Finite $N$ corrections to the superconformal index of toric quiver gauge theories

The superconformal index of quiver gauge theories realized on D3-branes in toric Calabi-Yau cones is investigated. We use the AdS/CFT correspondence and study D3-branes wrapped on supersymmetric cycles. We focus on brane configurations in which a single D3-brane is wrapped on a cycle, and we do not take account of branes with multiple wrapping. We propose a formula that gives finite $N$ corrections to the index caused by such brane configurations. We compare the predictions of the formula for several examples with the results on the gauge theory side obtained by using localization for small size of gauge groups, and confirm that the formula correctly reproduces the finite $N$ corrections up to expected order.


Introduction
The anti-de Sitter / conformal field theory (AdS/CFT) correspondence [1][2][3] has been extensively investigated, and a lot of evidence has been found. However, the majority of previous works have concerned the large-N limit. One reason for this is that as the parameter relation shows, if N is small the Planck length l p becomes comparable with the AdS radius L and the quantum gravity effect is expected to be relevant. Even so, we may be able to obtain non-trivial evidence for the AdS/CFT correspondence with finite N by analyzing quantities protected by supersymmetry. In this paper we use the superconformal index [4] as such a protected quantity, and study the AdS/CFT correspondence for finite N . Even if the quantum gravity correction does not affect the index there is another source of finite-N corrections. The parameter relation in Eq. three-dimensional lattice Z 3 with V I defined in it by V Z ∈ V = R 3 . V and V are dual spaces to each other and the inner product V ⊗ V → R is defined. The polyhedral cone in V spanned by V I is called the toric cone (see Fig. 1(a)).
The toric data expressed as the set of vectors V I ∈ V Z has two roles. One is to define the base manifold B. B is defined as the dual cone of the toric cone by B = {y ∈ V | V I · y ≥ 0 ∀I } (2.1) (see Fig. 1(b)). For each I we define the facet F I by Let R r with r = I + 1/2 be the ridge shared by the two adjacent facets F I and F I +1 . If a vertex I is not a corner but is on the side between two corners I 1 and I 2 , Eq. (2.2) gives not a plane but a line.
We can regard such an F I as a shrinking facet. The five-dimensional Sasaki-Einstein manifold SE 5 is the subset of the Calabi-Yau cone defined by ρ = 1. The radial coordinate ρ in the Calabi-Yau cone is given by ρ = b · y, where b ∈ V is a particular vector inside the toric cone called the Reeb vector. The cross section of B defined by b · y = 1 is a polygon P. The Sasaki-Einstein manifold SE 5 is given as the T 3 fibration over P. We denote the edge F I ∩ P and the corner R r ∩ P by E I and C r , respectively.
The fiber shrinks in a particular manner on the boundary of P. The second role of the toric data is to specify how the fiber shrinks on the boundary. At a generic point in P the fiber is T 3 , which is identified with V/V Z , and we can specify a cycle in the T 3 by a vector in V Z . On an edge E I the cycle V I shrinks, and at a generic point in E I the fiber is T 2 . At a corner C r two cycles V r±1/2 shrink at the same time and the fiber becomes S 1 . For each edge E I the fibration over E I gives a closed three-dimensional manifold, which we denote by S I .
If I 1 and I 2 are two adjacent corners of the toric diagram and k := I 2 − I 1 > 1 then there are k − 1 vertices between I 1 and I 2 , and the corresponding cycles S I are shrinking at the degenerate corners C I 1 +1/2 = · · · = C I 2 −1/2 . This means the existence of A k type singularity along the S 1 fiber over the degenerate corners. Each shrinking three-cycle S I (I 1 < I < I 2 ) is the direct product of the S 1 and a shrinking two-cycle at the singularity.

Quiver gauge theories
There is an algorithm [8,9] to obtain from the toric data the quiver diagram of the gauge theory realized on D3-branes placed at the apex of a toric manifold. We will not explain it in detail but comment only on facts relevant to our analysis. It is known that the number of anomaly-free U (1) global internal symmetries is the same as the perimeter d of the toric diagram. They include • one superconformal R-symmetry, 3/31 PTEP 2020, 043B09 R. Arai et al. • two mesonic symmetries, and • d − 3 baryonic symmetries.
The mesonic and baryonic symmetries are also called flavor symmetries. The R-symmetry and the mesonic symmetries act on the SE 5 as isometries. If the manifold has a non-Abelian isometry group they are the Cartan part of it.
An R-symmetry is defined as a symmetry that acts on the supercharges in a specific way. This condition, however, does not fix the R-charge, and there is an ambiguity to mix the other d −1 charges that do not act on supercharges. We can determine the R-charge appearing in the superconformal algebra by using a-maximization [12] (or, equivalently, volume minimization [13] on the gravity side.) For analysis of a general toric quiver gauge theory there is a convenient basis of d independent charges rather than the basis associated with the classification above. We denote them by R I (I = 1, . . . , d), and each of them corresponds to a vertex on the boundary of the toric diagram. We use a bipartite graph to connect a toric diagram and a corresponding quiver gauge theory, and on the gauge theory side the R I are defined with perfect matchings in the bipartite graph [14][15][16]. We normalize them so that all bi-fundamental fields carry charges 0 or +1. All the R I act on the supercharge in the same manner and hence they are all R-charges with unusual normalization; the supercharges carry R I = ±1/2. There is a prescription to associate each perfect matching on the graph with a vertex in the toric diagram. For each corner of the toric diagram there is a unique perfect matching and we can uniquely define the corresponding R I , while for other vertices on the boundary we have more than one perfect matching and we need to choose one of them to define R I . These d charges form a basis of the global symmetries, and all charges listed above are linear combination of R I in the form For an R-charge with the standard normalization d I =1 c I = 2, while for a flavor symmetry d I =1 c I = 0. An important and convenient property of these charges is that their geometric action on SE 5 is specified by V I . Therefore, for a baryonic symmetry, which has no geometric action, the coefficients satisfy d I =1 c I V I = 0. This is consistent with the fact that the rank of the baryonic symmetry is d − 3.
On the gravity side R I is the angular momentum associated with the geometric action V I . A wrapped D3-brane can carry angular momenta even if it stays still due to the coupling to the background RR 4-form potential field C 4 , and the values depend on the gauge choice of C 4 . We can specify the gauge choice by specifying the singular locus of the potential field just like the Dirac string of a Dirac monopole. The singularity of C 4 is expressed as a three-cycle in SE 5 . R I is defined with C 4 with the singularity on S I . With this definition we can show that a D3-brane wrapped over S I carries For a vertex I which is not a corner there exist more than one perfect matching, and hence the definition of R I is ambiguous. This is related to the ambiguity of choosing a basis of three-cycles at the singularity. At an A k−1 singularity there are k − 1 shrinking cycles. We can relate these cycles to simple roots of A k−1 algebra, and we have ambiguity in the choice of the simple roots from the root system. There are k! bases related by Weyl reflections. Once we fix a basis of shrinking two-cycles we can choose R I so that the charge relation in Eq. (2.4) holds. On the gauge theory side the baryonic symmetry is defined as the anomaly-free subgroup of the abelian group defined by replacing all gauge groups SU (N ) by U (1). It has, in general, the form where G disc is a discrete group. We neglect G disc in the main text for simplicity, and provide some analysis of G disc in Appendix C.

Superconformal index
We define the superconformal index by where the R I are the R-charges defined in the previous subsection and J and J are the Lorentz spins. The trace tr BPS is the summation over states (operators) saturating the BPS bound where Q is the supercharge with the quantum numbers For Eq. (2.6) to be the superconformal index the fugacities must satisfy Notice that to calculate the index in Eq. (2.6) we do not have to know the superconformal U (1) R charge r * .
In the numerical calculation we necessarily introduce a maximum order at which we cut off the infinite series. For this purpose we need to define "the order" for each term in the index as a linear combination of quantum numbers appearing in the index as exponents of fugacities. We can use the linear combination 3J + 3 2 r of J and an appropriately chosen R-charge r. A choice of r given by BPS operators. This requires c I ≥ 0, where the equality is allowed only for I corresponding to the shrinking cycles.
A natural choice for r with clear physical meaning is the superconformal U (1) R charge r * . In that case the order is 3J + 3 2 r * = H + J for BPS operators, and up to the spin J the order is identified with the dimension of operators. However, they are in general irrational, and then we need special treatment in the numerical calculation. To simplify the calculation we adopt r with rational coefficients.

Kaluza-Klein contributions
In the large-N limit the superconformal index is calculated on the gravity side as the index of the Kaluza-Klein modes, and given by where Pexp is the plethystic exponential defined in Eq. (A.2) and i KK is the single-particle index of the Kaluza-Klein modes. For the simplest example with SE 5 = S 5 corresponding to the N = 4 supersymmetric U (N ) Yang-Mills theory i KK was calculated in Ref. [4]: (To be precise, this is not the single-particle index of the N = 4 theory given in Ref. [4]. We subtracted the contribution of the diagonal U (1) part of the N = 1 vector multiplet while the diagonal parts of the three adjoint N = 1 chiral multiplets are left included because we treat adjoint fields as special bi-fundamental fields, for which we cannot define the diagonal part.) In the case of an abelian orbifold S 5 / , the contribution of the gravity multiplet is given by Eq. (3.2) again with v I replaced by appropriate powers of them. In addition, if the orbifold has singular loci, we need to include the contribution of tensor multiplets localized on the loci. The number of tensor multiplets is d − 3, and each tensor multiplet contributes to i KK by [17] with an appropriate choice of w. From these results it is natural to guess the following formula for a general toric manifold: where r is a label for edges of the toric diagram and w r are fugacities defined for each edge r. Indeed, the formula in the form of Eq. (3.4) was derived in Refs. [18,19], and the fugacities w r are defined in the following way. Let us focus on a ridge r of the dual cone shared by two adjacent facets F I and F I +1 with I = r−1/2. Let g r ∈ V Z be the primitive integer vector along the ridge. In general, a choice of a vector in V defines a coordinate in V through the inner product. We denote the coordinate defined with g r by ϕ r . The coordinate ϕ r has clear geometric meaning. By definition, g r is orthogonal to V I and V I +1 . Therefore, on the corresponding cycles in the T 3 fiber it takes a constant value, and it is well defined even if these cycles shrink on the ridge. Namely, it is a coordinate parameterizing the S 1 fiber on the 6  ridge. The primitivity of g r means that the coordinate ϕ r is normalized so that the period of S 1 is 1.
We define fugacities associated with the coordinates by (3.5) and these are the fugacities appearing in Eq. (3.4).

Wrapped D3-brane contributions
Let us discuss the contribution of D3-branes wrapped on three-cycles.
Analyses of orbifold theories [7,17] show that D3-branes wrapping on shrinking cycles do not contribute to the index. The large-N index is reproduced by taking account of only the Kaluza-Klein contribution of the gravity and tensor multiplets. This fact suggests that we have only to consider D3-branes wrapped on visible cycles corresponding to corners of the toric diagram.
In this paper we only focus on D3-branes with single wrapping, and leave the analysis of branes with multiple wrapping for future work. A technical remark is in order. If there are shrinking cycles, the definition of a basis for the shrinking cycles is ambiguous and the distinction between single wrapping and multiple wrapping becomes unclear. To make the following analysis simple, as much as possible we neglect the wrapping numbers for shrinking cycles by setting the corresponding fugacity to be 1. For more detailed explanation of this point see Appendix C.2, in which we show an example of analysis keeping such fugacities.
With this remark in mind, we propose the following formula for the index of a D3-brane wrapped over a visible cycle S I : The factor v N I is the classical contribution obtained from Eq. (2.4). i D3 S I is the single-particle index of the fluctuations on the wrapped D3-brane, and m I is a numerical factor representing the degeneracy associated with the U (1) holonomy on the wrapped D3-brane. In the following we explain how we can determine i D3 S I and m I from the toric data. Let S I be a visible three-cycle we are focusing on. This means I is a corner of the toric diagram. S I is a T 2 fibration over E I with the fiber shrinks at two ends. We can identify the T 2 fiber with L/L Z . (Remember that L is the plane with the toric diagram drawn on it and L Z is the integer lattice in L. ) We introduce two coordinates ϕ r and ϕ r with r = I − 1/2 and r = I + 1/2 to parameterize the plane L. The two sides sharing I are given by ϕ r = 0 and ϕ r = 0 (see Fig. 2). We regard the plane L as the covering space of the T 2 fiber on E I . The T 2 fiber is given as a lattice quotient of the plane as follows. Let I − k and I + k be the corners adjacent to I ; k and k are the lengths of the sides sharing the vertex I . We define m and n by and consider the parallelogram on L defined by ( Fig. 2 We regard this as a torus by identifying opposite sides. If there are no lattice points in this parallelogram except ϕ r = ϕ r = 0, this torus is nothing but the fiber at a generic point on E I . The cycle with constant ϕ r shrinks at the endpoint C r , while the cycle with constant ϕ r shrinks at the other endpoint C r . As a result, the three-cycle S I is topologically S 3 . If the area of the parallelogram is greater than 1 we should perform the orbifolding with , the discrete group defined by the lattice points in the parallelogram. Then the three-cycle S I is topologically S 3 / . The U (1) holonomy on the D3-brane wrapped on S I is specified by an element of the dual group = Hom( , U (1)), and the number of elements of is the same as | |, which is the same as the area of the parallelogram. Because we consider a single D3-brane, no fields on the brane couple to the U (1) holonomy and the existence of | | values of the holonomy affects the index simply as the overall numerical factor. This gives the degeneracy factor (3.9) We can also take I −1 and I +1 to define the parallelogram ( Fig. 2(b)). Namely, instead of Eq. (3.7) we define m and n by m = ϕ r | I +1 , n = ϕ r | I −1 . (3.10) If kk > 1 this gives a smaller parallelogram. With this minimal choice we obtain a smaller orbifold group , which is related to by This should give the same S I as above. Namely, the following relation should hold: This is checked as follows. From the relation in Eq. (3.11), S 3 / = (S 3 /(Z k × Z k ))/ and this is the same as S 3 / because Z k and Z k act on the coordinates ϕ r and ϕ r independently and Even if we use the expression S I = S 3 / , the multiplicity m I is still given by Eq. (3.9) because when we use the S 3 before the orbifolding has a type A k−1 singular locus if k > 1 and type A k −1 singular locus if k > 1. These singular loci cause the additional factors k and k in the multiplicity factor, and we again obtain m I = kk | | = | |.
Next, let us determine i D3 S I , the single-particle index of the fluctuations of the fields on a D3-brane wrapped on S I . In the case of SE 5 = S 5 it is [6] f (q, y, w r , where w r and w r are fugacities associated with the two ridges r = I − 1/2 and r = I + 1/2. We claim that if S I is topologically S 3 the formula in Eq. (3.13) gives the correct answer even for a general toric manifold for which S 3 is not always round. The reason is as follows. The index can be regarded as the partition function of the theory defined in S 3 × S 1 with appropriate background fields. In general, a supersymmetric partition function in a supersymmetric background depends only on a small number of parameters of the background fields [20]. In the case of spacetime with the topology S 3 × S 1 the parameters are nothing but the fugacities, and even for a deformed S 3 the index should take the same form as the one for the round S 3 with appropriately chosen fugacities depending on the background fields. The fugacities can be determined as parameters for the complex structure of the manifold and the moduli of the normal bundle following the detailed analysis in Ref. [20]. Fortunately, in the case of the toric theories we are studying here we can skip this analysis. In the basis of charges used in the definition in Eq. (2.6) of the index, all charges are quantized in units of 1/2 and quantum numbers of BPS states do not change under continuous deformations as long as the deformations respect the supersymmetry and the U (1) symmetries, and then the index is the same as that for the round sphere.
If S I is an orbifold S 3 / we need to perform the orbifold projection. Let (α g , β g ) be the coordinates (ϕ r , ϕ r ) at the lattice point in the parallelogram corresponding to g ∈ . The single-particle index on S I = S 3 / is given by where ω k ≡ exp 2π i k . This formula reproduces the results of the orbifold theories in Ref. [7]. As a simple consistency check let us confirm that two descriptions of S I with the two orbifold groups and related by Eq. (3.11) give the same single-particle index. We should confirm the relation P f (q, y, w r , w r ) = P f (q, y, w r , w r ). (3.15) Because of the relation in Eq. (3.11) we can decompose P as P = P P Z k ×Z k where P Z k ×Z k is defined by It is sufficient to show that

Examples
In this section we calculate the superconformal index on the gravity side according to the formula in Eq. (3.6) and compare the obtained results with those numerically calculated on the gauge theory side by the localization method summarized in Appendix A to confirm that our prescription does work correctly. Due to machine power limitations we take N = 2 and 3. We will show only the results for N = 2 in this section; the results for N = 3 are shown in Appendix B.   Fields First, we consider SE 5 = T 1,1 . T 1,1 is the base of the conifold, and the corresponding boundary theory is the so-called Klebanov-Witten theory [21]. The toric diagram, the bipartite graph, and the quiver diagram are shown in Fig. 3, and the matter contents of this theory are shown in Table 1.
Corresponding to four vertices of the toric diagram we have four R-charges R I (I = 1, 2, 3, 4). The conifold and T 1, The Cartan generators F A and F B of the SU (2) factors are The remaining linear combination of R I is the baryonic charge which does not act on T 1,1 geometrically. On the gauge theory side this can be regarded as the non-diagonal part of the U (1) × U (1) symmetry defined by replacing the SU (N ) gauge groups by U (1). Because the theory is non-chiral this symmetry is anomaly free. See Table 1 for the charge assignments for these generators. Corresponding to the charges F A , F B , and B, we introduce the fugacities u, v, and ζ by the relation The canonical variables v I are expressed with new variables by  The lowest-and higher-order contributions to the indices of the D3-branes with the wrapping numbers B ∈ Z, and the corresponding brane configurations for T 1,1 . S I + S J denotes the brane configuration consisting of a brane wrapped on S I and another brane wrapped on S J .

B
Lowest order, (config.) Higher order (config.) Due to the SU (2) A × SU (2) B flavor symmetry the index can be written in terms of SU (2) characters On the gauge theory side we define the index for the sector with a specific baryon number B by the expansion with respect to ζ : (4.7) Instead of defining I gauge B as the coefficients in the ζ expansion, we include ζ B in I gauge B . On the gravity side the baryonic charge corresponds to the wrapping number of D3-branes [22].
By using these we can determine brane configurations contributing to the index of a specific wrapping sector and the orders of their contributions. For example, let us consider the B = 1 sector. There are two single-wrapping brane configurations S 1 and S 3 with B = 1. They give O(q 3 4 N ) contributions to the index. Furthermore, we also have multiple-wrapping configurations like S 1 + S 2 + S 3 . If we naively assume that the contribution is given as the product of constituent contributions, this gives O(q 9 4 N ) terms. The expected orders of the corrections obtained in this way for different wrapping numbers are shown in Table 2.
Now, let us start the comparison of the results on the gravity side and those on the gauge theory side. We consider the N = 2 case here; see Appendix B for the results for N = 3. Let us first consider the B = 0 sector. The relation expected from the order estimation of the multiple-wrapping D3-brane contribution is On the gauge theory side the numerical analysis for N = 2 gives On the gravity side I KK was first calculated in Ref. [23], and is given by Eq. (3.4) with the fugacities  These two relations are not independent due to a symmetry. The toric diagram ( Fig. 3(a)) and the bipartite graph ( Fig. 3(b)) have the Z 4 rotational symmetry. The counter-clockwise π/2 rotation maps the vertex I to I + 1 and the charge R I to R I +1 . The charges r * , F A , F B , and B are mapped as (4.14) Therefore, the index is invariant under PTEP 2020, 043B09 R. Arai et al. (4.16) As mentioned above, this is written in terms of SU (2) characters χ u n and χ v n . On the gravity side there are two contributions: Although neither of them respects the SU (2) A flavor symmetry, the sum of two contributions becomes a linear combination of SU (2) A characters by the mechanism explained in detail in Ref. [6]. The result for N = 2 is

T 2,2 (complex cone over F 0 )
The next example we discuss is SE 5 = T 2,2 . As is obvious from the toric diagram shown in Fig. 4 this is a Z 2 orbifold of T 1,1 . The corresponding Calabi-Yau cone is a Z 2 orbifold of the conifold, which is often referred to as the complex cone over F 0 . The isometry group is ( Again, we have four R-charges R I (I = 1, 2, 3, 4) corresponding to the corners of the toric diagram. We define charges r * , F 1 , F 2 , and B in the same way as in the conifold case: The charge assignments are shown in Table 3.
Corresponding to the charges in Eq. (4.19) we define the fugacities u, v, and ζ by R. Arai et al. Table 3. Matter contents and charge assignments for the F 0 model.  These relations are identical to those in Eq. (4.5). The ridge fugacities are given by These are the squares of the corresponding variables in Eq. (4.11). This is a reflection of the fact that T 2,2 is the Z 2 quotient of T 1,1 . These are invariant under two Z 2 actions u → −u and v → −v corresponding to the quotient group Z 2 2 in the isometry group. According to the general rule there is one continuous baryonic U (1) symmetry. In addition, we have the non-trivial discrete factor G disc = Z 2 ×Z 2 . In this section we neglect this; see Appendix C.1 for some analysis with these discrete factors. Again, we define the index I gauge B of the sector with a specific baryonic charge B by the expansion (4.23) For small values of B we show in Table 4 the expected order of the corrections due to wrapped branes. We focus on the sectors with B = 0 and B = ±1. The expected relations are as follows: We check these relations for the N = 2 case. See also Appendix B for the analysis with N = 3. We use the SU (2) characters χ u n , χ v n , and χ J n defined in Sect. 4.1 to write down indices. On the gauge theory side the localization formula gives, for B = 0, On the gravity side the index for each supersymmetric cycle is given by (4.28) The sum of the two contributions I D3 S 1 and I D3 S 3 for the N = 2 case is We find that the last relation in Eq. (4.24) holds.

Y 2,1 (complex cone over dP 1 )
There is a family of SE 5 denoted by Y p,q . We consider Y 2,1 as the simplest example. The corresponding Calabi-Yau cone is the complex cone over the first del Pezzo surface (dP 1 ). The toric diagram, the bipartite graph, and the quiver diagram are shown in Fig. 5. Generally, the isometry of Y p,q is SU (2) × U (1) × U (1), and the toric diagram has d = 4. For Y 2,1 we define generators of the flavor symmetry by    Table 5. Matter contents and charge assignments for the dP 1 model.
for simplicity of the index calculation, rather than the one in the superconformal algebra [24]: The charge assignments are given in Table 5. We introduce new fugacities u, v, and ζ by The fugacities v I in terms of the new ones are We have one continuous baryonic U (1) symmetry. We can easily confirm on the gauge theory side that it is the full anomaly-free baryonic symmetry, and there is no discrete factor in this example. On the gauge theory side we define the index I gauge B of the sector with a specific baryonic charge B by the expansion On the gravity side the expected orders of corrections due to wrapped D3-branes are shown in Table 6. There are three sectors B = −1, −3, and 2 that receive corrections from single-wrapping brane configurations. We focus on these three sectors and the B = 0 sector. The expected relations for these sectors are  . The lowest-and higher-order contributions to the indices of the D3-branes with the wrapping numbers B ∈ Z, and the corresponding brane configurations for Y 2,1 .

B
Lowest order, (config.) Higher order, (config.) We check these relations for N = 2. See also Appendix B for the analysis with N = 3. For the B = 0 sector the localization formula with N = 2 gives We use the SU (2) characters χ u n = χ n (u) and χ J n = χ n (y). On the gravity side the contribution of the Kaluza-Klein modes is given by Eq. (3.4) with the fugacities The result is and we find the first relation in Eq. (4.36). For the B = −1 sector, the localization formula with N = 2 gives On the gravity side the corresponding brane contribution is On the gravity side we obtain We see that the third relation in Eq. (4.36) holds. For the B = 2 sector, the gauge theory result is On the gravity side the contributions of S 2 and S 4 are These are summed to = ζ 2 (. . . terms identical with Eq. (4.44) . . .)

L 1,2,1 (suspended pinch point)
The final example is L 1,2,1 , the base of the suspended pinch point. The toric diagram, the bipartite graph, and the quiver diagram are shown in Fig. 6. Because the toric diagram has a vertex on the boundary which is not a corner (vertex 1 in Fig. 6(a)), the manifold has the corresponding shrinking cycle. In this section we neglect the wrapping number on the shrinking cycle by setting the corresponding fugacity to be 1. See Appendix C.2 for an analysis with it taken into account. In this section we take account of only four charges, R 2 , R 3 , R 4 , and R 5 , PTEP 2020, 043B09 R. Arai et al.   Table 7. Matter contents and charge assignments for the L 1,2,1 model. The charges R 1 , R 1 , and B are related to the shrinking cycle and are defined in Appendix C.2.
associated with the corners of the toric diagram. For convenience we define the following four linear combinations: r is not the R-charge r * appearing in the superconformal algebra, which is given by [25][26][27] The charge assignments for the charges defined above are shown in Table 7.
In this example we have Z 2 symmetry acting on the toric diagram as the permutation of the vertices (1, 2, 3, 4, 5) → (1, 5, 4, 3, 2). On the gauge theory side this is the charge conjugation mapping a bi-fundamental field X ab to X ba ; r * and F 2 are even under this charge conjugation while F 1 and B are odd.
The gauge theory is non-chiral and the baryonic symmetry is obtained by replacing the SU (N ) gauge groups by U (1) and removing the diagonal U (1). There are two baryonic charges, and B defined in Eq. (4.48) is one of them. The other (shown as B in Table 7) is associated with the shrinking cycle and we neglect it in this section. We have no discrete factor in the baryonic symmetry.
We define new fugacities by R. Arai et al.
and v I are given in terms of the new variables by (4.51) v 1 is the fugacity associated with the shrinking cycle and is set to be 1. We define the index I gauge B of the sector with a specific baryonic charge B by the expansion The brane configurations and expected orders of their contribution are shown in Table 8 for −2 ≤ B ≤ 2. The expected relations obtained from the order estimation in Table 8 are On the gravity side, by using Eq. (3.4) with the fugacities where χ J n = χ n (y), and on the gravity side, and on the gravity side, = ζ 2 (. . . terms identical with Eq. (4.59) . . .)

Conclusions
We have investigated the superconformal index of N = 1 quiver gauge theories realized on D3branes in toric Calabi-Yau manifolds. The holographic dual of such a quiver gauge theory is type IIB string theory in AdS 5 × SE 5 , where SE 5 is a toric Sasaki-Einstein manifold. A D3-brane wrapped on a supersymmetric three-cycle in SE 5 , which corresponds to a baryonic operator in the gauge theory, contributes to the index as a finite-N correction. We proposed the formula in Eq. (3.6) for the correction to the index due to such a D3-brane and fluctuation modes of massless fields on the brane. Equation (3.6) is a natural generalization of similar formulas proposed in Ref. [6] for S-folds and in Ref. [7] for orbifolds. Similarly to these previous cases a wrapped D3-brane has topology S 3 / , where is an abelian group. A difference is that for a toric manifold the S 3 is in general not round. Even so, the formula is still quite simple thanks to the fact that the index depends on the background geometry through only a small number of parameters. The formula is applicable to general toric quiver gauge theories. Starting from the toric data of the SE 5 we can easily calculate the corrections induced by D3-branes with single wrapping. We did not take account of D3-branes with multiple wrapping. We confirmed that the formula works correctly for several examples (SE 5 = T 1,1 , T 2,2 , Y 2,1 , and L 1,2,1 ) by comparing the index obtained from 21/31 PTEP 2020, 043B09 R. Arai et al. the formula with the result of numerical calculation using the localization method. The errors are consistent with the interpretation that they are due to branes with multiple wrapping.
The formula consists of three factors: the degeneracy factor m I , the classical factor v N I , and the excitation factor Pexp i D3 S I . The degeneracy factor m I was interpreted as the degeneracy of states on the wrapped D3-brane due to the presence of different gauge holonomies. Because we considered only D3-branes with single wrapping the theory on the D3-brane is U (1) gauge theory consisting only of neutral fields. Then holonomies do not couple to any excitation modes on the brane, and different holonomies simply give an overall numerical factor m I .
Part of the degeneracy factor is associated with torsion cycles or shrinking cycles, and we can turn on fugacities coupling to them. In the main text we neglected wrapping on such cycles by setting the corresponding fugacity to be 1. See Appendix C for preliminary analyses in which we introduce fugacities to see the refined structure of the index in two examples.
PTEP 2020, 043B09 R. Arai et al. where the contribution of the SU (N ) a vector multiplet V a is adj , (A. 6) and the contribution of the chiral multiplet X ab is .
and χ is defined from this by replacing all fugacities with their inverse. χ (A.9) R. Arai et al.
where n i are the SU (N ) i instanton numbers. For an element (ζ 1 , ζ 2 , ζ 3 , ζ 4 ) to be anomaly free, the following relation must hold for arbitrary n i : Namely, ζ i must satisfy We can set ζ 4 = 1 with the decoupled diagonal U (1). Then the solution of Eq. (C.3) is where ζ ∈ U (1) and σ 1 , σ 2 = ±1. This means that the anomaly-free baryonic symmetry is U (1)×Z 2 2 . ζ is identified with the fugacity defined in Eq. (4.20). In addition, we can introduce Z 2 -valued fugacities σ 1 and σ 2 to define the index.
The action of each Z 2 symmetry on bi-fundamental fields is read off from Eq. (C.4). They act non-trivially only on baryonic operators. Corresponding to the four arrows in the quiver diagram in and these carry the Z 2 charges shown in Table C1. (We have neglected the superscripts of the bi-fundamental fields, which play no role here.) We expand the index by using two Z 2 -valued baryonic charges together with the integer baryonic charge: external lines are parallel to each other. In the conformal limit in which all the external lines meet at a point these parallel lines coincide, and the corresponding cycles shrink. To make these cycles visible we need to resolve the singularity by separating parallel lines from each other. The permutation group S k acting on these lines is identified with the Weyl group associated with the A k−1 singularity acting on the shrinking cycles at the singularity.
In the case of the L 1,2,1 model we have two parallel external lines corresponding to the A 1 singularity. We focus on the shrinking cycle S 1 and the visible cycles S 5 and S 2 intersecting with S 1 . Let n 5 , n 1 , and n 2 be the wrapping numbers for these cycles. In the web-diagram these numbers are interpreted as the numbers of open strings (Fig. C1(a)). Let us see what happens when we swap the two parallel external lines 1/2 and 1 + 1/2. This corresponds to the Weyl reflection of the A 1 . The wrapping numbers n 5 and n 2 are kept unchanged while n 1 is non-trivially transformed (Fig. C1(b)): n 1 → n 1 = n 5 − n 1 + n 2 . (C.17) We can easily check that this generates Z 2 . This relation is identical to the relation of R-charges in Eq. (C.16). The four charges r, F 1 , F 2 , and B defined in Eq. (4.48) do not include either R 1 or R 1 and are invariant under the Weyl reflection. As the additional charge associated with the shrinking cycle it is convenient to use the Cartan generator of the A 1 algebra, which is odd under the Weyl reflection. It is We define the corresponding fugacity ζ by (C.20) A D3-brane wrapped on the shrinking cycle S 1 carries B = 2 and its Weyl reflection carries B = −2. These correspond to the root vectors of A 1 .A D3-brane wrapped on the visible cycle S 2 carries B = −1 and its Weyl reflection S 2 +S 1 carries B = 1. Namely, these two form the fundamental representation of A 1 . Both these cycles contribute to I D3 S 2 in Eq. (4.58), and the refinement by introducing ζ splits it into two contributions, I D3 S 2 ∝ v N 2 ∝ ζ −1 and I D3 S 2 +S 1 ∝ v N 2 v N 1 ∝ ζ . Therefore, the degeneracy factor 2 in Eq. (4.58) is replaced by the A 1 character χ 1 ( ζ ) = ζ + 1/ ζ : This is also the case for the other adjacent visible cycle S 5 and its Weyl reflection S 5 + S 1 : By numerical calculation on the gauge theory side we can find the same refinement. Namely, the overall factor 2 on the gauge theory side is also replaced by the character ζ 1 , and the second and fifth relations in Eq. (4.53) still hold after the refinement. In general, if there are k parallel external lines, the factor k in the degeneracy factor is replaced by the fundamental A k−1 character for one of the adjacent cycles and by the anti-fundamental A k−1 character for the other adjacent cycle.