Operator Counting and Eigenvalue Distributions for 3D Supersymmetric Gauge Theories
PPUPT-2384
Operator Counting and EigenvalueDistributions for 3D Supersymmetric GaugeTheories
Daniel R. Gulotta, Christopher P. Herzog, and Silviu S. Pufu
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
Abstract
We give further support for our conjecture relating eigenvalue distributions of the Kapustin-Willett-Yaakov matrix model in the large N limit to numbers of operators in the chiral ringof the corresponding supersymmetric three-dimensional gauge theory. We show that therelation holds for non-critical R-charges and for examples with N = 2 instead of N = 3supersymmetry where the bifundamental matter fields are nonchiral. We prove that, fornon-critical R-charges, the conjecture is equivalent to a relation between the free energy ofthe gauge theory on a three sphere and the volume of a Sasaki manifold that is part of themoduli space of the gauge theory. We also investigate the consequences of our conjecture forchiral theories where the matrix model is not well understood.June 2011 a r X i v : . [ h e p - t h ] J un ontents N limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Flavored theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 δ . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Operator counting dependence on δ . . . . . . . . . . . . . . . . . . . . . . . 123.4 Matrix model and volumes of five-cycles . . . . . . . . . . . . . . . . . . . . 133.5 A Consistency condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 N = 2 deformations of the necklace quivers and matrix model . . . . . . . . 154.2 Operator counting for necklace quivers . . . . . . . . . . . . . . . . . . . . . 184.3 Flavored necklace quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Flavored N = 8 theory and its matrix model . . . . . . . . . . . . . . . . . . 204.5 Operator counting in flavored N = 8 theory . . . . . . . . . . . . . . . . . . 214.6 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 C / Z theory . . . . . . . . . . . . . . . . . . . . 235.3 Particular case: the cone over M , , / Z k . . . . . . . . . . . . . . . . . . . . 245.4 Missing operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 A F -Maximization for the necklace quivers 27B Further examples 29 B.1 Flavored ABJM theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29B.2 C / ( Z × Z ) theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 C Toric varieties in general 34D The Cone over Q , , / Z k Introduction
For those interested in superconformal gauge theories in three dimensions, the matrix modelof Kapustin, Willett, and Yaakov [1] provides a powerful tool. Using this matrix model,one can compute the partition function and the expectation values of supersymmetric Wil-son loops on a three sphere, even when the gauge theory is strongly interacting. The matrixmodel was derived through a localization procedure [2] that obscures the connection betweenmatrix model quantities and microscopic degrees of freedom in the gauge theory. Given thesuccess of the matrix model in post-dicting the N / large N scaling of the free energy of maximally supersymmetric SU ( N ) Yang-Mills theory at its infrared fixed point [3], itis a worthwhile goal to try to figure out the map between matrix model and gauge theoryquantities in greater detail. In ref. [4], we made some progress in understanding the relationbetween the eigenvalue distributions in the matrix model and the chiral ring of the super-symmetric gauge theory for the so-called necklace quivers, and we conjectured this relationwould hold more generally. In this paper, we work out further examples of field theoriesthat obey this conjecture. We restrict ourselves to field theories with M-theory duals of theFreund-Rubin type AdS × Y , where Y is a Sasaki-Einstein space. A calculation in 11-dsupergravity relates the field theory free energy to the volume of the internal space Y on thegravity side through the formula [5] F = N / (cid:115) π
27 Vol( Y ) + o ( N / ) , (1.1)where the normalization of the metric on Y used to compute Vol( Y ) is R mn = 6 g mn .Let us begin by describing the relation noticed in [4] between the eigenvalue distributionin the matrix model and the chiral ring for the necklace quiver gauge theories. These fieldtheories have N = 3 supersymmetry (SUSY), gauge group U ( N ) d , and associated Chern-Simons levels k a , a = 1 , . . . , d , such that (cid:80) a k a = 0. The matter sector consists of thebifundamental fields X a,a +1 and X a +1 ,a that connect the gauge groups together into a circle(see figure 1). The localization procedure [1] reduces the partition function to an integralover d constant N × N matrices σ a , where σ a is the real scalar that belongs to the same N = 2 multiplet as the gauge connection. In the large N limit, the matrix integral canbe evaluated in the saddle point approximation. As was shown in [5], at the saddle point,the real parts of the eigenvalues λ ( a ) j of σ a grow as N / while their imaginary parts stay of By “free energy” we mean minus the logarithm of the path integral on S , with an appropriate subtractionof UV divergences. N is taken to infinity. In addition, to leading order in N the real parts of theeigenvalues are the same for each gauge group. Therefore, in order to find the saddle pointone can consider the large N expansion λ ( a ) j = N / x j + iy a,j + . . . . (1.2)As one takes N → ∞ , the x j and y a,j become dense, and one can pass to a continuumdescription by defining the distributions ρ ( x ) = lim N →∞ N N (cid:88) j =1 δ ( x − x j ) , ρ ( x ) y a ( x ) = lim N →∞ N N (cid:88) j =1 y a,j δ ( x − x j ) . (1.3)The saddle point is then found by extremizing a free energy functional F [ ρ, y a ] under theassumption that ρ is a density, namely that ρ ( x ) ≥ (cid:82) dx ρ ( x ) = 1. It is convenient toenforce the latter constraint with a Lagrange multiplier µ that will appear in the formulaepresented below. In general, F [ ρ, y a ] may be a non-local functional because the eigenvaluescould interact with one another through long-range forces, and if this is the case the saddlepoint equations are usually hard to solve. The key insight in solving the saddle point equa-tions in [5] was that, luckily, in the continuum limit (1.3) the ansatz (1.2) leads to a local expression for F [ ρ, y a ] due to the cancellation of long-range forces. By solving the saddlepoint equations, it was shown in [5] that the distributions ρ ( x ) and ρ ( x )[ y a ( x ) − y b ( x )] canbe identified for any a and b with piecewise linear functions with compact support. Whilethe free energy F can be calculated by evaluating the functional F [ ρ, y a ] on the saddle pointconfiguration, it is also possible to calculate F by noticing that F [ ρ, y a ] satisfies a virialtheorem that gives F = 4 πµN / / X a,a +1 and X a +1 ,a fields and monopole operators modulo superpotential and monopolerelations. While one can define monopole operators that turn on any number of flux unitsthrough each U ( N ) gauge group, at large N the only relevant ones are the so-called “diagonalmonopole operators” that turn on the same number of units of flux through the diagonal U (1) subgroup of each U ( N ) gauge group. Operators in the chiral ring therefore have anassociated R-charge r and a (diagonal) monopole charge m . We can also introduce thethe function ψ X ab ( r, m ) that counts in the same way operators that don’t vanish when thebifundamental field X ab is set to zero. In [4] we found the following relation between the In our conventions, X ab transforms under the ( N a , N b ) representation of U ( N a ) × U ( N b ). ∂ ψ∂r ∂m (cid:12)(cid:12)(cid:12)(cid:12) m = rx/µ = rµ ρ ( x ) , (1.4a) ∂ ψ X ab ∂r∂m (cid:12)(cid:12)(cid:12)(cid:12) m = rx/µ = rµ ρ ( x )[ y b ( x ) − y a ( x ) + R ( X ab )] . (1.4b)In other words, the matrix model eigenvalue density ρ ( x ) and the quantity ρ ( x )[ y b ( x ) − y a ( x ) + R ( X ab )], which as mentioned above are linear functions of x , should be interpretedas derivatives of numbers of operators whose monopole charge to R-charge ratio is given by x/µ .One of the goals of the current paper is to provide further evidence for the conjectures(1.4) in superconformal theories with gravity duals that preserve only N = 2 supersymmetryas opposed to the N = 3 SUSY of the necklace quivers studied in [4]. In an N = 2 theory,the U (1) R-symmetry can mix with other Abelian flavor symmetries, so the matter fields canhave R-charges different from the canonical free-field value 1 /
2. The generalization of theKapustin-Willett-Yaakov matrix model to non-canonical R-charges was worked out in [6, 7].Furthermore, since the U (1) R symmetry can now mix with other Abelian flavor symmetries,it was conjectured in [6] that the correct R-symmetry in the IR can be found by extremizingthe free energy F as a function of all trial R-charges that are consistent with the marginalityof the superpotential. It has been seen in many examples [8–16] that this extremum is amaximum and that F is positive. We find that eqs. (1.4) are satisfied for more general quiver gauge theories where thebifundamental matter multiplets are non-chiral, meaning that they come in pairs of conjugaterepresentations of the gauge group. In the first half of section 4, we examine the necklacequiver gauge theories, this time with an arbitrary R-charge assignment consistent with themarginality of the superpotential. In the second half of section 4 and appendix B, weexamine theories where we add flavor (meaning N = 2 matter multiplets that transformin the fundamental or anti-fundamental representation of one of the gauge groups) to themaximally SUSY N = 8 theory and to the N = 6 ABJM theory of [18]. Lastly, in appendixB.2, we consider a theory that shares the same quiver with its (3 + 1)-dimensional cousinthat has a C / Z × Z moduli space (see figure 3). In all of these examples, eqs. (1.4) aresatisfied for any choice of trial R-charges.Another goal of this paper is to relate the conjecture (1.4) to the observation made It was suggested in [9] that F might be a good measure of the number of degrees of freedom even innon-supersymmetric field theories. See also [17].
4n [9,11] that, as checked in a number of examples, the relation (1.1) between the free energyand the volume of the internal space holds for any trial R-charges, and not just the ones thatextremize F . That this relation holds for any trial R-charges is surprising because only forthe critical R-charges does there exist a known 11-d supergravity background AdS × Y . Fornon-critical R-charges, measured geometrically in terms of the volume of some correspondingfive-cycles of Y , one can still identify a class of Sasakian metrics on Y and compute theirvolume. The volume Vol( Y ) is a function of the Reeb vector of Y , which parameterizes theway the U (1) R symmetry sits within the isometry group of Y . We show in section 3 that(1.4a) holds for some choice of trial R-charges if and only if eq. (1.1) holds for the samechoice of trial R-charges for the matter fields and a range of R-charges for the monopoleoperators. We also show an analogous result that relates (1.4b) to the volumes of five-dimensional sub-manifolds of Y . For a gauge invariant operator constructed from a closedloop of bifundamental fields X ab , it must be true that (cid:80) X ab ( y a − y b ) = 0. Given (1.4), thereis a geometric version of this sum that must also vanish. The last part of section 3 explainswhy.There are previously recognized difficulties, involving cancellation of long-range forces,in using the matrix model to study the large N limit of theories with chiral bifundamentalfields [9]. We do not surmount these difficulties, but we investigate in section 5 what (1.4a)and (1.4b) predict for a theory with a moduli space that is a fibration over C / Z (seefigure 2). We also study a field theory that was conjectured to be dual to AdS × Q , , / Z k in appendix D (see figure 5).The paper contains two heretofore unmentioned appendices. Appendix A proves that thecritical R-charges maximize F for the necklace quivers. Appendix C reviews how to countgauge invariant operators for an Abelian gauge theory with a toric branch of its modulispace. N limit To understand what it means to consider non-canonical (or non-critical) R-charges, let usintroduce some of the ideas developed recently in refs. [6, 7, 9]. Building on the work of [1],refs. [6, 7] used localization to reduce the path integral of any N = 2 Chern-Simons matter A similar relation between the anomaly coefficient a computed with a set of trial R-charges and thevolume of a 5-d Sasakian space Y is known to hold in theories with AdS duals [19, 20]. S to a matrix integral. By a Chern-Simons-matter theory we mean a theory constructedfrom some number d of N = 2 vector multiplets with gauge groups G a ( a = 1 , . . . , d ) andChern-Simons kinetic terms iπk a (cid:82) tr A a ∧ dA a + supersymmetric completion, as well as anynumber of N = 2 chiral superfields transforming in representations R i of the total gaugegroup G = (cid:81) da =1 G a . As mentioned in the introduction, one difference between theories with N = 2 supersymmetry and theories with more supersymmetry is that the R-charges ∆ i ofthe chiral fields at the IR superconformal fixed point are not fixed at the free field values∆ i = 1 /
2, so the free energy will generically depend on these R-charges. In fact, it wasproposed in [6] that a prescription for finding the correct R-charges in the IR is to calculatethe free energy F as a function of all possible R-charge assignments consistent with themarginality of the superpotential and to extremize F over the set of all such assignments.Let us focus on the case where all gauge groups are U ( N ) and index the gauge groups by a = 1 , . . . , d . Generalizing the techniques developed in [5], the authors of [9] used the saddlepoint approximation to evaluate the path integral on S for a class of N = 2 Chern-Simons-matter theories at large N that satisfy the following five conditions:1. The CS levels sum to zero: (cid:80) da =1 k a = 0.2. Any matter field X transforms either in the N a , or N b , or ( N a , N b ) representation forsome a and b .3. The total number of fundamental fields equals the total number of anti-fundamentalfields.4. For any bifundamental field X transforming in ( N a , N b ), there exists another bifunda-mental field ˜ X transforming in the conjugate representation ( N b , N a ).5. For each gauge group a we have (cid:88) X in ( N a , N b ) ( R [ X ] −
1) + (cid:88) ˜ X in ( N b , N a ) (cid:16) R [ ˜ X ] − (cid:17) = − . (2.1)This last condition is sufficient to guarantee the vanishing of the long-range forces on theeigenvalues in the saddle point approximation. Interestingly, this condition has appearedbefore in the context of superconformal (3 + 1)-dimensional gauge theories. The condition(2.1) would imply that the NSVZ beta function of gauge group a vanishes [11]. For quivergauge theories with a toric moduli space, bifundamental fields appear in exactly two terms6n the superpotential. Thus, if we sum (2.1) over a , we find the condition that( − ( . (2.2)In other words the quiver may give a triangulation of a torus where the faces of the trian-gulation are superpotential terms [21].If these five conditions are satisfied, one can take the N → ∞ limit as described ineqs. (1.2) and (1.3) in the introduction. The free energy is the extremum of the free energyfunctional F [ ρ ( x ) , y a ( x )] = 2 πN / (cid:90) dx xρ ( x ) (cid:32) d (cid:88) a =1 k a y a ( x ) + ∆ m (cid:33) + 2 πN / (cid:90) dx | x | ρ ( x ) (cid:88) X in N a (cid:18) − R [ X ]2 − y a ( x ) (cid:19) + (cid:88) X in N b (cid:18) − R [ X ]2 + 12 y b ( x ) (cid:19) + πN / (cid:90) dx ρ ( x ) (cid:88) X in ( N a , N b ) ( δy ab ( x ) + R [ X ])( δy ab ( x ) + R [ X ] − δy ab ( x ) + R [ X ] − , (2.3)where δy ab ( x ) ≡ y a ( x ) − y b ( x ). This formula was derived assuming the bifundamental fieldssatisfy 0 ≤ R [ X ] + δy ab ( x ) ≤
2. Extra care must be taken when R [ X ] + δy ab = 0 or 2 becausein these cases the discrete nature of the eigenvalues becomes important, and the equation ofmotion derived from varying (2.3) with respect to δy ab ( x ) need not hold.Generically, the functional 2.3 has many flat directions. The following d of them play animportant role in this paper because they correspond to changing the R-charges of the matterfields by linear combinations of the gauge charges with respect to the diagonal U (1) ∈ U ( N ) a : y a ( x ): y a ( x ) → y a ( x ) − δ ( a ) , chiral superfield X in N a : R [ X ] → R [ X ] + δ ( a ) , chiral superfield X in N b : R [ X ] → R [ X ] − δ ( b ) , chiral superfield X in ( N a , N b ): R [ X ] → R [ X ] + δ ( a ) − δ ( b ) , ∆ m : ∆ m → ∆ m + (cid:88) a k a δ ( a ) . (2.4)See [9] for a more detailed discussion of these flat directions and their AdS/CFT interpreta-tion. 7he ∆ m appearing in (2.3) is the bare R-charge of the “diagonal” monopole operator T (1) .A monopole operator T ( q a ) a turns on q a units of tr F a flux through a two-sphere surroundingthe insertion point. Diagonal monopole operators T ( m ) turn on the same number m oftr F a flux units in each gauge group. At large N , only the diagonal monopole operators areimportant.We will usually impose the constraint (cid:82) dx ρ ( x ) = 1 by introducing a Lagrange multiplier µ and defining the functional˜ F [ ρ, y a , µ ] = F [ ρ, y a ] − πN / µ (cid:18)(cid:90) dx ρ ( x ) − (cid:19) . (2.5)This functional should be extremized with respect to ρ ( x ), y a ( x ), and µ . In all gauge theories that we examine in this paper the fundamental and anti-fundamentalfields q α and ˜ q α appear in the superpotential as δW = (cid:88) α tr [ q α O α ˜ q α ] , (2.6)where O α are polynomials in the bifundamental fields. It was conjectured in [22, 23] that ifthis is case then the diagonal monopole operators T ( m ) satisfy the following OPE: T ( m ) T ( − m ) = (cid:32)(cid:89) α O α (cid:33) | m | . (2.7)This OPE was conjectured in part because a parity anomaly argument shows that themonopole operators have gauge charges g a [ T ( m ) ] = mk a + | m | (cid:88) α g a [ O α ] (2.8)with respect to the diagonal U (1) ⊂ U ( N ) a , and R-charges R [ T ( m ) ] = m ∆ m + | m | (cid:88) α R [ O α ] . (2.9)Using the fact that each term in (2.6) must be gauge-invariant and have R-charge two, wehave R [ q α ] + R [˜ q α ] + R [ O α ] = 2 and g a [ q α ] + g a [˜ q α ] + g a [ O α ] = 0 for any a . One can use these8elations to eliminate the sum over the flavor fields in (2.3): F [ ρ ( x ) , y a ( x )] = 2 πN / (cid:90) dx | x | ρ ( x ) (cid:32) R [ T (sgn x ) ] + d (cid:88) a =1 y a ( x ) g a [ T (sgn x ) ] (cid:33) + πN / (cid:90) dx ρ ( x ) (cid:88) X in ( N a , N b ) ( δy ab ( x ) + R [ X ])( δy ab ( x ) + R [ X ] − δy ab ( x ) + R [ X ] − . (2.10) We can relate the conjecture (1.4) to the observation that eq. (1.1) holds for any trial R-charges. In particular, we prove the following result: In a CS-matter theory dual to
AdS × Y fix a set of matter R-charges R [ X ] and a bare monopole charge ∆ m so that the conformaldimensions of all gauge-invariant operators satisfy the unitarity bound. Let ρ ( x ), µ , and ψ ( r, m ) be as defined in the introduction, and let’s assume ρ ( x ) has compact support. Thefollowing two statements are equivalent:A. The conjecture (1.4a) holds for the given R-charges R [ X ] and bare monopole charge ∆ m .B. For any δ in a small enough neighborhood of zero, we havelim N →∞ π N F = Vol( Y, δ ) , (3.1)where the free energy F of the CS-matter theory and the volume Vol( Y, δ ) of the internalspace Y are both computed assuming that the matter R-charges are R [ X ] and the baremonopole charge is ∆ m + δ .For notational convenience, let’s denote the LHS of eq. (3.1) by Vol m ( Y, δ ) and let’sintroduce the rescaled matrix model quantities:ˆ x = xµ , ˆ ρ (ˆ x ) = ρ ( x ) µ , ˆ y a (ˆ x ) = y a ( x ) . (3.2)9he equivalence between (A) and (B) follows from the following two equations:Vol m ( Y, δ ) = π (cid:90) d ˆ x ˆ ρ (ˆ x )(1 + ˆ xδ ) , (3.3)Vol( Y, δ ) = π (cid:90) d ˆ x lim r →∞ ψ (2 , ( r, r ˆ x ) /r (1 + ˆ xδ ) , (3.4)which we prove in sections 3.2 and 3.3, respectively.Assuming the eqs. (3.3) and (3.4) to be true, it is clear that the statement (A) implies(B). That (B) implies (A) follows from the fact that knowing Vol m ( Y, δ ) for δ in a smallneighborhood of zero, one can reconstruct ˆ ρ (ˆ x ), and analogously, from Vol( Y, δ ) one canreconstruct lim r →∞ ψ (2 , ( r, r ˆ x ) /r . Indeed, one can extend Vol m ( Y, δ ) to any complex δ as ananalytic function with singularities. We assume that ˆ ρ is supported on [ˆ x − , ˆ x + ] for some ˆ x − < < ˆ x + . We see from eq. (3.3) that the integral converges absolutely if δ ∈ ( − / ˆ x + , − / ˆ x − )or δ (cid:54)∈ R , so Vol m ( Y, δ ) can only have singularities on ( −∞ , − / ˆ x + ] ∪ [ − / ˆ x − , ∞ ).To relate the singularities of Vol m ( Y, δ ) to ˆ ρ (ˆ x ) we can perform two integrations by partsin (3.12) Vol m ( Y, δ ) = π δ (cid:90) d ˆ x ˆ ρ (cid:48)(cid:48) (ˆ x )ˆ x + δ (3.5)for any δ ∈ C \ (( −∞ , − / ˆ x + ] ∪ [ − / ˆ x − , ∞ )). Generically, eq. (3.5) shows that Vol m ( Y, δ )has two branch cuts, one on ( −∞ , − / ˆ x + ] and one on [ − / ˆ x − , ∞ ). From the discontinuitiesof Vol m ( Y, δ ) one can read off ˆ ρ (cid:48)(cid:48) ( − /δ ). Simple poles of Vol m ( Y, δ ) at δ = − / ˆ x (cid:48) correspondto contributions to ˆ ρ (cid:48)(cid:48) (ˆ x ) proportional to δ (ˆ x − ˆ x (cid:48) ); second order poles of Vol m ( Y, δ ) at δ = − / ˆ x (cid:48) correspond to δ (cid:48) (ˆ x − ˆ x (cid:48) ), etc. From the singularities of the analytic continuationof Vol m ( Y, δ ) one can therefore reconstruct uniquely ˆ ρ (cid:48)(cid:48) (ˆ x ), and hence ˆ ρ (ˆ x ), and similarly forVol( Y, δ ) and lim r →∞ ψ (2 , ( r, r ˆ x ) /r . If Vol m ( Y, δ ) and Vol(
Y, δ ) agree on an open set, then(A) holds.In our examples, Vol m ( Y, δ ) is a rational function of δ with poles of order at most three,so ˆ ρ (ˆ x ) is piecewise linear and it may have delta-functions. From the location and residuesof the poles one can first reconstruct ˆ ρ (cid:48)(cid:48) (ˆ x ), and then ˆ ρ (ˆ x ) by integrating ˆ ρ (cid:48)(cid:48) (ˆ x ) twice. Toperform this reconstruction starting with Vol m ( Y, δ ), one first decomposes Vol m ( Y, δ ) intopartial fractions, and then identifies the terms in ˆ ρ (cid:48)(cid:48) (ˆ x ) that give those partial fractions: if,10or example, Vol m ( Y, δ ) = π δ (cid:88) i a i x i δ − π δ (cid:88) i b i (1 + ˆ x i δ ) (3.6)for some ˆ x i , then ˆ ρ (cid:48)(cid:48) (ˆ x ) = (cid:88) i a i δ (ˆ x − ˆ x i ) + (cid:88) i b i δ (cid:48) (ˆ x − ˆ x i ) . (3.7) δ In this subsection we prove the result (3.3). As we have seen in the previous section, thematrix model generally takes the form˜ F [ ρ, y a , µ ] = (cid:90) dx ρ ( x ) f ( y a ( x )) − (cid:90) dx ρ ( x ) V ( x, y a ( x ))+ 2 πN / (cid:90) dx | x | ρ ( x ) R [ T (sgn x ) ] − πN / µ (cid:18)(cid:90) dx ρ ( x ) − (cid:19) , (3.8)for some functions f and V . While the explicit form of these function is given in (2.10),their precise form doesn’t matter. The only property of V that we will use is that it ishomogeneous of degree one in x , namely V ( λx, y a ( x )) = λV ( x, y a ( x )) for any λ >
0. Withthe rescaling (3.2), one can write ˜ F as˜ F [ ˆ ρ, ˆ y a , µ ] = − πN / µ + µ (cid:90) d ˆ x ˆ x (cid:20) ˆ ρ (ˆ x ) ˆ x f (ˆ y a (ˆ x )) − ˆ ρ (ˆ x )ˆ x V (ˆ x, ˆ y a (ˆ x ))ˆ x +2 πN / ˆ ρ (ˆ x ) | ˆ x | (cid:18) R [ T (sgn ˆ x ) ] − | ˆ x | (cid:19)(cid:21) . (3.9)The rescaling (3.2) is useful because now the equations of motion for ˆ ρ and ˆ y a do notinvolve µ . One can first solve these equations, and then µ can be found by integrating ˆ ρ :the normalization condition (cid:82) dx ρ ( x ) = 1 becomes (cid:90) d ˆ x ˆ ρ (ˆ x ) = 1 µ . (3.10)We now see that if we extremized (3.9) in the case where the monople R-charges were R [ T ( ± ], we could obtain the saddle point when they are R [ T ( ± ] ± δ ( ± through the trans-11ormation: ˆ ρ δ (ˆ x δ )ˆ x δ = ˆ ρ (ˆ x )ˆ x , x δ = 1ˆ x + δ (sgn ˆ x ) , ˆ y a,δ (ˆ x δ ) = ˆ y a (ˆ x ) , R [ T ( ± δ ] = R [ T ( ± ] ± δ ( ± . (3.11)Indeed, the equations of motion for ˆ ρ and ˆ y a are obtained by extremizing the expression inthe square brackets in (3.9), and this expression is invariant under (3.11). Given that ˆ ρ hascompact support, the transformations (3.11) make sense only when δ ( ± ) are small enough.For simplicity, from now on let’s restrict ourselves to the case δ (+1) = δ ( − = δ , eventhough one can make similar arguments for the case where δ (+1) and δ ( − are arbitrary orsatisfy a different relation. In [4], we showed that F = 4 πN / µ/
3, which implies thatVol m ( Y, δ ) = π µ δ = π (cid:90) d ˆ x δ ˆ ρ δ (ˆ x δ ) = π (cid:90) d ˆ x ˆ ρ (ˆ x )(1 + ˆ xδ ) . (3.12) δ We now prove the result (3.4). Let A be the chiral ring associated to the superconformalfield theory dual to AdS × Y in the Abelian case N = 1. A is also a vector space over C that is graded by the R-charge and monopole charge, meaning that one can define a basis ofoperators with well-defined R-charge and monopole charge. Let A m,r be the vector subspaceof elements of A with monopole charge m and R-charge r . We introduce the Hilbert-Poincar´eseries f ( t, u ) = (cid:88) m,r dim( A m,r ) t r u m . (3.13)Since the Abelian moduli space of the gauge theory is the Calabi-Yau cone over Y one canview the operators in the chiral ring as holomorphic functions on this cone. Martelli, Sparks,and Yau [24] show that Vol( Y, δ ) = π
48 lim t → (1 − t ) f ( t, t δ ) . (3.14)One can compute the Hilbert-Poincar´e series for Y in terms of ψ ( r, m ), the number ofoperators with R-charge at most r and monopole charge at most m . Approximating ψ by a12ontinuous function of homogeneous degree four, the definition (3.13) gives f ( t, u ) ≈ (cid:90) dr dm ψ (1 , ( r, m ) t r u m . (3.15)Since 1 − t ≈ − ln t for t ≈
1, we can use (3.14) and (3.15) to write Vol(
Y, δ ) asVol(
Y, δ ) ≈ π
48 (ln t ) (cid:90) dr dm ψ (1 , ( r, m ) t r + mδ = − π
48 (ln t ) (cid:90) dr dm ψ (2 , ( r, m ) t r + mδ = − π
48 (ln t ) (cid:90) dr d ˆ x rψ (2 , ( r, r ˆ x ) t r (1+ˆ xδ ) = π (cid:90) d ˆ x ψ (2 , ( r, r ˆ x ) /r (1 + ˆ xδ ) , (3.16)where in the second line we integrated by parts once, and in the third line we defined m = r ˆ x . For any gauge invariant operator X , we should have [25] R [ X ] = π Vol(Σ X ) / Y ), whereby Σ X we denoted the 5-d submanifold of Y defined by the equation X = 0. Using Vol( Y ) =( π / (cid:82) d ˆ x ˆ ρ (ˆ x ), one can rewrite this equation asVol(Σ X ) = π (cid:90) d ˆ x ˆ ρ (ˆ x ) R [ X ] . (3.17)For an operator X that is not gauge invariant, such as a bifundamental field that transformsin ( N a , N b ), R [ X ] is not invariant under baryonic symmetries (2.4), but R [ X ] + ˆ y a (ˆ x ) − ˆ y b (ˆ x )is. So we suspect thatVol(Σ X ) = π (cid:90) d ˆ x ˆ ρ (ˆ x )( R [ X ] + ˆ y a (ˆ x ) − ˆ y b (ˆ x )) . (3.18)We can think of this relation as a conjecture and prove the following result: If X is a chiraloperator transforming in ( N a , N b ), then for δ in a neighborhood of zero, letVol m (Σ X , δ ) = π (cid:90) d ˆ x δ ˆ ρ δ (ˆ x δ )( R [ X ] + ˆ y a,δ (ˆ x δ ) − ˆ y b,δ (ˆ x δ )) . (3.19)The following two statements are equivalent:13. The conjecture (1.4b) holds for the given R-charges R [ X ] and bare monopole charge∆ m .II. For any δ in a small enough neighborhood of zero, we haveVol m (Σ X , δ ) = Vol(Σ X , δ ) , (3.20)where the volume Vol(Σ X , δ ) is computed with the induced Sasakian metric on Y thatcorresponds to the matter R-charges R [ X ] and the bare monopole charge ∆ m + δ .The proof of this result is similar to that of the equivalence between (A) and (B) wediscussed above, so we skip most of the details. Using (3.11), one can check thatVol m (Σ X , δ ) = π (cid:90) d ˆ x ˆ ρ (ˆ x )( R [ X ] + ˆ y a (ˆ x ) − ˆ y b (ˆ x ))(1 + ˆ xδ ) . (3.21)Defining f X ( t, u ) to be the Hilbert-Poincar´e series for the ring of chiral operators obtainedfrom the chiral ring by setting X = 0, and using the Martelli, Sparks, and Yau result [24]Vol(Σ X , δ ) = π t → (1 − t ) f X ( t, t δ ) , (3.22)one can show as in section 3.3 thatVol(Σ X , δ ) = π (cid:90) d ˆ x lim r →∞ ψ (1 , X ( r, r ˆ x ) /r (1 + ˆ xδ ) . (3.23)Here, ψ X ( r, m ) denotes the number of chiral ring operators with X = 0, R-charge at most r , and monopole charge at most m , and can be approximated by a smooth function ofhomogeneous degree three. By an argument analogous to the one in section 3.1 it followsthat the statements (I) and (II) are equivalent. Note that gauge invariant operators in the quiver that have no monopole charge are con-structed from closed paths of bifundamental fields O α . A consequence of our conjecture(1.4b) is then that for a gauge invariant operator X = (cid:81) α O α with no monopole charge thefollowing sum vanishes: (cid:88) α (cid:20) ∂ ψ O α ∂r∂m − R [ O α ] ∂ ψ∂r ∂m (cid:21) = 0 . (3.24)14f course (cid:80) α R [ O α ] = R [ X ], and we can simplify this expression: (cid:88) α ∂ ψ O α ∂r∂m = R [ X ] ∂ ψ∂r ∂m . (3.25)We would like to show why (3.25) must hold from geometric considerations alone. The number of gauge invariant operators of fixed R-charge r and monopole charge m that do not contain the operator X is approximately ψ (1 , X ( r, m ) ≈ ψ (1 , ( r, m ) − ψ (1 , ( r − R [ x ] , m ) ≈ R [ X ] ψ (2 , ( r, m ) (3.26)when r (cid:29) R [ X ] is large.We can also use (3.22) to express the operator counts in terms of volumes. For coordinates x and y on a compact space, the volume of the set of points xy = 0 is the union of the setof points where x = 0 with the set of points where y = 0. That the volumes are additiveimplies the identity Vol(Σ X , δ ) = (cid:88) α Vol(Σ O α , δ ) . (3.27)From this identity at large r and the results in the earlier part of this section, we have ψ (1 , X ( r, m ) = (cid:88) α ψ (1 , O α ( r, m ) . (3.28)Combining (3.26) with (3.28) yields (3.25). N = 2 deformations of the necklace quivers and matrix model Our first field theory example consists of deformations of the necklace quiver gauge theorieswhose (undeformed) matrix models were also studied in [4, 5]. In N = 2 notation, the fieldcontent of the necklace quiver theories consists of d vector multiplets with Chern-Simonskinetic terms and coefficients k a , and chiral multiplets A a and B a connecting the gauge For a Calabi-Yau four fold, at large r and m we approximate ψ ( r, m ) and ψ O α ( r, m ) by homogeneouspolynomials of degree four and three respectively. d k k k k d − A d A A A B B B B d Figure 1: A necklace quiver gauge theory where the gauge sector consists of d U ( N ) gaugegroups with Chern-Simons coefficients k a and the matter content consists of the bifunda-mental fields A a and B a .groups into a necklace (see figure 1). The superpotential W = d (cid:88) a =1 k a tr( B a +1 A a +1 − A a B a ) (4.1)preserves N = 3 supersymmetry. For any given k a satisfying (cid:80) da =1 k a = 0, the field theoryis dual to AdS × Y where Y is a tri-Sasakian space, which is by definition the base of ahyperk¨ahler cone [26].While N = 3 SUSY restricts the R-charges of A a and B a to be 1 /
2, in this sectionwe examine what happens if we make more general R-charge assignments for the A a and B a fields that break N = 3 down to N = 2. These R-charge assignments are required topreserve the marginality of the superpotential (4.1). This condition implies that for genericvalues of the CS levels k a , namely if there are no cancellations between the various terms in(4.1), we must have R [ B a ] = 1 − R [ A a ]. The matrix model free energy functional is in this16ase ˜ F [ ρ, y a , µ ] = 2 πN / (cid:90) dx ρx d (cid:88) a =1 q a δy a + 2 π ∆ m N / (cid:90) dx ρx − πN / (cid:90) dx ρ d (cid:88) a =1 ( δy a − R [ A a ]) ( δy a + R [ B a ]) − πN / µ (cid:18)(cid:90) dx ρ − (cid:19) , (4.2)where δy a = y a − − y a , k a = q a +1 − q a . As per the discussion after eq. (2.3), the equations ofmotion for δy a following from (4.2) hold only when − R [ B a ] < δy a < R [ A a ]. It is possible tohave δy a = R [ A a ] or δy a = − R [ B a ], but in that case we should not impose the equation ofmotion for that particular δy a .Using the rescalings (3.2), one can write the solution of the equations of motion followingfrom (4.2) asˆ ρ (ˆ x ) = s L (ˆ x ) − s S (ˆ x ) , δ ˆ y a (ˆ x ) = R [ A a ] − R [ B a ]2 + 12 | s L (ˆ x ) + q a ˆ x | − | s S (ˆ x ) + q a ˆ x | s L (ˆ x ) − s S (ˆ x ) , (4.3)where s L (ˆ x ) ≥ s S (ˆ x ) are the two solutions of the equation s (ˆ x ) c + ˆ xc + d (cid:88) a =1 | s (ˆ x ) + ˆ xq a | = 2 , (4.4)with c ≡ d (cid:88) a =1 ( R [ A a ] − R [ B a ]) , c ≡ m + d (cid:88) a =1 q a ( R [ A a ] − R [ B a ]) . (4.5)The constraint imposed by varying ˜ F with respect to µ is (cid:82) d ˆ x ˆ ρ (ˆ x ) = 1 /µ .We have encountered a solution of this type in [4] in the case where R [ A a ] = R [ B a ] = 1 / m = 0. As in [4], one can think of eq. (4.4) as defining the boundary of a polygon P = (cid:40) (ˆ x, s ) ∈ R : sc + ˆ xc + d (cid:88) a =1 | s + ˆ xq a | ≤ (cid:41) . (4.6)The quantity ˆ ρ (ˆ x ) = s L (ˆ x ) − s S (ˆ x ) can then be interpreted as the thickness of a constant ˆ x slice P ˆ x through this polygon, ˆ ρ (ˆ x ) = Length( P ˆ x ). Consequently, (cid:82) d ˆ x ˆ ρ (ˆ x ) = Area( P ) andVol m ( Y ) = π µ = π
24 Area( P ) . (4.7)17ee appendix A for a proof that the N = 3 R-charge assignments minimize Vol m ( Y ) or,equivalently, maximize F . Just like ˆ ρ (ˆ x ), the quantities ˆ ρ (ˆ x ) δ ˆ y a (ˆ x ) can also be given geo-metrical interpretations:ˆ ρ (ˆ x )( δ ˆ y a (ˆ x ) + R [ B a ]) = Length ( P ˆ x ∩ { s + q a ˆ x ≥ } ) , ˆ ρ (ˆ x )( − δ ˆ y a (ˆ x ) + R [ A a ]) = Length ( P ˆ x ∩ { s + q a ˆ x ≤ } ) . (4.8)The equations above were written in a way that makes manifest the invariance underthe flat directions exhibited in (2.4). Indeed, while in writing the free energy functional(4.2) we assumed R [ A a ] and ∆ m to be independent, we see that the eigenvalue density ˆ ρ (ˆ x )and the quantities appearing on the LHS of (4.8) depend non-trivially only on the linearcombinations c and c that were defined in (4.5). These are the only linear combinationsof R [ A a ] and ∆ m that are invariant under all symmetries in (2.4). The reason why we wereable to find two such linear combinations at all is that the spaces Y have generically two U (1) isometries that commute with U (1) R . We now relate the matrix model quantities ˆ ρ (ˆ x ) and ˆ ρ (ˆ x ) δ ˆ y a from the previous section tonumbers of operators in the chiral ring of the gauge theory when N = 1. In [4] we providedsuch a relation in the case R [ A a ] = R [ B a ] = 1 / m = 0, and the argument presentedin that paper holds, with minor modifications, for the more general R-charge assignmentsconsidered in this paper. As explained in [4], gauge invariant operators can be constructedout of the bifundamental fields A a and B a and the diagonal monopole operators T ( m ) , andthey are O ( m, s, i, j ) = T ( m ) C mq + s C mq + s · · · C mq d + sd ( A B ) i ( A B ) j ,C mq a + sa ≡ A mq a + sa if mq a + s > B − mq a − sa if mq a + s < . (4.9)The labels m and s run over all integers, while i and j should be nonnegative integers.Let ψ ( r, m ) ( ψ ( r, m )) be the number of operators O ( m, s, i, j ) ( O ( m, s, , r and monopole charge at most m . In [4] we showed that at large r and m wehave ψ (2 , ( r, m ) ≈ ψ ( r, m ). This relation holds for the more general R-charge assignmentstoo because the only assumption needed to prove it was R [ A B ] = R [ A B ] = 1, which we18till assume. A simple computation yields R [ O ( m, s, , m ∆ m + (cid:88) k R [ C k ] | mq a + s | = 12 (cid:34) sc + mc + d (cid:88) a =1 | s + mq a | (cid:35) , (4.10)where c and c are as defined in (4.5). Using this formula one can check, as in [4], that ψ (2 , ( r, r ˆ x ) /r ≈ ψ (0 , ( r, r ˆ x ) /r is indeed given by the length of the slice P ˆ x through P . Wehave therefore verified explicitly eq. (1.4a) for the necklace quivers at non-critical R-charges.Let ψ X a ( r, m ) be the number of chiral operators with R-charge at most r and monopolecharge at most m that are nonzero when X a = 0. As in [4], we have that ψ (1 , X a ( r, m ) equalsthe number of operators of the form O ( m, s, ,
0) with R-charge at most r and monopolecharge at most m with the extra constraint that mq a + s ≤ X a = A a and mq a + s ≥ X a = B a . As argued in [4], these extra constraints imply that when r is large ψ (1 , X a ( r, r ˆ x ) /r is given by the length of the intersection between the slice P ˆ x and the half-plane s + q a ˆ x ≥ X a = B a or s + q a ˆ x ≤ X a = A a . Comparing with eq. (4.8) we see that the necklacequivers at arbitrary R-charges also obey our second conjecture (1.4b). The discussion in the previous two subsections can be generalized by including flavor fieldsthat interact with the existing matter fields through the superpotential δW ∼ d (cid:88) a =1 tr (cid:34) n a (cid:88) j =1 ˜ q ( a ) j A a q ( a ) j + m a (cid:88) j =1 Q ( a ) j B a ˜ Q ( a ) j (cid:35) . (4.11)Given that the A a transform in ( N a − , N a ) and the B a transform in the conjugate repre-sentation ( N a − , N a ), for eq. (4.11) to make sense we must take q ( a ) j , ˜ q ( a ) j , Q ( a ) j , and ˜ Q ( a ) j totransform in N a − , N a , N a − , and N a , respectively.We discussed a superpotential of this form at the end of section 2, where we found that theeffect of including the flavor fields was that the CS levels k a and ∆ m of the unflavored modelwere replaced by (sgn x ) g a [ T (sgn x ) ] and (sgn x ) R [ T (sgn x ) ], respectively. Eqs. (2.8) and (2.9)applied to our flavored necklace quivers give k a → (sgn x ) g a [ T (sgn x ) ] = k a + sgn x n a − m a − n a +1 + m a +1 ) , ∆ m → (sgn x ) R [ T (sgn x ) ] = ∆ m + sgn x (cid:88) a ( n a R [ A a ] + m a R [ B a ]) . (4.12)19rom k a = q a +1 − q a we further have q a → q a − sgn x n a − m a ) . (4.13)We believe that all the formulas presented in the previous two subsections continue to holdfor the flavored theory if one makes the above three replacements. In particular, the relationbetween the matrix model quantities and operator counting we conjectured in eq. (1.4)continues to hold, and the volume of the 7-d space Y is still proportional to the area of apolygon P of the type (4.6). N = 8 theory and its matrix model We broaden our scope of examples and verify (1.4) for maximally supersymmetric Yang-Millstheory to which we add flavor. The theory has one gauge group and three adjoint fields X i , i = 1 , , n + n + n pairs of fundamental fields through the superpotential W ∼ tr (cid:34) X [ X , X ] + n (cid:88) j =1 q (1) j X ˜ q (1) j + n (cid:88) j =1 q (2) j X ˜ q (2) j + n (cid:88) j =1 q (3) j X ˜ q (3) j (cid:35) . (4.14)The corresponding matrix model was solved in [9] in the large N limit. We review theirsolution for ρ ( x ). In the next subsection, we will compare ρ ( x ) with the distribution ofoperators in the chiral ring and show that (1.4a) holds. In this case, eq. (2.7) takes theform T (1) T ( − = X n X n X n [22, 23]. To keep the notation concise, we define ∆ i ≡ R [ X i ],∆ ≡ R [ T (1) ], and ˜∆ ≡ R [ T ( − ]. The matrix model free energy functional is then˜ F [ ρ ] = πN / (cid:20)(cid:90) dxρ (cid:16) ∆ ∆ ∆ ρ + (∆ + ˜∆) | x | + (∆ − ˜∆) x (cid:17) − µ (cid:18)(cid:90) dx ρ − (cid:19)(cid:21) . (4.15)As before, we define the hatted quantities (3.2). The eigenvalue density ˆ ρ (ˆ x ) isˆ ρ (ˆ x ) = − ˆ x ∆∆ ∆ ∆ if 0 < ˆ x < , x ˜∆∆ ∆ ∆ if − < ˆ x < , , (4.16)which agrees with (4.8) of [9]. 20 .5 Operator counting in flavored N = 8 theory The gauge-invariant operators built out of diagonal monopole operators and adjoint fieldsin this theory are tr[ T ( m ) X a X a X a ]. The R-charges of these operators are R [ T ( m ) X a X a X a ] = m ∆ + (cid:80) i =1 a i ∆ i m ≥ , − m ˜∆ + (cid:80) i =1 a i ∆ i m < . (4.17)Let ψ ( r, m ) be the number of operators with R-charge smaller than r and monopolecharge smaller than m . To match with ρ ( x ), we want to calculate ∂ ψ/∂r ∂m at large r . It is easiest to start by calculating the derivative ∂ψ/∂m which equals the number ofoperators with R-charge smaller than r and monopole charge equal to m . For m >
0, atlarge r the number of operators tr[ T ( m ) X a X a X a ] is approximately equal to the volumeof a tetrahedron with sides of length ( r − m ∆) / ∆ i ; similarly, for m <
0, the number ofoperators is equal to the volume of a tetrahedron with sides of length ( r + m ˜∆) / ∆ i . We thushave ∂ψ∂m = ( r − m ∆) ∆ ∆ if 0 < m < r ∆ , ( r + m ˜∆) ∆ ∆ if − r ˜∆ < m < , . (4.18)Taking two derivatives with respect to r , we find agreement with (4.16) and confirmation ofthe conjecture (1.4a). We presented flavored N = 8 in the main text because of its simplicity. One disadvantageof this example is that it possesses a single U ( N ) factor and so we could not computea δy and check (1.4b). To remedy this problem, in appendix B we consider two morecomplicated examples. The first of these is ABJM Chern-Simons theory (a theory with twogauge groups) [18] to which we add flavor. The second example has four gauge groups (seefigure 3). When a four-dimensional gauge theory has the field content of this second example,the Abelian moduli space is a Z × Z orbifold of C . Thus, with some abuse of notation,we refer to this second example as the Z × Z orbifold theory.The verification of (1.4) requires on the one hand calculating ρ ( x ) and δy ( x ) using thelarge N limit of the matrix model (2.3) and on the other counting operators in the chiral21ing. We have two methods at our disposal for this counting. One may count the operatorsdirectly as we did above. Because the moduli space is toric for these last three examples,the direct approach has some generic features which we review in appendix C. In section3, we presented an indirect counting method that involved calculating Vol( Y, δ ) (3.16) andVol(Σ X , δ ) (3.23) as a function of ρ ( x ) and δy ( x ). As noted in [9], the functional (2.3) does not appear to describe the large N limit of gaugetheories with chiral bifundamental fields. To derive (2.3), it was assumed that the long-rangeforces on the eigenvalues cancel. But for theories with chiral bifundamentals, there is no suchcancellation.The long-range forces at issue come from the interactions between the eigenvalues, bothwithin a vector multiplet and between vector multiplets connected by a bifundamental field X ab [9]: F ( a ) i, self = (cid:88) j (cid:54) = i coth π ( λ ( a ) i − λ ( a ) j ) ,F ( a,b ) i, inter = (cid:88) j (cid:34) R [ X ab ] − − i ( λ ( b ) i − λ ( a ) j )2 (cid:35) coth π (cid:16) λ ( b ) i − λ ( a ) j − i (1 − R [ X ab ]) (cid:17) ,F ( b,a ) i, inter = (cid:88) j (cid:34) R [ X ba ] − i ( λ ( b ) i − λ ( a ) j )2 (cid:35) coth π (cid:16) λ ( b ) i − λ ( a ) j + i (1 − R [ X ba ]) (cid:17) . (5.1)If (cid:12)(cid:12)(cid:12) λ ( a ) i − λ ( b ) j (cid:12)(cid:12)(cid:12) (cid:29)
1, then we may approximate coth x ≈ sgn Re x . The long-range forces arethe forces (5.1) with coth replaced by sgn Re. For theories with non-chiral bifundamentalsand equal ranks, the long-range forces cancel out when Re λ ( a ) i = Re λ ( b ) i for all i, a, b and(2.1) is satisfied. In general, the long-range forces on λ ( a ) i cancel out only when (cid:88) b ( R [ X ab ] − y b,j ) + (cid:88) b ( R [ X ba ] − − y b,j ) = − . (5.2)Thus the free energy functional (2.3) is correct for theories with chiral bifundamentals onlyif the y a ( x ) satisfy some constraints. 22 .2 Operator counting for the C / Z theory To investigate what the matrix model for a chiral theory should give in the large N limit, westudy the U ( N ) Chern-Simons theory described by the quiver in figure 2. Let the Chern-Simons coefficients be ( k , k , k ) such that k + k + k = 0. We will assume k > k < k <
0. The moduli space is a K¨ahler quotient of C with weights ( ( k + + k − ) , ( k + + k − ) , ( k + + k − ) , − k + , − k − ), where we define k − = k − k and k + = k − k .There is a superpotential of the form W ∼ tr [ (cid:15) ijk A ,k A ,j A ,i ] , (5.3)and a monopole relation T (1) T ( − = 1. We let R [ A ij, ] = ∆ x , R [ A ij, ] = ∆ y , R [ A ij, ] = ∆ z ,with ∆ x + ∆ y + ∆ z = 2 as any other choice of R-charges may be transformed into this choiceby a transformation of the form (2.4). We denote R [ T (1) ] = − R [ T ( − ] = ∆.The gauge invariant operators have the form T ( m ) 3 (cid:89) i =1 3 (cid:89) j =1 ( A i ( i +1) ,j ) n i ( i +1) ,j . (5.4)To be gauge invariant, for m ≥ (cid:80) j n ,j = mk + s , (cid:80) j n ,j = mk + mk + s and (cid:80) j n ,j = s and for m < (cid:80) j n ,j = mk + s , (cid:80) j n ,j = mk + mk + s , and (cid:80) j n ,j = s . Given the R-charge assignments, it is convenient tointroduce n j = (cid:80) i n i ( i +1) ,j . Each gauge invariant operator corresponds to a quadruple( n , n , n , m ) such that (cid:80) j n j = mk sgn( m ) + 3 s and m is bounded between − (cid:80) j n j /k − and (cid:80) j n j /k + .Given the description of the gauge invariant operators, it is now a straightforward taskto count them by either the direct method described in appendix C or the indirect methoddescribed in section 3. For ∆ x ≥ ∆ y ≥ ∆ z a piecewise expression for ˆ ρ (ˆ x ) is:ˆ ρ (ˆ x ) = , ˆ x ≤ − k − ∆ z − ∆ , k − ∆ z − ∆)ˆ x z (∆ x − ∆ z )(∆ y − ∆ z ) , − k − ∆ z − ∆ ≤ ˆ x ≤ − k − ∆ y − ∆ , (∆ x − ∆ y − ∆ z )(1 − ∆ˆ x ) − ∆ y ∆ z k − ˆ x x − ∆ y )(∆ x − ∆ z )∆ y ∆ z , − k − ∆ y − ∆ ≤ ˆ x ≤ − k − ∆ x − ∆ , − ∆ˆ x x ∆ y ∆ z , − k − ∆ x − ∆ ≤ ˆ x ≤ k + ∆ x +∆ , (∆ x − ∆ y − ∆ z )(1 − ∆ˆ x )+∆ y ∆ z k + ˆ x x − ∆ y )(∆ x − ∆ z )∆ y ∆ z , k + ∆ x +∆ ≤ ˆ x ≤ k + ∆ y +∆ , − ( k + ∆ z +∆)ˆ x z (∆ x − ∆ z )(∆ y − ∆ z ) , k + ∆ y +∆ ≤ ˆ x ≤ k + ∆ z +∆ , , k + ∆ z +∆ ≤ ˆ x . (5.5)23e note three odd things about (5.5): 1) If ∆ x = ∆ y = ∆ z , ˆ ρ has a delta function at − k − ∆ x − ∆ and k + ∆ x +∆ . 2) In contrast to nonchiral examples, ˆ ρ (ˆ x ) while still piecewise linearis no longer a convex function of ˆ x . 3) The matrix model (2.3) gives the same result for ˆ ρ inthe central region despite the fact that the long range forces do not cancel. (In other regionsand for δ ˆ y ab , the matrix model results are different.)Now we set A , to zero. The nonzero operators are those with no A i ( i +1) , fields. As apiecewise function:ˆ y (ˆ x ) − ˆ y (ˆ x ) = − ∆ z , − k − ∆ z − ∆ ≤ ˆ x ≤ − k − ∆ y − ∆ , ∆ y ∆ z (1+( k − ∆ x − ∆)ˆ x )(∆ x − ∆ y − ∆ z )(1 − ∆ˆ x ) − ∆ y ∆ z k − ˆ x , − k − ∆ y − ∆ ≤ ˆ x ≤ − k − ∆ x − ∆ , , − k − ∆ x − ∆ ≤ ˆ x ≤ k + ∆ x +∆ , ∆ y ∆ z (1 − ( k + ∆ x +∆)ˆ x )(∆ x − ∆ y − ∆ z )(1 − ∆ˆ x )+∆ y ∆ z k + ˆ x , k + ∆ x +∆ ≤ ˆ x ≤ k + ∆ y +∆ , − ∆ z , k + ∆ y +∆ ≤ ˆ x ≤ k + ∆ z +∆ . (5.6)Finally, we set A , to zero. The nonzero operators are those with no A i ( i +1) , ’s, andthose with m ≥ , n x + n y + n z = k + m . As a piecewise function:ˆ y (ˆ x ) − ˆ y (ˆ x ) = − ∆ z , − k − ∆ z − ∆ ≤ ˆ x ≤ − k − ∆ y − ∆ , ∆ y ∆ z (1+( k − ∆ x − ∆)ˆ x )(∆ x − ∆ y − ∆ z )(1 − ∆ˆ x ) − ∆ y ∆ z k − ˆ x , − k − ∆ y − ∆ ≤ ˆ x ≤ − k − ∆ x − ∆ , , − k − ∆ x − ∆ ≤ ˆ x ≤ k + ∆ x +∆ , − ∆ y ∆ z (1 − ( k + ∆ x +∆)ˆ x )(∆ x − ∆ y − ∆ z )(1 − ∆ˆ x )+∆ y ∆ z k + ˆ x , k + ∆ x +∆ ≤ ˆ x ≤ k + ∆ y +∆ , z , k + ∆ y +∆ ≤ ˆ x ≤ k + ∆ z +∆ . (5.7)The result for ˆ y (ˆ x ) − ˆ y (ˆ x ) follows by taking the difference of (5.6) and (5.7). We havechecked that the operator counts where we set each of the remaining seven bifundamentalfields to zero in turn yield the same results for the differences in the ˆ y ’s. M , , / Z k Consider the case where the internal space Y is M , , / Z k . It was proposed in [27–29] thatthe dual field theory is the one in figure 2 with CS levels k = 2 k and k = k = − k , so k + = k − = 3 k . As a function of the trial R-charges, the volume of Y isVol( Y ) = π (cid:90) d ˆ x ˆ ρ (ˆ x ) = 3 k π (∆ + 9 k (∆ x ∆ y + ∆ x ∆ z + ∆ y ∆ z )8 (9 k ∆ x − ∆ ) (cid:0) k ∆ y − ∆ (cid:1) (9 k ∆ z − ∆ ) . (5.8)24 k k A ,i A ,i A ,i Figure 2: The quiver for the C / Z theory. When the CS levels are (2 k, − k, − k ) this fieldtheory is believed to be dual to AdS × M , , / Z k .Under the constraint ∆ x +∆ y +∆ z = 2, this expression is maximized for ∆ x = ∆ y = ∆ z = 2 / π / (128 k ), which is the volume of M , , / Z k [25].For the critical R-charges, our predicted eigenvalue density isˆ ρ (ˆ x ) = 98 θ (cid:18) k − | ˆ x | (cid:19) + 932 k δ (cid:18) ˆ x + 12 k (cid:19) + 932 k δ (cid:18) ˆ x − k (cid:19) , ˆ ρ (ˆ x ) (ˆ y (ˆ x ) − ˆ y (ˆ x )) = − k δ (cid:18) ˆ x + 12 k (cid:19) − k δ (cid:18) ˆ x − k (cid:19) , ˆ ρ (ˆ x ) (ˆ y (ˆ x ) − ˆ y (ˆ x )) = − k δ (cid:18) ˆ x + 12 k (cid:19) + 38 k δ (cid:18) ˆ x − k (cid:19) . (5.9)The volumes of the five-cycles corresponding to the bifundamental fields areVol(Σ A ,a ) = π (cid:90) dx ˆ ρ (ˆ x ) (cid:18) ˆ y (ˆ x ) − ˆ y (ˆ x ) + 23 (cid:19) = 3 π k , Vol(Σ A ,a ) = π (cid:90) dx ˆ ρ (ˆ x ) (cid:18) ˆ y (ˆ x ) − ˆ y (ˆ x ) + 23 (cid:19) = 21 π k , Vol(Σ A ,a ) = π (cid:90) dx ˆ ρ (ˆ x ) (cid:18) ˆ y (ˆ x ) − ˆ y (ˆ x ) + 23 (cid:19) = 21 π k . (5.10)Let us understand how these volumes are related to the volumes of the divisors computedin [25]. The cone over M , , is a K¨ahler quotient of C by a U (1) that acts with weights(2 , , , − , −
3) on the coordinates ( u , u , u , v , v ) parameterizing C . The Z k orbifoldused to produce the quiver in figure 2 acts by the identification ( v , v ) ∼ ( v e πi/k , v e − πi/k )leaving the u i coordinates untouched. It is natural to identify A ,a with u a , A ,a with u a v ,and A ,a with u a v . Using the explicit metric on M , , the authors of [25] calculated the25olumes of the five-cycles corresponding to either u a = 0 or v b = 0 in M , , to beVol(Σ u a ) = 3 π , Vol(Σ v b ) = 9 π . (5.11)We see that these equations are consistent with (5.10): we have k Vol(Σ A ,a ) = Vol(Σ u a ) aswell as k Vol(Σ A ,a ) = Vol(Σ u a ) + Vol(Σ v ) and k Vol(Σ A ,a ) = Vol(Σ u a ) + Vol(Σ v ). Thefactor of k in these formulas comes from the fact that the cycles whose volumes are given in(5.10) belong to a Z k orbifold of M , , .For those interested in another simple example of a theory with chiral bifundamentalfields, we describe our predictions for a theory with the cone over Q , , as its Abelianmoduli space in appendix D. There is a difference between the matrix model and operator counting that manifests itselfin chiral theories. The matrix model depends explicitly on the bifundamental fields, and a δy saturates when it reaches minus the R-charge of a bifundamental field. In the absenceof flavors, the saturation of the δy is responsible for all of the corners in ρ and ρδy . In the C / Z example, ρ has a corner at ˆ x = k + ∆ x +∆ . We might expect that there exists somebifundamental field A ij,k so that δy + R [ A ij,k ] becomes zero at ˆ x = k + ∆ x +∆ , or equivalentlythat ψ (1 , A ij,k ( r, r ˆ x ) becomes zero at ˆ x = k + ∆ x +∆ . There is no such field. However, if we considerthe density ψ (1 , A , ,A , ,A , ) of operators when we set A , = A , = A , = 0, then thisdensity does become zero at ˆ x = k + ∆ x +∆ . So it appears to be important to allow arbitrarysets of bifundamental fields to be set to zero. A more geometric way of saying this is thatthe important objects in the operator counting formula are not the bifundamental fields butrather five-cycles in the Sasaki-Einstein manifold. In the C / Z theory, there seems to be nooperator constructed from bifundamental fields that corresponds to a five-brane wrappingthe cycle A , = A , = A , = 0. We might say that we are missing some operators. Wenote that the problem could be resolved if we added an operator A , /A , , since the cycle A , = 0 is the sum of the cycles A , = 0 and A , = A , = A , = 0. The problemnever arises in non-chiral non-flavored theories because these theories do have an operatorfor every cycle.The flavored N = 8 and flavored ABJM model also have missing operators. At x = 0, Unlike setting A , = A , = A , = 0 where there are no non-vanishing operators, when we set A , = A , = A , = 0, the number of non-zero operators ψ ( A , ,A , ,A , ) scales as r , indicating thepresence of a 5-cycle in the geometry. δy ab saturating at the R-charge of some bifundamental field X . Instead, the corner comes from the q fields. From theoperator counting perspective, this corner can be explained by the fact that ψ (1 , T becomeszero at ˆ x = 0. Acknowledgments
We would like to thank F. Benini, D. Jafferis and I. Klebanov for discussion. DG, CH,and SP were supported in part by the NSF under Grant No. PHY-0756966. DG and CHwere also supported in part by the US NSF under Grant No. PHY-0844827, and SP byPrinceton University through a Porter Ogden Jacobus Fellowship. CH also thanks the SloanFoundation for partial support. A F -Maximization for the necklace quivers We would like to show that to leading order in N the free energy of the necklace quiverswith arbitrary R-charges studied in section 4.1 is maximized when R [ A a ] = R [ B a ] = 1 / m = 0. We can only show this if the gauge groups are SU ( N ). In the U ( N ) d case,the symmetries (2.4) imply that the free energy has flat directions, but we can neverthelessshow that the free energy is maximized when the invariant combinations c and c defined ineq. (4.5) are set to zero. The critical R-charges correspond to the case where there is N = 3supersymmetry as opposed to just N = 2.The essential ingredient of the proof is the observation that the polygon P , which dependson (cid:126)c , is the polar dual of a polygon Q that does not depend on (cid:126)c about the unit circle centeredat ( − (cid:126)c/ (cid:126)β a = (1 , q a ) and (cid:126)c = ( c , c ) be vectors in R . The polygon Q is the Minkowskisum Q = (cid:40) d (cid:88) a =1 u a (cid:126)β a ∈ R : u a ∈ ( − / , / (cid:41) (A.1)of the vectors (cid:126)β a . Indeed, one can rewrite P as the intersection of half-planes P = (cid:40) (cid:126)t ∈ R : 12 (cid:126)t · (cid:126)c + d (cid:88) a =1 (cid:126)t · (cid:16) u a (cid:126)β a (cid:17) ≤ , ∀ u a ∈ ( − / , / (cid:41) . (A.2)The boundaries of these half-planes are precisely the polar duals of the points in Q about27he unit circle centered at ( − (cid:126)c/ (cid:126)v i be the vertices of Q ordered so that the line segment between (cid:126)v i and (cid:126)v i +1 is partof the boundary of Q . The line passing through (cid:126)v i and (cid:126)v i +1 is polar dual to a vertex (cid:126)w i,i +1 of P . Polar duality implies (cid:126)w i,i +1 · ( (cid:126)v i + (cid:126)c/
2) = (cid:126)w i,i +1 · ( (cid:126)v i +1 + (cid:126)c/
2) = 1, so (cid:126)w i,i +1 = ∗ ( (cid:126)v i +1 − (cid:126)v i )( ∗ ( (cid:126)v i +1 + (cid:126)c/ · ( (cid:126)v i + (cid:126)c/ , (A.3)where ∗ denotes the Hodge dual in R . By splitting P into triangles we can write the areaof P as Area( P ) = (cid:88) i Area( (cid:126)w i − ,i , (cid:126)w i,i +1 ,
0) = (cid:88) i | (cid:126)w i − ,i · ( ∗ (cid:126)w i,i +1 ) | , (A.4)where we denoted the area of a triangle whose vertices are given by the vectors (cid:126)α , (cid:126)β , and (cid:126)γ by Area( (cid:126)α, (cid:126)β, (cid:126)γ ). Using eq. (A.3), eq. (A.4) becomesArea( P ) = 14 (cid:88) i Area( (cid:126)v i − , (cid:126)v i , (cid:126)v i +1 )Area( (cid:126)v i , (cid:126)v i − , − (cid:126)c/
2) Area( (cid:126)v i +1 , (cid:126)v i , − (cid:126)c/ . (A.5)As long as − (cid:126)c/ Q , the Hessian matrix of each term in this sum, seenas a function of (cid:126)c , is positive definite, so Area( P ) is a convex function of (cid:126)c . (To compute theHessian it is easiest to work in a coordinate system where (cid:126)c is parametrized by the distancefrom two neighboring sides of the polygon to − (cid:126)c/ Q is symmetric about the origin as can be easily seen from eq. (A.1). Con-sequently, Area( P ) is an even function of (cid:126)c , and we have just shown that it is also convex.It follows that Area( P ) is minimized for (cid:126)c = 0. Equivalently, the free energy is maximizedwhen (cid:126)c = 0. Using the F -maximization conjecture of [6], we have thus shown that the correctR-charges in the necklace quivers with superpotential (4.1) satisfy c = c = 0. That’s all onecan say about the U ( N ) d theory. If the gauge groups are instead SU ( N ), the tracelessnessconstraints (cid:82) dx ρ ( x ) δy a ( x ) = 0 imply (when (cid:126)c = 0) (cid:90) dx ρ ( x ) δy a ( x ) = R [ B a ] − R [ A a ]2 = 0 , (A.6)so R [ A a ] = R [ B a ] = 1 /
2. From c = 0 we also get ∆ m = 0.28 Further examples
For notational convenience we set T (1) = T and T ( − = ˜ T . B.1 Flavored ABJM theory
We consider the flavored ABJM model with the superpotential W ∼ tr (cid:34) (cid:15) ij (cid:15) kl A i B k A j B l + n a (cid:88) j =1 q (1) j A ˜ q (1) j + n a (cid:88) j =1 q (2) j A ˜ q (2) j + n b (cid:88) j =1 Q (1) j B ˜ Q (1) j + n b (cid:88) j =1 Q (2) j B ˜ Q (2) j (cid:35) . (B.1)When N = 1, the superpotential is supplemented by the relation (2.7) which in this case is T ˜ T = A n a A n a B n b B n b [22, 23]. The corresponding matrix model was solved in the large N limit in [9]. Our strategy is the same as for the flavored N = 8 theory. In this section,we will review the solution for ρ ( x ) and δy ≡ y − y . In the next section, we will comparethese results with the distribution of operators in the chiral ring.We define R [ A i ] ≡ ∆ A i , R [ B i ] ≡ ∆ B i , ∆ ≡ R [ T ], and ˜∆ ≡ R [ ˜ T ]. Without loss ofgenerality, we will assume that ∆ A < ∆ A and ∆ B < ∆ B . To keep the notation concise,we also define k ± ≡ k ±
12 ( n a + n a − n b − n b ) , ∆ ≡ ∆ A ∆ A − ∆ B ∆ B , ∆ ≡ ∆ A ∆ A (∆ B + ∆ B ) + ∆ B ∆ B (∆ A + ∆ A ) . (B.2)Taking the marginality constraints on the R-charges into account, in the large N limit,the matrix model free energy functional is˜ F [ ρ, δy ]2 πN / = (cid:90) dx ρ (cid:34)
12 ( k + + k − ) x δy − ρ (cid:18) ( δy ) + ∆ δy −
12 ∆ (cid:19) + 12 (∆ − ˜∆) x + 12 | x | (cid:16) ∆ + ˜∆ + ( k + − k − ) δy (cid:17)(cid:35) − µ (cid:18)(cid:90) dx ρ − (cid:19) . (B.3)29he eigenvalue density has four regions: − R [ ˜ T A k − ] < ˆ x < − R [ ˜ T A k − ] : ˆ ρ = 1 + ˆ xR [ ˜ T A k − ]∆ + 2∆ A ∆ − A , δ ˆ y = − ∆ A ; (B.4) − R [ ˜ T A k − ] < ˆ x < ρ = 2 + 2ˆ x ˜∆ + ˆ xk − ∆ ∆ + 2∆ ,δ ˆ y = k − ˆ x ∆ − (1 + ˆ x ˜∆)∆ x ˜∆ + ˆ xk − ∆ ; (B.5)0 < ˆ x < R [ T B k + ] : ˆ ρ = 2 − x ∆ + ˆ xk + ∆ ∆ + 2∆ ,δ ˆ y = k + ˆ x ∆ − (1 − ˆ x ∆)∆ − x ∆ + ˆ xk + ∆ ; (B.6)1 R [ T B k + ] < ˆ x < R [ T B k + ] : ˆ ρ = 1 − ˆ xR [ T B k + ]∆ − B ∆ − B , δ ˆ y = ∆ B ; (B.7)As in (3.2), we have introduced the rescaled variables x = ˆ xµ and ρ ( x ) = ˆ ρ (ˆ x ) µ . Operator counting
There are operators containing ˜ T − m for m < T m for m >
0. Theytake the form T m A α A α B β B β and ˜ T − m A α A α B β B β , where gauge invariance demands α + α − β − β = − mk ± . If we wanted to count operators that don’t vanish when, forexample, A = 0, then we just set α = 0.We counted the operators using a slightly modified version of the method outlined inappendix C. Having written the operators in terms of both T and ˜ T , it is simpler to use twodifferent coordinate systems on the cone C , one when m > m <
0. Thecoordinate systems are related by (2.7). The operator counts reproduce (B.4), (B.5), (B.6),and (B.7) via our conjecture (1.4).Here are some of the details for the calculation of ˆ ρ (ˆ x ) when m >
0. The density ofoperators is given by ∂ ψ∂r∂m = (cid:90) dα dα dβ dβ δ ( α + α − β − β + mk + ) × δ ( r − m ∆ − α ∆ A − α ∆ A − β ∆ B − β ∆ B ) . (B.8)This integral gives the area of a slice of a tetrahedron. The slice is either a triangle or aquadrilateral (which may be regarded as a triangle with another triangle cut out). We find30 k A k k A A A A A A A A A A A Figure 3: The quiver for C / ( Z × Z ). There are four U ( N ) gauge groups with Chern-Simons coefficients k a . The matter content consists of the 12 bifundamental fields A ab for a (cid:54) = b , transforming under the fundamental of the b th gauge group and the antifundamentalof the a th gauge group.for m ≥ ∂ ψ∂r∂m = (cid:88) j =1 ( r − mR [ T B k + j ]) θ ( r − mR [ T B k + j ])2(∆ − B j ∆ − B j ) . (B.9)Taking a derivative of this expression with respect to r yields (B.6) and (B.7). B.2 C / ( Z × Z ) theory Let’s examine the field theory in figure 3. It has four gauge groups with CS levels k a , a = 1 , . . . ,
4, and twelve bifundamental fields A ab transforming in ( N a , N b ), one for everyordered pair ( a, b ) with a (cid:54) = b . The superpotential is W = tr (cid:34) (cid:88) a =1 (cid:15) abcd A db A cd A bc (cid:35) . (B.10)The superpotential relations are supplemented by the monopole OPE (2.7) T ˜ T = 1. Wedefine R [ A ab ] ≡ R ab and R [ T ] = − R [ ˜ T ] ≡ ∆.The superpotential contains eight distinct terms that impose the relations R ab + R bc + R ca = 2 for any triplet ( a, b, c ) of pairwise distinct gauge groups. These eight equationsimply the long-range force cancellation (2.1). Only seven of these equations are linearlyindependent, leaving five independent R-charges out of the twelve R ab .Even though for given k a the matrix model depends on 6 R-charges (∆ and the five31inearly independent R ab ), the dependence on three of these parameters is trivial because ofthe flat directions (2.4). We can use these symmetries to reduce the number of independentR-charges to three: ∆ x , ∆ y and ∆ where we pick R = R = R = R = ∆ x ,R = R = R = R = ∆ y ,R = R = R = R = 2 − ∆ x − ∆ y ≡ ∆ z . (B.11)The matrix model is then F [ ρ, y a ] = 2 πN / (cid:90) dx ρx d (cid:88) a =1 k a y a + 2 π ∆ N / (cid:90) dx ρx + πN / (cid:90) dx ρ (cid:88) ( a,b,c ) ( y b − y a + R ab ) ( y c − y b + R bc ) ( y a − y c + R ca ) . (B.12)For simplicity, let’s focus on the case k = − k = k = − k = k > y ≥ ∆ x .The saddle point eigenvalue distribution splits into three regions where ρ is linear:1∆ − k ∆ x < ˆ x < − k ∆ y : ˆ ρ (ˆ x ) = 1 − ˆ x (∆ − k ∆ x )4∆ x (∆ y − ∆ x )∆ z , ˆ y − ˆ y = − ∆ x , − k ∆ y < ˆ x <
1∆ + 2 k ∆ y : ˆ ρ (ˆ x ) = 1 − ˆ x ∆4∆ x ∆ y ∆ z , ˆ y − ˆ y = 2 k ˆ x ∆ x ∆ y − ˆ x ∆ ,
1∆ + 2 k ∆ y < ˆ x <
1∆ + 2 k ∆ x : ˆ ρ (ˆ x ) = 1 − ˆ x (∆ + 2 k ∆ x )4∆ x (∆ y − ∆ x )∆ z , ˆ y − ˆ y = ∆ x . (B.13)In all three regions, ˆ y = ˆ y and ˆ y = ˆ y . Operator counting
Without the monopole operators, the ring of functions A ab modulo superpotential relationsis the ring of functions on C / ( Z × Z ). This ring consists of polynomials in x, y, z with theconstraint that the numbers of x, y, z in each term must be either all even or all odd. We call A , A , A , A “ x fields”, A , A , A , A “ y fields”, and A , A , A , A “ z fields”.We can get a gauge invariant operator by taking a combination of two x fields (e.g. A A ),two y fields ( A A ), two z fields ( A A ), or one of each type of field ( A A A ). Thegauge invariant operators with m = 0 are those with an even number of each of x, y, z , oran odd number of each of x, y, z .Adding back the monopole operators yields a ring of functions on a four-dimensional cone.32) C x n y nA B b) D x nn y A B C
Figure 4: The area of the polygonal regions
ABC and
ABCD is proportional to ∂ ψ/∂r∂m for the C / Z × Z quiver: a) r − ∆ m − k | m | ∆ y <
0; b) r − ∆ m − k | m | ∆ y > , − , , −
1) from T can be cancelled out by two x ’s ( A A ) or two y ’s ( A A ), but not by z ’s. So, if we have an operator of the schematic form T m x n x y n y z n z for m > T − m x n x y n y z n z for m <
0, then we have the constraint n x + n y ≥ | m | . Theoperator density is then ∂ ψ∂r∂m = 14 (cid:90) dn x dn y dn z θ ( n x + n y − k | m | ) δ ( r − ∆ x n x − ∆ y n y − ∆ z n z − ∆ m ) . (B.14)The factor of comes from the constraint that the numbers of x, y, z must be all even or allodd.Performing the integral over n z introduces an overall factor of 1 / ∆ z . The remainingintegral reduces to the area of a polygonal region satisfying the constraints n y > n x > n x + n y > k | m | , and ∆ x n x + ∆ y n y < r − ∆ m . For small | m | , the polygonal region isa quadrilateral while for large | m | , the region is a triangle (see figure 4). Assuming that∆ y > ∆ x , we find ∂ ψ∂r∂m = z ( r − ∆ m − k | m | ∆ x ) ∆ x (∆ y − ∆ x ) if r − ∆ m − k | m | ∆ y < , z (cid:104) ( r − ∆ m ) ∆ x ∆ y − (2 k | m | ) (cid:105) if r − ∆ m − k | m | ∆ y > . (B.15)Taking an additional derivative with respect to r , we can easily check that this formulaagrees with (B.13).Now, in order to compute ˆ y (ˆ x ) − ˆ y (ˆ x ), we count gauge invariant operators with A set to zero. Because of the superpotential relations, all operators with a z are set to zero.The factor of 1/4 remains the same because now we may only consider operators with evennumbers of x and y fields. The expression for ∂ψ /∂m is given by the area of the same33olygonal region that governs ∂ ψ/∂r∂m , but we lose the factor of ∆ z because we drop theintegral over n z : ∂ ψ ∂r∂m = ∆ z ∂ ψ∂r ∂m . (B.16)Therefore, we have ˆ ρ (ˆ x )(∆ z + ˆ y (ˆ x ) − ˆ y (ˆ x )) = ˆ ρ (ˆ x )∆ z , and hence ˆ y (ˆ x ) = ˆ y (ˆ x ). A similarcalculation shows ˆ y (ˆ x ) = ˆ y (ˆ x ).Finally we count the operators with A set to zero. Most operators with an x willbecome zero. However, fields containing only T , A , A , A , A , and the z fields are notset to zero by the superpotential relations. So the nonzero fields are those with n x = 0 andan even number of y and z fields, or m ≥ n x + n y = 2 km with an even number of z fields. After a little work, we find ∂ ψ ∂r∂m = m/r < − (2 k ∆ y − ∆) − r − ∆ m +2 km ∆ y y ∆ z if − (2 k ∆ y − ∆) − < m/r < (2 k ∆ y + ∆) − r − ∆ m − km ∆ x z (∆ y − ∆ x ) if m/r > (2 k ∆ y + ∆) − (B.17)This result matches ˆ y (ˆ x ) − ˆ y (ˆ x ) computed from (B.13). C Toric varieties in general
By toric moduli space we mean more specifically that the moduli space for the Abelian gaugetheory is an eight-dimensional toric Calabi-Yau cone V . That V is toric means it is a T torus fibration over a four-dimensional rational polyhedral cone C . This polyhedral cone isthe set of points satisfying C = { y ∈ R : y · v a ≥ } , (C.1)where v a ∈ Z , a = 1 , . . . , n , are inward pointing vectors normal to the faces F a of the cone: F a = { y ∈ C : y · v a = 0 } . (C.2)The fact that V is Calabi-Yau implies that the end-points of the vectors v a lie in a commonhyperplane R .One convenient aspect of this construction is that lattice points in C correspond tooperators in the chiral ring of the Chern-Simons theory. The coordinates of a lattice pointare the U (1) global charges of the operator. The vector b that measures the R-charge isoften called the Reeb vector where the R-charge is then r = y · b . The vectors v a correspond34o other global charges, q a = y · v a , and we can introduce additional charges as well. In thegauge theories considered in this paper, the monopole charge m played an important role.Let us introduce t as the vector that measures monopole charge.We introduced previously the function ψ ( r, m ) as the number of operators with R-chargeless than r and monopole charge less than m . From the toric perspective, this function inthe large r and m limit is the volume of a four-dimensional polytope: C r,m = C ∩ { y · b ≤ r } ∩ { y · t ≤ m } , (C.3)where ψ ( r, m ) = Vol( C r,m ).We would like to understand geometrically how to compute derivatives of ψ ( r, m ). Thevalue of ψ ( r, m ) is a four-dimensional integral we can write as ψ ( r, m ) = (cid:90) C r,m d y . (C.4)To take a derivative of ψ with respect to r , we can rotate the coordinate system so that oneof the y ’s points in the direction of b and replace d y with d y dr/ | b | where | b | is the Jacobianfactor from the change of variables. The derivative is then related to the three-dimensionalvolume of the polyhedron D r,m = C ∩ { y · b = r } ∩ { y · t ≤ m } (C.5)where ∂ψ/∂r = Vol( D r,m ) / | b | . Similarly, we can visualize ∂ ψ/∂r∂m as the area of a two-dimensional polygon P r,m : P r,m = C ∩ { y · b = r } ∩ { y · t = m } . (C.6)Now we rotate our coordinate system so that two of the y ’s lie in the plane spanned by b and t . The Jacobian factor is | t ∧ b | = (cid:112) t b − ( t · b ) . Geometrically, the second partial is ∂ ψ∂r∂m = Area( P r,m ) | t ∧ b | . (C.7)The function ψ X ( r, m ) has a toric interpretation as well. In the examples we considered, This last expression may seem strange because the right hand side seems to depend on a metric whilethe left hand side depends only on a volume form on C . Interpreting Vol( D r,m ) as a three form instead of anumber, we could rewrite this expression in a manifestly metric independent way: ( ∂ψ/∂r ) t = (cid:63) Vol( D r,m ). corresponds to an integer linear combination of the v a . Let us consider the simple casewhere X a corresponds to a single v a . Operators with no X a are contained in the face F a ⊂ C .This fact suggests a relation between ψ X a ( r, m ) and a generalization of ψ ( r, m ) involving athird charge q a , ψ ( r, m, q a ). In particular, it is true that ψ X a ( r, m ) = ψ (0 , , ( r, m, . (C.8)Operators with no X a and fixed m and r lie along a line L a,m,r ⊂ F a : L a,m,r = F a ∩ { y · b = r } ∩ { y · t = m } . (C.9)Generalizing the argument used to derive (C.7) to one more charge, we find ∂ψ X a ∂r∂m = ψ (1 , , ( r, m,
0) = Length( L a,m,r ) | t ∧ b ∧ v a | . (C.10)Eqs. (C.7) and (C.10) provide a convenient starting point for counting chiral operators inthe examples in the text. D The Cone over Q , , / Z k As another example with chiral bifundamental fields, we can examine the square quiver infigure 5 with CS levels ( k, k, − k, − k ) and matter fields A i , B i , C i , and D i , with i = 1 , W ∼ tr (cid:2) (cid:15) ij (cid:15) kl D i C k B j A l (cid:3) (D.1)this quiver is thought to be dual to AdS × Q , , / Z k [29, 30]. The quiver has two flavor SU (2) symmetries, one under which A i and C i transform as doublets, and one under which B i and D i transform as doublets, so one expects the R-charges of the fields belonging to thesame edge of the quiver to be equal when F is maximized. Using the flat directions (2.4)and taking into account the marginality of the superpotential (D.1), one can then set theR-charges of all the bifundamental fields equal to 1 / m = 0. With this choice one can36 − kk − kD i B i A i C i Figure 5: Quiver gauge theory believed to be dual to
AdS × Q , , / Z k .go through the operator counting exercise in the Abelian theory and predict thatˆ ρ (ˆ x ) = θ (cid:18) k − | ˆ x | (cid:19) + 14 k δ (cid:18) k + ˆ x (cid:19) + 14 k δ (cid:18) k − ˆ x (cid:19) , ˆ ρ (ˆ x ) (ˆ y (ˆ x ) − ˆ y (ˆ x )) = − k δ (cid:18) k + ˆ x (cid:19) − k δ (cid:18) k − ˆ x (cid:19) , ˆ ρ (ˆ x ) (ˆ y (ˆ x ) − ˆ y (ˆ x )) = 38 k δ (cid:18) k + ˆ x (cid:19) − k δ (cid:18) k − ˆ x (cid:19) , ˆ ρ (ˆ x ) (ˆ y (ˆ x ) − ˆ y (ˆ x )) = − k δ (cid:18) k + ˆ x (cid:19) − k δ (cid:18) k − ˆ x (cid:19) . (D.2)As a consistency check, one can compute the volumesVol( Y ) = π (cid:90) d ˆ x ˆ ρ (ˆ x ) = π k , Vol(Σ A i ) = π (cid:90) d ˆ x ˆ ρ (ˆ x ) (cid:18) ˆ y (ˆ x ) − ˆ y (ˆ x ) + 12 (cid:19) = π k , Vol(Σ B i ) = π (cid:90) d ˆ x ˆ ρ (ˆ x ) (cid:18) ˆ y (ˆ x ) − ˆ y (ˆ x ) + 12 (cid:19) = π k , Vol(Σ C i ) = π (cid:90) d ˆ x ˆ ρ (ˆ x ) (cid:18) ˆ y (ˆ x ) − ˆ y (ˆ x ) + 12 (cid:19) = π k , Vol(Σ D i ) = π (cid:90) d ˆ x ˆ ρ (ˆ x ) (cid:18) ˆ y (ˆ x ) − ˆ y (ˆ x ) + 12 (cid:19) = π k . (D.3)Since Vol( Q , , ) = π /
16 [25], we see that Vol( Y ) matches that of a Z k orbifold of Q , , . Asfor M , , , we can relate the volumes of the five-cycles in (D.3) to those computed in [25].The cone over Q , , is a U (1) K¨ahler quotient of C with weights (1 , , − , − , ,
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