4] we propose the following equation for the finite-N index The second term in the parentheses in (15) gives the finite-N corrections due to wrapped M5-branes. C runs over "the representative configurations" of wrapped M5-branes specified in the following, and I M5 C is the contribution of each configuration C.
We determine the representative configurations C by a preliminary analysis of a rigid M5-brane, an M5-brane wrapped on a large S 5 in S 7 . Let us introduce complex coordinates z a (a = 1, 2, 3, 4) to describe the S 7 by 4 a=1 |z a | 2 = 1. The R-symmetry su(4) ⊂B acts on these coordinates in the natural way. For a rigid M5-brane to be BPS with respect to the chosen superchargeQ the worldvolume must be given by the holomorphic equation [16] were c a are homogeneous coordinates in P 3 . The collective motion of the M5brane can be treated as a particle in the moduli space P 3 . By the analysis of the coupling of the brane and the background flux we find the wave function Ψ of a rigid M5-brane is a section of the line bundle O(N) over P 3 . We can give Ψ as a homogeneous function of the coordinates c a of degree N. States described by such wave functions belong to the su(4) representation with Dynkin labels [N, 0, 0]. On the gauge theory side these states are identified with baryonic type operators in the ABJM theory [13]. The corresponding index isq The characters of the fundamental and the anti-fundamental representations are given by Now let us remember the Weyl's character formula. It gives q where "permutations" represents three terms obtained from the first term by cyclic permutations ofû a . From the quantum mechanical point of view, the first term can be interpreted as the partition function of the system with the ground stateq 1 2 NûN 4 and three bosonic excitationsû 1 /û 4 ,û 2 /û 4 , andû 3 /û 4 . We define the representative configuration as the M5-brane corresponding to the ground state. For the first term in (18) it is given by z 4 = 0. Corresponding to the other terms obtained by the permutations there are three more representative configurations z a = 0 (a = 1, 2, 3).
The main idea in [4] is that we can obtain the finite-N corrections to the index by ornamenting the Weyl's formula (18) with all other fluctuation modes by replacing the zero-mode contributionû 1 /û 4 +û 2 /û 4 +û 3 /û 4 by the complete singleparticle index of the theory on the worldvolume of the M5-brane. In addition, to obtain the complete corrections, we need to take account of representative configurations including more than one branes [9]. Namely, the general form of C is given by where a multiple zero is understood as coincident branes. (n 1 , n 2 , n 3 , n 4 ) = (0, 0, 0, 0) is excluded because it corresponds to the first term in the parentheses in (15). The contribution of each configuration C is factorized into two factors I ground C and I excitations C . Each wrapped brane contributes N/2 to the energy (in the unit ofL −1 ) and the ground state of C includes the factorq 1 2 nN with n = n 1 + n 2 + n 3 + n 4 . I ground C is given as the product of the ground state contribution of each brane: I excitations C is the contribution of excitations on C. If n ≥ 2 the theory on C is interacting and it is not so easy to calculate I excitations C . In this work we only consider four configurations with n = 1 given by z a = 0 (a = 1, 2, 3, 4). Then, the theory on C is free and I excitations C is given by where i M5 za=0 is the single-particle index of the fluctuation modes on the worldvolume of an M5-brane wrapped on z a = 0.
Let us calculate the single-particle index i M5 za=0 for each representative configuration. In the following we consider the configuration z 4 = 0. The other three are obtained by the permutations of the fugacitiesû a . We start with the analysis of the scalar modes. If we neglect the self-dual potential field and fermion fields on the worldvolume the M5-brane action is given as the sum of the Nambu-Goto action S NG and the Chern-Simons term S CS : where G ab is the induced metric and A 6 is the background 6-form potential satisfying dA 6 = (2πN/V 7 )vol(S 7 ). We use the following AdS 4 × S 7 metric: ds 2 =L 2 (− cosh 2 ρdt 2 + dρ 2 + sinh 2 ρdΩ 2 2 ) +r 2 (cos 2 θdΩ 2 5 + dθ 2 + sin 2 θdφ 2 ).
We consider an M5-brane wrapped on R × S 5 defined by ρ = θ = 0. There are 5 scalar fields corresponding to transverse directions of the M5-brane: three in AdS 4 and two in S 7 . To describe fluctuations in AdS 4 we introduce a threedimensional unit vector n and rewrite dΩ 2 2 as dn 2 . We define fluctuation fields by By neglecting higher order terms and using the relations in (8) we obtain where ∇ is the derivative on the unit S 5 . The constant term gives the energy E = 1 2 N of the wrapped M5-brane. By solving the equations of motion we can easily obtain the spectrum of fluctuation modes. (See Table 1.) We have six Table 1: Scalar fluctuation modes on an M5-brane wrapped on z 4 = 0. ℓ = 0, 1, 2, . . . is the angular momentum on S 5 .
zero-modes of z * at ℓ = 1 and three of them are BPS. They correspond to three excitationsû 1 /û 4 ,û 2 /û 4 , andû 3 /û 4 appearing in the Weyl's formula (18). We also have one BPS tachyonic mode of z * at ℓ = 0. A few comments on the tachyonic mode are in order. First, the existence of the tachyonic mode does not cause the instability of the system. The tachyonic mode carries the R-chargeR 78 = −1, and a tachyonic particle is always created together with an anti-particle withR 78 = +1. As is shown in Table 1 such an anti-larticle, which corresponds to the ℓ = 0 mode of z, carries the energy E = 5/2, and the pair creation raises the total energy of the system. Another comment is about the consistency with the BPS bound. Ordinary, a particle with negative energy is against the BPS bound E ≥ 0. In the theory on the wrapped brane, however, we do not have such a bound. An M5-brane wrapped on z 4 = 0 breaks the half supersymmetries. Among 32 supercharges only 16 that commute with the generatorẐ are preserved. The algebra of the preserved symmetry iŝ The central factor u(1)Ẑ is generated byẐ. The bosonic subalgebra ofĈ is The fluctuation modes on the M5-brane form a representation of the unbroken algebraĈ. The HamiltonianĤ appears inĈ only throughĈ, and the bound obtained from the algebra is notĤ ≥ 0 butĈ ≥ 0. The tachyonic mode satutates this bound.
In principle, we can calculate the complete single-particle index i M5 za=0 by carrying out the mode expansion of the tensor and the fermion fields. However, there is an easy way to obtain the index from the known 6d superconformal index of the tensor multiplet.
We are interested in the theory of a tensor multiplet living on R × S 5 , the worldvolume of a wrapped M5-brane. This system is similar to the system of a tensor multiplet living on the boundary of AdS 7 . In the next section we investigate the six-dimensional system living on the AdS boundary R × S 5 , on which the (2, 0) superconformal algebraǍ acts. The two free theories, the theory on a wrapped M5-brane in AdS 4 × S 7 and the theory on the boundary of AdS 7 , are in fact the same theory, at least at the linearized level, and we can obtain the index of the former from the index of the latter by a simple variable change of fugacities.
We first establish the relation between the symmetry algebras. Namely, we need to find an isomorphism between the unbroken algebra on the wrapped M5brane (27) and a subalgebra ofǍ. There is an ambiguity of the choice of the subalgebra ofǍ. A convenient one is the symmetry (93) realized on a wrapped M2-brane studied in the next section. It is isomorphic to (27); The explicit relations between the two sets of the bosonic generators are as follows.
We can relate two systems not only at the level of the symmetry but also at the level of the Lagrangians. The boundary metric of AdS 7 is For distinction fromt used in (25) we useť for the time coordinate. The Lagrangian of the five scalar fields φ I (I = 1, . . . , 5) living on this background is where the last term is the conformal coupling to the background curvature. We simply relate the triplet fields by X ∝ (φ 3 , φ 4 , φ 5 ), while in the relation between z and φ 1,2 we need to apply the time-dependent phase rotation corresponding to the relation of two Hamiltonians 2Ĥ =Ȟ − 3Ř 12 obtained from the last two equations in (30). In addition, we rescale the time coordinate bŷ t = 2ť to match the background metric (31) and the metric on the wrapped M5-brane obtained from (23) by the restriction ρ = θ = 0. Then, we obtain the Lagrangian in (25) from (32). We can extend the relations (30) to fermionic generators. An important fact is that the supercharges used to define the superconformal indices on two sides are related byQ and the relation∆ = 2∆ immediately follows from this. This implies that the superconformal indices defined on two sides are essentially the same. Indeed, we can rewrite the six-dimensional index (82) to the three-dimensional index (13) by using the map (30) and the variable changě Applying the variable change (36) to the index i M5 bdr in (100) of the free tensor multiplet we obtain the following single-particle index for the excitations on an M5-brane wrapped on z 4 = 0: The first few terms in the expansion correspond to the tachyonic modes and rigid motion modes obtained in the analysis of scalar fluctuations.

Comparison with known results
In the last subsection we obtained the following hypothetical formula where the first term in the right-hand side is defined by and the second term O(q 1 2 (2N +δ) ) is the expected error due to the neglect of the multiple-wrapping configurations with the tachyonic shift δ. Based on the experience in the D3-brane case we expect δ is independent of N, and this is directly confirmed below for small N.
Let us first give the results on the gauge theory side. If N = 1 we do not have to use the ABJM theory. Instead, we can use free theory of scalar fields and fermions living on an M2-brane. The index is given by I ABJM N =1 = Pexp i M2 bdr with the single-particle index [14] i M2 bdr =q For N ≥ 2 we need to use the ABJM theory with the Chern-Simons level k = 1, and sum up contributions of monopole operators according to [15]. See Appendix A for the explicit formula. The results for N = 1, 2, 3 are In this section we only show the results withû a = 1 for readability. Refer to Appendix B for the full expressions. Let us first compare these results with the Kaluza-Klein index We find the finite-N corrections appear atq 1 2 (N +1) . These are consistent with the contributions of a single wrapped brane with one tachyonic mode. (39) gives the following results for N = 1, 2, 3.
We find nice agreement. The error appears at the orderq 1 2 (2N +δ) with δ = 6. As is expected δ is N-independent. At present we have no explanation for this specific value of δ.

Z k orbifold
It is easy to extend our formula (39) to the orbifold AdS 4 × S 7 /Z k with k ≥ 2 defined by the orbifold action On the gauge theory side this is described by the ABJM theory with the Chern-Simons level k ≥ 2. The Kaluza-Klein contribution I Z k KK is given by where P k is the Z k projection operator defined for a function g of su(4) fugacitieŝ u a by The representative configurations C are given by (19), and again we focus on the four single-wrapping configurations z a = 0 (a = 1, 2, 3, 4). Due to the Z k orbifolding, the worldvolume of the M5-brane becomes S 5 /Z k , and the excitation is described by the projected single-particle index P k i M5 za=0 . Then, the projected index I M5 za=0 is given by Because of the non-trivial five-cycle homology H 5 (S 7 /Z k ) = Z k we can classify states by the topological wrapping number B ∈ Z k of M5-branes, and we can calculate the index for each sector with specific B. If a configuration C is given by equation h(z) = 0 the function h(z) must have a specific Z k charge for consistency with the Z k orbifolding. Namely, it must satisfy with some B ∈ Z k . Then, B is the topological wrapping number of the worldvolume. Among the four representative configurations with n = 1, z 1 = 0 and z 2 = 0 carry B = +1, and z 3 = 0 and z 4 = 0 carry B = −1.
On the ABJM theory side k is the Chern-Simons level, and a wrapped M5brane with B = 0 corresponds to a baryonic operator carrying Z k baryonic charge B. In the ABJM theory with the gauge group U(N) k × U(N) −k this baryonic symmetry is a part of gauge symmetry, and baryonic operators are not gauge invariant. In order to calculate the index with the contribution of baryonic operators we need to use the ABJM theory with the gauge group (U(N) k ×U(N) −k )/Z k where the Z k quotient acts on the diagonal U(1) symmetry [17,18]. In the index calculation this quotient changes the quantization of monopole charges.
The index of ABJM theory is calculated by summing up contribution of different monopole charges [15]. The monopole charges are labeled by 2N GNO charges: (m 1 . . . . , m N ) for U(N) k and ( m 1 , . . . , m N ) for U(N) −k . In the U(N) k × U(N) −k theory all charges are integers, while in the (U(N) k ×U(N) −k )/Z k theory the quantization condition is given by The index of the B = 0 sector is the same as the index of the U(N) k × U(N) −k ABJM theory, while B = 0 sector gives the index for baryonic operators, which corresponds to the contribution of M5-branes with topological wrapping number B on the gravity side. In the following we calculate the indices for k = 2 and k = 3 on both sides of the duality, and confirm the agreement up to the expected order ofq. We use the notations I for the indices calculated on the two sides of the duality.

k = 2
In the case of k = 2 there are two sectors labeled by B ∈ Z 2 .
Let us first calculate the index of the B = 0 sector. The indices for N = 1, 2, 3 Let us compare these with the Kaluza-Klein contribution.
We find the corrections appear at orderq 1 2 (2N +2) . They are interpreted as contributions of two-brane configurations, which belong to the B = 0 sector. Hence, it exceeds our scope.
Next, let us consider the index of B = 1 sector: On the gravity side we need to consider wrapped M2-brane with B = 1. Because B is Z 2 -valued B = +1 and B = −1 are identified, and all four configurations z a = 0 (a = 1, 2, 3, 4) contribute to the index; The results for N = 1, 2, 3 are In all cases the leading term is of orderq 1 2 N , and there is no tachyonic shift. This is because the Z 2 projection removes the tachyonic term from the singleparticle index. This is consistent with the fact that the branes are wrapped on topologically non-trivial cycles. The error between the ABJM index and (61) appears atq 1 2 (3N +6) . This is consistent with the fact that only brane configuration with odd n contribute to the index of the B = 1 sector and the error is due to n = 3 configurations.

k = 3
The Z k orbifolding with k ≥ 3 breaks the N = 8 supersymmetry down to N = 6.
We consider k = 3 case and there are three sectors specified by B ∈ Z 3 . Let us first consider the B = 0 sector. The ABJM index is given for N = 1, 2, 3 as follows.
Let us compare these with the Kaluza-Klein index We find the corrections atq 1 2 (2N +2) . We can interpret these corrections as the contributions of brane configurations with n = 2 consisting of a brane with B = +1 and another brane with B = −1.
Next, let us consider baryonic sectors with B = ±1. These two sectors are related by the charge conjugation symmetry B → −B we focus only on the B = +1 sector. The ABJM index is given as follows for N = 1, 2, 3.
On the gravity side we take only two single-wrapping configurations z 1 = 0 and z 2 = 0 into account because the other two carry B = −1.

6d N = (2, 0) superconformal theories
In this section we consider six-dimensional N = (2, 0) superconformal theories realized on a stack of M5-branes. The gravity dual is M-theory in AdS 7 × S 4 . The AdS 7 radiusĽ, the S 4 radiusř, and the M2-brane tension T M2 satisfy the following relations similar to (2) and (8): The ratioĽ/ř = 2 gives the dimension of a free scalar field in six-dimension, and the second relation suggests that wrapped M2-branes in S 4 are responsible for the finite-N corrections in the superconformal index.

Wrapped M2-branes
Let I (2,0) N be the superconformal index of the theory realized on the stack of N M5-branes. The large-N limit I (2,0) N =∞ is given by the Kaluza-Klein index of AdS 7 × S 4 . It is given by I KK = Pexp i KK with the single-particle index [14] i where χ m (ǔ) is the su(2) character of the spin m/2 representation and χ [a,b] (y) is the su (3) character of the representation with Dynkin labels [a, b]. χ [1,0] for the fundamental representation and χ [0,1] for the anti-fundamental representation are For the theory on a finite number of M5-branes we propose the formula I M2 C is the contribution of an M2-brane configuration C. The sum of C runs over representative configurations, which are determined shortly in a parallel way to the three-dimensional case. Let us introduce Cartesian coordinates x 1 , . . . , x 5 and describe S 4 by 5 a=1 x 2 a = 1. We also introduce the complex coordinates The subalgebra su(2) ⊂ so(5) of the R-symmetry commuting withQ transforms these complex coordinates as a doublet. For a rigid M2-brane wrapped on a large S 2 in S 4 to preserve the supersymmetryQ, the M2-brane worldvolume must be given by the holomorphic equation [16] where (a 1 , a 2 ) are homogeneous coordinates of the moduli space P 1 of the rigid brane. Due to the coupling to the background flux the wave function Ψ of the rigid brane is a section of O(N) line bundle over P 1 . Namely, Ψ can be given as a homogeneous polynomial of (a 1 , a 2 ) of degree N. There are N + 1 such linearly independent polynomials belonging to the (N + 1)-dimensional representation of su(2) acting on P 1 . The corresponding index iš As in the case of wrapped M5-branes the two terms are interpreted as the contribution of two representative configurations of M2-brane, z 1 = 0 and z 2 = 0, respectively. The general representative configurations are given in the form C : z n 1 1 z n 2 2 = 0, n 1 , n 2 ∈ Z ≥0 , (n 1 , n 2 ) = (0, 0), and the corresponding contribution I M2 C is given by where n = n 1 + n 2 . For C with n ≥ 2 it is difficult to calculate I excitations C , while for n = 1 configurations z a = 0 (a = 1, 2) the theory on the wrapped brane is free and given by I excitations za=0 is the single-particle index on an M2-brane wrapped on z a = 0.
As is explained in the last section this is isomorphic to the symmetry preserved by a wrapped M5-brane in (27). By using the isomorphism map (30), we can obtain i M2 z 1 =0 from i M2 bdr in (40) by a simple variable change. The inverse of (36) iŝ q =qǔ − 1 2 ,û 1 =q and by substituting these relations into (40) we obtain The index i M2 z 2 =0 for the other configuration z 2 = 0 is obtained from (96) by the Weyl reflectionǔ →ǔ −1 .
It is of course possible to calculate the index directly by the mode expansion of fields on the wrapped brane. We show the results for scalar fields in Table 2. There is one BPS tachyonic mode withȞ = −2 and one BPS zero mode. These correspond to the first two terms in theq expansion of i M2 z 1 =0 :
By removing the contribution of the free tensor multiplet we obtain the index of the A N −1 theory: Explicit forms of I A N−1 for small N obtained by using (99) are as follows.
See Appendix C for the index of each irreducible representation. We exploited the notation for representations used in [19] to denote the corresponding indices. These results support the correctness of the formula (99). In addition, the expansion of I A 1 seems to be exceptionally simple. In particular, as was pointed out in [19] the D[0, 4] representation is absent in the A 1 theory.

Schur-like index
As shown in (90) a generic representative configuration consists of M2-branes wrapped on two cycles z 1 = 0 and z 2 = 0. We can simplify the problem by taking a special limit in which only one of these two cycles, say, z 1 = 0, contributes to the index. For M2-branes wrapped on z 2 = 0 not to contribute to the index we need to tune the fugacities so that an extra supersymmetry which is broken by the M2-brane wrapped on z 2 = 0 is preserved by the definition of the index (82).
, which is the Weyl reflection of the second term in the numerator of (96), and it consists of three terms To make the definition of the index (82) respect this supercharge we impose the following condition on the fugacities.
Then the first term in (113) becomes −1, and its plethystic exponential vanishes.
As the result, only configurations consisting of M2-branes wrapped on z 1 = 0 contribute to the index. We adopt the following parametrization of fugacities satisfying (115) (andy 1y2y3 = 1).
New fugacitiesq ′ ,x ′ ,y are unconstrained variables. With this specialization the index (82) becomes where∆ (117) is nothing but the Schur-like index studied in [20]. 5 In fact, the analytic result of the index for M5-brane theories was obtained from five-dimensional U(N) SYM [21,20]: By expanding this with respect toq ′2N we obtain 4 The Z k symmetry (48) acts on the first two terms and the last term in different ways and this causes inequality between the third one and the others. We should not take the third term to define the Schur-like limit because the corresponding supercharge is non-perturbative in the sense that it is not manifest in the ABJM Lagrangian and is generated dynamically. 5 The fugacities in this paper are related to those in [20] byq ′ = q 1 2 andy = s.
where F n (q ′ ) are rational functions ofq ′ . The functions for n = 1, 2, 3 are Let us compare (120) with the hypothetical relation (86), which reduces in the Schur-like limit to the following relation: where I M2 n is the Schur-like index of the theory realized on a stack of n M2-branes wrapped around the cycle z 1 = 0. The agreement in the large-N limit is easily confirmed: The agreement of finite-N corrections requires I M2 n (q ′ ,y) = F n (q ′ ) n = 1, 2, 3, . . . .
For n = 1, the single-wrapping contribution, we can easily confirm (126) by using the Schur-like limit of i M2 For n ≥ 2 we expect that F n is the index of the ABJM theory realized on S 2 ⊂ S 4 . It is straightforward to write down the integral form if the index. A non-trivial point is how we should choose the integration contours. Although at present we have not completely understood it we found that with a certain prescription we can reproduce the first few terms in F 2 and F 3 . See Appendix A for details.

Summary and Discussions
In this paper we investigated the superconformal index of theories on M2-branes and M5-branes using the AdS/CFT correspondence. We proposed formulas that give finite-N corrections to the superconformal indices of these theories as the Table 3: Results of the comparison for the ABJM theory. k, B, and N are the Chern-Simons level, the baryonic charge, and the rank of the gauge group.
The column "single" shows the orders of the correction due to single-wrapping configurations. The column "multiple" shows the orders of the errors due to multiple-wrapping configurations. k B N single multiple 1 -1, 2, 3q contribution of wrapped M-branes. We only included single-wrapping brane configurations, and the contributions of multiple branes are left for future work. For M2-brane theories we proposed the formula (39). We compared the results with the results of direct calculation using the ABJM theory. The results of the comparison are summarized in Table 3. We found complete agreement up to errors due to multiple branes. It would be difficult to calculate the contribution of multiple wrapping on the gravity side because we need to deal with multiple M5-branes. Conversely, it may be possible to obtain some information about multiple M5-branes from the higher order corrections in the ABJM index. For example, we found that δ does not depend on N at least for N = 1, 2, 3. This may suggest that the theory on multiple M5-brane does not couple to the background flux.
For M5-brane theories we proposed the formula (99). We determined the tachyonic shift δ = 3 by using the result for N = 1. Under the assumption of N-independence of δ, the multiple brane contributions are of orderq 2(2N +3) , and our formula should give correct index below the order. We showed the explicit form of the index up to orderq 10 . As a consistency check we confirmed that the indices can be decomposed into the contributions of superconformal irreducible representations. In particular, the decomposition of A 1 theory is exceptionally simple, and it seems to match the expectation that the A 1 theory is the minimal N = (2, 0) theory.
There are some proposals about the superconformal index of the (2, 0) theories. The index was related to the partition function or index of five-dimensional supersymmetric Yang-Mills theories in [22,21], and a relation to topological strings was investigated in [23]. It is important task to compare their results and ours.
We also discussed the Schur-like limit of the 6d superconformal index, for which an analytic formula is known. We reported the preliminary result that with a certain prescription for pole selection we could reproduce the first few terms of multiple-wrapping contributions F 2 and F 3 in (122) and (123).
We constructed the formulas by extending the Weyl's character formula. It would be interesting to search for an algebraic structure behind our formulas. If there exist a large algebra which includes creation and annihilation operators of not only fluctuation modes on wrapped branes but also wrapped branes themselves then it might be possible to regard the whole spectrum of a boundary theory as an irreducible representation of such an algebra.
There are many ways of extension. We can consider more general 3d and 6d theories whose gravity duals are known. For example, it is easy to extend the formula to Chern-Simons quiver gauge theories realized on M2-branes in toric Calabi-Yau fourfolds and 6d N = (1, 0) theories realized on M5-branes in orbifolds. It is also important problem to derive the explicit formula for the multiple brane contributions, together with the pole selection rules in the gauge fugacity integral. We hope we could return these issues in near future.

Acknowledgments
The work of R. A. was supported by the Sasakawa Scientific Research Grant from The Japan Science Society.

A Index of the ABJM theory
In Section 2 we calculated the superconformal index of the ABJM theory as the boundary theory. The index is given by summing up contributions of monopole sectors, which are labeled by monopole charges quantized by The contribution from each monopole sector is given by where i is the single-particle index The gauge fugacity integral gives non-vanishing value only if the monopole charges satisfy In Section 3 we discussed Schur-like index of the theory on M2-branes wrapped on the cycle z 1 = 0. The integral form giving the Schur-like index of each monopole sector I mα, mα is obtained from (129) by setting k = 1 and the variable changê These are compositions of (95) and (116). The single-particle index (130) reduces to where we leaveq to keep the expression simple. As expected this isy-independent. Although the Schur-like index must bex ′ -independent the single-particle index depends onx ′ throughq =q ′x′2 . This is because the above formula is derived by deforming the Lagrangian byQ-exact terms, which does not respect the extra superchargeQ ′ used in the definition of the Schur-like index. If we regard I ma, ma as a function ofq ′ andq, we can easily factor out thě q ′ -dependence by the replacement and obtain I mα, mα =q ′2mtot × (function ofq).
Furthermore, thex ′ -independence of the Schur-like index guarantees that the function ofq is in fact aq-independent constant.
In order to carry out the gauge fugacity integrals we need to choose integration contours. Although we have not yet completely understood how we should do it, we found a prescription that reproduces the known results after some trial and error. We express the integrand as the expansion  (17).
In the case of k = 2 the system still has N = 8 supersymmetry, and the index can be expanded in terms of su(4) characters.
I B-type representations appear. We used in the main text the notations in [19]. They correspond to those used in [24] as follows.
For D[0, 4] and D [1,4] the RS procedure works well, and we obtain no RS trial weights with negative coefficients. For D [3,2] we obtain many weights with negative coefficients. In [24] it is proposed that such weights should be simply eliminated. However, we found that this procedure givesx-dependent result. Namely, the elimination spoils the Bose-Fermi degeneracy of states with∆ = 0. Fortunately the elimination affects terms of orderq 12 or higher, and the lowest order of thex-dependent terms isq For this representation we obtain many weights with negative coefficients. We again found that the elimination of them causes thex-dependence of the result. The elimination affects the terms of orderq 10 or higher, and thex-dependence appears atq 32 3 . (170) is the index after the elimination. Fortunately, terms shown in (170) do not depend onx.
We also calculated (170) in another way. For n ≥ 1 the primary null state of B[2, 0] n appears at level ℓ = 1, and the procedure is much simpler than the case of n = 0 for which the level of the primary null state is ℓ = 3. The RS procedure works well for such representations and all generated weights have positive coefficients. To obtain B[2, 0] 0 we simply substitute n = 0 in the general formula for n ≥ 1. Although we have no justification for this "continuation," this kind of continuation reproduces correct results in many cases. Indeed, we obtained the result whose first few terms agree with (170), and this strongly suggests the correctness of (170).