Charges and holography in 6d (1,0) theories

We study the recently proposed AdS$_7$/CFT$_6$ dualities for a class of 6d $\mathcal{N} = (1,0)$ theories that flow on the tensor branch to long linear quiver gauge theories. We find a precise agreement in the symmetries and in the spectrum of charged states between the 6d SCFTs and their conjectured AdS$_7$ duals. We also confirm a recent conjecture that a discrete $S_N$ symmetry relating the baryons in the quiver theories is in fact gauged.


Introduction and summary
The study of quantum field theory becomes harder as one increases the dimension of spacetime: familiar interacting models become non-renormalizable and require new UV completions. However when such completions can be found, they provide interesting windows on non-perturbative physics, whose lessons can be useful in lower dimensions too. For supersymmetric models, the renormalization group (RG) fixed points at high energies should be superconformal field theories (SCFTs), which can only exist up to d = 6. Over the years, many realizations for such theories have been proposed using various string-theoretic techniques.
In this paper we will be interested in a class of 6d SCFTs with (1, 0) supersymmetry [1,2], which have effective descriptions in terms of linear quiver gauge theories with gauge group Π i SU (r i ), matter fields charged in bi-fundamental representations, and tensor fields. Based on their brane engineering, these SCFTs were suggested in [3] to have AdS 7 holographic duals in Type IIA supergravity, which have been classified [4] and written down explicitly [5,6]. A concrete check of this duality was performed in [6,7], where the a anomaly was compared on both sides and found to agree in full generality. While that test was reassuring, one would like to have further quantitative tests to probe in more detail the non-perturbative physics of these SCFTs. 1 Here we will compare in detail the symmetries and the spectrum of charged operators. It has been known already that the non-abelian symmetries are realized on the D8-branes and D6-branes present in the gravity solutions, but the U (1) symmetries are more subtle. On the field theory side, they should acquire masses through a Stückelberg mechanism [2]; but some of them are in fact anomalous (the analysis becoming particularly interesting in presence of SU (2) gauge groups). On the gravity side, these U (1)'s are particular combinations of the RR gauge field and of the gauge fields that live on branes; we match them with the non-anomalous combinations on the field theory side. Amusingly, the matching involves a Stückelberg mechanism on the supergravity side.
There are in general two types of charged operators in the field theories we consider. The first are composed as strings of matter fields which we refer to as string-mesons, and the second consist of antisymmetrized products of matter fields which we refer to as baryons. These operators carry non-abelian flavor symmetry charges as well as U (1) charges. Interestingly, the different baryons all have the same charges and dimension. The baryons and string-mesons are expected to be related by chiral-ring relations. On the gravity side, we identify the string-mesons with strings connecting the D-branes in the solution, and the baryons with D0-branes. We also provide a realization of the chiral-ring relation through an instantonic process: namely, D0-branes can turn into fundamental strings (and vice versa) via the nucleation of a Euclidean D2-brane which wraps the internal space of the dual gravity background.
When the D0-brane is in a region with non-zero Romans mass, it develops a tadpole that should be canceled by strings ending on it; this corresponds nicely with the structure of the field theory baryons, which are built from rectangular matrices and hence need to include sequences of fields charged under neighboring gauge groups. Taking this into account, however, we find that the D0-brane mass has a single minimum. This suggests that the field theory baryons should all be identified in the SCFT, in stark contrast with most theories in lower dimensions. This is consistent with their charges being equal, and with a recent conjecture [13] that these SCFTs have an S N gauge symmetry. Below we will provide a field-theoretic argument supporting this conjecture which exploits the fact that, for 6d theories with an M5-brane origin, the S N symmetry can be identified with a subgroup of the gauge group in a certain duality frame (or, in other words, upon performing a different reduction to Type IIA). Furthermore, we are able to match exactly and in general the charges and dimension of this single baryon to the mass of the D0-brane (with strings attached) at its minimum. This paper is organized as follows. In section 2 we present the gauge theory description of the 6d (1, 0) SCFTs, highlighting their global symmetries and spectrum of gauge-invariant charged BPS operators. In section 3 we review the construction of the AdS 7 gravity duals; we identify the non-abelian gauge fields as well as the nonanomalous abelian ones coming from the reduction of the ten-dimensional Type IIA theory on the internal space; we construct the charged states and compute their masses and charges, matching them with those of the dual operators. In sections 2 and 3 we concentrate on the case in which the 6d quiver gauge theory has a segment in which the gauge group ranks do not change, namely a plateau, and the dual Type IIA background has a region with a vanishing Romans mass. In section 4 we briefly discuss the case in which the gauge theory does not have a plateau, and the dual Type IIA background does not have a massless region, mainly emphasizing the differences with the case containing a plateau. In section 5 we work out three explicit examples, including a plateau-less case. In two appendices we include a supersymmetry analysis of D0-branes and strings, and some details related to the computation of their U (1) charges. Figure 1: The general 6d linear quiver theory.
2 Field theory

Linear quivers
The general structure of the theories we are interested in is shown in Fig. 1. These are linear quiver gauge theories with N − 1 gauge nodes SU (r i ), where i = 1, . . . , N − 1, a bi-fundamental hypermultiplet for each SU (r i ) × SU (r i+1 ) pair, and n i fundamental flavor hypermultiplets for SU (r i ). These theories are free of gauge anomalies provided that each gauge group SU (r i ) has effectively 2r i flavors, namely that There exists a simple brane construction for this theory in Type IIA string theory [1,2]. It consists of N parallel NS5-branes separated along one coordinate, with r i D6branes stretched between the ith NS5-brane and the (i + 1)st NS5-brane, and a stack of n i D8-branes intersecting the D6-branes between the ith and (i + 1)st NS5-brane, Fig. 2a. The first and last D8-brane stacks can be traded for semi-infinite D6-branes via a Hanany-Witten transition, as shown in Fig. 2b. For each gauge node there is also a tensor multiplet corresponding to the NS5-brane degrees of freedom. The anomaly cancellation condition (2.1) is seen as a tadpole cancellation condition on the NS5-brane worldvolumes. For our purposes we will assume that the flavored gauge nodes are well separated, so that the quiver theories take the form of Fig. 3. We label the flavors by b = 1, . . . , s, and the flavored nodes are labelled by j b . We assume that the first and last nodes are flavored, namely that j 1 = 1 and j s = N − 1. The anomaly condition (2.1) implies that r j is a convex function of j, and therefore that it either has a maximal plateau in some segment between consecutive flavors j p ≤ j ≤ j p+1 , or that it is maximized at one of the ends of the quiver. In the remainder of this section, as well as in the next section, we will assume that there exists a plateau between the pth and (p + 1)st flavors. In section 4 we will briefly summarize the results for the plateau-less case.
The anomaly condition (2.1) can be solved by imposing the "boundary conditions" r 0 = r N = 0, to give where a R,L (j) labels the flavor nearest the jth node on the right and left, respectively, and r max is the maximal gauge group rank, which is given by x j2 x 1 n 1 n 2 n 3 n 4 n s Figure 3: The general 6d linear quiver theory with s sets of flavors. We denote by x i ,x i the two complex scalar components of the bi-fundamental hypermultiplets, and by y a ,ỹ a the two complex scalar components of the fundamental hypermultiplets.

Global symmetry
The gauge nodes are SU (r i ) rather than U (r i ), since the the U (1) gauge bosons acquire a mass via a Stückelberg coupling to a scalar in the tensor multiplet [2]. The U (1) symmetries are therefore global symmetries in the low energy theory. However in general they are anomalous. Let us denote by J µ i , with i = 1, . . . , N − 2, the U (1) currents associated to the bi-fundamental fields x i , and by I µ b , with b = 1, . . . , s, the U (1) currents associated to the flavor fields y b . The anomalies are given by Anomaly-free currents are found by summing from flavor to flavor with appropriate coefficients. There are s − 1 independent conserved currents given by summing from the bth flavor to the (b + 1)st flavor as follows (up to an overall normalization) The global symmetry of the 6d theory is therefore generically U (1) s−1 × s b=1 SU (n b ). 2 In the next section we will see that this agrees with the gauge symmetry of the proposed dual AdS 7 background.

SU (2) node
There are a number of special cases that deserve our attention. These occur when there is an SU (2) factor. The U (1) anomaly is absent for SU (2) since TrF 3 SU (2) = 0. This would suggest that there are additional U (1) symmetries in these cases. However this is not the case as we will now explain. The most general situations with an SU (2) factor are depicted in Fig. 4, which shows an edge of the quiver containing an SU (2) gauge node with n flavors, where n = 0, . . . , 4. There are actually two different theories with this gauge field and matter content. Fig. 4a shows the minimal theory with n flavors, and Fig. 4b shows the theory which has an extra tensor multiplet associated to the edge. This is represented as an "SU (1)" gauge node which is really a flavor. The corresponding brane configuration is shown in Fig. 5b, where one of the D8-branes is traded for an extra NS5-brane on which a single D6-brane ends. In the first case the U (1) symmetry associated to y is anomaly-free by itself, and in the second case the two U (1) symmetries associated to x 1 and y are anomaly-free by themselves. However these symmetries are actually absent in the SCFT.  Let us begin with the edge without the extra tensor multiplet, Fig. 4a. The relevant cases are n = 0, 1, 2 and 4. The case with n = 3 is equivalent to the case with n = 4 of Fig. 4b, which we will discuss below. For n = 4 this is just an SU (2) theory with 4 flavors. The classical global symmetry of this theory is SO (8). However it has been argued that the symmetry is reduced in the SCFT to SO(7) [13,14]. The other cases can be obtained by gauging an SU (4 − n) subgroup of the global symmetry. Classically this would leave a U (1) global symmetry acting on the bi-fundamental field x 1 for n = 0, and two U (1) global symmetries acting on x 1 and on the flavor y for n = 1. For n = 2 it would leave an SU (2) 3 global symmetry, with two SU (2) factors acting on the flavors and one acting on the bi-fundamental field. However since the global symmetry of the n = 4 theory is actually SO(7) these conclusions are modified. For n = 0 the edge has no global symmetry, for n = 1 it only has a single U (1) corresponding precisely to the first two terms in (2.6), and for n = 2 it has SU (2) 2 . 3 We conclude that the formula for the anomaly-free U (1) currents (2.6) holds also in the cases with an SU (2) edge of the form shown in Fig. 4a with n = 0, 1, and that there are no extra U (1) symmetries. For n = 2 the full quiver is actually fixed to be 2 + SU (2) N −1 + 2. The classical global symmetry is SU (2) N +2 , but by repeating the above analysis at each of the nodes one can show that this reduces to just SU (2) 3 [13]. Now consider the SU (2) edge with the extra tensor multiplet, Fig. 4b. Let us start with the case n = 4, namely an SU (2) theory with 4 flavors and an extra tensor multiplet. It has been argued that in this case the global symmetry is reduced to G 2 [13]. As before the other cases correspond to gauging an SU (4 − n) subgroup, now with n = 1, 2. 4 For n = 1 we get no remaining global symmetry, and for n = 2 we are left 4 For n = 3 this is an SU (2) theory with 4 flavors and two extra tensor mutiplets. This theory has a global symmetry SU (3).

Charged operators
Our main interest is in the spectrum of gauge-invariant charged BPS operators. There are two types of such operators in the theory. The first are given by strings of matter fields beginning and ending with a flavor field, which we will refer to as string-mesons. There are s − 1 independent string-mesons which we can take as beginning at the bth flavor and ending at the (b + 1)st flavor: This operator has a scaling dimension ∆ M b = 2(j b+1 − j b + 2), and transforms in the bi-fundamental representation of SU (n b ) × SU (n b+1 ). From (2.6) we also deduce that it carries charges under at most three of the U (1) symmetries: The first charge is zero for b = 1, whereas the third is zero for b = s − 1.
The second type of operators are given by antisymmetrized products. We will therefore refer to them as baryons. There are N independent baryon operators B j with j = 0, . . . , N − 1, given by where all products are antisymmetrized. All of these have the same scaling dimension ∆ B j = 2r max , they are all singlets under the non-abelian part of the global symmetry, and all carry only U (1) p charge: (2.10) The string-meson operators and the baryon operators are not completely independent. The product of the N baryons can be expressed in terms of the s−1 string-mesons in a chiral-ring-like relation: with appropriate antisymmetrization of the products on the RHS. Here we are omitting neutral mesonic factors of the form x ixi and y bỹb , and presenting only the charged components of the chiral-ring relation. Note in particular that the RHS is a singlet under all the non-abelian factors of the global symmetry. The first term on the RHS transforms as (n p ) rmax−npjp asym of SU (n p ), the second term as (n p ,n p+1 ) rmax asym of SU (n p ) × SU (n p+1 ), and the third term as (n p+1 ) rmax−n p+1 (N −j p+1 ) asym of SU (n p+1 ). Antisymmetrizing the product of these gives a singlet.

A discrete (gauge) symmetry
In addition to the continuous global symmetries discussed in section 2.2, the linear quiver theories also appear to have an emergent discrete symmetry S N , originally identified in [13]. 6 This symmetry is not seen at the level of the action, but becomes apparent at the level of the gauge-invariant operators discussed above. Since the N baryon operators B i all have the same scaling dimensions and charges under the continuous global symmetries, there is an S N symmetry that permutes them. From the point of view of the Type IIA brane construction this corresponds to the permutation of the N NS5-branes.
It has been conjectured in [13] that this symmetry is actually gauged. This was argued using an auxiliary three-dimensional N = 4 quiver gauge theory, the so-called magnetic quiver, whose Coulomb branch is conjectured to be identical to the Higgs branch of the 6d electric quiver. The Coulomb branch of the three-dimensional theory is given by the closure of a certain nilpotent orbit of the global symmetry. When the N NS5-branes (or M5-branes) are brought together, the new nilpotent orbit is the S N quotient of the one for the separated branes.
There is another way to see that the S N symmetry is gauged, at least for the 6d theories on M5-branes, as we will now explain. Some, but not all, 6d SCFTs are realized in terms of M5-branes in various M-theory backgrounds. These theories have two different gauge theory deformations. The first is a 6d quiver gauge theory of the type we have been discussing, corresponding to the low energy effective theory on the tensor branch of the 6d SCFT. From the point of view of string/M-theory this corresponds to separating the M5-branes and reducing along a direction transverse to them. For example, for the 6d (2, 0) theory of N M5-branes in flat space this gives a Type IIA brane configuration with a single infinite D6-brane intersecting N NS5branes, namely a theory of N free hypermultiplets and N free tensor multiplets. As we explained above, this description exhibits an emergent global S N symmetry. The second gauge theory deformation is a 5d supersymmetric gauge theory obtained by compactifying the 6d SCFT on a circle, namely reducing along one of the M5-brane directions. This is the theory living on N D4-branes in Type IIA string theory in a background corresponding to the reduction of the M-theory background. For example the reduction of the (2, 0) theory gives the 5d N = 2 SYM theory with gauge group U (N ). In this case the permutation group S N is part of the gauge symmetry: it is the Weyl group of U (N ). While the two gauge theories are clearly different, they both correspond to deformations of the same 6d SCFT. We conclude from this that the S N symmetry of the 6d quiver theory is in fact gauged, at least in the case of M5-brane theories. While this argument is not directly applicable to 6d quiver theories that are not deformations of M5-brane theories, since these do not have an obvious 5d gauge theory reduction, it is plausible that the conclusion continues to hold, given that these theories can often be related to ones that are deformations of M5-brane theories via Higgsing (see for example section 5.3).
We will confirm that S N is gauged in all of these theories using holography. A gauged S N implies that there is only one physical baryon operator corresponding to the sum of the N baryons. Our analysis will show that there is indeed a unique bulk state dual to this operator.

Solutions
We begin this section by briefly reviewing the Type IIA AdS 7 solutions of [4] that are conjectured to be dual to 6d (1, 0) theories of the type described in the previous section [3,5,15]. The ten-dimensional geometry is a warped product AdS 7 × M 3 , with the internal space M 3 having the topology of S 3 , given by an S 2 fibered over an interval. There is NSNS 3-form flux on M 3 and RR 2-form flux on the S 2 fiber, and in general also a non-vanishing RR 0-form flux F 0 , the so-called Romans mass. For F 0 = 0 the Type IIA background lifts to the M-theory background The Type IIA background is defined by a single piecewise smooth function on the interval α(z), with z ∈ [0, N ], satisfying the differential equation 7 with appropriate boundary conditions corresponding to the asymptotics of the brane configuration. Let us divide the interval into N equal segments with j ≤ z ≤ j + 1, where j = 0, . . . , N − 1, corresponding to the N NS5-branes in the Type IIA brane construction. The Romans mass F 0 (z) is a piecewise constant function given in the segment z ∈ (j, j + 1) by where r j is the rank (plus 1) of the jth gauge group in the dual quiver theory. Recall that we impose the boundary conditions r 0 = r N ≡ 0. The D8-branes present in the original brane construction remain as sources localized at z = j b , with b = 1, . . . , s, across which the Romans mass jumps by 2π∆F 0 = n b . The plateau is the region j p ≤ z ≤ j p+1 in which F 0 = 0. We will refer to this as the massless region. The case without a massless region will be discussed in the next section. It follows from (3.2) and (2.2) that The solution to (3.1) for j ≤ z ≤ j + 1 takes the form [6] α The first-order supergravity equations reduce to this single ODE.
where α j and y j are fixed in terms of r j by the boundary and continuity conditions (see [6,7] for the general expressions). The ten-dimensional metric is explicitly given by where ds 2 AdS 7 is the unit-radius metric on AdS 7 , ds 2 S 2 is the unit-radius metric on S 2 with coordinates {θ, φ} (and volume form dΩ 2 = sin θdθ ∧ dφ), and the warp factor is The dilaton is given by The background is in general singular at the two poles of M 3 , reflecting the presence of the semi-infinite D6-branes, or equivalently the D8-branes at z = j 1 = 1 and z = j s = N − 1. The gauge-invariant NSNS and RR field strengths are given by , (3.11) and the corresponding fluxes are The latter corresponds to the so-called Maxwell D6-brane charge, which is gaugeinvariant but not quantized. A quantity that is more closely related to the number of D6-branes is the Page charge, defined in general byF = F ∧ e B 2 , and given in this case byF 2 = F 2 + F 0 B 2 = dC 1 . Page charge is quantized but not invariant under gauge transformations of the NSNS potential B 2 . It is convenient to work in a gauge in which B 2 only has components along the S 2 , and vanishes at the two poles of M 3 . This requires performing a large gauge transformation between the poles of M 3 such that B 2 → B 2 − πN dΩ 2 . We will assume that this gauge transformation is performed at a point in the massless region, namely at z = z 0 where j p < z 0 < j p+1 . The gauge 1-form function is therefore given by Λ 1 (z) = πN Θ(z − z 0 ) cos θdφ, and we have (3.14) The D6-brane Page charge in the segment j < z < j + 1 is subsequently given by where in the second equality we used (2.2) and (2.3). This has a discontinuity across the location of each D8-brane stack at z = j b that is accounted for by a D6-brane charge carried by the D8-brane stack itself. This is sourced by a worldvolume magnetic flux, which for a D8-brane in the bth stack is 8 In particular the D6-brane charges carried by the first D8-brane stack at z = j 1 = 1 and by the last D8-brane stack at z = j s = N − 1 account for the singularities at the two poles. For large N these are effectively at z = 0 and z = N .

Symmetries
As usual the global symmetries of the SCFT should correspond to gauge symmetries in the bulk, namely to massless vector fields. The 6d conformal symmetry is of course dual to the isometry of AdS 7 . The SU (2) R symmetry is realized in the bulk as the isometry group of the S 2 fiber of M 3 . In particular the U (1) R Cartan subgroup corresponds to the U (1) isometry associated to the 1-form cos θ dφ. These are the only symmetries arising from the geometry. Additional symmetries arise from ten-dimensional supergravity gauge fields and from the source D8-branes. There is a U (1) gauge field given by the RR 1-form C 1 , and U (n b ) gauge fields coming from the D8-brane worldvolumes. 9 This appears to amount to U (1) s+1 × s b=1 SU (n b ). However not all of the U (1) vector fields are massless in this background. This can be seen from the appearance of Stückelberglike terms in the seven dimensional theory on AdS 7 .
For the RR 1-form such a term comes from the reduction on M 3 of Type IIA supergravity This is a seven-dimensional Stückelberg term, where the Stückelberg scalar field corresponds to the dual of C 5 . There are also Stückelberg terms for the D8-brane worldvolume gauge fields that arise from the worldvolume CS terms. The general form of the worldvolume CS action is given by where C is the formal sum of all RR potentials, and we have defined the modified RR potentials asC such that dC =F . The terms relevant for us are The first term gives the Stückelberg terms where a b is the diagonal U (1) gauge field of the bth D8-brane stack, and the Stückelberg scalar is given by the dual of the reduction ofC 7 on S 2 . These terms imply that the combination b n b a b is massive and therefore not part of the low energy spectrum. The second term in (3.20) gives the Stückelberg terms which together with the C 1 Stückelberg term in (3.17) seem to imply that the combination N C 1 + b µ b n b a b is also massive. 10 However this not completely correct. The modified RR potentials defined by (3.19) are not invariant under B 2 gauge transformations, and so are affected by the large gauge transformations that we used above. In particular when we cross the plateauC 7 undergoes the transformatioñ giving an additional contribution to (3.22) from (3.21). The net effect, given (3.16), is to replace µ b by j b , So the second massive gauge field is given by We are left with s − 1 massless U (1) gauge fields A c , with c = 1, . . . , s − 1, that can be parameterized as where the coefficients β We conclude that the gauge symmetry in the AdS 7 supergravity background is U (1) s−1 × s b=1 SU (n b ), in agreement with the global symmetry of the 6d SCFT.

Charged states
The states charged under the symmetries identified above, and dual to the operators described in the previous section, are described by open strings and D0-branes.

Open strings
The string-meson M b is dual to an open string between a D8-brane in the bth stack and one in the (b + 1)st stack (and located at the origin of AdS 7 ). Let us refer to this state as F1 b . The mass of this string is given by (with α = 1) where P [G] ws denotes the pullback of the metric in (3.5) onto the worldsheet, spanning time in AdS 7 and z. This agrees at large N with the dimension ∆ M b = 2(j b+1 − j b + 2) of the string-meson operator M b we gave below (2.7). The non-abelian charges also agree, since the open string is charged in the bi-fundamental representation of SU (n b )× SU (n b+1 ). The U (1) c charges will be discussed shortly.
We will confirm that this is a BPS state by computing its R-charge. The charge under U (1) R is given by the coupling of the string to a fluctuation of the 1-form associated to U (1) R δ(cos θ dφ) = A R . (3.29) The relevant terms in the worldsheet action are given by with Σ b = [j b , j b+1 ] × R. From (3.14) we see that the bulk term contributes an amount N/2, but only for b = p, namely for the open string in the massless region. The boundary terms contribute an amount given by (µ b+1 − µ b )/2. Adding the two contributions and using (3.16) we find for all b Comparing with the mass in (3.28) we see that this is a BPS state. (This is also confirmed by solving the Killing spinor equation -see appendix A.3.)

D0-brane (and strings)
The baryon operators B j are dual to states containing a D0-brane. The mass of a D0-brane at a point z on the interval (and at the origin of AdS 7 ) is given by with G the ten-dimensional metric in (3.5). Generically the D0-brane is located in a massive region with F 0 (z) = 0. This requires the presence of 2πF 0 strings between the D0-brane and D8-branes. From (3.3) it follows that if the D0-brane is to the left of the massless region the strings end one at a time on each of the D8-branes to the right up to z = j p , and if the D0-brane it is to the right of the massless region they do so on each of the D8-branes to the left down to z = j p+1 . If the D0-brane happens to be in the massless region there are no strings attached. The mass of the D0-brane-strings combination is therefore given by (3.33) The BPS states dual to B j will correspond to minimizing this mass. While the functional form of M D0 (z) in (3.32) looks rather daunting, minimizing the mass of the D0-brane-strings combination turns out to be quite simple by realizing that the derivative of M D0 (z) simplifies at the point whereα(z) = 0, This minimizes uniquely the mass of the D0-brane-strings combination (3.33). Using the explicit form of the solution (3.4) we then find where the last equality follows from (2.2). This agrees with the dimension of the baryon operators B j we gave below (2.9), namely ∆ B j = 2r max . Importantly this also confirms the conjecture that there is only a single baryon operator in the SCFT, namely that the S N symmetry permuting the N baryons B j of the quiver gauge theory is gauged. The non-abelian charges are trivial since in every stack of n b D8-branes there is either a string ending on every D8-brane, giving an n b -fold antisymmetric representation of SU (n b ) which is of course a singlet, or no strings at all. The U (1) c charges will be discussed shortly.
To confirm that this is a BPS state we again compute its R-charge. The U (1) R charge will get contributions from the coupling of the D0-brane to the RR 1-form C 1 , as well as from the coupling of the string ends to the D8-brane gauge fields. The contribution of the RR field can be read off from (3.15) and is given by (3.36) The contribution of the string ends can be read off from (3.16): (3.37) Adding the two contributions and using (2.2) gives confirming that the BPS bound is saturated.

Matching the U (1) charges
Recall that there are s b . This is a rather cumbersome calculation, the outline of which we sketch in appendix B. Of course this by itself does not provide a test of the duality. Now let us consider the D0-brane. There are in general three contributions to the U (1) c charge of this state. The first is the RR charge of the D0-brane, which contributes an amount γ (c) . The second is the D8-brane worldvolume charge at the endpoint of the D0-D8 string, which contributes an amount β for each D8-brane located to the left of the D0-brane. 11 The sum of the last two contributions is independent of the position of the D0-brane. The total charge is given by where in the second equality we used the fact that the charge is independent of the position of the D0-brane to set j p < j < j p+1 , and in the third equality we used (3.26). Now comes a non-trivial test of the duality. Given the values of β In this case condition (3.26) reduces to n 1 β 1 + n 2 β 2 = 0 (3.41) and then condition (3.27) reduces to In addition, (2.3) reduces to r max = j 1 n 1 = (N − j 2 )n 2 (3.43) 11 This is due to the D8-brane worldvolume coupling C 7 ∧ f b = F 8 ∧ a b . The D8-brane captures half of the total F 8 flux sourced by the D0-brane, and the sign depends on the relative orientation of the D8-brane and the D0-brane. In fact the string creation effect can be understood from the requirement that the worldvolume charge remains invariant as the D0-brane crosses the D8-brane. 12 In this case c = 1, so we can drop this index from β b and γ. so j 1 n 1 + j 2 n 2 = N n 2 . (3.44) Matching the string-meson charge (2.8) reduces to We then find and thus . (3.47) The charge of the D0-brane-string combination is then given by in precise agreement with the charge of the dual baryon operator (2.10).

Holographic chiral-ring relation
The chiral-ring relation (2.11) also has a nice geometrical description. Consider a Euclidean D2-brane that wraps the entire internal space M 3 . The coupling of the background fluxes to the D2-brane worldvolume induces tadpoles which must be cancelled by attaching both D0-branes and strings to the D2-brane. The NSNS flux on M 3 induces a worldvolume magnetic charge via the DBI action, whose expansion contains the term M 3 B 2 ∧ * f = M 3 ϕ H 3 , where ϕ is the D2-brane worldvolume magnetic-dual scalar potential. The amount of magnetic charge is given by the NSNS flux on M 3 , namely N . This is a pointlike charge in the compact threedimensional Euclidean worldvolume of the D2-brane, and constitutes a tadpole which must be cancelled by having N D0-brane worldlines end on the D2-brane at a point.
The RR flux on S 2 induces a worldvolume electric charge via the worldvolume CS coupling M 3 C 1 ∧ f = M 3F 2 ∧ a, where a is the D2-brane worldvolume gauge potential. The amount of charge in a given segment j b < z < j b+1 is given by the RR flux on S 2 in that segment, i.e. (3.15). This charge also constitutes a tadpole, which must be cancelled by open string worldsheets ending on the D2-brane along the above segment, where the number of open strings F1 b is given by the electric charge in the corresponding segment. This gives (3.49) The wrapped Euclidean D2-brane therefore describes a process in which N D0branes turn into a collection of open strings, with N F1 b strings between the bth and (b + 1)st D8-brane stack, or vice versa; see Fig. 6. This is precisely the content of the chiral-ring relation in (2.11). The number of open strings N F1 b appears as the total power of the string-meson M b on the RHS. The neutral meson factors, which we omitted, account for the deficit or excess of energy in the process.

Plateau-less cases
For a linear quiver without a plateau the rank is maximized at the edge. We will assume this to be the right edge, and therefore r max = r N −1 . Imposing r 0 = r N = 0 in this case we find and The expression for the non-anomalous U (1)'s is the same as before (2.6). The expressions for the string-meson operators and their charges are also the same. The baryon operators are slightly different in this case. There are again N baryons, now given by and All of these have a scaling dimension ∆ B j = 2r max , they all transform under SU (n s ) in the (n s − r max )-fold antisymmetric representation, and they all carry only U (1) s−1 charge: The chiral-ring relation is also a little different in the plateau-less case, and is given, modulo neutral meson factors, by In this case both sides transform in the N (n s − r max )-fold antisymmetric representation of SU (n s ). The supergravity solutions dual to the plateau-less quivers do not have a massless region, and the Romans mass for j < z < j + 1 is given by (4.7) A convenient gauge choice for the NSNS field in this case is namely the large gauge transformations are performed in the very last segment beyond the last stack of D8-branes. The D6-brane Page charge in the segment j < z < j + 1 is consequently given by 9) and the associated D8-brane worldvolume magnetic fluxes are The analysis of the symmetries is unchanged, as is the analysis of the open strings dual to the string-meson operators.
The results for the D0-brane are slightly different. The D0-brane has 2πF 0 strings attached, where F 0 is given by (4.7). It follows from (4.7) that for j s − 1 < z < j s there are n s − r max strings, and n b additional strings for each stack of D8-branes that the D0-brane crosses as it moves to the left. Therefore, unlike in the solutions containing a massless region, this state is charged under the last flavor symmetry SU (n s ), transforming in the (n s − r max )-fold antisymmetric representation, in agreement with the dual baryon operators.
The U (1) c charges are given by where the first term is the contribution of the D0-brane RR charge, the second term is the contribution of the induced D8-brane worldvolume charges, and the third term is the contribution of the string end charges. Establishing that these agree with the U (1) c charges of the baryon operators is again a tedious, yet straightforward exercise. We will do it for a specific example with s = 2 below. The mass of the D0-brane-strings combination is given by This again has a unique minimum whenα = 0, in which case we again find that 13) in agreement with the dimension of the baryon operators, and confirming again that the baryon permutation symmetry is gauged in the field theory. The geometric description of the chiral-ring relation (4.6) is similar to the case with a massless region. A Euclidean D2-brane wrapping M 3 contains N units of magnetic charge induced by the NSNS flux on M 3 , andF 2 2π units of electric charge induced by the RR flux on S 2 in a given segment. As before, these charges must be canceled by having N D0-brane worldlines and N F1 b string worldsheets between z = j b and z = j b+1 end on the D2-brane, where in this case j a n a . (4.14) This is precisely the content of the chiral-ring relation (4.6).

A uniform quiver: M5-branes on C 2 /Z k
The simplest class of examples corresponds to N M5-branes on C 2 /Z k , which at a generic point on the tensor branch is given by the uniform linear quiver gauge theory shown in Fig. 7. The global symmetry is SU (k) × SU (k) × U (1), where the single anomaly-free U (1) current is given by The charged operators consist of a single string-meson 2) and N baryons 3) whose properties are summarized in Table 1. The chiral-ring relation takes the simple form The dual supergravity background has F 0 = 0 everywhere, and the solution is given by [4] 13 Note that this is invariant under z → N − z, which is consistent with the reflection symmetry of the quiver. The singularities at the poles z = 0 and z = N correspond to the semi-infinite D6-branes, or equivalently to two stacks of k D8-branes at z = 1 and at z = N − 1, with worldvolume magnetic fluxes µ 1 = 1 and µ 2 = −1, respectively. The RR flux on S 2 is independent of z and given by The SU (k) × SU (k) symmetry is realized as the non-abelian part of the D8-brane worlvolume gauge symmetry. There is also a single massless U (1) gauge field A parameterized by the two diagonal D8-brane worlvolume U (1) gauge fields and the RR gauge field as The string-meson operator M is dual to an open string connecting the two poles, or equivalently between the two D8-brane stacks. This transforms in the bi-fundamental representation of SU (k) × SU (k), and given (5.8), it carries N units of U (1) charge, in agreement with the dual operator. The mass is given by (3.28), which in this case is which agrees for large N with the operator dimension. Note that the mass of a string between the two poles is 2N , in precise agreement with the operator dimension. This is consistent with the large N equivalence of the two descriptions of the sources at the poles.
The baryon operator is dual to the D0-brane. This is a singlet of SU (k) × SU (k), and its U (1) charge is given by in agreement with the charge of the baryon operator. The BPS conditionα = 0 has a unique solution that fixes the position of the D0-brane at z min = N/2, and its mass is 11) in agreement with the dimension of the dual baryon operator. The geometric description of the chiral-ring relation (5.4) reduces in this example to an instantonic D2-brane mediated process turning N D0-branes into k strings.

A uniformly rising quiver
As our second example we will take a quiver with a uniformly increasing rank, Fig. 9. This is an example of a plateau-less quiver. For n = 0 this reduces to the previous example of the uniform quiver. The global symmetry is SU (k) × SU (k + N n) × U (1), except if k = 0 in which case it is just SU (N n). The anomaly-free U (1) current is now given by For k = 0 the first term is absent and there is no anomaly-free U (1) current.
… k x 1 x 2 x N 2 k + n k + 2n k + (N 1)n k + N n y 0 y Figure 9: A uniformly rising quiver.
The charged BPS operators include a string-meson of the same form as (5.2), now transforming as a bi-fundamental of SU (k) × SU (k + N n), and N baryons which transform in the n-fold antisymmetric representation of SU (k + N n). The properties of these operators are summarized in Table 2. The chiral-ring relation is given by where both sides transform in the N n-fold antisymmetric representation of SU (k+N n).   The RR flux in this case is also independent of z and given by The D8-branes realize the SU (k) × SU (k + N n) part of the gauge symmetry, and for k = 0 there is again a single massless U (1) gauge field given in this case by Note that this reduces to the parameterization of the massless U (1) of the uniform quiver (5.8) for n = 0. The string-meson operator is again dual to an open string between the two D8-brane stacks, and has the same mass as before. This state is charged in the bi-fundamental representation of SU (k) × SU (k + N n), and carries N units of U (1) charge, as is easily read off from (5.21). For k = 0 this state is absent.
The baryon operator is dual to a D0-brane, which now has n strings connecting it to the south pole, or equivalently to n of the k + N n D8-branes at z = N − 1.
Since the massless spectrum of the D0-D8 string consists of a single fermion, this state transforms in the n-fold antisymmetric representation of SU (k + N n), in agreement with the baryon operator. The U (1) charge of this state is given by

A simple symmetric quiver
Our third and final example is the quiver shown in Fig. 11. For n = 0 this reduces Figure 11: The symmetric quiver.
again to the uniform quiver. For q = 1 it also reduces to the uniform quiver with k replaced by k + n. More generally this theory can be viewed as a Higgs branch deformation of the uniform quiver with k replaced by k + qn [3]. The global symmetry is SU (k) 2 × SU (n) 2 × U (1) 3 , where the three U (1) currents are given by For the special case of k = 0 there is only the second U (1) symmetry. The spectrum of charged BPS operators includes three string-mesons x i ·ỹ 2 (5.25) and N baryons whose properties are summarized in Table 3. The chiral-ring relation is given by (1,k, 1, n) (5.33) These have all the same properties as the other N − 2 baryons. The chiral-ring relation in this case reduces to The background dual to this theory has three regions, with 2πF 0 = −n, 0, n, see Fig. 12. The solution in the three regions takes the form ) and the RR flux on S 2 is given by (5.36) For k = 0 there are singularities at the two poles. As before, these are usefully described as k D8-branes with µ 1 = 1 at z = 1 and k D8-branes with µ 4 = −1 at z = N − 1. In addition there are n D8-branes with µ 2 = q at z = q, and n D8-branes with µ 3 = −q at z = N − q. The latter are visible in Fig. 12.
The non-abelian part of the symmetry is again realized directly by the D8-branes. There are now three independent massless U (1) gauge fields A c with c = 1, 2, 3, parameterized by the four diagonal D8-brane U (1) gauge fields and the RR gauge field as (5.38) One can easily verify that these satisfy the conditions in (3.26) and (3.27). The three string-mesons are dual to open strings between the corresponding pair of neighboring D8-brane stacks. The non-abelian charges under SU (k) 2 × SU (n) 2 are precisely those of the dual operators. The U (1) charges can be read off from (5.38), and are also in complete agreement with those of the dual operators (see Table 3). For example the charges of the open string dual to M 1 are The masses of the open strings are given by which are in agreement, for large N and q, with the dimensions of the dual string-meson operators. The baryon is dual to a D0-brane, possibly with n strings attached if it is located in one of the two massive regions. By symmetry it is clear that the BPS state is described by a D0-brane at the center z = N/2, with no strings attached. The mass is in general given by 2r max , which in this case is M (D0) = 2(k + nq) , (5.43) in agreement with the dimension of the baryon. The U (1) charges are given by , and yields the charge-conjugate spinor ζ c ≡ Bζ * (with * denoting complex conjugation).

A.3 Probe F1-strings
We now consider fundamental strings stretched along the z direction. The BPS condition reads Γ 0ẑ 1 = 1 , Γ 0ẑ 2 = − 2 , where χ S 2 is the Killing spinor on the internal S 2 . This is written in a frame where γẑ = σ 3 , which is also the chiral gamma in the S 2 direction. With this, we see that (A.17) requires the component of χ S 2 with negative chirality to vanish. This happens at the north pole of the S 2 . Thus we have found that an F1 stretched along z can be BPS at the locus θ = 0.

B Abelian charges of operators and dual string states
In this appendix we identify a consistent set of numbers {β (c) b , γ (c) } such that the U (1) charges of the open string given in (3.39) coincide with those of the string-meson given in (2.8), and the U (1) charge of the D0-brane-strings combination given in (3.40) coincides with that of the baryon given in (2.10).
For later convenience, let us repeat here our conventions. In the generic brane configuration with plateau, there are s ≥ 0 D8 stacks (i.e. flavor nodes) labeled by b = 1, . . . , s, each containing n b > 0 branes and located at z = j = j b ∈ [0, N ] along the base interval. (Remember that the n b may not be all independent, since once we specify all the ranks r j then we must have n b = 2r j b − r j b −1 − r j b +1 .) Accordingly we can define s − 1 string-mesons M b in field theory, beginning in the bth and ending in the (b + 1)st flavor.
The plateau is the subinterval [j p , j p+1 ] ⊂ [0, N ], i.e. it begins at the last D8 stack in the left region located at j = j p , and ends at the first stack on the right region located at j = j p+1 .