Simple-Sum Giant Graviton Expansions for Orbifolds and Orientifolds

We study giant graviton expansions of the superconformal index of 4d orbifold/orientifold theories. In general, a giant graviton expansion is given as a multiple sum over wrapping numbers. It has been known that the expansion can be reduced to a simple sum for the ${\cal N}=4$ $U(N)$ SYM by choosing appropriate expansion variables. We find such a reduction occurs for a few examples of orbifold and orientifold theories: $\mathbb{Z}_k$ orbifold and orientifolds with $O3$ and $O7$. We also argue that for a quiver gauge theory associated with a toric Calabi-Yau $3$-fold the simple-sum expansion works only if the toric diagram is a triangle, that is, the Calabi-Yau is an orbifold of $\mathbb{C}^3$.


Introduction
AdS/CFT correspondence [1,2,3] provides a useful window to investigate quantum gravity. Although we have not yet understood how to directly deal with quantum effects of gravity, we can obtain information of such effects through the correspondence from analyses in the boundary theories. The superconformal index [4,5,6] is an important and useful quantity for quantitative investigation of the correspondence. The index can be calculated on the gauge theory side as long as the theory is Lagrangian, and we are also able to calculate the index on the gravity side in an appropriate parameter region. In the strict large N limit, which means N is much larger than the energy scale (or the order in the Taylor expansion of the index) which we are focusing on, the index obtained on the boundary side can be reproduced semi-classically on the gravity side as the contribution from massless fields living in the AdS background [5]. To access the quantum gravity effects via AdS/CFT correspondence, we should consider parameter regions out of the strict large N limit.
One interesting region is the one with the energy scale of order N 2 with large N. On the gravity side such a region is described by classical blackhole solutions, and it was found that the superconformal index of large N gauge theory can correctly reproduces the Beckenstein Hawking entropy by taking appropriate limit of the index [7,8,9]. This discovery is important because it indicates that the Boson-Fermion cancellation does not occur for the majority of states in the Hilbert space and the index can be used as the thermal partition function by taking appropriate values of fugacities.
Another important parameter region, which we focus on in this work, is the one with the energy comparable to N. In the q expansion of the index we find the deviation from the large N limit around this order. On the gauge theory side, this is related to the existence of additional operators or additional constraints due to the finiteness of the rank of the gauge group. On the gravity side, this can be interpreted as the contribution of extended branes. The first example of such a brane was found in the orientifold model, in which D3-branes wrapped on the topologically non-trivial three-cycle in S 5 /Z 2 correspond to Pfaffian operators [10]. Because Pfaffian operators are BPS operators contributing to the index, the corresponding wrapped branes must also contribute to the index. In such an example it is natural to expect the finite N corrections can be given in the form of expansion with respect to the wrapping numbers associated with nontrivial cycles. Even if there are no such non-trivial cycles, there exist stable extended brane configurations called giant gravitons [11,12,13,14]. They are BPS configurations, and should also contribute to the index.
Direct analyses of the contributions of wrapped branes to the superconformal index were carried out for N = 4 SYM [15,16,17] and many other examples [18,19,20,21,22,23]. Essentially the same expnsions were also studied on the gauge theory side in [24,25], and named giant graviton expansions. See also [5,26,27,28,29] for earlier works for the giant graviton contribution to indices and supersymmetric partition functions.
In the analysis of finite N corrections to the superconformal index we need to include extended branes regardless of whether the brane wrapped on topologically non-trivial cycles. Although the term "giant gravitons" originally means extended branes without topological wrappings, in this work we call general extended branes giant gravitons regardless of whether they have topological wrapping or not.
Let us consider the N = 4 U(N) SYM, whose dual geometry is AdS 5 ×S 5 . The superconformal index is defined by where J 1 and J 2 are angular momenta and R x , R y , and R z are R-charges.
The fugacities for these generators, q, p, x, y, and z are constrained by to respect one of the supercharges. We can calculate the index I U (N ) of the N = 4 U(N) SYM by the localization formula dg Pexp(f vec χ U (N ) where Pexp is the plethystic exponential, U (N ) dg is the gauge group integral with the Haar measure, χ We use the notation i[R] for the index of an irreducible superconformal representation R, and we adopt the notation in [30] for R.
In the large N limit, the U(N) integral in (3) can be easily evaluated with the saddle point method, and the result is [5] I U (∞) = Pexp f sugra , where f sugra is the letter index of the supergravity multiplet in AdS 5 × S 5 .
This is obtained by summing up contributions from modes in AdS 5 × S 5 given in [31,32]. If N is finite, we have finite N corrections, and are given by the giant graviton expansion. Let us introduce three complex coordinates X, Y , and Z such that the S 5 is given by |X| 2 + |Y | 2 + |Z| 2 = 1. We take account of giant gravitons wrapped around three cycles X = 0, Y = 0, and Z = 0, and the giant graviton expansion of the index is given by the triple sum [15,17] I U (N ) I U (∞) = ∞ mx,my,mz=0 x mxN y myN z mz N F mx,my,mz , where m x , m y , and m z are wrapping numbers associated with three threecycles in S 5 : X = 0, Y = 0, and Z = 0, respectively.  with the letter index f X=0 of the U(1) vector multiplet living on X = 0. Because the worldvolume of the giant graviton is S 3 × R, and is the same as the AdS boundary, the theory on the worldvolume is essentially the same as the N = 4 SYM. An important difference is the action of symmetry generators, and the generators acting on the boundary and those acting on the cycle X = 0 are related by the involution map [15]: where A is the generator of U(1) R of type IIB supergravity normalized so that A ∈ Z/2. 2 Correspondingly, we can obtain f X=0 from f vec by a simple variable change. This is also the case for the other two-cycles, and the variable changes to obtain the letter indices for three cycles are given by [15,24]. σ x : (q, p, x, y, z) → (y, z, x −1 , q, p), σ y : (q, p, x, y, z) → (z, x, p, y −1 , q), σ z : (q, p, x, y, z) → (x, y, q, p, z −1 ). (10) 2 The U (1) R symmetry acts on the two three-form flux fields non-trivially, and is broken to Z 4 generated by e πiA (for a generic value of the axiodilaton field) due to the flux quantization. Similar to angular momenta J i and R-charges R a , A also related to the fermion number F by e 2πiA = (−1) F . We use a convention with the quantum numbers (J 1 , J 2 , R x , R y , R z , A) = (− 1 2 , − 1 2 , + 1 2 , + 1 2 , + 1 2 , + 1 2 ) for the supercharge respected by the definition of the superconformal index.
(We use σ I (x = x, y, z) for both the involutions acting on the generators and variable changes for the fugacities.) With these variable changes, we can give i I [m I ] (I = x, y, z) as follows.
The contribution from bi-fundamental fields denoted by dots in (8) can be obtained by directly analyzing the open string states. For example, the contribution from the intersection of cycles X = 0 and Y = 0 is f xy χ (mx,my) with and χ (m,m ′ ) is the bi-fundamental character Although we can write down the integrand in (8), there is a difficulty in carrying out the gauge integral. To obtain the functions F mx,my,mz that correctly reproduce the known index we have to carefully choose contours in the integrals and pick up correct poles. Although a set of rules for the pole selection for N = 4 U(N) SYM was proposed in [17], its derivation and rules for more general theories have not yet been known. Although the rules for the functions associated with a single cycle like F m,0,0 are simple and natural, treatment of bi-fundamental fields is involved and calculation of F mx,my,mz for intersecting giant gravitons is complicated.
We can avoid this problem if we can somehow remove the contributions from intersecting giant gravitons. Surprisingly, this is possible. Gaiotto and Lee [24] proposed a giant graviton expansion with simple-sum: The reason of the reduction of the triple-sum expansion to the simple sum is explained with a special behavior of the functions F mx,my,mz , which is referred to as "the wall-crossing" in [24]. Namely, functions F mx,my,mz are not analytic on some walls in the space parametrized by the fugacities, and by choosing an appropriate chamber, some of the functions become identically zero, and they are decoupled from the index calculation. In other words, by choosing appropriate expansion variables, we can decouple some contributions and we can simplify the giant graviton expansion [33].
To clarify what is happening in functions F mx,my,mz , let us first consider a simple toy model.
If we Taylor expand this function around q = 0, this gives the following non-trivial expansion.
However, if we regard F (q) as a function of s = q −1 and perform the sexpansion (expansion around q = ∞), we obtain Indeed, the function F (q) has a singular wall along the unit circle |q| = 1, and it is a non-trivial function inside the wall, while it is trivial outside the wall.
Let us return to the functions F mx,my,mz . We can explain the relation between two expansions, one with the simple sum and the other with the triple sum, by different choices of the expansion variables. Now we have five fugacities, q I = (q, p, x, y, z) constrained by (2). To specify expansion variables we introduce four independent auxiliary variables t i (i = 1, . . . , 4) and write five fugacities in terms of t i as follows Then, we carry out t 1 -expansion first, and then sequentially perform t 2 , t 3 , and t 4 -expansions in that order. This multiple expansion is specified by the set of constants d I,i . Actually, we focus only on the first expansion specified by d I,1 . Let us denote t 1 by t and d I,1 by d I . The first expansion with respect to t(= t 1 ) is equivalently performed by the t expansion after the replacement We can regard t as a fugacity for the operator In the following we call the constant d I "the degree" assigned to the fugacity q I and denote it by d I = deg(q I ). The consistency with the constraint (2) requires the degrees satisfy In [17], the triple-sum expasnion (7) with the degrees deg(q, p, x, y, z) = ( 3 2 , 3 2 , 1, 1, 1) was studied. Then the expansion variable t is the fugacity for the operator In this case all F mx,my,mz give non-trivial contributions. The degrees adopted in the reference [24], which proposed the simple-sum expansion (14), are deg(q, p, x, y, z) = (1, 1, 0, 1, 1) corresponding to the charge With the degrees (24) the contributions with m y + m z ≥ 1 decouple. Let us see how the decoupling occurs with the degrees in (24). In the toy model with (17), the expansion with deg(q) = +1 gives the non-trivial expansion (16), while deg(q) = −1 gives the trivial one. This occurs as follows. Each term q k in the letter index gives the factor 1/(1 − q k ) in the plethystic exponential, and if d = deg(q) is negative, the t-expansion of this factor starts with −t |d| q −1 . Namely, each negative-degree term in the letter index gives positive power of t in the plethystic exponential, and if we have infinitely many such terms, the result becomes t +∞ = 0. Based on this, we obtain the following simple criterion for the decoupling: • The decoupling criterion: If the letter index includes infinitely many negative-degree terms with positive coefficients, its plethystic exponential is trivial and the contribution decouples. 3 Let us apply the criterion to F mx,my,mz for N = 4 U(N) SYM and show the decoupling for m y + m z ≥ 1. As we explained, F mx,my,mz is given in (8) with the adjoint contributions (11). In particular, if m y ≥ 1, i y [m y ] includes corresponding to the constant term (the Cartan part) in χ U (my) adj . If we adopt the degrees in (24), this letter index contains infinitely many negative-degree terms of the form x k y −1 (k = 0, 1, 2, . . .). Therefore, the contributions with m y ≥ 1 decouple. This is also the case for i z [m z ] with m z ≥ 1. This does not happen to i does not contain negative-degree terms and F mx,0,0 with m x ≥ 1 give nontrivial contributions. An advantage of the simple-sum expansion is that we can calculate F mx,0,0 much more easily than general contributions from intersecting branes. By using the relation f X=0 = σ X f vec we can relate F m,0,0 and I U (m) by Therefore, the expansion (14) can be written as A purpose of this paper is to discuss generalization of the simple-sum expansions to orbifold and orientifold theories. We will not give comprehensive analysis. We demonstrate in a few examples that the decoupling occurs and the triple-sum expansion reduces to the simple-sum expansion. We use the decoupling criterion above as a main tool to check the decoupling and we numerically test that the simple-sum expansion actually gives the correct index.
An interesting point of the simple-sum GG expansion is that not only the LHS in (29) but also the RHS is given in terms of the superconformal index of four-dimensional theories labeled by the rank m of the gauge group. We can thus consider the large m limit. In fact, the theory also has the holographic dual, and we can apply the giant graviton expansion to the theory again. We will show in some examples that the expansion of the "dual" theory gives the original theory. Namely, the relation is mutual and invertible. For N = 4 U(N) SYM, this relation is "self-dual", but in general two theories may be different. In the following sections we demonstrate how we can obtain the "dual" theory from the original one.
Another interesting point of the orbifold and orientifold theories is that three fugacities x, y, and z may not be symmetric. As we mentioned above, the degrees (24) give simple-sum expansion associated with the cycle X = 0. Let us call such an expansion "the X-expansion". Similarly, we can also define the Y -expansion and the Z-expansion associated with Y = 0 and Z = 0, respectively. For N = 4 SYM this does not give anything new because of the symmetry among fugacities. However, in more general cases with less supersymmetries, the three simple-sum expansions may give different expansions for a single theory.
It is natural to ask if the simple-sum expansion works for more general examples like AdS 5 × SE 5 , where SE 5 is a Sasaki-Einstein fivefold. It is hard to believe that the simple-sum expansion works for such a case because the expansion is based on the analysis of fluctuation modes on branes wrapped around a specific supersymmetric three-cycle in SE 5 , and the global structure of the manifold cannot be captured. We will discuss the decoupling of supersymmetric cycles for toric SE 5 based on the decoupling criterion above, and show that the simple-sum expansion works only for SE 5 whose toric diagram is a triangle. This means that the SE 5 needs to be an orbifold of S 5 for the simple-sum expansion to work. This paper is organized as follows. In section 2, we study giant graviton expansions for Z k orbifold with N = 2 supersymmetry. In section 3, we discuss O3 orientifold models with N = 4 supersymmetry. In section 4, we consider another orientifold projection with O7-plane. In section 5, we discuss extenion to toric quiver gauge theories. Section 6 is devoted for discussion.

S 5 /Z k
In this section we consider N = 2 quiver gauge theory realized on probe D3-branes in C 2 /Z k background, which we call T N 1 ,...,N k . The AdS/CFT correspondence of these theories was studied in [34,35,36]. See [29] for analytic results for the Schur limit of the index.
We will find that the decoupling works for X-expansion. By the Xexpansion we obtain a dual theory as the theory on GG, which we call T m 1 ,...,m k . We also find that the X-expansion of T

Projection
We consider the boundary theory obtained from the N = 4 U(N) SYM by the Z k orbifold projection with the generator The field contents are obtained from that of the N = 4 U(N) SYM by picking up the Z k invariant degrees of freedom [37]. The insertion of U k in the trace of the superconformal index is realized by the following Z k action on the fugacities.
In addition, the action on the Chan-Paton factor is realized by the following Z k action on the gauge fugacities: h a are holonomy variables and each component takes value in Z/(kZ) = {1, . . . , k − 1, k}, and without loosing generality we can assume h 1 ≤ h 2 ≤ · · · ≤ h N (by using the Weyl group). In other words, holonomy variables are given by where N i are non-negative integers constrained by N 1 + N 2 + · · · + N k = N. The gauge group U(N) is broken by the orbifolding to G UV = U(N 1 ) × · · · × U(N k ), and flows in the IR to G IR = SU(N 1 ) × · · · × SU(N k ). The resulting theory is the quiver gauge theory shown in Figure 2, which we denote by T N 1 ,N 2 ,...,N k . In the IR the diagonal subgroups U(1) i ⊂ U(N i ) become global baryonic symmetries. We will put off the discussion of the baryonic charges, and Figure 2: A part of the circular quiver diagram of the Z k orbifold theory we here focus on the sector with vanishing baryonic charges. In the index calculation this is equivalent to carrying out the gauge fugacity integral not for G IR but for G UV including U(1) i .
In the following we first consider the case with N i = N (and so N = kN). The superconformal index of the non-baryonic sector of T N ,...,N is given by where the superscript '0' indicates this is the index for the non-baryonic sector and P k denotes the projection associated with the Z k action (31) and (32) on the fugacities: Note that we do not remove the contribution from the k IR-free U(1) vector multiplets in (34). The large N limit of the theory T ∞ = lim N →∞ T N ,...,N is dual to AdS 5 × (S 5 /Z k ) where the Z k action on S 5 is given by (30). The fixed locus of the orbifold is AdS 5 × S 1 . The index is The letter index consists of two parts, the contribution from the supergravity multiplet in the ten-dimensional bulk and the contribution from k − 1 tensor multiplets living on the six-dimensional fixed locus. f AdS 5 ×S 1 tensor is the letter index of a single tensor multiplet. See B.1 for a derivation of f AdS 5 ×S 1 tensor .

Decoupling
As in the S 5 case, the index of the theory on a system of giant gravitons with wrapping numbers m x , m y , and m z is given by (8) with different i I [m I ] (and different terms represented by dots). In the orbifold case i I [m I ] are given by where P k is defined by (35). Let us focus on the constant term in the adjoint characters. For each cycle we obtain Let us first consider the Z-expansion defined with the degrees deg(q, p, x, y, z) = (1, 1, 1, 1, 0). We can easily see that none of three functions in (38) contains negative-degree terms for k ≥ 2. This means all three cycles give non-trivial contributions, and the triple-sum expansion does not reduce to the simplesum expansion.
Next, let us consider X-expansion defined with the degrees deg(q, p, x, y, z) = (1, 1, 0, 1, 1). In this case, the functions P k [σ y f vec ] and P k [σ z f vec ] contain infinitely many negative-degree terms. Then, the contributions with m y +m z ≥ 1 become trivial, and we obtain the simple-sum expansion.
As we will discuss in detail in the next subsection, the gauge group U(m) on m coincident giant gravitons is broken down to G UV = U(m 1 )×· · ·×U(m k ) due to non-trivial holonomies on the giant gravitons. We denote the theory by T m 1 ,m 2 ,...,m k . The baryonic charges in T N ,...,N are related to the ranks m k of the unbroken gauge groups, and vanishing baryonic charges correspond to the gauge groups with equal ranks: m 1 = m 2 = · · · = m k =: m. We denote such a theory by T m,...,m . This means that only the terms with wrapping number m = km contribute to the index of non-baryonic sector. Therefore, the simple-sum expansion takes the form where the index appearing on the RHS is that for non-baryonic sector because the baryonic charges in T m,...,m are related to the ranks N i , and we consider the special case with all N i being the same.

Projection
Theory on giant gravitons on the cycle X = 0 is also Z k orbifold theory, and Z k action is obtained from the action in the original theory by the operator map (9): where U k is the Z k generator in (30). This Z k generator contains J 1 nontrivially acting on the boundary coordinates, and the boundary becomes The theory is locally the N = 4 U(m) SYM and Z k identification breaks the gauge symmetry to The insertion of the operator U k in the trace of the index is realized by the following Z k action on the fugacities: The gauge fugacities are also transformed by where h a are holonomy variables which take values in Z/(kZ) = {1, . . . , k}.
Without loosing generality we assume where m i are non-negative integers constrained by m 1 + · · · + m k = m. The holonomy breaks the U(m) gauge symmetry to G UV = U(m 1 ) × · · · × U(m k ).
As we mentioned at the end of the previous subsection the vanishing baryonic charges in T N ,...,N correspond to holonomies with m 1 = · · · = m k =: m. With such a choice of the holonomy, the index is given by where P k is the projection associated with the Z k actions (41) and (42), and is explicitly defined by The index of the large m limit T ∞ = lim m→∞ T m,...,m is given by On the gravity side this is reproduced as the contributions from the gravity multiplet in the ten-dimensional bulk and k − 1 tensor multiplets on the sixdimensional fixed locus AdS 3 × S 3 . See B.2 for the explicit form of f AdS 3 ×S 3 tensor .

Decoupling
Let us consider the triple-sum expansion of I T m,...,m . The functions F mx,my,mz appearing in the expansion take the form (8) with i I [m I ] given by which are similar to (37) but P k is replaced by P k . Let us focus on the constant terms in the adjoint characters χ U (m I ) adj . For each cycle we have Let us first consider whether simple-sum Y -expansion works. With the degrees deg(q, p, x, y, z) = (1, 1, 1, 0, 1), only P k σ z f vec contains infinitely many negative-degree terms, and the decoupling works only partially. Therefore, we cannot obtain simple-sum expansion associated with the cycle Y = 0 (and it is also the case for Z = 0).
On the other hand, with the degrees deg(q, p, x, y, z) = (1, 1, 0, 1, 1) P k [σ y f vec ] and P k [σ z f vec ] contain infinitely many negative-degree terms, and the corresponding cycles decouple. Therefore, the triple sum reduces to the simple sum in the form On the right hand side the index of the original theory T N ,...,N appears because σ x is an involution and applying σ x on U k gives the original Z k generator U k .

Baryonic charges
In the analysis in the previous subsections we focused on the non-baryonic sector, and obtained (39) and (49). Let us generalize the relations by including states with non-vanishing baryonic charges.
We can introduce baryonic charges on the both sides of the relation. We use B i and B i to denote the charges in T N ,...,N and those in T m,...,m , respectively. As we will explain shortly, baryonic charges on one side are related to the ranks of the unbroken gauge symmetries on the other side. Namely, B i are identified with m i and B i are identified with N i up to certain equivalence relations. In general, if SU factors had different ranks, the spectrum of gauge invariant baryonic operators would be complicated. Therefore, in the following we turn on only one of B i and B i for simplicity.
Let us first discuss the baryonic charges B i in T N ,...,N . The index of the baryonic sector with charges B i can be obtained by the insertion of the background charges in (34). These are defined so that the baryonic operators det X i+1,i and det Y i,i+1 carry B i = +1 and −1, respectively. Because (50) This means that only the differences B i − B j are physical quantities.
To extend the GG expansion (39), we should also modify I 0 The giant graviton expansion for the baryonic sector is given by where (m)∈[B] is the summation over non-negative integers m i belonging to the equivalence class [B i ]. The large N limit appearing in the denominator on the left hand side in (52) is the same as before. The inverse expansion of (52) is given by We can also consider the baryonic charges B i in T m,...,m . Then the corresponding ranks N i change. For arbitrary ranks m i the spectrum of gauge invariant baryonic operators is complicated. So, we consider the simple case with equal ranks m 1 = m 2 = · · · = m k =: m. On the T N 1 ,...,N k side this corresponds to restricting the states to the non-baryonic sector. The GG expansion (49) becomes and its inverse expansion is

Numerical tests
First let us test the expansion (39), whose right hand side is an infinite sum over the wrapping number m. We introduce a cutoff m max , and see how the changes as we increase m max .
To realize the X-expansion we take the degrees deg(q, p, x, y, z) = (1, 1, 0, 1, 1). To reduce the computational cost, we take the unrefined parametrization (q, p, x, y, z) = (tx and treat the index as a function of two variables t and x. We first expand with respect to t, and then we perform the x-expansion. For example, the ratio I T 2,2 /I T∞ is If we subtract m = 0 and m = 1 contributions many terms are canceled, and further subtraction of the m = 2 contribution removes more terms appearing in (58).
We show the cancellation graphically by using two-dimensional plots in Figure 3.

X-expansion
Let us numerically test the expansion (49) for T m,...,m . We introduce a cutoff N max and calculate the error  Figure 3: I T 2,2 /I T∞ and ∆ T 2,2 (m max ) with m max = 1, 2 are shown as twodimensional plots. A term t nt x nx with non-vanishing coefficient in the Taylor expansion is expressed as a dot at the coordinates (n t , n x ), and the coefficient of the term is shown beside the dot.
We show the ratio I 0 /I T∞ and the errors ∆ T 2,2 (N max ) with N max = 1 and 2 as two-dimensional plots in Figure 4. /I T∞ and the errors ∆ T 2,2 (N max ) with N max = 1 and 2 are shown.

Baryonic sector
Let us numerically test (52). As a simple case we consider k = 2 and (61) Figure 5 shows the ratio I  We test the expansion (55) for k = 2 and (N 1 , N 2 ) = (3, 2). We define the error function Numerical results for I 0 T 3,2 /I T∞ and ∆ 0 T 3,2 (m max ) with m max = 1 and 2 are shown in Figure 6.

O3-Dsystem
In this section we discuss N = 4 SYM realized by O3-D3 systems. The AdS/CFT correspondence of the model was first studied in [10]. We will mainly discuss the case with O3 − plane, and we denote the corresponding O(2N) SYM by T O(2N ) . We will find the X-expansion works for T O(2N ) , and the theory on giant gravitons, which we denote by T O(2m) , is another orientifold theory. The inverse X-expansion also works, and it gives the original theory T O(2N ) as the theory on giant gravitons (Figure 7).

Projection
Let us consider N = 4 SYM with orthogonal and symplectic gauge groups realized by orientifolds. We first consider the case with O3 − -plane, which gives the orthogonal gauge groups.
The orientifold is defined with the O3 flip operator commuting with all generators in the N = 4 superconformal algebra. Table 1 shows the directions of D3-branes, O3-planes, and the worldvolume of giant gravitons. The gauge group (O or Sp) should be chosen according to the relative positions of D-branes and O3-plane. Table 1: Extended directions of branes are shown. Directions of D3 and O3 are shown by using the coordinates in the flat ten-dimensional spacetime. In the description of the directions for giant gravitons the time directions are treated as the radial direction in the XY Z space. The column O/Sp shows the gauge group realized on D-branes when the orientifold plane is O3 − . For O3 + all the gauge groups become Sp.
For the following analysis it is convenient to refine the superconformal index by introducing a Z 2 fugacity η = ±1 for the operator S.
Then, the orientifold projection operator P O3 acting on letter indices is defined by where we introduced the notation The introduction of the Z 2 fugacity η modifies the variable changes in (10) as follows [15]: To describe the orientifold action on the Chan-Paton factor it is convenient to define the refined character χ where χ and for g ∈ Sp(N/2) ⊂ U(N) (70) make sense only for even N. The letter index of the orientifold theory with O3 − is the Z 2 -invariant part of the refined U(N) letter index When we consider orientifold with O3 + we should replace χ . With the letter index (71), the full index is given by See Appendix A for the explicit forms of the Haar measure and the character. The large N limit of the index I T O(N) is given by (73) f sugra is the Z 2 -refined index of supergravity Kaluza-Klein modes in AdS 5 ×S 5 defined by where the Z 2 twisted index f sugra is given by ( (75) is obtained by directly calculating the alternating sum over n.)

Decoupling
Let us discuss whether we can decouple some cycles by assigning appropriate degrees. The giant graviton contribution again takes the form (8) with Let us assign the degrees deg(q, p, x, y, z) = (1, 1, 0, 1, 1). The letter index We can easily check that for both signs [σ y f vec ] ± contains infinitely many negative-degree terms. It is the case also for [σ z f vec ] ± associated with the cycle Z = 0. Therefore, F mx,my,mz with m y + m z ≥ 1 decouple, and the GG expansion reduces to the simple sum. If the gauge group is not SO(2N) but O(2N), Pfaffian operators are not gauge invariant. This means only contributions with even wrapping number should be included. The simple-sum giant graviton expansion is given by where we denote the theory on m coincident giant gravitons by T O(m) .

Projection
The where S is defined in (63). The directions of the worldvolumes of D-branes and the O-plane are shown in Table 2.
(79) non-trivially acts on the AdS boundary, and the orientifold gives a theory in R t × S 3 /Z 2 . It is locally the same as the N = 4 U(m) SYM, but non-trivial holonomy breaks the gauge symmetry down to O(m). The fixed locus is an O3-plane, which is localized at the center of AdS 5 and wrapped around S 3 ⊂ S 5 given by X = 0.
Therefore, the orientifold projection operator acting on letter indices is defined by With this projection, the index I T O(m) is given by The large m limit of the index I T O(m) is given by

Decoupling
The index for giant gravitons can be easily obtained by combining the variable changes and the U O3 -projection. The functions F mx,my,mz are given by (8) with ] (for I = y, z).
These are explicitly given by We can easily check that both i y [m y ] with m y ≥ 1 and i z [m z ] with m z ≥ 1 contain infinitely many negative-degree terms provided we assign the degrees deg(q, p, x, y, z) = (1, 1, 0, 1, 1). Hence, the X-expansion reduces to the simple-sum expansion of the form adj . The decoupling again works, and instead of (78) and (86) we obtain the expansion and the inverse expansion The gauge group O(2N) consists of two disconnected components. One of them is SO(2N), and we denote the other component by SO (2N). Correspondingly, the index for O(2N) splits into two parts: Two contributions correspond to the two elements of Z 2 = O(2N)/SO(2N). This Z 2 symmetry couples to Pfaffian operators. In O(2N) gauge theory this Z 2 is gauged, and Pfaffian operators do not contribute to the index. This is analogous to the baryonic U(1) symmetries studied in Section 2. For the latter we can extract the contribution of states with non-vanishing baryonic charges by inserting the factor (50) in the gauge fugacity integral in (72). This is also possible for the Z 2 charge. By inserting the factor det g (g ∈ O(2N)) in the integral (72) we obtain the index of the Z 2 -odd states.
Because a giant graviton, which correspond to a Pfaffian operator, is Z 2 odd, only configurations with odd wrapping numbers contribute to (90), and the expansion (78) is changed to The sum of (78) and (91) gives the expansion for SO(2N) gauge group Two expansions (78) and (91) are similar to the expansion (52) for the baryonic sector in the sense that the number of giant gravitons is constrained according to the value of the Z 2 charge.
We can interchange the roles of the Z 2 charge and the number of branes, and obtain the following expansion similar to (55).
where I is defined with the det g insertion into (82), and given in a similar way to (90). The left hand side of (93) should be identical with the left hand side of (87) due to the Montonen-Olive duality. The consistency requires non-trivial relation between I T Sp(m) and I appearing in the expansions. In fact, we can numerically confirm the following relation holds: This is the counterpart of the Montonen-Olive duality on the giant graviton side.

X-expansion of I T O(2N)
Let us check the expansion of (78) for T O(2N ) . Here and in following numerical tests, we again employ the unrefined parametrization (57). We introduce a cutoff m max , and calculate the error function below for N = 1, The results are shown in Figure 8 as two-dimensional plots.

X-expansion of I T O(2N)
Let us check the inverse expansion for T O (2N ) , namely, the expansion of I T O(2N) (86). We calculate the error function for N = 1 with cutoff m max = 1 and 2. The results are shown in Figure 9. Firstly, let us check the expansion of I T Sp(N) (87) and its inverse expansion (88). We introduce cutoff m max and calculate the error functions for N = 1. The results are shown in Figure 10 and Figure 11. Secondary, we check the expansion of (91). We calculate the error function  for N = 1. The results are shown in Figure 12. Finally, we check the expansion of (93), by calculating the error function with m max = 1 and 2 for N = 1. The results are shown in Figure 13.

O7-D3 system
In this section we discuss N = 2 SCFT realized on D3-branes probing the O7-plane background, which we denote by D 4 [N]. We will find all X-, Y -, The O7 − worldvolume is space-filling in AdS 5 and wraps around S 3 ⊂ S 5 given by Z = 0. To keep the conformal invariance we need to introduce four D7-branes (and their mirror images) coincident with the O7-plane.
The orientifold projection with (101) breaks the SO(6) R symmetry down to SU(2) R × U(1) R × SU(2) F , with Cartan operators R x + R y , R z , and R x − R y . The theory realized on the worldvolume of N D3-branes is an N = 2 Sp(N) superconformal field theory [38], which we call D 4 [N]. It has SO(8) flavor symmetry realized on the 7-branes, and we can refine the index by introducing SO(8) fugacities.
With this projection operator, the index of D 4 [N] is given by The first term in the letter index is the contribution from D3-D3 open strings, and is explicitly given by The second term in the letter index is the contribution of D3-D7 open strings. f hyp is the letter index of the N = 2 hypermultiplet.
See Appendix A for the explicit forms of the Haar measure and the characters. 4 The holographic dual of D 4 [N] is AdS 5 ×(S 5 /Z 2 ), where Z 2 action on S 5 is given by (101), and the worldvolume of the O7-plane and the four coincident D7-branes is AdS 5 × S 3 [39,40]. The large N index can be calculated on the gravity side as the contribution from massless fields. There are two contributions: ]. (107) One is the contribution from the gravity multiplet

Decoupling
Again, the contribution from the system of giant gravitons with wrapping numbers m x , m y , and m z is given by (8). We again focus only on the three terms i I [m I ]. They are given by With these letter indices, we can check that all the X-, Y -, and Z-expansions work.
To obtain Z-expansion, we adopt the degrees deg(q, p, x, y, z) = (1, 1, 1, 1, 0). Then, we can show that both i x [m x ] with m x ≥ 1 and i y [m y ] with m y ≥ 1 contain infinitely many negative-degree terms, and they decouple. We obtain the simple-sum GG expansion In this case, the theory on the m giant gravitons is again D 4 [m]. This is because σ z maps the orientifold operator U O7 to itself. The X-expansion and the Y -expansion are essentially the same with each other and they are related by the Weyl reflection x ↔ y of the SU(2) R symmetry. Let us consider the X-expansion for concreteness. To obtain Xexpansion, we adopt the degrees deg(q, p, x, y, z) = (1, 1, 0, 1, 1). Then, we can easily confirm that both i y [m y ] with m y ≥ 1 and i z [m z ] with m z ≥ 1 contain infinitely many negative-degree terms, and the cycles Y = 0 and Z = 0 decouple. As the result, we obtain the simple-sum giant graviton expansion where we denote the theory realized on giant gravitons by D 4 [m]. This will be defined in the next subsection.
The fixed locus of U O7 is the O7-plane (together with four coincident D7branes) along different directions from the one for U O7 . See Table 4. The Table 4: Extended directions of branes are shown.
Sp worldvolume theory on D3-branes probing the 7-brane background is locally the N = 4 U(m) SYM, and the Z 2 identification breaks the U(m) gauge symmetry to O(m).
The insertion of U O7 in the Z 2 refined index (64) is equivalent to the variable change (q, p, x, y, z, η) → (q, −p, −x, −y, −z, −η). (113) Correspondingly, the projection operator P O7 acting on letter indices is defined by With this projection operator, the index is given by where the first term in the letter index is the contribution of D3-D3 open strings, and is rewritten as The second term is the contribution of chiral fermions arising from D3-D7 open strings. They are living on the J 2 -fixed locus in the boundary S 3 , and quantum numbers carried by them are J 1 and SO (8) flavor charges. f ψ is given by The index of the theory D 4 [∞] in the large N limit is where is the contribution from the supergravity multiplet in the ten-dimensional bulk, and the second term is the contribution from D7-D7 open strings.

Decoupling
In the triple-sum GG expansion each contribution with a specific set of wrapping numbers (m x , m y , m z ) takes the form (8). The contribution from each cycle i I [m I ] is given by Because P O7 is symmetric under the permutations among x, y, and z, X-, Y -, and Z-expansions are essentially the same. Let us consider X-expansion with the degrees deg(q, p, x, y, z) = (1, 1, 0, 1, 1). We need to show the decoupling of the cycles Y = 0 and Z = 0. The letter index associated with Y = 0 is We can easily confirm that this contains infinitely many negative-degree terms if m y ≥ 1, and hence the cycle Y = 0 decouples. It is the case for Z = 0 cycle, too, and the triple-sum expansion reduces to the simple-sum expansion of the form

Numerical tests
Let us numerically test the expansion (111). We introduce a cutoff m max and define the error function We calculate the ratio I D 4 [1] /I D 4 [∞] and the error ∆ D 4 [1] (m max ) for the unrefined fugacities (57) and the results with m max = 1, 2, and 3 are shown in Figure 15.

X-expansion of I D 4 [m]
Let us test the expansion of I D 4 [m] in (122). We introduce a cutoff N max and define the error function The ratio I D 4 (1) /I D 4 (∞) and the errors ∆ D 4 [1] (N max ) with N max = 1, 2 and 3 are calculated, and the results are shown in Figure 16. (1)  (3) Figure 16: The ratio I D 4 (1) /I D 4 (∞) and the errors ∆ D 4 [1] (N max ) with N max = 1, 2, and 3 are shown.

Z-expansion of I D 4 [N ]
Let us numerically test the Z-expansion of I D 4 [N ] in (110). We introduce a cutoff m max and define the error function The ratio I and the results are shown in Figure 17. 5 Toric quiver gauge theories

General rules
Let us consider a toric quiver gauge theory associated with a toric Calabi-Yau cone. There is a systematic prescription to determine the gauge theory from the toric data of the Calabi-Yau [41,42,43,44,45,46,47,48]. Let (x I , y I ) (I = 1, . . . , d) be the lattice points on the boundary of the corresponding toric diagram labeled in the counter-clockwise order. The gauge theory has d U(1) symmetries corresponding to the d boundary points. Let R I be the U(1) generators normalized so that R I = ± 1 2 for supercharges and R I = 0 or R I = 1 for scalar component fields in chiral multiplets. The action of R I on the toric fibers of X is given by the vector V I = (x I , y I , 1). If d > 3 V I are not linearly independent, and d−3 linear combinations of V I vanish. Let B a (a = 1, . . . , d − 3) be the corresponding linear combinations of R I . B a generate non-geometric symmetries, which are often called baryonic symmetries.
The superconformal index of the N = 1 toric quiver gauge theory is defined by Supersymmetric three-cycles on which D3-brane can wrap and which contribute to the multiple-sum giant graviton expansion can be defined for each I as the fixed locus of R I , and we denote them by S I . If a boundary point I of the toric diagram is an internal point of a side, the fixed locus S I is a one-dimensional circle, which can be interpreted as a shrinking three-cycle, and the cycle corresponding to a corner point has finite size. For this reason the distinction between the corner points and the others is important. We define the set of all boundary points I = {1, . . . , d} and the subset I C ⊂ I corresponding to the corner points. The complement of I C in I is denoted by I * C = I\I C . As we will see below, wrapping numbers m I for I ∈ I * C are related to holonomies on branes wrapped around finite-size three-cycles.
For a toric diagram with perimeter d, the giant graviton expansion is d-ple sum: where we used short-hand notation m I = {m 1 , . . . , m d }, v m I I = I∈I v m I I , and F m I = F m 1 ,...,m d . However, because the sum over m I * C is the holonomy sum, it is natural to divide m I into the genuine wrapping numbers m I C and holonomy variables m I * C , and we rewrite (128) as We can regard the sum in the parentheses as the index of the theory on the GG system with wrapping numbers m I C . In this sense the GG expansion is |I C |-ple sum, and the GG expansion of the Z k orbifold theory studied in Section 2 is triple-sum for generic degree assignment in this sense. We want to reduce the sum by an appropriate choice of the degrees. The index F m I is given by χ (m I ,m I ′ ) are the bi-fundamental characters in (13). P I and f I in the first term of the letter index are the projection and the letter index associated with the finite-size cycle S I . Note that P I is the projection operator acting on both f I and χ U (m I ) adj , and the action on χ U (m I ) adj depends on the holonomy variables m I * C . I,I ′ in the second term is the summation over pairs of adjacent corners (I, I ′ ), and f I,I ′ is the letter index associated with the intersection S I ∩ S I ′ .
In the second term we should also take account of the holonomy dependence, which is omitted in the following for simplicity.
f I and f I,I ′ can be obtained according to the toric structure encoded in the toric diagram as follows.
As we mentioned above, the Taylor expansion of the index can be expressed as the formal sum of a set of weighted lattice points in the ddimensional lattice. In the large N limit, operators with baryonic charges do not contribute, and B a [u] = 0 are satisfied for all terms u in the Taylor expansion of the index. This define the 3-dimensional sublattice in the ddimensional lattice. In addition, BPS bounds guarantee R I [u] ≥ 0. (Namely, R I [u] ≥ 0 for all I ∈ I.) These inequalities define a cone in the threedimensional lattice.
The large N index I ∞ is given by [49,50] 5 where w I for a specific I is the primitive fugacity satisfying The primitive fugacity is the one such that all fugacities satisfying the same conditions are given as positive powers of it. For each corner point I let (I −1 , I 0 = I, I +1 ) be the three consecutive corner points. We define the three non-baryonic fugacities w I,α (α = 0, ±1) for each I as the dual basis of R Iα by the conditions We can show w I,−1 w I,0 w I,+1 = d I=1 v I = qp. In the case of the orbifold S 5 /Γ the toric diagram is a triangle, and there are three corner points corresponding to R x , R y , and R z . If R I = R x , R I +1 = R y , and R I −1 = R z , the corresponding fugacities are w I,0 = x, w I,+1 = y, and w I,−1 = z. In fact, the letter index f I for a giant graviton wrapped around a finite-size cycle S I is obtained simply by replacing x, y, and z in σ x f vec by w I,0 , w I,+1 , and w I,−1 , respectively. .
If the lattice generated by the three vectors V Iα is not the whole threedimensional lattice but its sub-lattice, then some of R J [w I,α ] are fractional. Such terms must be removed, and this is realized by the projection P I in (130). The letter index of bi-fundamental fields can also be written with these fugacities. Let I and I ′ (= I +1 ) be two consecutive corners. We introduce fugacities w I,α and w I ′ ,α according to (133). If one or both of the intersecting cycles S I and S I ′ is orbifolded, we need to take account of the coupling with holonomies. Let us consider the simple case with neither S I nor S I ′ being orbifolded. This is the case when both V Iα and V I ′ α span the whole 3d lattice. Then, fugacities w I,−1 and w I ′ ,+1 are the same. Let us denote them by w. This is the fugacity associated with the U(1) symmetry shifting the intersection. The letter index of the bi-fundamental fields along the intersection S I ∩ S I ′ is obtained from (12) by replacing z with w.
Now, let us consider degree assignment. We assign d q , d p , and d I to q, p, and v I , respectively. This means that we introduce auxiliary variable t by (q, p, v I ) → (t dq q, t dp p, and perform the t-expansion first. t is the fugacity for the charge and the degree of a fugacity u is nothing but deg(u) = Q t [u]. R t is a linear combination of R I . Because we are not interested in the baryonic charges, it is enough to know its action on the toric fibers, which is expressed as the linear combination of V I . With the normalization d I=1 d I = 1, it can be expressed as a point D = d I=1 d I (x I , y I ) in the toric diagram, and if the position of D is specified, degrees of non-baryonic fugacities are fixed. In particular, the degrees of the fugacities w I,α are determined by the relative positions of D and three corners I α . If D is inside the triangle made by the three corners I α , deg(w I,α ) > 0, while if D and a corner I α are on the opposite sides of the line passing through the other two corners, then deg(w I,α ) < 0. Remark that D is always in the |I C |-gon of the toric diagram (including its boundary), and deg(w I,± ) are always non-negative while deg(w I,0 ) may be negative.
Let us apply the decoupling criterion. For decoupling of a finite-size cycle S I , the letter index f I with the projection P I applied needs to include infinitely many terms with Q t < 0. Because J 1 and J 2 (the exponents of q and p) are non-negative for all terms in the expansion of (134), there must be infinitely many terms with R t < 0. This can be true if the following conditions hold: deg(w I,0 ) > 0 and (deg(w I,+1 ) = 0 or deg(w I,−1 ) = 0).
This means where (I −1 , I 0 ] is the segment between I −1 and I 0 with I −1 excluded and I 0 included. [I 0 , I +1 ) is similarly defined. Note that (139) is a necessary condition, and to show the decoupling of a cycle S I we need to confirm that infinitely many negative-degree terms remain after the projection P I . Obviously, the condition (139) is satisfied at most for two corners I, and the |I C |-ple sum giant graviton expansion at best reduces to (|I C | − 2)-ple sum expansion. For an orbifold S 5 /Γ, the toric diagram is a triangle with |I C | = 3, and as we discussed in Section 2 it may be possible to obtain simple-sum expansion. Unfortunately, this is not possible for the case with |I C | ≥ 4.

Klebanov-Witten theory
As a simplest example of non-orbifold toric Calabi-Yau, let us consider the conifold. The toric diagram is shown in Figure 18. It is a square, and the four corners are labeled by I = 1, 2, 3, 4.
There are a set of prescriptions to read off information of the corresponding quiver gauge theory from the toric diagram [43,44]. It is convenient to use the bipartite graph associated with the toric diagram. The graph is drawn on the torus, and is called the brane tiling. In the brane tiling, faces, edges, and vertices correspond to gauge groups, bi-fundamental chiral multiplets, and terms in the superpotential, respectively.
The brane tiling and the quiver diagram for the conifold are shown in Figure 19. The quiver gauge theory is called the Klebanov-Witten theory [51], which we denote by KW [N].
We can also read off charge assignment for R I from the brane tiling by using perfect matchings. All perfect matchings in the tiling is shown in Figure  20. Each perfect matching is associated with an internal or boundary lattice P 1 P P P point of the toric diagram. Let µ I be a perfect matching associated with a boundary point I. µ I is a subset of edges, and defines a subset of the chiral multiplets. We assign R I = +1 to them, and R I = 0 to the others. There is the unique perfect matching for each I ∈ I C , and charge assignment for R I (I = 1, 2, 3, 4) is uniquely determined as shown in Table 5.  N 1 The global symmetry of the theory is and the U(1) r charge r, the SU(2) A Cartan generator F A , and SU(2) B Cartan generators F B are given by U(1) B is the baryonic symmetry. Corresponding to the linear relation of four vectors The normalization of B is chosen so that det A 1 carries B = +1. The degrees for the point D shown in Figure 18 are up to the ambiguity for baryonic charge. With these degrees the cycles S 3 and S 4 decouple, and the expansion becomes double-sum associated with S 1 and S 2 : The fugacities w I,α are given by The letter index f I for a giant graviton wrapped on S I is and the letter index for the intersection modes on S I ∩ S I+1 is

Z 2 orbifold
Let us re-consider the Z 2 orbifold theory T N,N discussed in 2.1 as an example of toric quiver gauge theories. The toric diagram is shown in Figure 21. cycles S 1 , S 3 , and S 4 , which are respectively referred to as X = 0, Y = 0, and Z = 0 cycles in Section 2, and R I (I ∈ I C ) are related to the R-charges used in Section 2 by The brane tiling and the quiver diagram for the Z 2 orbifold theory are shown in Figure 22. In terms of N = 1 multiplets,  Table 6 and two U(N) vector multiplets V 11 and V 22 .
As in the previous example we can determine charge assignment of R I by using perfect matchings. All perfect matchings for the Z 2 theory are shown in Figure 23. Important difference from the previous example is that we have the vertex I = 2 which is not a corner. In general, there are more than one perfect matchings associated with a point I ∈ I * C , and we should choose one of them to define the corresponding charge R I . This ambiguity affects the definition of the baryonic charges. In the case of the Z 2 orbifold theory T N,N , we have two perfect matchings µ  Table 6. Table 6: Chiral multiplets in the Z 2 orbifold theory T N,N and their charges.
The global symmetry of the T N,N theory is where SU(2) R × U(1) R is the R-symmetry of the N = 2 superconformal algebra, SU(2) F is the flavor symmetry, and U(1) B is the baryonic symmetry. SU(2) R × U(1) R × SU(2) F is the geometric symmetry in the sense that it is realized as the isometry of the background geometry. The SU(2) R Cartan generator R SU (2) , the U(1) R charge R U (1) , and the SU(2) F Cartan generator F are given by The baryonic charge is determined from the linear dependence V 1 −2V 2 +V 3 = 0 as The normalization of B is chosen so that det X 12 carries B = +1. The cross in the toric diagram in Figure 21 shows the point D that gives the degrees (up to ambiguity for the baryonic charge). These are the same as the degrees for the X-expansion used in Section 2 up to normalization. According to the rule (139) two cycles S 3 and S 4 decouple, and the giant graviton expansion becomes the double sum m 1 is the wrapping number for the finite-size cycle S 1 , and runs over all non-negative integers. m 2 , the wrapping number for the shrining cycle S 2 , can be regarded as the holonomy variable, as we will explain below. In the analysis in Section 2 we consider the index I By substituting (52) into I [B,0] T N,N we obtain Let us confirm that (153) and (155) are the same. The relation between two sets of fugacities (x, y, z, b) and (v 1 , v 2 , v 3 , v 4 ) can be read off from With these relations we can rewrite (155) as follows.
By comparing the prefactors in (153) and (157) we obtain These relations imply that m 2 is the parameter specifying the symmetry breaking pattern U(m 1 ) → U(m 1 − m 2 ) × U(m 2 ). In other words, m 2 is the parameter specifying the holonomy on the m 1 coincident giant gravitons. This interpretation require 0 ≤ m 2 ≤ m 1 . The upper bound m 2 ≤ m 1 would be interpreted as a kind of s-rules.
We can also confirm that F m 1 ,m 2 ,0,0 in (153) reproduces the result in Section 2. Because the cycles S 3 and S 4 decouple (130) gives the letter index The fugacities w 1,α are given by With these fugacities we can see f 1 = σ x f vec and (160) agrees with i x [m x ] in (37) with m x = m 1 and the holonomy (159).

RG flow
The orbifold theory T N,N flows to the Klebanov-Witten theory KW[N] by the deformation with the superpotential Because the RG flow does not change the superconformal index we can relate I T N,N and I KW[N ] by the RG flow. The deformation breaks the global symmetry G T N,N of the UV theory to G flow = U(1) r ×SU(2) F ×U(1) B , and at the IR fixed point G flow is enhanced to G KW . Therefore, I T N,N = I KW[N ] holds only for restricted values of fugacities corresponding to G flow . Let r, f , and b be the fugacities for U(1) r , SU(2) F , and U(1) B in G flow , respectively. The restriction on the fugacities of the UV theory is given by For the restricted values the fugacity for the mass terms (162), which carry Concerning the symmetry G KW , only the diagonal subgroup of SU(2) A × SU(2) B is preserved in G flow . The restricted fugacities v ′ I of the IR theory are (In this subsection we use the primed variables v ′ I for KW[N] for distinction from v I for T N,N .) With the localization formula on the gauge theory side we can easily check that the two indices agree for the restricted values of fugacities. The letter index of the Z 2 orbifold theory is given by with χ Φ shown in Table 6, where f v and f c are the letter index for the vector multiplet and the chiral multiplet: The letter index of the Klebanov-Witten theory is also given by (166) with χ Φ shown in Table 5. Indeed, we can easily check We can easily confirm that the prefactor of the orbifold theory in (153) and the prefactor of the Klebanov-Witten theory in (144) agree provided we identify (m ′ , m ′′ ) on the T N,N side with (m 1 , m 2 ) on the KW[N] side: We can also confirm the agreement of the letter index for giant gravitons. In the GG expansion of T N,N the letter index for m 1 coincident giant gravitons on the X = 0 cycle with the holonomy (159) is For the Klebanov-Witten theory, the letter index is given by where f 1 and f 2 are given in (146) and f 12 is given in (147). Two letter indices (171) and (172) agree for the restricted values of fugacities.

Discussion
In this paper we discussed the reduction of the multiple-sum giant graviton expansions to the simple-sum expansions. For orbifold and orientifold examples the triple-sum expansion for generic degrees can be reduced to the simple-sum expansion by assigning appropriate degrees. All theories studied in this paper are Lagrangian theories, and we can calculate the superconformal index directly by using the localization formula. This is in general not the case. For example, M2-brane theory (ABJM theory [52]) and M5-brane theory (six-dimentional (2, 0) theory) are related by the simple-sum giant graviton expansions [33]. In this case, the calculation of the index of (2, 0) theory is only possible with indirect methods [53,54,55,56,57]. We can use the giant graviton expansion as another convenient method to calculate the index of (2, 0) theory by using the ABJM index [58]. The orbifold version of the relation between M2-theory and M5-theory is also interesting because it may enable us to calculate the index of six-dimensional (1, 0) theories by using orbifolds of ABJM theory.
Another interesting class of non-Lagrangian theories includes Argyres Douglas and Minahan Nemeschanski theories realized on D3-branes probing 7-brane backgrounds with constant axiodilation. They are labeled by the type of the 7-brane G = H 0 , H 1 , H 2 , D 4 , E 6 , E 7 , E 8 and the rank N. The AdS/CFT correspondence of these theories in the large N limit was studied in [39,40], and it was confirmed in [22] that the multiple-sum expansion works well at least for the leading giant graviton contributions. If we can apply the simple-sum expansion to this class of theories it gives interesting relations among the indices. Actually, the D 4 [N] theory studied in Section 4 is a special case of the general theories G[N]. We confirmed that the Zexpansion of the D 4 [N] theory works and is self-dual. If this is also the case for general G[N], the following relation should hold In the orbifold and orientifold cases, we can consider three expansions: X, Y , and Z-expansions. Although not always the decoupling occurs, for some examples we can perform the expansion in more than one ways, and we can consider "the web of GG expansions", by applying GG expansions repeatedly. In the case of N = 4 U(N) SYM all the three expansions work. In this case the giant graviton expansion is "self-dual" in the sense that the theory on the giant graviton is also N = 4 U(N) SYM, and application of the giant graviton expansions gives just different frames. Each step of giant graviton expansions is specified by one of the variable changes σ x , σ y , and σ z , and a frame is specified by the composition of the variable changes. Combining three variable changes σ x , σ y , σ z and permutations in (x, y, z) and (q, p) associated with the Weyl groups of SU(4) R and SO(4) spin symmetries we can generate many frames. All these frames gives N = 4 U(N) SYM. However, in more general orbifolds and orientifolds, it may be possible to generate the web including different theories. By applying a projection P , we can obtain an orbifold theory, and in a frame specified by the variable change σ the projection is replaced by σP σ −1 , and in general this is different from the original one. Therefore, the web consists of theories with different orbifold actions. Although in the orbifold case all theories in the web are Lagrangian theories, in more general cases (like G[N] and orbifolds of Mbrane theories) the web may contain both Lagrangian and non-Lagrangian theories, and then it will be convenient to analyze theories that are difficult to analyze directly. This is analogous to the S-duality of type IIB string theory. The duality takes the type IIB theory to the same theory (with the different coupling constant). Let us take Z 2 projection with the worldsheet parity Ω, which gives type I theory. Then the other side of the duality is also Z 2 projected theory. However, this Z 2 is not Ω but (−1) F R , which gives the SO(32) heterotic string. In this way, by applying a projection, we can generate new duality from the duality for the theories before the projection. We can consider a similar situation for the web of giant graviton expansions.
As we have emphasized, the simple-sum expansion is much more easy to calculate because it does not have issues of integration contours. This is because the contribution from single cycle is obtained by a simple variable change from the standard index, for which we can adopt the standard choice of the contours. In Section 5 we saw that the giant graviton expansions of the orbifold theory The invertibility means that a similar expansion of F m gives the original functions F N . Namely, if the functions F ′ N appearing in the expansion of F m are the same as F N , then we can say the expansion is invertible. Unfortunately, we have not yet succeeded in proving (or disproving) this fact. We will only comment on what happen if we naively substitute (175) to (174). It gives Let us suppose x is a generic phase factor x = e 2πiα with irrational α. Then with an appropriate regularization we obtain where N is a large number depending on the regularization. Then we obtain If the product of F ∞ and σ x F ∞ gave N −1 we would obtain expected result F N = F ′ N . Because the large N index F ∞ and F ∞ are given as the plethystic exponential of the letter indices f ∞ and f ∞ of the massless fields in AdS, this can be formally calculated as follows. ( Interestingly, in all examples of pairs of theories related by the X-expansion, we find the relation holds. Therefore, naively, (179) becomes Pexp(−1) = 0. This is nice because we want to obtain N −1 as the result of the regularization. Of course the above manipulation is so naive and formal that it does not make sense as it is. It would be nice if we can improve the derivation.
For G = U(N) the adjoint character is given by The overall factor 1/N! can be fixed by the normalization condition G dµ = 1.
The characters for the symplectic and orthogonal groups are obtained from U(N) characters by appropriate restrictions and projections. To obtain Sp(N) and SO(2N) characters we start from U(2N), and impose the following constraints on the 2N fugacities ζ a (a = 1, . . . , 2N): ζ a+N = 1 ζ a (a = 1, . . . , N).
Then the fundamental and the anti-fundamental characters in (182) (with N replaced by 2N) become the same. The adjoint characters for Sp(N) and SO(2N) are given by taking the symmetric and anti-symmetric products of two copies of the fundamental character: Both these characters include constant term N, and the Haar measures are given by The adjoint character and the Haar measure of SO(2N + 1) are obtained in a similar way starting from U(2N + 1) and imposing the constraints ζ a+N = 1 ζ a (a = 1, . . . , N), ζ 2N +1 = 1.
Remark that the value ζ 2N = −1 should be set after the calculation of the plethystic exponential. For example, Pexp ζ 2N = 1/(1−ζ 2N ) = 1/2. To avoid the confusion with fermionic terms with negative coefficients, we introduce an auxiliary variable θ which is set to be −1 after the calculation of the plethystic exponential. The adjoint character is and the normalized Haar measure is The fixed locus of the Z k generator (30) The term including δ ℓ,1 is the contribution from the equation of motion of the fermion. By summing up all contributions we obtain for a single tensor multiplet.
By summing up all contributions, we obtain B.4 Vector multiplet on AdS 3 × S 5 The fixed locus of O7-flip (112) is AdS 3 × S 5 . Let us use the notation to represent the quantum numbers. R SU (4) is given by the Dynkin labels. The mode expansion in AdS 3 × S 5 gives the conformal representations with the following primaries. [ Dynkin labels [R 1 , R 2 , R 3 ] correspond to the following R-charges.
Underlined conformal representations contribute to the index.
Summing up all contributions, we obtain