Strings on conifolds from strong coupling dynamics: quantitative results
aa r X i v : . [ h e p - t h ] N ov NSF-KITP-07-193NI-07076arXiv:0711.3026 [hep-th]
Strings on conifolds from strong couplingdynamics: quantitative results
David E. Berenstein ♯ and Sean A. Hartnoll ♭♯ Department of Physics, University of CaliforniaSanta Barbara, CA 93106-9530, USA ♯ Isaac Newton Institute for Mathematical SciencesCambridge CB3 0EH, UK ♭ KITP, University of CaliforniaSanta Barbara, CA 93106-4030, USA [email protected], [email protected]
Abstract
Three quantitative features of string theory on
AdS × X , for any (quasi)regular Sasaki-Einstein X , are recovered exactly from an expansion of field theory at strong couplingaround configurations in the moduli space of vacua. These configurations can be thoughtof as a generalized matrix model of (local) commuting matrices. First, we reproducethe spectrum of scalar Kaluza-Klein modes on X . Secondly, we recover the precisespectrum of BMN string states, including a nontrivial dependence on the volume of X .Finally, we show how the radial direction in global AdS emerges universally in thesetheories by exhibiting states dual to AdS giant gravitons. Introduction
The AdS/CFT correspondence [1] provides, in principle, a nonperturbative approach toquantum gravity in asymptotically Anti-de Sitter space. A traditionally thorny issue inquantum gravity is the emergence of spacetime and gravitons in a semiclassical limit. InAdS/CFT, addressing this question requires us to directly tackle the dual strongly coupledconformal field theory, in the large N limit. This is a different sort to problem to muchwork that has been done in AdS/CFT, in which protected quantities, or integrable sectorsof the theory, are computed at weak and strong coupling and compared directly.A program aimed at understanding the emergence of semiclassical quantum gravityfrom field theory was initiated in [2]. The starting point is a guess concerning the effective,semiclassical, degrees of freedom which characterize the ground state and dominate the lowenergy physics of the strongly coupled theory, together with a proposal for their dynamics.We will review aspects of this proposal below. Using this effective low energy theory,various non protected quantities were computed and successfully compared with the dualstring theory [3, 4, 5]. Furthermore, the proposal was extended from the original case of N = 4 Super Yang-Mills theory to orbifolds of this theory in [6, 7].It was recently argued [8] that the original proposal, which was for the N = 4 theory, canbe generalized to a large class of conformal field theories with only N = 1 supersymmetry.In particular to the theories arising on N D3 branes at the tip of a Calabi-Yau cone. Thisis a substantial generalization, as there are many such theories. In fact, these theories arein one to one correspondence with the space of five dimensional Sasaki-Einstein metrics [9].In this paper we will use and extend the recent proposal [8] to derive various quantitiesin the strongly coupled N = 1 theories. These will be non-BPS quantities and they willreproduce in detail the dual, spacetime, AdS gravity results. We start in sections 3 and4 by obtaining the ground state of the effective theory and showing that it describes theemergence of the dual Sasaki-Einstein geometry. In sections 5, 6 and 7 we study fluctuationsabout the ground state, reproducing the spacetime spectrum of scalar Kaluza-Klein har-monics and of BMN string states. Finally, in section 8 we show, by considering excitationsdual to giant gravitons, that the radial direction of AdS emerges universally, i.e. orthog-onally to and independently of the internal Sasaki-Einstein manifold. In the concludingdiscussion we emphasize the many computations that remain to be done in order to fleshout this framework in detail. 1 Summary of the computational framework
The upshot of the detailed arguments in [8] will now be summarized, together with somenew statements which we will expand upon in later sections. The objective is to describethe low energy physics of the strongly coupled superconformal theories arising on N D3branes at the tip of a Calabi-Yau cone. The field theory is on a spatial S and hence dualto global AdS space. • The degrees of freedom which dominate the large N low energy dynamics are configu-rations of scalar fields that explore the moduli space of vacua of the field theory. Thescalar fields are uniform on the S , that is, only the s-wave modes are excited. Locallyon the moduli space this is very similar to N = 4 SYM, where the configurations aregiven by six commuting N × N matrices. Thus in the first instance we have integratedout the higher harmonics on the spatial S , all the gauge fields and fermions, and allthe (generalized) off diagonal modes. • The N eigenvalues of these matrices, { x i } , are valued on a Calabi-Yau cone over aSasaki-Einstein manifold X . This would be the moduli space of the theory on R .Placing the theory on S lifts the moduli space. Firstly because of the conformalcoupling mass term. Secondly because there is an enhanced symmetry U (1) → U (2) when two eigenvalues coincide, the measure terms arising from this degenerationinduces a repulsion between eigenvalues. The competition between these two effectsis captured by the Hamiltonian H = X i (cid:20) − µ ∇ i · ( µ ∇ i ) + K i (cid:21) . (1)This expression is a sum of single particle Hamiltonians, labeled by the subscript i ,except for the measure factor µ which depends on the locations of all the eigenvalues.Here K is the K¨ahler potential of the Calabi-Yau. • The measure factor µ requires an inspired guess. In the N = 4 theory we haveaccess to a weakly coupled regime in which the measure factor can be directly de-termined [2]. This is also possible in the case of orbifolded theories [6, 7]. One canthen use nonrenormalization theorems for the BPS sector of the theory to argue thatthe expression remains valid at strong coupling. The conjectured form we use here,generalizing a property of the measure in the N = 4 case, is that µ = e − P i = j s ij , (2)2here s ij is the Green’s function of a sixth order differential operator on the Calabi-Yau cone −∇ s ( x, x ′ ) = 64 π δ (6) ( x, x ′ ) . (3)As we will discuss below, this expression has the virtue of automatically localizing thelarge N eigenvalue distribution on a hypersurface in the Calabi-Yau cone and thusleading to an emergent geometry. • Given the Hamiltonian (1), one can find the ground state. We do this in section 3below. The answer is simply ψ = e − P i K i . (4)In section 4 below we show that in this state in the large N limit the eigenvalues forman X at fixed radius r in the cone, which we compute. This is to be interpreted asthe X of the dual geometry, which has emerged from the matrix quantum mechanics. • Given the ground state (4), one can find the spectrum of low lying excitations. Thereare three types of excitations to consider. The first are those given by operators madefrom the six matrices that appear in the matrix quantum mechanics. The energies ofthese states are given by the spectrum of the Hamiltonian (1). An important set ofeigenstates, that we will consider, are of the form ψ = ψ Tr f ( x ) , (5)where f ( x ) is some function of the six matrices that has polynomial growth.Secondly, there are excitations of the off diagonal modes of the six matrices, whichcommute in the ground state. These require additional input. It was argued in [8]that the physics of off diagonal modes connecting nearby eigenvalues is the same asthat of the N = 4 theory, with an effective, N = 4 coupling g eff. . In particular, thisimplies that the energy of the mode connecting nearby eigenvalues x i and x j is [2, 3] E ij = s g π | x i − x j | , (6)where the distance is given by the metric on the Calabi-Yau cone. We require moreoverthat g N is large in the sense of ’t Hooft.Thirdly, there are excitations of the fields that have been set to zero in the quantummechanics: the higher harmonics of fields on S , the gauge fields and the fermions.These modes remain largely unexplored, although see [4, 10].3ith this framework, the strategy for computing quantities is as follows. Firstly wecompute the ground state wavefunction of the Hamiltonian. We can then compute theenergies of excitations about the ground state. These will not in general be BPS. We showthat the spectrum of various excitations matches that computed in supergravity and stringtheory, providing evidence for the calculational recipe just presented. The eigenvalue dynamics takes place on the six dimensional cone over a five dimensionalcompact manifold X ds = dr + r ds . (7)Denote the coordinates on the five dimensional manifold by θ . As we have mentioned, animportant ingredient for writing down the Hamiltonian for these eigenvalues is the Green’sfunction on the cone satisfying −∇ s ( r, r ′ , θ, θ ′ ) = − (cid:18) r ddr r ddr + 1 r ∇ (cid:19) s ( r, r ′ , θ, θ ′ ) = 64 π δ (6) ( r, r ′ , θ, θ ′ ) . (8)This Green’s function appears in the measure that is necessary to write the Hamiltonian asa differential operator. See equations (1) and (2) above. We will now motivate the use ofthis Green’s function.In the case of N = 4 SYM the measure arising in going to an eigenvalue description canbe calculated, and is given by a generalized Vandermonde determinant µ = Y i 3. Let K be the K¨ahler potential of the Calabi-Yau.The conjectured Hamiltonian [8] is H = X i − g a ¯ b ( z i , ¯ z i )2 µ h ∇ z ai (cid:16) µ ∇ ¯ z ¯ bi (cid:17) + ∇ ¯ z ¯ bi (cid:0) µ ∇ z ai (cid:1)i + K ( z i , ¯ z i ) ! , (12)where the measure factor is µ = e − P i = j s ij . (13)Here s ij = s ( z i , ¯ z i , z j , ¯ z j ) is the Green’s function. We have suppressed the a index in places.The ground state wavefunction for the Hamiltonian (12) will now be shown to be ψ = e − P i K i , (14)where K i = K ( z i , ¯ z i ). Acting on this state with the Hamiltonian (12) gives Hψ = X j K j + 3 − g a ¯ bj h ( ∇ z aj K j ) ∇ ¯ z ¯ bj + ( ∇ ¯ z ¯ bj K j ) ∇ z aj i ( K j + 2 X k = j s kj ) ψ . (15)Now, for an arbitrary Calabi-Yau cone with metric (7) we have that K = r . (16)This can be derived from a short argument starting with the observation that the K¨ahlerform is homogeneous with degree two in r , see e.g. [14]. It follows that the vector appearingin (15) is the Euler vector of the cone g a ¯ bj h ( ∇ z aj K j ) ∇ ¯ z ¯ bj + ( ∇ ¯ z ¯ bj K j ) ∇ z aj i = r ∂∂r . (17)The scaling (11) then implies that X i r i ∂∂r i X j = i s ij = − N ( N − π Vol( X ) . (18)6utting the above statements together we obtain Hψ = (cid:18) N + N ( N − π Vol( X ) (cid:19) ψ ≡ E ψ . (19)Thus ψ is an eigenstate as claimed. The lack of dependence on the angular coordinates θ suggests that it is the ground state. The two key ingredients here were the relation betweenthe K¨ahler potential and the Euler vector (17), and the scaling behaviour of the Green’sfunction (11). Any scaling function would have given the same results. In the large N limit, the ground state wavefunction (14) describes an emergent semiclassicalgeometry [2]. This occurs because a specific configuration of eigenvalues dominates thematrix integral.The probability of the eigenvalues being in some particular distribution is given bythe square of the wavefunction multiplied by the measure factor (13) needed to make theHamiltonian (12) self-adjoint. That is µ | ψ | = e − P i r i − P j = i s ij ≡ e − S . (20)In the large N limit, we expect a particular configuration to dominate. This will be givenby minimizing the effective action S = Z d xρ ( x ) r x + Z d xd yρ ( x ) ρ ( y ) s ( x, y ) , (21)where we have introduced the large N eigenvalue density, ρ ( x ), which satisfies Z d xρ ( x ) = N . (22)The notation we are using here is that x runs over the six coordinates on the cone, whichwe denote r x and θ x .The saddle point equations are r x = − Z dr y dθ y r y q g ( θ y ) ρ ( r y , θ y ) ∂s ( r x , r y , θ x , θ y ) ∂r x , (23)0 = Z dr y dθ y r y q g ( θ y ) ρ ( r y , θ y ) ∂s ( r x , r y , θ x , θ y ) ∂θ x , (24)where we have explicitly separated the dependence on the r and θ coordinates.7ith the Green’s function discussed above, one can prove that the density of eigenvaluesis not smooth. This is a generalization of an argument found in [2]. The basic idea is thatwe can also write the saddle point equations as K ( x ) + Z d yρ ( y ) s ( y, x ) = C , (25)where C is a Lagrange multiplier enforcing the constraint in equation (22) and K is theK¨ahler potential. From here, if ρ is smooth, the operation of differentiating with respectto x commutes with the integral. We can act with the Laplacian associated to the metric(7) three times on both sides of the equation. On the left hand side we find that ∇ K isconstant, and further applications of ∇ give zero. Inside the integral, we would act threetimes with ∇ on the Green’s function, and then we would use the defining equation ofthe Green’s function itself to find that ρ = 0. This is incompatible with the constraint, sothe assumption that ρ has smooth support is wrong. The simplest solution that one couldimagine with singular support will have some δ function distribution in it.Using the formulae in Appendix A for s , equations (92) - (94), and the fact that R dθ √ g Θ ν ( θ ) = 0 for ν > 0, i.e. that the higher harmonics on X integrate to zero,it is straightforward to see that if ρ ( r, θ ) has no θ dependence, then all the θ x dependencedrops out of the equations of motion once the θ y integrals are done. Solving the saddle pointequations reduces to a purely radial problem. Moreover, since we know that the density ofeigenvalues has singular support, we can make a simple guess to solve the problem.The eigenvalues are found to fill out an X : ρ ( x ) = N δ ( r x − r ) r x Vol( X ) , (26)at the constant radius r = r N s π Vol( X ) . (27)This expression reduces to the previously known S of radius r = p N/ S ) = π . It is a solution to the equations of motion for all basemanifolds X , it does not depend on the manifold being homogeneous. Thus we see thatpart of the AdS × X geometry has emerged from the eigenvalue quantum mechanics. Weobtain X together with its Sasaki-Einstein metric, because of the requirement that themetric on the conical target space of the eigenvalues is Calabi-Yau [8].This X eigenvalue distribution is to be understood as the large N ground state ofthe theory, where quantum mechanical measure effects have repelled the eigenvalues awayfrom their classical origin at r = 0. It is a self consistent starting point for studying the8ow energy dynamics. All off diagonal fluctuations are massive [2, 3] as are all the higherharmonics on S (this second point follows from the analysis in [15, 10]). In the remainderof this paper we study three particular excitations about this ground state. We will seethat they reproduce quantitative features of strings and D branes in the dual spacetime. N = 4 In the N = 4 SYM theory, the spectrum of gravity multiplets can be deduced from thehalf BPS states. The half BPS primary fields corresponding to single graviton states aregiven by single-trace operators of the form Tr z n , with z = x + ix . These are holomorphichighest weight states of SO (6), for a symmetric traceless tensor representation of SO (6).In the commuting matrix model of strong coupling, as we reviewed in section 2 above,the wave functions of these states are conjectured to be ψ = ψ Tr z n , (28)where ψ is the ground state wave function of the matrix model. In the N = 4 matrixmodel, on R , the SO (6) symmetry is manifestly part of the dynamics, and ψ is an SO (6)singlet. It is natural to expect that the wave functions of other states that are not half BPSwith respect to the same half of the supersymmetries as z n are given by ψ = c i ...i n Tr( x i . . . x i n ) ψ , (29)where c i ...i n is symmetric and traceless.Since the matrices commute, the trace is just a sum over eigenvalues, and we findourselves with a one-particle wave function problem. The resulting symmetric tracelesspolynomials of six variables are characterized by the property that ∇ ( c i ...i n x i . . . x i n ) ∼ c i i i ...i n x i . . . x i n = 0 , (30)this is, they are harmonic functions on R .These states have energy n , and thus the dual operators have dimension n . We can re-cover this result by considering the one-particle wave function problem for a six-dimensionalharmonic oscillator H = − ∇ + 12 ~x . (31)9his Hamiltonian differs from the full multi matrix model Hamiltonian (12) for the N = 4problem by the absence of the measure, which mixes the eigenvalues. We show in AppendixB, for the general conifold, that the measure may be neglected for these states to leadingorder at large N . Thus in this limit it is sufficient to investigate the spectrum of (31). Theabsence of mixing between eigenvalues allows us to focus on a single eigenvalue, hence wehave dropped the i index in (31).We take our wave function to be ψ = e − ~x / c i ...i n x i . . . x i n . (32)When we calculate ∇ ψ , there are three types of terms that appear. The terms with twoderivatives acting on the exponential are cancelled by the term x in the Hamiltonian.The terms with two derivatives acting on the polynomial vanish because of equation (30).We are left with terms with one derivative acting on the exponential and one derivativeacting on the polynomial. If we write the Laplacian in spherical coordinates, we find thatthese terms give c i ...i n e − r / (cid:18) r ddr (cid:0) r x i . . . x i n (cid:1) + r ddr (cid:0) x i . . . x i n (cid:1)(cid:19) . (33)Now, x i = rf i ( θ ), for some f i ( θ ), so we need to evaluate12 r ddr r n +6 + r ddr r n = ( n + 3) r n . (34)Via this exercise, we find that the wavefunction written down in equation (32) is an eigen-function of the one particle Hamiltonian (31), and that its energy is n units greater thanthe energy of the vacuum state. The same value of n is the dimension of the correspondingoperator in the conformal field theory. This calculation provides a link between the energyof a state in the harmonic oscillator problem, and the dimension of the corresponding statein supergravity. We should also notice that what matters for this computation is that thepolynomial we considered was a homogeneous function (it is a scaling function under thevector r∂ r ), and that the energy obtained is exactly this scaling dimension.In the case of N = 4 SYM, all of these symmetric traceless functions are obtainedby acting with rotations on z n , and therefore they are in some sense locally holomorphicwith respect to a suitable choice of complex coordinates. This is characterized exactly byhaving a harmonic function. We will now extend this calculation on the moduli space of a‘single brane’ in N = 4 SYM to the case of a ‘single brane’ in the case of a conformal fieldtheory associated to a general conifold. As we noted, the term mixing the eigenvalues in10he Hamiltonian, the measure, will again not be important to leading order at large N forthis problem. As we have discussed, for the general conifold the eigenvalue dynamics is locally given by N = 4 SYM. The wave function is a global object, but the property of being a harmonicfunction is something that one can check locally, as it is governed by a second order differ-ential equation. It seems natural to take the same ansatz for this more involved case as wedid for N = 4 SYM. We consider wave functions of the form ψ = ψ Tr h ( x ) , (35)where h ( x ) is a harmonic function on the Calabi-Yau cone over X and ψ = e − r / , as wefound above.The one particle problem (i.e. without the measure, see Appendix B) now correspondsto the Hamiltonian H = − r ddr r ddr − r ∇ + r . (36)One can separate variables in θ (the coordinates on X ) and r , and hence consider harmonicfunctions of the form h ( x ) = h ( r )Θ( θ ), where Θ is an eigenfunction of the Laplacian on theSasaki-Einstein space. That is −∇ Θ( θ ) = ν Θ( θ ) . (37)Harmonicity now requires solving the following differential equation for h ( r ) (cid:18) − r ddr r ddr + ν r (cid:19) h ( r ) = 0 . (38)This is solved by h ( r ) = r λ , where λ satisfies λ ( λ + 4) − ν = 0 , (39)or equivalently λ = − p ν , (40)where we chose the root that makes the wavefunction nonsingular at the origin. The samemanipulations that told us in the case of N = 4 SYM that the energy of the harmonicfunction of weight n multiplying the ground state wave function has energy n , now show usthat the energy of the single-particle wave function (35) is given by λ .11he equation (39) is familiar from supergravity in AdS [16, 17] (see also [18]), whereone associates a scaling dimension λ to a scalar particle in five dimensions that originatesfrom perturbations mixing the graviton and the self-dual five-form field strength. We seethat the scaling dimensions of the operators are controlled by harmonic analysis on theSasaki-Einstein space, recovering exactly the spectrum of some of the scalar fluctuations inthe dual gravity theory. In particular, for all holomorphic wave functions we recover theexact scaling dimension predicted by the chiral ring. Most of these harmonic functions arenot holomorphic, however, so we are recovering universally the spectrum of a large familyof non-BPS Kaluza-Klein harmonics of the dual supergravity theory.In Appendix C we discuss the possibility of building coherent states using these singletrace states. These appear to be dual to classical geometries, as one would expect forcoherent states of gravitons. The off diagonal modes connect pairs of eigenvalues. For small separations, ∆ z ij = z j − z i ,the energies of these modes are given by their mass term plus the distance between the twoeigenvalues, see [3, 8] and section 2 above, E ij = 1 + g π g i a ¯ b ∆ z aij ∆¯ z ¯ bij . (41)Recall that g eff. is the effective N = 4 coupling which controls the masses of off diagonalmodes connecting nearby eigenvalues. The z i are all at constant radius r given by (27).Thus we have E ij = 1 + λ eff. π π Vol( X ) | ∆ θ ij | g ,i , (42)where λ eff. = g N , ∆ θ ij is the separation in X , and g is the metric on X .We would like to write down an operator that describes these off diagonal fluctuations.The operators that do the trick [2, 3, 4] are strong coupling realisations of the BMN [19]operators O k,J = J X n =1 Tr h z n β † z J − n ˜ β † i e πink/J . (43)In this expression k is an integer, J is the R charge of the operator, β † and ˜ β † are creationoperators for off diagonal modes, and z is a complex coordinate on the conical moduli spacewith a fixed scaling dimension c . 12he wavefunction corresponding to this operator is ψ k,J = O k,J ψ . (44)In principle, the inclusion of the operator O k,J will backreact on the dominant eigenvaluedistribution, in a way similar to that described below in probing the radial direction. How-ever, here we wish to take J large, but not of order N . In this case the effect of the insertionof O k,J in (44) on the eigenvalue distribution is subleading in 1 /N . Thus we can take theeigenvalues z i to lie on the ground state solution (26).Invariance under the unbroken U (1) N symmetry requires that β † and ˜ β † carry oppositecharges. Thus if we take β † to connect the i th and j th eigenvalues, then ˜ β † must connectthe j th to the i th. This is implemented automatically by the trace in (43). The operator(43) may be written O k,J = X i,j J X n =1 z ni z J − nj β † ij ˜ β † ji e πink/J . (45)At large J , there is a dominant contribution to this sum [3, 4]. Firstly, the dominantlycontributing eigenvalues maximize | z | . This does not fix the location along the angle ψ dualto the R charge, as z is a chiral operator and hence | z | is invariant under R charge rotations.More specifically, on the locus where | z | is maximized we may write z i ∝ r c e icψ i . (46)The exponent follows from two observations. Firstly, because z has conformal dimension c , we have r∂ r z = cz . Secondly, see for instance [12, 13], ∂ ψ = J ( r∂ r ), where J is thecomplex structure on the Calabi-Yau. Therefore ∂ ψ z = icz , as implied by (46). Now doinga saddle point approximation to the sum over n in (45) we find ψ i − ψ j = − πkcJ . (47)This is a crucial relation which says that for given k and J , the dominant contributionto the operator O k,J comes from two off diagonal modes connecting a pair of eigenvaluesseparated according to (47). It follows from our previous expression (42) for the off diagonalenergies that E O k,J − cJ = 2 E ij = 2 s λ eff. π Vol( X ) k c J , (48)where we included the contribution to the energy from z J in (43). Conveniently, we didnot need to find the point on the remaining directions in X at which | z | is maximized, as13 ( ∂ ψ , ∂ ψ ) = 1 is in fact constant over the Sasaki-Einstein space, see for instance [12, 13].We will now see that this result is precisely the spectrum of excitations about a rapidlyrotating BMN string in the dual spacetime. The spacetime dual to the superconformal field theory is AdS × X . The metric may bewritten as ds = L ds AdS + L (cid:2) ( dψ + σ ) + ds KE (cid:3) . (49)Here ds KE is a four dimensional K¨ahler-Einstein metric and dσ is proportional to theK¨ahler two form corresponding to this metric. We restrict ourselves here to (quasi)regularSasaki-Einstein manifolds, in which the fibre coordinate ψ has a finite periodicity.States with large angular momentum about the ψ direction, corresponding to large R charge, are captured by the Penrose limit of this background [19]. This limit was computedin [20] – Penrose limits of the special case of X = T , were also computed in [21, 22] – togive ds = − dx + dx − − | x | ( dx + ) + dx , (50)where x + = 12 ( t + ψ ) , x − = L t − ψ ) . (51)Note that (50) is just the maximally supersymmetric plane wave background [19, 23].The conformal dimension and R charge are given by∆ = i∂ t , cJ = − i∂ ψ . (52)Where for ease of comparison with the previous subsection, we denote the total R chargeby cJ . Therefore from (51)2 p − = i∂ x + = i ( ∂ t + ∂ ψ ) = ∆ − cJ , (53)2 p + = i∂ x − = iL ( ∂ t − ∂ ψ ) = 1 L (∆ + cJ ) . (54)Quantising the string excitations [19, 24] in the plane wave background (50) gives thespectrum of excitations 2 δp − = s k α ′ ( p + ) . (55)14sing the expressions for the momenta (53) and working to leading order at large J , butwith L /α ′ J fixed, one obtains ∆ − cJ = r L k α ′ c J . (56)The supergravity background has a Ramond-Ramond five form F (5) = N √ π X ) (vol AdS + vol X ) . (57)The solution to the supergravity equations specifies a relation between the units of flux, N ,and the AdS radius, L , in string units L α ′ = 4 πg s N π Vol( X ) , (58)To further relate this expression to our previous results, note that the local effective N = 4coupling, g eff. , must be related to the expectation value of the dilaton in the usual way g = 4 πg s . (59)This follows, for instance, by noting that these quantities transform in the correct wayunder S duality. Hence we obtain from (56)∆ − cJ = s λ eff. π Vol( X ) k c J . (60)Recalling that the eigenvalue Hamiltonian is in fact the conformal dimension, H = ∆, wehave precisely reproduced the matrix quantum mechanics result (48). We need to multiply(60) by two because we are considering two excitations. Note that we nontrivially matchthe appearance of the volume factor Vol( X ). In the previous two sections we have shown how off diagonal modes connecting ground stateeigenvalues are dual to string excitations about the AdS × X background in the BMNlimit. In this section we return to purely eigenvalue excitations, no off-diagonal modes, butwith a larger R charge, J ∼ N . We will see how these excitations move a single eigenvalueinto the radial direction, and are dual to AdS giant gravitons. To familiarise ourselves withthe procedure, we will consider the N = 4 case first.15 .1 Probing the radial direction in N = 4 In the N = 4 case, the cone is over S , i.e. the total space is just R . We will use cartesiancoodinates ~x to denote the matrices, rather than the ‘polar’ coordinates r, θ .Consider the wavefunction ψ = ψ Tr z J , (61)where z = x + ix and ψ is the ground state wavefunction. In Appendix B we showthat in the large N limit, this is an eigenfunction of the Hamiltonian (12) with eigenvalue E = E + J . The probability density is µ | ψ | = e − P i ~x i + P i = j log | ~x i − ~x j | +log P i,j ( x i + ix i ) J ( x j − ix j ) J , (62)where we used the explicit Green’s function on R to write µ = Q i 1. We may thus approximate the last termlog X i,j (cid:0) x i + ix i (cid:1) J (cid:0) x j − ix j (cid:1) J → J log (cid:2) ( x N ) + ( x N ) (cid:3) . (63)This is the assumption that one eigenvalue will be moved away from the others.The large N semiclassical support of the wavefunction is found by extremising theexponent in (62). The equations of motion are x Ai = X i = j x Ai − x Aj | ~x i − ~x j | + J δ iN (cid:2) δ A x N + δ A x N (cid:3) ( x N ) + ( x N ) . (64)We look for a solution to these equations which is given by the ground state before theinsertion, an S of radius squared r = N/ N/π r , together with a singleeigenvalue ~x N separated from the sphere. There will be an S worth of such solutions, wherethe S lies in the x − x plane. Without loss of generality, we can take the eigenvalue tomove off in the x direction x AN = x N δ A . (65)For i = N , the equation of motion (64) is satisfied to leading order in N , because theequation of motion is just that corresponding to the ground state wavefunction which issolved by the S . The effect of the extra eigenvalue is subleading. The equation for i = N ,however, gives a nontrivial equation for x N . Using the integral8 N π Z π dθ sin θ x N − r cos θx N + r − x N r cos θ = N (6 x N − x N r + r )6 x N , (66)16igure 1: A single eigenvalue is removed from the X to a distance x N ∼ √ J .the equation of motion becomes, using r = N/ x N − ( J + N ) x N + N x N − N 24 = 0 . (67)If we set x N = d N , J = jN , (68)then the solution to the (cubic) equation (67) is d = 1 + j j ( j + 2)3 p ( j ) / + p ( j ) / , (69)where p ( j ) = 12 h j + 48 j + 16 j + p j + 672 j + 288 j i . (70)This gives us the distance of the x N eigenvalue from the origin as a function of the angularmomentum J . We see that J ∼ N is indeed large as required. Figure 1 illustrates theconfiguration we have just obtained.Taking the further limit that the eigenvalue is far away from the sphere in √ N units,i.e. j ≫ 1, gives the result x N = √ J + · · · ( J/N ≫ . (71)The association of an object with large R charge to a radial motion is strongly reminis-cent of AdS giant gravitons. This will shortly lead us to identify the radial direction of theCalabi-Yau cone outside of the X occupied by the ground state with the radial directionof global AdS . The argument goes through essentially unchanged for the case of a general cone over X .We make the assumption that one eigenvalue will have a larger modulus than the others17 z N | > | z i | , for all i = N . Thus in the limit J ≫ µ | ψ | = µ | ψ Tr z J | = e − P i r i − P i = j s ij + J log | z N | . (72)As we note in Appendix B, the holomorphic coordinate z must be a power of r multipliedby a harmonic function on X z = r c F c ( θ ) . (73)The large N semiclassical equations of motion following from (72) are therefore r i + X j = i ∂s ij ∂r i = cJ δ iN r N , X j = i ∂s ij ∂θ i = J δ iN ∂ θ N | F c ( θ N ) || F c ( θ N ) | . (74)In the large N limit, as for the case of S above, the equations of motion for theeigenvalues i = N are unaffected by the insertion of Tr z J , as the motion of the singleeigenvalue z N away from the ground state configuration is a subleading effect. The equationsof motion for r N and θ N however are nontrivial. Recall the observation we made in section4: that the independence of the ground state eigenvalue density on θ implies that anyintegral of the form R dθ x ρ ( θ x ) s ( θ x , θ y ) kills the θ y dependence. This fact, together withthe expression for s in equation (92) and the integral Z ∞ dλλ J ( √ λr ) r ∂∂r N J ( √ λr N ) r N = − r N r + 6 r N + r r N , (75)leads to the following equations r N − (cid:16) cJ + Nπ Vol( X ) (cid:17) r N + (cid:16) Nπ Vol( X ) (cid:17) r N − (cid:16) Nπ Vol( X ) (cid:17) = 0 , (76) ∂ θ N | F c ( θ N ) | = 0 , (77)where we also used the radius of the ground state X in (27).The immediate observation we can make from these equations is that the radial andangular parts have completely decoupled. We can interpret this as the fact that the radialdirection in the bulk geometries emerges universally. It does not depend on where theeigenvalue is sitting in X . This reflects the direct product structure of the dual geometry: AdS × X .The equation (77) for θ says that | z N | is maximized given its fixed radius r N . This,together with the fact that we will find r N > r , guarantees that our assumption that | z N | > | z i | for i = N is consistent. As in the case for S , there will not be a unique solutionto (77). Rather there will be an S worth of solutions, corresponding to the R symmetrycircle. 18f we make the definitions r N = d N π Vol( X ) , cJ = j N π Vol( X ) , (78)then we find that the radial equation (76) is exactly the same as the one we found in thecase of S , with solution (69). Thus (69) describes how the eigenvalue z N moves out in theradial direction as a function of the R charge cJ . From the equation (76) we see that thegeneral relation between r N and cJ depends on the volume of X . However, in the limit j ≫ r N = √ cJ + · · · ( J/N ≫ . (79) In the N = 4 case, at weak coupling, the wavefunction dual to an AdS giant graviton withangular momentum J along the equator of the S is [25] ψ = ψ χ S J ( z ) , (80)where χ S J is the Schur polynomial corresponding to the totally symmetric representationof rank J . In terms of the eigenvalues of zχ S J ( z ) = X ≤ i ≤···≤ i J ≤ N z i · · · z i J . (81)We would like to approximate χ S J ( z ) with Tr z J , so that we can use the results of theprevious section to evaluate the semiclassical wavefunction. This will be valid providedthat the largest eigenvalue | x N /x p | ≫ p = N , which requires j ≫ 1. In thislimit d = j in (69). Furthermore, it is unclear that the Schur polynomials (81) will beorthogonal at strong coupling. On the other hand, we have shown in Appendix B that thestates ψ Tr z J are eigenstates to leading order at large N , with different eigenvalues, andtherefore are orthogonal.In the bulk, the AdS giant gravitons are D S expands to a finiteradius in AdS , due to their angular momentum about the R symmetry direction. Withangular momentum J the radius is [26, 27] given by r = J/N . Here we are using globalcoordinates in AdS × X L ds = − (1 + r ) dt + dr r + r d Ω S + d Ω X . (82)Comparing this bulk result with our matrix model result (79), and absorbing the factorof c into the definition of J , now the total R charge, we obtain r N √ N = r giant for r giant ≫ . (83)19hus the radial direction in the space of eigenvalues, in units of √ N , is exactly equal tothe radial direction of AdS in the large radius limit. This is a nice result, but it is also notclear why the particular coordinate r that we chose in (82) should have been singled out inthis way.We can unambiguously draw the following conclusions from this section, however: Theradial direction of the Calabi-Yau cone outside of the X occupied by the eigenvalues is tobe topologically identified with the radial direction in global AdS . This direction emergesindependently of and orthogonally to the manifold X . The radial coordinate may beprobed using operators with large R charge and identifying the dual string theory states.It is of great interest to obtain more information using this approach, such as the warpingof spacetime and the redshift of AdS as a function of radius. One main objective of this paper has been to show that a framework now exists for per-forming precise computations in many strongly coupled N = 1 conformal field theories.This is of interest because these theories are dual to compactifications of type IIB stringtheory on Sasaki-Einstein spaces. We have seen how the Sasaki-Einstein manifold emergesas the semiclassical limit of the ground state of a commuting matrix model. We have thenfound that the spectrum of certain non-BPS supergravity and stringy excitations may bereproduced exactly as excited states in the matrix quantum mechanics.The basic setup has exploited a connection that all these field theories have an effective(local) N = 4 SYM description on moduli space, and that one should copy strategiesthat worked in N = 4 SYM by analogy and a use of local concepts on moduli space. Inparticular, the formalism used in this paper required the introduction of a measure that wasdetermined by solving a differential equation on the moduli space. If one can find a closedform expression for the corresponding measure in various cases (let us say the conifold), itwould be possible to test this proposal further.A very important result that follows from our proposed measure is that a particularSasaki-Einstein slice of the Calabi-Yau cone is singled out by a saddle point calculation.We checked that the volume of this manifold is properly normalized in field theory units:we had no additional free parameters in matching the BMN limits. These volumes are alsorelated to the gravitational calculation of the conformal anomalies of the field theory.These matchings show that the conjectured framework can precisely capture quantitative20spects of strongly coupled theories. The ultimate objective of this research program is toprovide a description of situations where no other approach seems feasible, such as when thedual spacetime develops a region of high curvature. However, before reaching that point,more computations should be done.It is clear that the calculations that have been done here can be improved further andone might be able to go beyond BMN limits to capture more information about stringmotion in these geometries. Ideally, one would want to derive that the string motion shouldobey the equations of motion associated to a non-linear sigma model on the correspondingAdS dual geometry.It is also important to understand more precisely to what extent the approximationsthat we have described are applicable, and when they break down. Acknowledgments This research was supported in part by the National Science Foundation under Grant No.PHY05-51164, and by the DOE under grant DE-FG02-91ER40618. A Logarithmic scaling of the Green’s function In this appendix we show that the Green’s function satisfying −∇ s ( r, r ′ , θ, θ ′ ) = − (cid:18) r ddr r ddr + 1 r ∇ (cid:19) s ( r, r ′ , θ, θ ′ ) = 64 π δ (6) ( r, r ′ , θ, θ ′ ) , (84)has a logarithmic scaling under r, r ′ → αr, αr ′ , as advertised in the main text. Recall that r is the radial direction in the cone (7), whereas the θ are coordinates on the five dimensionalbase manifold.One could find the Green’s function using a standard partial wave expansion for thisLaplace-like equation. However, the symmetry r ↔ r ′ , crucial for our purposes, may bekept manifest as follows. Consider the eigenmodes of the related equation −∇ φ λ ( r, θ ) = λφ λ ( r, θ ) . (85)These modes give a complete basis of functions. There is an infinite degeneracy for eachvalue of λ given by the modes φ λ ( r, θ ) = Φ λ,ν ( r )Θ ν ( θ ) , (86)21here −∇ Θ ν ( θ ) = ν Θ ν ( θ ) , Z dθ √ g Θ ∗ ν ( θ )Θ ν ′ ( θ ) = δ ν,ν ′ . (87)The eigenvalues ν are discrete and the lowest is ν = 0, corresponding to a constant modeon the base of the cone. For each value of ν , the radial functions are normalised as Z drr Φ λ,ν ( r )Φ λ ′ ,ν ( r ) = δ ( λ − λ ′ ) . (88)The delta function may now be written δ (6) ( r, r ′ , θ, θ ′ ) = X ν Z ∞ dλ Φ ∗ λ,ν ( r )Φ λ,ν ( r ′ )Θ ∗ ν ( θ )Θ ν ( θ ′ ) . (89)Solving the equation (85) for the radial part of the mode (86) and imposing the normal-isation (88), one obtains the Bessel functionΦ λ,ν ( r ) = J √ ν ( √ λ r ) √ r . (90)Note that this expression is real. We may now use this expression to solve for the Green’sfunction in (84). Na¨ıvely, we would like to write the following s na¨ıve ( r, r ′ , θ, θ ′ ) = X ν Z ∞ π dλλ Φ λ,ν ( r )Φ λ,ν ( r ′ )Θ ∗ ν ( θ )Θ ν ( θ ′ ) . (91)Although this expression formally solves the equation (84), it is divergent. The divergencearises as λ → ν = 0 term. This problem is entirely expected, dueto the fact that the sixth order equation (84) has zero modes. Specifically, the six modesare: { , log r, r ± , r ± } . We can deal with this divergence as follows.Firstly, regularize the divergent part of the na¨ıve expression (91): s ǫ = 32 π Vol( X ) Z ∞ ǫ dλJ ( √ λ r ) J ( √ λ r ′ ) r r ′ λ + s ν> , (92)where s ν> contains the terms in (91) with ν > s ν> = X ν> Z ∞ π dλr r ′ λ J √ ν ( √ λ r ) J √ ν ( √ λ r ′ )Θ ∗ ν ( θ )Θ ν ( θ ′ ) . (93)The integral over λ in this last expression can be performed to obtain a hypergeometricfunction. Performing the integral will break the symmetry r ↔ r ′ however, as the resultdepends on which of r and r ′ is bigger.We can now obtain a finite ‘renormalised’ Green’s function via a minimal substraction s = lim ǫ → (cid:18) s ǫ + π log ǫ X ) (cid:19) . (94)22he expression (94) solves the equation for the Green’s function (84) and is manifestlysymmetric in r ↔ r ′ . However, we need to check that it is regular as r → r ′ fixed.Taking r ≪ r ′ we obtain s ǫ ( r ≪ r ′ ) = 4 π Vol( X ) Z ∞ ǫ r ′ dλJ ( √ λ ) λ + O ( r/r ′ ) , (95)which via (94) leads to s ( r ≪ r ′ ) = − π log r ′ Vol( X ) + const. + O ( r/r ′ ) , (96)where the constant is unimportant, as the Green’s function is only defined up to a constantin any case. This expression is manifestly finite as r → r ′ − Z B r ′ drdθ √ g ∇ s = π " r ddr (cid:18) r ddr r ddr (cid:19) log r r ′ = 64 π , (97)as required by (84).From (94) or (96) it is easy to see that the Green’s function obeys the logarithmic scalingadvertised in (11) s ( αr, αr ′ , θ, θ ′ ) = s ( r, r ′ , θ, θ ′ ) − π log α Vol( X ) . (98) B Holomorphic polynomials are eigenfunctions at large N In this appendix we show that wavefunctions of the form ψ = ψ Tr P ( z ) = X i P ( z i ) e − P j K j , (99)for P ( z ) a holomorphic polynomial in z with all terms of degree J , which in turn is a holo-morphic coordinate on the Calabi-Yau cone with fixed conformal dimension c , are eigen-functions of the Hamiltonian (12) to leading order at large N . Some, but not all, of thesearguments essentially appear in appendix A of [5]. These arguments go through if P is notholomorphic, but simply a harmonic function on the Calabi-Yau cone.Holomorphy implies ∇ P ( z ) = 0 and scaling dimension c of z implies r∂ r z = cz .Straightforward algebra then shows that Hψ = ( E + cJ ) ψ + ψ X i ∇ i P ( z i ) · X j = i ∇ i s ( z i , z j ) . (100)23e now show that the last term vanishes to leading order at large N .In the continuum large N limit, the last term in (100) is proportional to Z d xρ ( x ) ∇ x P ( z x ) · Z d yρ ( y ) ∇ x s ( z x , z y ) . (101)If P ( z ) is a polynomial with not too high degree, as in the case P ( z ) = z J we considered insection 8 above, then the backreaction of P ( z ) onto the eigenvalue distribution is subleadingat large N . Therefore in (101) we may take the ρ ( x ) to be the ground state (26). Inparticular, this distribution adds no extra dependence on the coordinates θ of X . Theintegral over d y includes an integral over X . From the fact that R dθ √ g Θ ν ( θ ) = 0 for ν > s ( z x , z y ) that is independent of both θ x and θ y survives the θ y integral. Thus (101) is proportional to Z d θ √ g ∂ r P ( z ) . (102)The final step is now to show that P ( z ), and hence also ∂ r P ( z ), is a nontrivial eigenfunctionof the Laplacian on X , and therefore the integral (102) vanishes. From holomorphy wehave ∇ P ( z ) = (cid:18) r ddr r ddr + 1 r ∇ (cid:19) P ( z ) = 0 . (103)The scaling dimension of z implies that each monomial in P ( z ) is of the form P J ( z ) = r cJ F J ( θ ). It is immediately seen that (103) implies that −∇ F J ( θ ) = cJ ( cJ + 4) F J ( θ ) . (104)Therefore P J ( z ) is a harmonic of the Laplacian on X , as we required.The upshot of the preceding paragraph is that (101) does indeed vanish and hence theholomorphic polynomial does give an eigenfunction, as claimed. C Coherent states and orthogonality In the large N limit, single trace operators are supposed to be related to single string states.To the extent that these are free, one can build coherent states of these traces. Formally,we would want to consider a coherent state as an exponential of a raising operator. In ouridentification, we have said that Tr h ( x ) is a single graviton state, so a coherent state ofgravitons would be described formally by ψ coh ∼ e α Tr h ( x ) ψ . (105)24e can try to understand the distribution of particles on the cone that is associated to thiswave function. We do this by thinking of α as a formal parameter (usually h ( x ) will growfaster at infinity than the decay of ψ , which is just gaussian decay).If we replace Tr h ( x ) by R ρ ( x ) h ( x ), as is required for the large N limit, we can repeatthe arguments made in studying equation (25) to show that once again the dominant semi-classical density of eigenvalues is a singular distribution. It was suggested in [2] that havingsingular distributions of particles in the saddle point limit is exactly the type of situationthat leads to classical gravity solutions. This supports the proposal made above for thewave functions associated to non-BPS gravitons.Unfortunately, it seems that the corresponding wave functions are not eigenfunctionsof the full effective Schr¨odinger equation with the measure added. This has already beenseen for the case of N = 4 SYM [5]. There is no new effect that shows up in this moregeneral case that is not there in the case of maximal supersymmetry. Moreover, as shownin appendix B, they become eigenstates to leading order in the large N limit.One can also show that these single trace wave functions (including the measure) areapproximately orthogonal to the ground state and to each other. One would need to evaluatethe overlap Z e − r µ Tr h ( x )Tr h ( x ) . (106)The idea to show approximate orthogonality is that the overlap is dominated by the saddleof the ground state. Then the dependence of µ on eigenvalue i , that is written as µ i = exp( − Z ρ ( x ) s ( x i , x )) , (107)can be approximated by a function that depends only on the radial variable r i , but not onthe angular variables. 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