The Veneziano Limit of {\cal N}=2 Superconformal QCD: Towards the String Dual of {\cal N}=2 SU(N_{c}) SYM with N_{f}=2N_{c}

# The Veneziano Limit of N=2 Superconformal QCD: Towards the String Dual of N=2SU(Nc) SYM with Nf=2Nc

C.N. Yang Institute for Theoretical Physics,
Stony Brook University,
Stony Brook, NY 11794-3840, USA
###### Abstract:

We attack the long-standing problem of finding the AdS dual of superconformal QCD, the super Yang Mills theory with gauge group and fundamental hyper multiplets. The theory admits a Veneziano expansion of large and large , with and kept fixed. The topological structure of large diagrams motivates a general conjecture: the flavor-singlet sector of a gauge theory in the Veneziano limit is dual to a closed string theory; single closed string states correspond to “generalized single-trace” operators, where adjoint letters and flavor-contracted fundamental/antifundamental pairs are stringed together in a closed chain. We look for the string dual of superconformal QCD from two fronts. From the bottom-up, we perform a systematic analysis of the protected spectrum using superconformal representation theory. We also evaluate the one-loop dilation operator in the scalar sector, finding a novel spin chain. From the top-down, we consider the decoupling limit of known brane constructions. In both approaches, more insight is gained by viewing the theory as the degenerate limit of the orbifold of SYM, as one of the two gauge couplings is tuned to zero. A consistent picture emerges. We conclude that the string dual is a sub-critical background with seven “geometric” dimensions, containing both an and an factor. The supergravity approximation is never entirely valid, even for large , indeed the field theory has an exponential degeneracy of exactly protected states with higher spin, which must be dual to a sector of light string states.

preprint: YITP-SB-09-48

## 1 Motivation

How general is the gauge/string correspondence? ’t Hooft’s topological argument [1] suggests that any large gauge theory should be dual to a closed string theory. However, the four-dimensional gauge theories for which an independent definition of the dual string theory is presently available are rather special. Even among conformal field theories, which are the best understood, an explicit dual string description is known only for a sparse subset of models. In some sense all examples are close relatives of the original paradigm of super Yang-Mills [2, 3, 4] and are found by considering stacks of branes at local singularities in critical string theory, or variations of this setup, e.g. [5, 6, 7, 8, 9, 10, 11].444We should perhaps emphasize from the outset that our focus is on string duals of gauge theories. There are strongly coupled field theories that admit gravity duals with no perturbative string limit, see e.g. [12, 13]. Conformal field theories in this class can have lower or no supersymmetry, but are far from being “generic”. Some of their special features are:

1. The and conformal anomaly coefficients are equal at large [14].

2. The fields are in the adjoint or in bifundamental representations of the gauge group. (Except possibly for a small number of fundamental flavors – “small” in the large limit – as in [15]).

3. The dual geometry is ten dimensional.

4. The conformal field theory has an exactly marginal coupling , which corresponds to a geometric modulus on the dual string side. For large the string sigma model is weakly coupled and the supergravity approximation is valid.555In some cases, as in SYM, the opposite limit of small corresponds to a weakly coupled Lagrangian description on the field theory side. In other cases, like the Klebanov-Witten theory [8], the Lagrangian description is never weakly coupled.

The situation certainly does not improve if one breaks conformal invariance – the field theories for which we can directly describe the string dual remain a very special set, which does not include some of the most relevant cases, such as pure Yang-Mills theory. Many more field theories, including pure Yang-Mills, can be described indirectly, as low-energy limits of deformations of SYM (as e.g. in [16] for SYM) or of other UV fixed points, not necessarily four-dimensional (as in [17] for YM or [18, 19] for SYM). These constructions count as physical “existence proofs” of the string duals, but if one wishes to focus just on the low-energy dynamics, one invariably encounters strong coupling on the dual string side. In the limit where the unwanted UV degrees of freedom decouple, the dual appears to be described (in the most favorable duality frame) by a closed-string sigma model with strongly curved target. This may well be only a technical problem, which would be overcome by an analytic or even a numerical solution of the worldsheet CFT. The more fundamental problem is that we lack a precise recipe to write, let alone solve, the limiting sigma model that describes only the low-energy degrees of freedom.

To break this impasse and enlarge the list of dual pairs outside the SYM universality class, we can try to attack the “next simplest case”. A natural candidate for this role is SYM with gauge group and flavor hypermultiplets in the fundamental representation of . The number of flavors is tuned to obtain a vanishing beta function. We refer to this model as superconformal QCD (SCQCD). The theory violates properties (i) and (ii) but it still has a large amount of symmetry (half the maximal superconformal symmetry) and it shares with SYM the crucial simplifying feature of a tunable, exactly marginal gauge coupling . (The theory also exhibits -duality [20, 21, 22], though this will not be important for our considerations, since we will work in the large limit, which does not commute with -duality.)

The large expansion of SCQCD is the one defined by Veneziano [23]: the number of colors and the number of fundamental flavors are both sent to infinity, keeping fixed their ratio ( in our case) and the combination . Which, if any, is the dual string theory? And what happens to it for large ?

## 2 The Veneziano Limit and Dual Strings

### 2.1 A general conjecture

To understand in which sense we should expect a dual string description of a gauge theory in the Veneziano limit, we start by reviewing general elementary facts about large counting, Feynman-diagrams topology, and operator mixing. At this stage we have in mind a generic field theory that contains both adjoint fields, which we collectively denote by , with , and fundamental fields, denoted by , with . We can consider the theory both in the ’t Hooft limit of large with fixed, and in the Veneziano limit of large .

, fixed

Let us first recall the familiar analysis in the ’t Hooft limit [1], where the number of colors is sent to infinity, with and the number of flavors kept fixed. In this limit it is useful to represent propagators for adjoint fields with double lines, and propagators for fundamental fields with single lines – the lines keep track of the flow of the type (color) indices. Vacuum Feynman diagrams admit a topological classification as Riemann surfaces with boundaries: each flavor loop is interpreted as a boundary. The dependence is , for the genus and the number of boundaries.

The natural dual interpretation is then in terms of a string theory with coupling , containing both a closed and an open sector – the latter arising from the presence of explicit “flavor” branes where open strings can end. Indeed this is the familiar way to introduce a small number of flavors in the AdS/CFT correspondence [24]: by adding explicit flavor branes to the bulk geometry (the simplest examples is adding D7 branes to the background). Since , the backreaction of the flavor branes can be neglected (probe approximation).

According to the standard AdS/CFT dictionary, single-trace “glueball” composite operators, of the schematic form (where Tr is a color trace) are dual to closed string states, while “mesonic” composite operators, of the schematic form , are dual to open string states. At large , these two classes of operators play a special role since they can be regarded as “elementary” building blocks: all other gauge-invariant composite operators of finite dimension can be built by taking products of the elementary (single-trace and mesonic) operators, and their correlation functions factorize into the correlation functions of the elementary constituents.666Note that in this discussion we are not considering baryonic operators, since they have infinite dimension in the strict large limit. Baryons are interpreted as solitons of the large theory; as familiar, in AdS/CFT they correspond to non-perturbative (D-brane) states on the string theory side [7]. This factorization is dual to the fact for the string Hilbert space becomes the free multiparticle Fock space of open and closed strings.

Flavor-singlet mesons, of the form , mix with glueballs in perturbation theory, but the mixing is suppressed by a factor of , so the distinction between the two classes of operators is meaningful in the ’t Hooft limit. On the dual side, this translates into the statement that the mixing of open and closed strings in subleading since each boundary comes with a suppression factor of .

We can now repeat the analysis in the Veneziano limit of large and large with and fixed. In this limit it is appropriate to use a double-line notation with two distinct types of lines [23]: color lines (joining indices) and flavor lines (joining indices). A propagator decomposes as two color lines with opposite orientations, while a propagator is made of a color and a flavor line (Figure 1). Since , color and flavor lines are on the same footing in the counting of factors of . It is natural to regard all vacuum Feynman diagrams as closed Riemann surfaces, whose dependence is , for the genus. At least at this topological level, by the same logic of [1], we should expect a gauge theory in the Veneziano limit to be described by the perturbative expansion of a closed string theory, with coupling . More precisely, there should be a dual purely closed string description of the flavor-singlet sector of the gauge theory.

This point can be sharpened looking at operator mixing. It is consistent to truncate the theory to flavor-singlets, since they close under operator product expansion. The new feature that arises in the Veneziano limit is the order-one mixing of “glueballs” and flavor-singlet “mesons”. For large , the basic “elementary” operators are what we may call generalized single-trace operators, of the form

 Tr(ϕk1Mℓ1ϕk2…ϕknMℓn),Mab≡Nf∑i=1qai¯qib. (1)

Here we have introduced a flavor-contracted combination of a fundamental and an antifundamental field, , which for the purpose of the large expansion plays the role of just another adjoint field. The usual large factorization theorems apply: correlators of generalized multi-traces factorize into correlators of generalized single-traces. In the conjectural duality with a closed string theory, generalized single-trace operators are dual to single-string states.

We can imagine to start with a dual closed string description of the field theory with , and first introduce a small number of flavors by adding flavor branes in the probe approximation. As we increase to be , the probe approximation breaks down: boundaries are not suppressed and for fixed genus we must sum over worldsheets with arbitrarily many boundaries. The result of this resummation – we are saying – is a new closed string background dual to the flavor-singlet sector of the field theory. The large mixing of closed strings and flavor singlet open strings gives rise to new effective closed-string degrees of freedom, propagating in a backreacted geometry. This is the string theory interpretation of the generalized single-trace operators (1).

In stating the conjectured duality we have been careful to restrict ourselves to the flavor-singlet sector of the field theory. One may entertain the idea that “generalized mesonic operators” of the schematic form (with open flavor indices and ) would map to elementary open string states in the bulk. However this cannot be correct, because generalized mesons and generalized single-trace operators are not independent – already in free field theory they are constrained by algebraic relations – so adding an independent open string sector in the dual theory would amount to overcounting.

### 2.2 Outline of the paper

In this paper we focus on the concrete example of SCQCD and look for a closed string theory description of its flavor-singlet sector. We work at the superconformal point (zero vev for all the scalars) and thus look for a string background with unbroken isometry. We attack the problem from two fronts: from the bottom-up, using the weakly-coupled Lagrangian description, and from the top-down, studying brane constructions in string theory. Correspondingly, the paper is divided into two main parts. The field theory analysis occupies sections 3-5, the string theory analysis sections 7-8. Section 6 provides a bridge, a first attempt to put together the clues of the field theory analysis and guess features of the dual string theory. In the field theory sections we pose and answer in rigorous detail a well-defined question: what is the protected spectrum of SCQCD in the generalized single-trace sector? The string theory analysis is more qualitative and our program not yet complete. We review brane constructions and argue that the decoupling limit leads to a sub-critical string background. We carry the analysis far enough to see that the string dual, which is largely constrained by symmetry, matches several field theory expectations, but we leave the determination of the precise non-critical background for future work.

In both the bottom-up and top-down approaches it is very useful to view SCQCD as part of an “interpolating” superconformal field theory (SCFT) that has product gauge group and correspondingly two exactly marginal couplings and . For one finds SCQCD plus a decoupled vector multiplet, while for one finds the orbifold of SYM. The orbifold theory has a well-known closed string dual, type IIB on , and changing amounts to changing the period of the NSNS -field through the blow-down cycle of the orbifold. As we are going to discuss in detail, the flavor-singlet operators of SCQCD are a subsector of the operators of the interpolating SCFT. So in a sense we are guaranteed success: we know a priori that the flavor-singlet sector of SCQCD must be described by the closed string theory obtained by following the limit in the bulk. This is however a rather subtle limit, and making sense of it will occupy us in the second part of the paper.

In a companion paper [25] we have taken the next step of the bottom-up analysis. We have evaluated the planar one-loop dilation operator in the scalar sector of SCQCD, as well as of the interpolating SCFT, and written it as the Hamiltonian of a spin-chain system. The spin-chain for SCQCD is novel, since the chain is of the “generalized single-trace” form (1). The dynamics of magnon excitations is quite interesting. In particular it is amusing to see how the flavor-contracted fundamental/antifundamental pairs arise as by a process of “dimerization” of the magnons of the interpolating SCFT. Some results of [25] will be an input in section 4 to the analysis of the protected spectrum of SCQCD.

A more detailed outline of the rest of paper is as follows. We begin in section 3 with a review of the Lagrangian and symmetries of SCQCD and of the interpolating SCFT that connects it to the orbifold of SYM. In sections 4 and 5 we study the protected spectrum of short supermultiplets777We use the word “short” casually, to denote a multiplet that obeys any of type of shortening condition, unlike some authors who distinguish between “short” and “semi-short”. We use the precise notation for multiplets reviewed in appendix A when we need to make such distinctions. of SCQCD and its relation with the spectrum of the interpolating SCFT. This turns out to be a rather intricate exercise in superconformal representation theory. A part of the protected spectrum of SCQCD is easy to determine, namely the supermultiplets built on primaries made of scalar fields: (19) is the complete list of such primaries, as shown in [25] using the one-loop spin-chain. In section 4 we follow in detail the evolution of the protected states of the interpolating SCFT, starting at the orbifold point where the complete protected spectrum is easily determined. In the limit we recover (19) as the subsector of protected primaries of the interpolating SCFT that are flavor singlets. Now there are many more protected states in SCQCD than there are for generic in the interpolating SCFT: the extra protected states arise from long multiplets of the interpolating SCFT that split into short multiplets at . In section 5 we use the superconformal index to demonstrate the existence of these extra protected states. We show that the number of extra states grows exponentially with the conformal dimension. We also characterize the quantum numbers of the first few of them using a “sieve” algorithm; this characterization is up to a certain intrinsic ambiguity of the superconformal index, which can only determine “equivalence classes” of short multiplets, as we review in detail. Still, we have enough information to unambiguously demonstrate the existence of higher-spin protected states in the generalized single-trace sector, in sharp contrast with SYM.

In section 6 we use the clues offered by the protected spectrum to argue that the dual of SCQCD should be a sub-critical string background, with seven “geometric” dimensions, containing both an and an factor. There must be a sector of light string states, with mass of the order of the AdS scale for all , dual to the higher-spin protected states detected by the superconformal index – so even for large the supergravity approximation cannot be entirely valid. We suggest that there is also a separate sector of heavy string states, with for . We have in mind a scenario where in the interpolating SCFT there are two effective string lengths and , corresponding to the two ‘t Hooft couplings and : for and fixed , the string length is associated with the massive sector, while is associated with the light sector. In section 7 we review brane constructions of the interpolating SCFT and of SCQCD. The most useful construction is the Hanany-Witten setup with D4 branes suspended between NS5 branes. We argue that the relevant dynamics is captured by a sub-critical brane setup, with color D3 and flavor D5 boundary states in the exact IIB worldsheet CFT . We identify the dual of SCQCD with the backreacted background, where the D-branes are replaced by flux. We do not yet know the precise background, but it is largely constrained by symmetries. In section 8 we show that just assuming a solution exists, the results of the top-down approach are in nice qualitative agreement with the bottom-up expectations. A useful tool is the spacetime “effective action” of the non-critical theory, which we identify as the seven-dimensional maximal supergravity with the (non-standard) gauging. We conclude in section 9 with a brief discussion.

Several technical appendices supplement the text. In appendix A we review the shortening conditions of the superconformal algebra. In appendix B we review the chiral ring of SCQCD and of the interpolating SCFT. In appendix C we evaluate the superconformal index for various combinations of short multiplets. In appendix D we review the Kaluza-Klein reduction on of the tensor multiplet, with a new detailed treatment of the zero modes. In appendix E we review the sub-critical IIB background and its spectrum. We make a new claim about the “effective action” describing the lowest plane-wave states, which we identify with maximally supersymmetric -gauged supergravity.

### 2.3 Relation to previous work

The idea that sub-critical string theories play a role in the gauge/gravity correspondence is of course not new. Polyakov’s conjecture that pure Yang-Mills theory should be dual to a string theory, with the Liouville field playing the role of the fifth dimension, predates the AdS/CFT correspondence (see e.g. [26, 27, 28]). In fact one of the surprises of AdS/CFT was that some supersymmetric gauge theories are dual to simple backgrounds of critical string theory. General studies of AdS solutions of non-critical spacetime effective actions include [29, 30]. Non-critical holography has been mostly considered, starting with [31, 32], in the supersymmetric case, notably for super QCD in the Seiberg conformal window, which is argued to be dual to non-critical backgrounds of the form with string-size curvature. There is an interesting literature on the RNS worldsheet description of these non-critical backgrounds and their gauge-theory interpretation, see e.g. [33, 34, 35, 36]. Non-critical RNS superstrings were formulated in [37, 38] and shown in [39, 40, 41, 41, 42, 43] to describe subsectors of critical string theory – the degrees of freedom localized near NS5 branes or (in the mirror description) Calabi-Yau singularities. Non-critical superstrings have been also considered in the Green-Schwarz and pure-spinor formalisms, see e.g. [44, 45, 46, 47, 48].

Our analysis in sections 6 and 7 for SCQCD will be in the same spirit as the analysis of [33, 36] for super QCD. We will use the double-scaling limit defined in [42, 43] and further studied in e.g. [49, 50, 51]. One of our points is that the supersymmetric case should be the simplest for non-critical gauge/string duality. On the string side, more symmetry does not hurt, but the real advantage is on the field theory side. Little is known about the SCFTs in the Seiberg conformal window, since generically they are strongly coupled, isolated fixed points. By contrast SCQCD has an exactly marginal coupling , which takes arbitrary non-negative values. There is a weakly coupled Lagrangian description for , and we can bring to bear all the perturbative technology that has been so successful for SYM, for example in uncovering integrable structures.888 SQCD at the Seiberg self-dual point admits an exactly marginal coupling (the coefficient of a quartic superpotential), which however is bounded from below – the theory is never weakly coupled. At the same time we may hope, again in analogy with SYM, that the string dual will simplify in the strong coupling limit .

There are also interesting approaches to holography for gauge theories with a large number of fundamental flavors in critical string theory/supergravity, see e.g. [52, 53, 54, 55, 56, 57, 58, 59, 60]. The critical setup inevitably implies that the boundary gauge theory will have UV completions with extra degrees of freedom (e.g. higher supersymmetry and/or higher dimensions).

## 3 Field Theory Lagrangian and Symmetries

In this section we briefly review the structure and symmetries of SCQCD, and its relation to the orbifold of SYM. Much insight is gained by viewing SCQCD, which has one exactly marginal parameter (the gauge coupling ), as the limit of a two-parameter family of superconformal field theories. This is the family of theories with product gauge group999The ranks of the two groups coincide, , but it will be useful to always distinguish graphically with a “check” all quantities pertaining to the second group . and two bifundamental hypermultiplets; its exactly marginal parameters are the two gauge-couplings and . For one recovers SCQCD plus a decoupled free vector multiplet in the adjoint of . At , the second gauge group is interpreted as a subgroup of the global flavor symmetry, . For , we have instead the familiar orbifold of SYM. Thus by tuning we interpolate continuously between SCQCD and the universality class.

The and anomalies are constant, and equal to each other, along this exactly marginal line: at the end point , the vector multiplets decouples, accounting for the “missing” in SCQCD.

### 3.1 N=2 Scqcd

Our main interest is SYM theory with gauge group and fundamental hypermultiplets. We refer to this theory as SCQCD. Its global symmetry group is , where is the R-symmetry subgroup of the superconformal group. We use indices for , for the flavor group and for the color group .

Table 1 summarizes the field content and quantum numbers of the model: The Poincaré supercharges , and the conformal supercharges , are doublets with charges under . The vector multiplet consists of a gauge field , two Weyl spinors , , which form a doublet under , and one complex scalar , all in the adjoint representation of . Each hypermultiplet consists of an doublet of complex scalars and of two Weyl spinors and , singlets. It is convenient to define the flavor contracted mesonic operators

 MIaJb≡1√2Q aJ i¯QI i b, (2)

which may be decomposed into the singlet and triplet combinations

 M1≡MIIandMI3J≡MIJ−12MKKδIJ. (3)

The operators decompose into adjoint plus singlet representations of the color group ; the singlet piece is however subleading in the large limit.

### 3.2 Z2 orbifold of N=4 and interpolating family of SCFTs

SCQCD can be viewed as a limit of a family of superconformal theories; in the opposite limit the family reduces to a orbifold of SYM. In this subsection we first describe the orbifold theory and then its connection to SCQCD.

As familiar, the field content of SYM comprises the gauge field , four Weyl fermions and six real scalars , where are indices of the R-symmetry group. Under , the fermions are in the representation, while the scalars are in (antisymmetric self-dual) and obey the reality condition101010The indicates hermitian conjugation of the matrix in color space. We choose hermitian generators for the color group.

 X†AB=12ϵABCDXCD. (4)

We may parametrize in terms of six real scalars , ,

 XAB=1√2⎛⎜ ⎜ ⎜ ⎜⎝0X4+iX5X7+iX6X8+iX9−X4−iX50X8−iX9−X7+iX6−X7−iX6−X8+iX90X4−iX5−X8−iX9X7−iX6−X4+iX50⎞⎟ ⎟ ⎟ ⎟⎠ (5)

Next, we pick an subgroup of ,

 1+2−3^+4^−⎛⎜ ⎜ ⎜ ⎜⎝SU(2)R×U(1)rSU(2)L×U(1)∗r⎞⎟ ⎟ ⎟ ⎟⎠. (6)

We use indices for (corresponding to ) and indices for (corresponding to ). To make more manifest their transformation properties, the scalars are rewritten as the singlet (with charge under ) and as the bifundamental (neutral under ),

 Z≡X4+iX5√2,XI^I≡1√2(X7+iX6X8+iX9X8−iX9−X7+iX6). (7)

Note the reality condition . Geometrically, is the group of rotations and the group of rotations. Diagonal transformations () preserve the trace, , and thus correspond to rotations.

We are now ready to discuss the orbifold projection. In R-symmetry space, the orbifold group is chosen to be with elements . This is the well-known quiver theory [61] obtained by placing D3 branes at the singularity , with and and invariant. Supersymmetry is broken to , since the supercharges with indices are projected out. The symmetry is broken to , or more precisely to since only objects with integer spin survive. The factors are the R-symmetry of the unbroken superconformal group, while is an extra global symmetry under which the unbroken supercharges are neutral.

In color space, we start with gauge group , and declare the non-trivial element of the orbifold to be

 τ≡(INc×Nc00−INc×Nc). (8)

All in all the action on the fields is

 Am→τAmτ,ZIJ→τZIJτ,λI→τλIτ,XI^I→−τXI^Iτ,λ^I→−τλ^Iτ. (9)

The components that survive the projection are

 Am = (10) λI = (λaIb00ˇλˇaIˇb)λ^I=⎛⎝0ψa^Iˇa~ψˇb^Ib0⎞⎠ (11) XI^I = (12)

The gauge group is broken to , where the factor is the relative111111Had we started with group, we would also have an extra diagonal , which would completely decouple since no fields are charged under it. generated by (equ.(8)): it must be removed by hand, since its beta function is non-vanishing. The process of removing the relative modifies the scalar potential by double-trace terms, which arise from the fact that the auxiliary fields (in superspace) are now missing the component. Equivalently we can evaluate the beta function for the double-trace couplings, and tune them to their fixed point [62].

Supersymmetry organizes the component fields into the vector multiplets of each factor of the gauge group, and , and into two bifundamental hypermultiplets, and . Table 2 summarizes the field content and quantum numbers of the orbifold theory.

The two gauge-couplings and can be independently varied while preserving superconformal invariance, thus defining a two-parameter family of SCFTs. Some care is needed in adjusting the Yukawa and scalar potential terms so that supersymmetry is preserved. We find

 LYukawa(gYM,ˇgYM) = i√2Tr[−gYMϵIJ¯λI¯λJϕ−ˇgYMϵIJ¯ˇλI¯ˇλJˇϕ (13) +gYMϵ^I^J~ψ^Iϕψ^J+ˇgYMϵ^I^Jψ^Jˇϕ~ψ^I +gYMϵ^I^J~ψ^JλIQI^I+ˇgYMϵ^I^JQI^IˇλI~ψ^J −gYMϵIJ¯Q^JIλJψ^J−ˇgYMϵIJψ^JˇλI¯Q^JJ]+h.c.
 V(gYM,ˇgYM) = g2YMTr[12[¯ϕ,ϕ]2+MII(ϕ¯ϕ+¯ϕϕ)+MJIMIJ−12MIIMJJ] (14) +ˇg2YMTr[12[¯ˇϕ,ˇϕ]2+ˇMII(ˇϕ¯ˇϕ+¯ˇϕˇϕ)+ˇMIJˇMJI−12ˇMIIˇMJJ] +gYMˇgYMTr[−2QI^Iˇϕ¯Q^II¯ϕ+h.c.]−1NcVd.t.,

where the mesonic operators are defined as121212Note that .

 MIaJb≡1√2QaJ^Jˇa¯Q^JIˇab,ˇMIˇaJˇb≡1√2¯Q^JIˇaaQaJ^Jˇb, (15)

and the double-trace terms in the potential are

 Vd.t. = g2YM(Tr[MJI]Tr[MIJ]−12Tr[MII]Tr[MJJ]) +ˇg2YM(Tr[ˇMIJ]Tr[ˇMJI]−12Tr[ˇMII]Tr[ˇMJJ]) = (g2YM+ˇg2YM)(Tr[MJI]Tr[MIJ]−12Tr[MII]Tr[MJJ]).

The symmetry is present for all values of the couplings (and so is the R-symmetry, of course). At the orbifold point there is an extra symmetry (the quantum symmetry of the orbifold) acting as

 ϕ↔ˇϕ,λI↔ˇλI,Am↔ˇAm,ψ^I↔~ψ^I,QI^I↔−ϵIJϵ^I^J¯QJ^J. (17)

Setting , the second vector multiplet becomes free and completely decouples from the rest of theory, which happens to coincide with SCQCD (indeed the field content is the same and susy does the rest). The symmetry can now be interpreted as a global flavor symmetry. In fact there is a symmetry enhancement : one sees in (13, 14) that for the index and the index can be combined into a single flavor index .

In the rest of the paper, unless otherwise stated, we will work in the large limit, keeping fixed the ‘t Hooft couplings

 λ≡g2YMNc≡8π2g2,ˇλ≡ˇg2YMNˇc≡8π2ˇg2. (18)

The normalizations of and are convenient for the perturbative calculations of [25], in this paper it is just important to keep in mind that they are (square roots of) the ’t Hooft couplings. We will refer to the theory with arbitrary and as the “interpolating SCFT”, thinking of keeping fixed as we vary from (orbifold theory) to ( SCQCD extra free vector multiplets).

## 4 Protected Spectrum of the Interpolating Theory

In the present and in the following section we will study the protected spectrum of SCQCD at large , in the flavor singlet sector, and its relation with the protected spectrum of the interpolating SCFT. We have argued that in the large Veneziano limit, flavor singlets that diagonalize the dilation operator take the “generalized single-trace” form (1). We will look for the generalized single-trace operators belonging to short multiplets of the superconformal algebra. These are the operators expected to map to the Kaluza-Klein tower of massless single closed string states, so they are the first place to look in a “bottom-up” search for the string dual.

The determination of the complete list of short multiplets of SCQCD in this “generalized single-trace” sector turns out to be more subtle than expected. A class of short multiplets is relatively easy to isolate, namely the multiplets based on the following superconformal primaries:

 TrM3=(Qai¯Qia)3,Trϕℓ+2,Tr[Tϕℓ],ℓ≥0. (19)

Here . We hasten to add that this will turn out to be only a small fraction of the complete set of protected operators. The set (19) is the complete list of one-loop protected primaries in the scalar sector, as we show in [25] by a systematic evaluation of the one-loop anomalous dimension of all operators that are made out of scalars and obey shortening conditions. The operators correspond to the vacuum of the spin-chain studied in [25], while the correspond to the limit of a gapless magnon of momentum .

The operators and obey the familiar BPS condition , where is the spin and the charge, and they are generators of the chiral ring (with respect to an subalgebra), see appendix B.131313 Incidentally, the analysis of the chiral ring extends immediately to flavor non-singlets. The only chiral ring generator which is not a flavor singlet is the triplet bilinear (20) in the adjoint of . The conserved currents for the flavor symmetry belong to the short multiplet with bottom component . Similarly the current for the baryon number belongs to the multiplet. By contrast obey a “semi-shortening” condition and it may be missed in a naive analysis; in these operators there is a large mixing of “glueballs” and “mesons” and the idea of considering “generalized single-traces” is essential. The multiplet plays a distinguished role since it contains the stress-energy tensor and -symmetry currents.

Protection of the operators (19) can be understood from the viewpoint of the interpolating SCFT connecting SCQCD with the orbifold of SYM, as follows. The complete spectrum of short multiplets at the orbifold point is well-known. We will argue, using superconformal representation theory [63], that the protected multiplets found at the orbifold point cannot recombine into long multiplets as we vary , so in particular taking they must evolve into protected multiplets of the theory

 {N=2SCQCD⊕decoupledSU(Nˇc)vectormultiplet}. (21)

The list (19) is precisely recovered by restricting to singlets. Remarkably however, the superconformal index of SCQCD, evaluated in the next section, will show the existence of many more protected states. The extra protected states arise from the splitting long multiplets of the interpolating theory into short multiplets as .

We will make extensive use of the the list given by Dolan and Osborn[63] of all possible shortening conditions of the superconformal algebra. We summarize their results and establish notations in appendix A.

### 4.1 Protected Spectrum at the Orbifold Point

At the orbifold point () the state space of the field theory is the direct sum of an untwisted and a twisted sector, respectively even and odd under the “quantum” symmetry (17).

#### 4.1.1 Untwisted sector

Operators in the untwisted sector of the orbifold descend from operators of SYM by projection onto the invariant subspace. Their correlators coincide at large with correlators [64, 65]. In particular the complete list of untwisted protected states is obtained by projection of the protected states of . We will be interested in single-trace operators; as is well-known, the only protected single-trace operators of belong to the BPS multiplets , built on the chiral primaries , with , in the representation of (symmetric traceless of ) The decomposition of each BPS multiplet into multiplets reads [63]

 B12,12[0,p,0] ≃ (p+1)^B12p⊕Ep(0,0)⊕¯E−p(0,0) (22) ⊕(p−1)^C12p−1(0,0)⊕p(D12(p−1)(0,0)⊕¯D12(p−1)(0,0) ⊕p−2⨁k=1(k+1)(B12k,p−k(0,0)⊕¯B12k,k−p(0,0)) ⊕p−3⨁k=0(k+1)(C12k,p−k−2(0,0)⊕¯C12k,k−p+2(0,0)) ⊕p−