Double-trace spectrum of N = 4 supersymmetric Yang-Mills theory at strong coupling

The spectrum of IIB supergravity on AdS 5 × S 5 contains a number of bound states described by long double-trace multiplets in N ¼ 4 super Yang-Mills theory at large ‘ t Hooft coupling. At large N these states are degenerate and to obtain their anomalous dimensions as expansions in 1 N 2 one has to solve a mixing problem. We conjecture a formula for the leading anomalous dimensions of all long double-trace operators which exhibits a large residual degeneracy whose structure we describe. Our formula can be related to conformal Casimir operators which arise in the structure of leading discontinuities of supergravity loop corrections to four-point correlators of half-BPS operators.


I. INTRODUCTION
Recently much progress has been made in understanding the structure of the spectrum of double-trace operators in N ¼ 4 super Yang-Mills theory at large N and large 't Hooft coupling λ ¼ g 2 N [1]. Based on these results, operator product expansion (OPE) and bootstrap techniques have been applied in [2,3] to obtain closed form expressions for supergravity loop corrections of certain holographic correlators, uncovering novel and rich structure (see [4,5] for related approaches to such loop corrections). Here we complete the picture for the double-trace spectrum and conjecture a general formula for the leading anomalous dimensions of all long double-trace operators of any twist, spin, and suð4Þ representation.
In the regime N → ∞ and λ ≫ 1, the theory is in correspondence with classical IIB supergravity on AdS 5 × S 5 [6]. The graviton and the Kaluza-Klein multiplets are dual to protected half-BPS operators in the ½0; p; 0 representation of suð4Þ, where Φ i¼1;…;6 are the elementary scalar fields, the complex vector ⃗ y is null, and the ellipsis stands for 1=N-suppressed multitrace terms (for p ≥ 4), whose precise nature will be described in Sec. II. At leading large N and for any value of λ, we may consider spin l long double-trace superconformal primary operators of the form ðp ≤ qÞ: ð2Þ In the large N limit, the operators O pq are orthogonal and have dimension Δ 0 ¼ τ þ l, hence τ coincides with the twist in the limit N → ∞. For fixed τ and suð4Þ labels ½a; b; a, there are d allowed values of the pair ðp; qÞ. We denote this set by D long τ;l;a;b and we parametrize it as follows: The operators O pq are in long multiplets, but in the strict large N limit their dimensions are protected. At order 1=N 2 they acquire anomalous dimensions and mix among themselves and with other long operators. In the supergravity regime λ ≫ 1, operators corresponding to massive string excitations should decouple from the spectrum leaving only those corresponding to supergravity states, e.g., the single-particle states O p and the two-particle bound states O pq . At leading order in large N the O pq just mix among themselves to produce the true scaling eigenstates, which we denote by K pq . Mixing with higher multiparticle states will only occur at higher orders in the 1=N expansion. Analysis of the OPE in the tree-level supergravity regime (see Sec. III) leads us to the following conjecture, generalizing results in [1][2][3].

A. Main conjecture
Up to order 1=N 2 , the dimensions of the operators K pq are given by where ð…Þ 6 is the Pochhammer symbol, and Note that for μ > 1 and t > 2 some dimensions exhibit a residual degeneracy because they are independent of q. We display this property with an illustration of D long τ;l;a;b (see Fig. 1). The dots connected by vertical lines in the ðp; qÞ plane represent operators of common anomalous dimension.

II. HOLOGRAPHIC CORRELATORS
The correlators hO p 1 O p 2 O p 3 O p 4 i ≡ hp 1 p 2 p 3 p 4 i may be written as a free part plus an interacting part, The factor P carries the conformal and suð4Þ weights and assuming (without loss of generality) p 21 ≥ 0, p 43 ≥ 0, and p 43 ≥ p 21 , it takes the form P ¼ N where p ij ¼ p i − p j and g ij ¼ ðy i · y j Þ=x 2 ij . The quantities I and H are functions of the variables x,x, y,ȳ, related to the conformal and suð4Þ cross ratios u, v, σ, τ via In terms of these variables we have The decomposition into free and interacting parts in (7) reflects the property of "partial nonrenormalization" [7], i.e., the statement that all the dependence on the coupling appears in the function H. Here we consider the leading contribution to H at large λ. In the OPE of ðO p 1 × O p 2 Þ and ðO p 3 × O p 4 Þ, the free term contributes both a protected sector and a long sector. Identifying the sectors is nontrivial due to possible semishort multiplet recombination at the unitarity bound [8,9].
At leading order in the 1=N 2 expansion, a correlator is determined by disconnected contributions to the free part. These only exist for hppqqi and cases related by crossing, ð11Þ At the next order in 1=N 2 in the supergravity regime, treelevel Witten diagrams contribute both the free theory connected diagrams and the first contribution to H.

A. Supergravity states and free theory
It was noticed in [10] that the connected part of hp 1 p 2 p 3 p 4 i free , generated via tree-level Witten diagrams, disagrees with free theory four-point functions of single-trace half-BPS operators. The resolution is that single-particle supergravity states are not dual to singletrace half-BPS operators, rather they are uniquely defined as those orthogonal to all multitrace operators. From this property we can identify multitrace contributions to TrΦ p for p ≥ 4. The presence of multitrace admixtures was also discussed in [11,12]. Consider e.g., O 4 , the condition With this identification of O 4 the free theory computation of h2244i agrees with that of supergravity [10]. The correct identification of the operators O p is also necessary for the "derivative relation" of [13] to hold, as can be directly observed for the cases h22nni. More generally, connected free theory diagrams where, e.g., O p 3 is joined only to O p 4 (see Fig. 2

B. Tree-level dynamics
The conjecture of [14] is a simple Mellin integral for the leading term in H In the sum i, j, k ≥ 0 and we use the notatioñ The coefficients a ijk are given by The conjecture agrees with all known supergravity computations ( [15] and refs. therein). The precise integration contour and the assumptions which led to (13) are spelled out in [12].
C. Determining N p 1 p 2 p 3 p 4 from the lightlike limit The normalization N is not determined in [14]. Here we fix it using the following nontrivial statement: The limit u, v → 0 with (u=v) fixed corresponds to taking the points x 1 , x 2 , x 3 , x 4 to be sequentially lightlike separated.
Examining both the free theory and interacting contributions to the lhs of (16) above, we find that it takes the form P M r¼1 A r ðuτ=vÞ r where The first term in (17) comes from hp 1 p 2 p 3 p 4 i free =P and arises from the diagrams in Fig. 3. The normalization of each of these diagrams in the planar limit can be simply obtained by counting the number of inequivalent planar embeddings. Cyclic rotation on each vertex leaves the diagram unchanged, hence the factor p 1 p 2 p 3 p 4 . Additionally, the diagonal propagators can be drawn inside or outside the square, giving and upon residue integration will produce a term proportional to ðuσÞ i ðu=vÞ 1þj τ j . Since I ¼ τ þ Oðu; vÞ, the contribution to A r comes from taking the simple poles with i ¼ 0 in (15). The residue is Crucially the j dependence cancels between a 0jk =ðj!k!Þ and Γ p 1 p 2 p 3 p 4 and hence A r is in fact independent of r. Now the statement (16) is clearly equivalent to the statement A r ¼ 0 for all r. Rearranging (17) we thus obtain the result for N p 1 p 2 p 3 p 4 , The result combines neatly with the coefficients a ijk , Note that the expression (20) is consistent with the results for N ppqq and N p;pþ1;q;qþ1 obtained in [1,3].

D. Proof of lightlike vanishing
The lightlike limit projects the common OPE of ðO p 1 × O p 2 Þ and ðO p 3 × O p 4 Þ onto operators with large spin and naive twist τ ≤ p 43 þ 2M, i.e., τ < minðp 1 þ p 2 ; p 3 þ p 4 Þ. To justify the statement (16) let us consider the various contributions to the OPE expected in the supergravity regime. First of all we have single-particle states corresponding to half-BPS superconformal primary operators. Such operators have spin zero and do not contribute in the limit v → 0 which receives contributions from large spin. Next we have (both protected and unprotected) double-trace operators of the form ½O p □ n ∂ l O q or mixtures thereof. The leading large N contribution to three-point functions of the form hO p O q ½O p 0 □ n ∂ l O q 0 i ∼ OðN pþq Þ arises when p ¼ p 0 and q ¼ q 0 when the three-point function factorizes into a product of two-point functions. The twist τ of the doubletrace operator therefore must obey τ ≥ p þ q, otherwise the three-point function will be suppressed by 1=N 2 . The exchanged operators surviving the lightlike limit (16) all have twist less than both p 1 þ p 2 and p 3 þ p 4 and hence the contributions will be suppressed by at least 1=N 4 and will not contribute to the lhs of (16). Higher multitrace operators are even more suppressed and we conclude that no operators in the supergravity spectrum can contribute in the lightlike limit, justifying (16).

III. UNMIXING EQUATIONS
We now describe how the system of relations implied by the OPE describes an eigenvalue problem which allows us to determine the anomalous dimensions of the true doubletrace eigenstates K pq . In particular, we consider the long multiplet superconformal partial wave (SCPW) expansion of the correlators hp 1 p 2 p 3 p 4 i, in which the pairs ðp 1 ; p 2 Þ and ðp 3 ; p 4 Þ both run over the set D long τ;l;a;b described in (3). The result is a symmetric (d × d) matrix whose partial wave expansion reads Terms of order 1=N 2 which are analytic at u ¼ 0, i.e., without a factor of log u, have been dropped on the rhs. The matrix A τ;l a;b in (22) is determined by disconnected free theory and is diagonal due to the form of the disconnected contributions (11). The matrix M τ;l a;b is obtained from the discontinuity around u ¼ 0 of H RZ . For completeness, we recall the explicit expression [16,17] of a long superblock of naive twist τ, spin l, and suð4Þ rep R ¼ ½n − m; 2m þ p 43 ; n − m, This structure is the simplest among the determinantal superconformal blocks [9], since it factorizes into an ordinary conformal block B sjl ðx;xÞ [18], where JP stands for a Jacobi polynomial. The matrices A and M contain conformal field theory data for the operators K pq , Here the (d × d) matrix C, indexed by pairs ðp 1 ; p 2 Þ and ðq 1 ; q 2 Þ running over D long τ;l;a;b , is given by and η ¼ diagðη pq Þ is a (d × d) diagonal matrix where η pq is (half) the anomalous dimension of the operator K pq for ðp; qÞ ∈ D long τ;l;a;b , APRILE, DRUMMOND, HESLOP, and PAUL PHYS. REV. D 98, 126008 (2018) 126008-4 The eigenvalue problem (26) is well defined as a consequence of the equality Let us comment on the structure of the matrices A and M. The SCPW expansion of disconnected free theory has the following compact expression: where the function F is given by The SCPW of matrix elements in M τ;l;a;b has the form where P d ðlÞ is a polynomial in l of degree d ¼ minðp 1 þ p 2 ; p 3 þ p 4 Þ − ðp 43 − p 21 Þ − 4, and r labels ðp 3 ; p 4 Þ. We determine this polynomial case by case and solve the eigenvalue problem following [1][2][3]. We have verified that our conjecture (5) holds systematically in the suð4Þ channels ½a; b; a with 0 ≤ a ≤ 3, 0 ≤ b ≤ 6 up to twist 24 for both even and odd spins. In particular, we have been able to perform nontrivial tests on the pattern of residual degeneracies. It would be fascinating to understand whether higher order corrections lift the pattern of residual degeneracies observed at order 1=N 2 or whether they remain due to some as yet unknown symmetry.