Structure of large spin expansion of anomalous dimensions at strong coupling

The anomalous dimensions of planar N=4 SYM theory operators like tr(Phi D^S Phi) expanded in large spin S have the asymptotics \gamma= f ln S + f_c + 1/S (f_11 ln S + f_10) + ..., where f (the universal scaling function or cusp anomaly), f_c and f_mn are given by power series in the `t Hooft coupling \lambda. The subleading coefficients appear to be related by the so called functional relation and parity invariance (or reciprocity) property of the function expressing \gamma in terms of the conformal spin of the collinear group. Here we study the structure of such large spin expansion at strong coupling via AdS/CFT, i.e. by using the dual description in terms of folded spinning string in AdS_5. The large spin expansion of the classical string energy happens to have the same structure as that of \gamma in the perturbative gauge theory. Moreover, the functional relation and the reciprocity constraints on the coefficients are also satisfied. We compute the leading string 1-loop corrections to the coefficients f_c, f_11, f_10 and verify the functional/reciprocity relations at subleading \lambda^{-1/2} order. This provides a strong indication that these relations hold not only in weak coupling (gauge-theory) but also in strong coupling (string-theory) perturbative expansions.


Introduction and summary
Recent advances in the study of duality between planar N = 4 SYM theory and free AdS 5 × S 5 superstring theory which utilise their integrability property led to important insights into the structure of dependence of anomalous dimensions of gauge-invariant operators on the quantum numbers like spin and on the t'Hooft coupling. While there was a remarkable recent progress in understanding the asymptotic large spin limit in which the compactness of the spatial direction of the world sheet may be ignored [1,2], 1 it is important to study corrections to this limit.
Here we shall consider the famous example [6] of folded spinning string in AdS 5 dual to a minimal twist gauge theory operator like tr(ΦD S + Φ). Starting with the classical string energy for the solution of [7,6] and expanding it in large semiclassical spin parameter S one finds (see [8] and below) with a 0 = 1 π , a c = 1 π (ln 8π − 1), etc. 2 That means that in the semiclassical string theory limit in which one first takes the string tension √ λ 2π to be large for fixed S and then expands in large S, the corresponding string energy can be written as ( where to leading order in 1 Here we refer to the integral equations that describe the minimal anomalous dimension in the band [3,4].
These equations were obtained from the all-loop Bethe ansatz by taking a special scaling limit [5,3] which describes a condensation of magnons and holes at the origin. 2 Note that the small S behaviour of the energy is quite different [9]: E = √ 2S[h0 + h1S + ...].
Following the analysis of quantum corrections to the folded string solution in [10], one may conclude that this structure of the large S expansion is preserved by the α ′ ∼ 1 √ λ corrections, with the coefficients f, f c , f 11 , ... being promoted to power series in 1 √ λ , i.e. f mk ∼ n b mk,n ( √ λ) n . Indeed, as we shall find below, the 1-loop corrections for leading coefficients in (1.4) Here c is a constant that we were not able to determine with the method for evaluation of 1-loop string correction we used below. Equivalently, in (1.3) we get the same f , f 11 and (1.11) The (at first surprising) vanishing of the first two terms in f ′ 10 in (1.8) is a consequence of the relation f 10 − fc f f 11 = 0 (here verified at first two leading orders, i.e. to order O(λ 0 ) in f 10 ). In fact, since we also see that in (1.5),(1.7) f 11 = 1 2 f 2 , this relation is equivalent to f 10 − 1 2 f c f = 0. As we shall see below, these relations are consequences of the "functional relation" and reciprocity at strong coupling. Note that these conditions thus determine 2 our of 4 coefficients in the part of E up to order O( ln 2 S S 2 ). We thus led to expect that f ′ 10 = 0 should be true in to all orders in the strong coupling expansion, suggesting the advantage of the form of E in (1.4) over (1.3) and the importance of the functionf c .
Reversing the usual logic, we may then conjecture that structurally same large spin expansion should appear also at weak coupling, i.e. in the perturbative expressions for the corresponding gauge theory anomalous dimensions. This is not, a priori, guaranteed since the limit taken on the gauge theory side is different from the above string-theory limit: there one first expands the anomalous dimension in small λ at fixed S and then takes S large in each of the λ n coefficients.
Yet, remarkably, expanding in large S the known 2-, 3-and 4-loop perturbative anomalous dimensions of twist 2 and twist 3 operators in SYM theory one does find [11,12,8,13,15,16,17,18] the expression of the form (1.3) with the coefficients given by power series in λ, i.e. f mk ∼ n a mk,n λ n . 4 Assuming that the expansion (1.3)  applies for any λ, and given the important role of the universal scaling function or cusp anomalous dimension f (λ) [19], one may raise the question about the interpretation of other "interpolating" functions f mk (λ) in (1.3).
From the gauge theory point of view, the function f (λ) appears in the asymptotics of anomalous dimensions of gauge invariant operators as well as (for twist 2) in the IR asymptotics of gluon scattering amplitudes related to UV cusp anomaly of light-like Wilson loops (for a review and references see, e.g., [20]). On string side that corresponds, respectively, to the closed string [6] and the open string [21,22] sectors. They are connected in the strict large S limit since then the ends of the folded spinning string reach the boundary of AdS 5 and thus the associated world surface has a Wilson line interpretation [23]. This open string sector interpretation should not be expected to apply to other subleading coefficients f mk (λ) since for finite S the end points of the folded string no longer touch the boundary (cf., however, [24,25] where the gauge theory interpretation of the constant f c is discussed).
In fact, many of the f mk coefficients in (1.3), (1.12) are not actually independent, as was first observed at few leading orders in weak coupling expansion and then given a general interpretation in [15,16]. According to [15], these coefficients are constrained by (i) the so called "functional relation" suggested by the conformal invariance (which relates the leading f mm functions to powers of the scaling function f and thus implies their universality) and also by (ii) the "parity preserving relation" or "reciprocity" [12,15,16,26,18] (which relates some subleading nonuniversal coefficients, e.g., f 10 to f c , f 32 to f 21 , etc.).
Our aim here will be to investigate the presence of such relations at strong coupling, i.e.
in the semiclassical string theory expansion for the spinning string states, extending earlier observations made in [15]. 6 4 The 4-loop prediction for twist 2 and 3 anomalous dimension at finite S [17] so far was not based on direct gauge theory computation. 5 Here, as in (1.2),(1.3), we suppress the dependence on finite twist J. 6 Let us note also that the fine structure of the constant term fc in (1.3) for "non-minimal" operators and the dual string states was studied in [3,4] and in [27].
Before proceeding let us add an important clarification. While for low twists J = 2, 3 in weak-coupling expansion the powers of ln S in e m in the anomalous dimension (1.12) appear to be positive, 7 for J > 3 one finds also terms with negative powers like J k ln m S . Namely, for J > 3 the leading term in E − S − J appears to be e 0 = k 1 (J) + k 2 (J) ln 2 S + ..., where for k 1 = k 11 J + k 10 , k 2 = k 23 J 3 + k 22 J 2 + k 21 J + k 20 and k 2 vanishes for J = 2, 3. 8 The presence of such terms was first observed in [3] (in the 1-loop approximation in the sl(2) sector) in the limit of large J and S with j ≡ J ln S fixed and small, i.e. e 0 = (k 11 j + k 23 j 3 + ...) ln S, and they are likely to be present also for finite J with S ≫ 1. 9 At strong coupling, i.e. in the string-theory semiclassical expansion, one cannot distinguish between finite values of J and J = 0; for large J one finds (see [3] and section 2.2 below) that in a similar limit of large S with ℓ ≡ J √ λ ln S fixed e 0 = (n 1 ℓ 2 + n 2 ℓ 4 + ...) ln S. Thus the dependence on J and S is different at strong and weak coupling and to relate the two expansions one needs a non-trivial resummation rather than simple interpolation of coefficients in λ [40,2].
Let us now review in more detail what is known at weak coupling (see [15] and references there). The results of explicit higher-loop planar gauge-theory computations of anomalous dimensions γ(S, J, λ) of operators like tr(D S + Φ J ) (S is the Lorentz spin and J is the twist) were interpreted in [15] in the following way (see also [16]). Observing that such Wilson-type operators can be classified according to representations of the collinear SL(2, R) subgroup of the SO(2, 4) conformal group [30] which are labeled by the conformal spin s = 1 2 (S + ∆) one may argue that the anomalous dimension γ = ∆ − S − J should be a function of S only through its dependence on the conformal spin s. Since the scaling dimension ∆ is 10 ∆ = S + J + γ(S, J) , (1.13) that then leads to the following "functional relation" for γ γ(S, J) = f(s; J) = f S + 1 2 J + 1 2 γ(S, J); J . (1.14) Without further information, this relation is nothing more than a change of variable, since, at least in perturbation theory, it is always possible to compute the function f in terms of the 7 Here we have in mind the minimal anomalous dimension in the band [3]; the J 3 ln 2 S terms appear [28] in non-minimal dimensions even for low twists. We thank G. Korchemsky for this clarification. 8 We thank A. Rej for a discussion of the structure of these terms at 1-loop order in sl (2) sector (see also [3]). 9 There is a numerical evidence for the presence of k 2 (J ) ln 2 S term for J ≥ 4 from the analysis of the corresponding 1-loop Baxter equation (A. Rej, private communication). For a general method to derive higher order terms in 1/S expansion at fixed J see [29]. Let us mention also that coefficients in large S expansion beyond cusp anomaly one may be also controlled by integral equations like the BES [1] one (see in this connection [54]). 10 Here we assume that Φ in the operator tr(D S + Φ J ) is a scalar field (as is the case in the sl(2) sector of SYM theory). The relation between the notation used in [15] and ours is: N → S, L → J, J → C and j → s. anomalous dimension γ(S, J). Nevertheless, the above reasoning suggests that f could be more fundamental than γ. 11 This is what we shall assume below when referring to the functional relation.
Suppressing the dependence on J in γ and f we may write this functional relation simply as 12 At weak coupling γ(S) = ∞ n=1 γ n (S)λ n ; expanding the coefficients γ n in large S (for fixed J) one finds that for all explicitly known perturbative gauge-theory results one gets the same expansion as in (1.3) where the coefficients f, f c , f 11 , ... are power series in λ. Remarkably, the structure of this expansion turns out to be perfectly consistent with the functional relation (1.15): the function f starts with a logarithmic term (and is "simpler" than f , i.e. has no ln n S S n terms in it) so that the coefficients of the leading ln m S S m terms are all determined by the scaling function f [15,26,4] The universality (i.e. twist and flavor independence) of the scaling function or cusp anomalous dimension f thus implies the universality of all of the coefficients f mm in (1.16) as they are simply proportional to f m+1 , These should be understood as relations between the functions f mm (λ) and f (λ) defined as power series in λ.
Let us note that anomalous dimensions of operators with twist higher than two occupy a band [3], the lower bound of which is the minimal dimension for given S and J. The relation 11 Apart from conformal invariance, for twist 2 case, this conclusion is also expected on the basis of the QCD origin of the functional relation (1.14) [13]. In that context, the function f for twist 2 operators is closely related to a special reformulation of the parton distribution functions evolution equation which aims at treating symmetrically the space-like channel of deep inelastic scattering and the time-like crossed channel describing e + e − annihilation [14]. In particular, the functional relation (1.14) turns out to be predictive within the large S expansion (1.12) because the function f happens to be simpler than γ and, in particular, it does not contain ln m S/S m terms [16,15,26]. 12 In the approach of [13,16], which recently received a nice confirmation in [31], this relation follows from a suitable modification of the evolution equations governing the renormalization of the twist operators.
(1.17) is expected to apply for the minimal dimension in the band. Interestingly, as was found at weak coupling, a similar relation also holds for the excited trajectories [4]. This is also what we shall see at strong coupling on the example of the spiky string in section 2.3 (see (2.32)).
In Appendix F below we will summarize the known perturbative expansions for the minimal anomalous dimensions of twist 2 and twist 3 operators of various flavors, as obtained from the asymptotic Bethe ansatz of [32]. These results are indeed consistent with (1.17),(1.18) and thus with the universality of the f mm coefficients in (1.16).
As for the subleading ( ln k S S m , k < m) terms in (1.16), their coefficients are, at least partially, controlled by special properties of the function f in the functional relation (1.15). Indeed, it was observed on many examples that the function f should satisfy a "parity preserving relation" or reciprocity property. 13 This property implies that the large S expansion of f(S) should run in the inverse even powers of the quadratic Casimir of the collinear SL(2, R) group, namely, [15] f(S) = ∞ n=0 a n (ln C) C 2n , (1.19) where C is the "bare" quadratic Casimir defined in terms of the "canonical" value of the con- The reciprocity property (1.19) of the function f in the relation (1.15) then imposes constraints on some of the coefficients of the subleading terms in the expansion (1.16): where dots stand for similar expressions for f 31 , f 30 , f 55 , ..., f 50 , etc. 15 Again, these equations relate functions f mk (λ) defined as power series in λ. For twist J = 2 we get simply The name reciprocity has its origin in the the formulation of this property, for twist 2 case, in terms of the Mellin transform: . For twist larger than 2 the "parity preserving relation" was suggested in [15] as a more appropriate name. Here for simplicity we shall not make this distinction and will often refer to reciprocity when implying the "parity invariance" property in (1.19), (1.20) below.
These so called MVV [12] relations were first observed for twist 2 QCD anomalous dimensions up to 3 loops. The large S expansions for the known twist 2 and twist 3 SYM anomalous dimensions that we will present in Appendix F are indeed consistent with these relations, i.e.
with the reciprocity property of the function f. 16 It is natural to expect that the functional relation and the reciprocity property should hold also at higher orders in small λ expansion. Since the planar perturbation theory should be convergent, they should then also be visible at strong coupling [15], i.e. in the large spin expansion of the corresponding semiclassical string energies.
One may also wonder if the reciprocity property may apply to higher twist operators above the lower bound of the band [3,4]. If that were the case, it could then be checked also at strong coupling on the example of the spiky string solution of [34].
The agreement in the structure of the large S expansion found in perturbative gauge theory and in perturbative string theory is already quite remarkable. This agreement is non-trivial since, as was already mentioned, the gauge-theory and string-theory perturbative expansions are organized differently: the gauge-theory limit is to expand in small λ at fixed S and then expand the λ n coefficients in large S, while the semiclassical string-theory limit is to expand in large λ with fixed S = S √ λ and then expand the 1 ( √ λ) n terms in E in large S. Even assuming these limits commute (which so far appears to be verified only for the leading universal ln S term) the reason for the validity of the functional relation (1.15) and, moreover, of the reciprocity property (1.19) is obscure on the semiclassical string theory side.
The functional relation (1.15) for the anomalous dimensions of Wilson-type operators on the gauge theory side was argued [15] to follow from the invariance under the collinear SL(2, R) subgroup of the conformal SO(2, 4) group. Given that this argument is based on the conformal symmetry, one may think that it should then apply also on the string theory side. However, as we will review in Appendix A, the realization of the conformal group on states represented by classical spinning string solutions in global AdS 5 coordinates is a priori different from the one used on the gauge-theory side (which is based on the collinear subgroup), so that the direct connection is not obvious. The reason for the reciprocity property on the string theory side is even far less clear. 16 Three-loop tests of reciprocity for QCD and for the universal twist 2 supermultiplet in N = 4 SYM were discussed in [15,16]. A four-loop test for the twist 3 anomalous dimension in the sl(2) sector was performed in [26]. The case of twist 3 gauge field strength operators was analyzed in [33] (at three loops) and in [18] (at four loops). In the latter paper it was also proved that even the wrapping-affected four loop result for the twist two operators [17] is reciprocity respecting. As we shall discuss below (extending earlier observations in [8,15]), not only the structure of the large spin expansion on the string theory side happens to be the same as on the gauge theory side but also its coefficients are indeed consistent with the functional relation and the reciprocity for the minimal dimension case represented by the folded spinning string. This will be demonstrated at the classical as well as 1-loop string theory level.
We shall also show that the functional relation but not the reciprocity appears to apply also to the case of the classical spiky string solution.
The rest of this paper is organized as follows.
In section 2 we shall first consider the large spin expansion of the classical energy of folded spinning string in AdS 5 and show that the large spin expansion has the structure (1.2) and the functional and reciprocity relations between the coefficients are satisfied. We shall then include (in section 2.2) the dependence on the angular momentum J in S 5 in the "long string" limit (J ≪ S). In section 2.3 we shall study the same large spin expansion for a spiky string in AdS 5 ; in this case we shall find that the reciprocity condition is violated which should be related to the fact that the corresponding operator has higher than minimal dimension for a given spin.
In section 3 we shall return to the case of the folded spinning string in AdS 5 (i.e. assume that J is negligible compared to S) and compute the 1-loop correction to the energy expanded in large S, determining corrections to several leading coefficients. As result, we shall verify that the string 1-loop corrections preserve the structure (1.3) of the large spin expansion and, moreover, that the reciprocity condition is satisfied beyond the string tree level.
In Appendix A we shall make some comments on relation between different realizations of conformal group. In Appendix B and C we shall review the folded spinning string solution and discuss long-string or large-spin expansions used in the 1-loop computation in section 3. In Appendix D we shall give details of large spin expansions for (S, J) string considered in section 2.
In Appendix E we shall discuss some consequences of the functional relation and the reciprocity at strong coupling, pointing out a subtlety in the definition of the latter in the semiclassical string expansion. In Appendix F we shall summarize the known weak coupling planar SYM results for the large spin expansion of twist 2 and 3 anomalous dimensions up to 4-loop order in the 't Hooft coupling.
2 Large spin expansion: classical string theory 2.1 Folded spinning string with J = 0 We shall start with a discussion of the limit when the S 5 momentum J of the string state can be ignored, i.e. we shall concentrate only on the AdS 5 spin S dependence of the string energy.
This is the limit when the twist of the gauge theory operator is sufficiently small compared to the Lorentz spin.
We review the folded spinning string solution [6] in Appendix B. The integrals of motion are the energy E = √ λ E and the spin S = √ λ S, which can be expressed in terms of the elliptic functions E and K of an auxiliary variable η 17 To find the energy in terms of the spin one is to solve for η. Here we are interested in the large spin expansion which corresponds to the long string limit (when the string ends are close to the boundary of AdS 5 ). For such long string one has η → 0. Solving (2.2) for small η and substituting it into (2.1) one finds for E as a function of S Furthermore, it is possible to verify that the expansion of E − S also satisfies the reciprocity property (1.19), (1.22). The large S expansion of the function f (its leading term in the strongcoupling limit) is much simpler than that of the anomalous dimension E −S in (2.1) and contains only even powers of Equivalently, we find that the MVV-like relations (1.22) are satisfied. 18 A more systematic analysis of the reciprocity (parity invariance) property of the function f is possible with the help of an integral representation for it. Using that (1 where the contour Γ encircles the pole of the integrand and prime stands for derivative. 19 It is natural to replace the variable ω in (2.6) with the expression (2.2) for the semiclassical spin is the "conformal spin" expressed in terms of the semiclassical quantities. The integral then gives the functionγ evaluated at the zero of the denominator; this is the same as the statement that the anomalous dimension as a function of the Lorentz spin is, effectively, a function of the conformal spins.
To verify the reciprocity property of the functionf(S) in (2.7) it is useful to redefine the variable η as 20 η → −1 + 16η + 1 + 256 η 2 and examine the large S or small η limit of the expressions. One finds thatγ(η) is a series in even powers of η γ(η) = − 1 + ln η π + 4(ln η + 12) π η 2 − 6(62 ln η + 777) π η 4 + ... , (2.8) while the expression for the conformal spin runs in odd powers of η s(η) = 1 8πη From the equation for the pole of the integrand in (2.7),s − S = 0, one can find the parameter η in terms of the spin S, concluding that it is given by a power series in odd negative powers of 18 The definition of reciprocity condition in string semiclassical expansion is discussed in Appendix E. 19 The expression that multipliesγ in the integrand has residue 1, so that the integral isγ evaluated at the pole ω = S − 1 2γ . Then defining x = S − 1 2f (S) we have 2S − 2x =γ which coincides with the equation for the pole with x = ω. Note that assuming f exists, one can formally reconstruct it from γ using [15] This relation also arises by expanding the denominator in (2.6) in smallγ and integrating the resulting series. 20 This choice is not unique. An analogous transformation was used in [15].
S. As a result,f(S), which is same asγ(η) evaluated at the pole, should also run only in even negative powers of S or C = C √ λ (cf. (1.20)). The above discussion has a straightforward generalization to the multifolded spinning string case. The leading terms in the large S expansion of the energy of a string with m folds are (see In this case it is possible to show again that the large S expansion is consistent with the reciprocity property.

Folded spinning string with J = 0
Let us now consider the case when the S 5 angular momentum of the string is not negligible compared to S, i.e. when the string state is dual to an operator with large spin S and large twist J. The corresponding charges are the energy E = √ λ E and the two angular momenta 10,35]: Here κ and ω (or η) are parameters of the classical solution which should we eliminated to find E as a function of S and J .
We will be interested in large S expansion with S ≫ J since only in this case the expansions like (1.16),(1.19), i.e. going in the inverse powers of S with the coefficients being polynomials in ln S, will apply (see also [10,15]).
In the large S ≫ J or long string limit, when η ≪ 1, one should distinguish between "small" or "large" J cases [10,36]. In the "slow long string" approximation (corresponding to taking S to be large with ℓ ≡ J ln S fixed and then expanding in powers of ℓ) the leading terms in the semiclassical energy read (cf.
whereS ≡ 8πS, and dots stand for higher order corrections depending on J . 21 In the case of "fast long string", when ln S ≪ J ≪ S, the corrections to the energy read Dots in the square brackets indicate corrections in 1/Ŝ, corrections in 1/ lnŜ can be added in the round brackets and terms like ln(lnŜ) have been neglected.
The leading terms here can be summed up as [3] where ln S J ≪ 1 plays the role of an expansion parameter. Notice that in contrast to the slow long string case where the expansion (2.13) has the same structure as in (1.12), in the fast long string case (2.14) we get higher powers of ln S not suppressed by S, and so this case (cf. also its discussion in Appendix D) is somewhat outside our main theme here.
To study the properties of the subleading corrections, one may again make use of the integral representation for the functional relation as in (2.6). The discussion will apply to both the "slow" and the "fast" long string limits. Here the "conformal spin" iss = 1 2 (S + E) = S + 1 2 J + 1 2γ , while the "semiclassical" value of the Casimir operator in (1.20) is C = C √ λ ≈ S + 1 2 J . Then the integral in (2.7) can be written as After a redefinition of η one can then show that the expansion of f in large C runs only in even negative powers of C. Some details are given in Appendix D. In the kinematic region of "fast" long strings, with 1 ≪ ln S ≪ J ≪ S, this parity invariance property was already demonstrated in a closely related way in [15].

Large spin expansion of energy of a spiky string in AdS 5
Let us now consider the spiky spinning string in AdS 5 [34], and find corrections to the leading ln S term in its large spin expansion.
The integrals of motion here are the difference between the position of the spike and of the middle of the valley between the two spikes, the spin and the energy [34] 22 where n is the number of the spikes and The string is rigidly rotating with the radial coordinate being ρ = ρ(σ), with ρ 0 and ρ 1 as its minimal and maximal values (positions of the bottom of the valley between the spikes and the spikes themselves). ρ 0 and ρ 1 are related by the condition (2.17). Solving for the remaining free parameter gives E = E(S, n).
The large spin limit corresponds to ρ 1 → ∞, i.e. to the case when the ends of the spikes approach the boundary of AdS 5 . Let us set and expand in y → 0. Then, at leading order, ∆θ = π n = arcsin 1 u 0 +O(y) implies u 0 = cosh 2ρ 0 = csc π n and i.e.
This is the result already found in [34], which reduces to the case of the folded string when n = 2. 23 Expanding further near y ≃ 0 one gets In the case of the multiply folded string with n spikes multiplying formulas one should multiply (2.19) by the number m of the folds, and use that in this case ∆θ = π n m . As a result, one is simply to substitute n → n m. 23 For n = 2 we have ∆θ = π 2 (i.e. the angle between spikes is π), and thus ρ0 = 0 or u0 = 1.

36)
24 Note also that the form of p1 is consistent with the interpretation of the subleading term in the energy in [27].
where q 5 , q 6 are non-zero for n = 2. The expansion (2.33), even if considerably simpler compared to the energy (2.26), is not parity invariant under S → −S. It is interesting though that higher powers of ln S appear to cancel in the subleading terms in (2.33). 25 The parity invariance is restored in the case of the folded string when n = 2, where indeed (2.33) coincides with (2.5).
This breakdown of parity invariance for a string with n > 2 spikes is not totally surprising, as such spiky string should correspond to an operator with non-minimal anomalous dimension for a given spin, while the reciprocity was checked at weak coupling only for the minimal anomalous dimensions. Indeed, anomalous dimensions of operators of twist higher than two with trajectories close to the upper boundary of the band also do not respect the reciprocity as was seen recently in the twist three case at weak coupling in [4].
It is interesting that our strong-coupling result (2.32),(2.33) has close similary with weakcoupling one found for n = 3 in [4]: the functional relation (2.32) is still satisfied, and the parity invariance is broken at level 1/S. Interestingly, the 1/S coefficient ∼ nq 2 in (2.33) (cf. (2.27)) is proportional, for n = 3, to √ 3, which is, suprisingly, the same factor appearing also in the corresponding expression at weak coupling [4]. 26 In general, this coefficient should be a function of λ interpolating from weak to strong coupling but its dependence on n might be the same for any λ.

Large spin expansion of folded string energy: 1-loop order
Let us now go back to the folded spinning string case of section 2.1 and compute the leading 1-loop corrections to its energy (2.3) in the large S expansion. We shall follow the general approach for computation of quantum string corrections developed in [10] where the 1-loop shift of the ln S term was found. 27 We shall find the 1-loop corrections to the subleading terms in (2.3) by applying a perturbative procedure similar to the one used in [9] in the small spin expansion case.
Our aim will be to verify that: (i) the structure of the large spin expansion (2.3) remains the same also with the 1-loop corrections included, and (ii) the constraints on the coefficients imposed by the functional relation and the reciprocity remain to be satisfied at the 1-loop order.
The fluctuation action in the conformal gauge expanded to quadratic order in fluctuations 25 This feature of thef-function is in a marked contrast with the anomalous dimension, whose large S expansion includes growing powers of ln S in the coefficients of 1/S n terms. This reduction of singularity of the large S expansion off was observed also at weak coupling [26,18,4]. 26 We thank G. Korchemsky for this observation. 27 The 2-loop correction to the scaling function was found in [38,39]; a generalization to non-zero J was considered in [36,40] (parallel results from the Bethe ansatz were found in [41,42]).
near the folded spinning string solutionS = − √ λ 4π dτ 2π 0 dσL has the following bosonic part (see [10] and Appendix B) where and can be interpreted as describing a system of 4+4 2d Majorana fermions with σ-dependent mass ρ ′ . As explained in [9], after squaring the corresponding "Dirac" operator, the fermionic contribution to the 1-loop partition function can be represented as In the leading-order computation in the long-string limit that we are going to discuss below the term ±ρ ′′ in the effective fermionic mass squared term in (3.4) can be ignored: as follows from (B.12), ρ ′′ = O(η) and since according to (3.4) half of fermions has +ρ ′′ and half −ρ ′′ in their mass term, the leading O(η) contribution to the partition function can come only from the ρ ′2 term in ∆ F ± . We shall assume this when writing the fermionic contribution below.
Switching to euclidean signature (τ → iτ ), the 1-loop correction to the energy can be found from the 2d effective action Since the spinning string solution is stationary, both the bosonic and the fermionic fluctuation Lagrangians do not depend on τ ; thus, as in [9], we may compute the relevant 2d functional determinants by reducing them to 1d functional determinants using where m 2 is a generic mass term which may depend on σ.
Given that ρ(σ) is a complicated function (see (B.4)), we are unable to determine the fluctuation spectrum exactly, and, as in [9], we will resort to perturbation theory in 1 S or in parameter η determining the maximal string length (see Appendix B and C). In (3.5) we have from (B.6) Γ 1 will also be expected to have expansion in powers of η ∼ 1 S (see (B.9)) with the coefficients containg powers of ln η.
To proceed, we need to expand the fluctuation Lagrangian in small η corresponding to large S. Some relations needed below can be found in Appendix B. Let us first perform (as in [36]) the following rotation (t,φ) → (ξ, χ): Then the fluctuation Lagrangian takes the form The reason for this rotation is that in the subsequent small η expansion the bosonic fluctuation Lagrangian at order O(η 0 ) will become σ-independent, i.e. will have constant coefficients as at the leading order in long-string expansion considered in [10,36]. 28 Expanding the solution for ρ(σ) and the parameters κ and w in small η (see Appendix B), the bosonic fluctuation Lagrangian becomesL B =L 0 + ηL 1 + ..., wherẽ and As already mentioned, the 1-loop effective action can be expressed in terms of 1d functional determinants (with ∂ 0 → iω, see (3.6)). We shall denote the quadratic fluctuation operator in 28 As discussed below, we shall ignore the contribution of the turning points at σ = π 2 and 3π 2 and will treat the fluctuation problem separately on each "quarter-string" interval. the coupled (χ, ξ,ρ) sector as Q ω . SinceL 0 has constant coefficients, the leading part of the fluctuation operator coming fromL 0 can be written as (3.12) The 1-loop correction to the effective action is then (3.14) ω is the next to leading order coupled operator from (3.11) (3.15) Here and we performed the Fourier transform in σ, i.e. replaced ∂ 1 → in, n = 0, ±1, ..., as appropriate for fluctuation fields which are 2π periodic in σ.
Our aim will be to determine the 1-loop correction to string energy to order η by computing As in [9] the first, second and fourth terms in (3.13) can be computed to order O(η) using that While in [9] a similar contribution to the effective action happened to vanish since it was proportional to the sum of squares of fluctuation masses, 29 here this leading term is no longer zero as in the present case the expansion is around a nontrivial string background with different propagators for different string fluctuations.
In (3.11) we used the expansion (B.11) of the solution ρ(σ) in small η. As discussed in Appendices B and C, this expansion breaks down at the turning points where subleading terms are of the same order as the leading term. As in the computation of the leading order in [10], here we shall assume that one can ignore the contributions from the turning points. The classical folded string solution is built out of four parts making up the closed string (e.g., the expansion (B.11) used in (3.11) is defined for 0 ≤ σ ≤ π 2 ). 30 The closed string fluctuations by definition must be periodic in 0 ≤ σ ≤ 2π.
We shall assume that we can treat the problem "piece-wise" also at the fluctuation level.
Direct implementation of this may effectively bring back the turning-point contributions. We shall parametrize our current lack of control of such terms by including the possible contribution with an arbitrary coefficient in the final result.
We shall split the integral over σ as follows Considering the first interval (0, π 2 ), the order η contribution of the decoupled boson β u in (3.1) and (3.10),(3.11) can be obtained as Similarly, for the fermionic contribution (the first term in (3.13)) we get (3.23) 29 The mass sum rule implies the 1-loop UV finitness of the superstring; it was proven in general for any string solution in [43]. 30 Note that for the second and the fourth σ intervals where ρ decreases we need to use the minus sign in (B.2).
For the coupled part one finds .

(3.24)
The contributions of the other three intervals of σ are the same.
Collecting the above results we observe that the final expression for the order η term in the effective action Γ 1 is UV finite. Moreover, the part that does not contain the sinh(πκ 0 ) factor is IR finite, i.e. the non-trivial potentially IR divergent contributions of the two unphysical The coefficient of the part proportional to sinh(πκ 0 ) given by n C n is UV finite but formally has an IR singular contribution. 33 This term should be an artifact of our computational procedure related to the problem with expansion in η in (B.11) near the turning points (see Appendices B and C). Insisting on omitting the turning point contributions means that we should drop this IR singular ∼ sinh(πκ 0 ) term, and this is what we will do below. 34 We believe that in a more systematic treatment that consistently treats the turning point contributions such terms will be automatically absent (equivalently, in our present form of the expansion, such terms should resum away, see also the discussion in Appendix C). 31 Their flat-space contribution is cancelled against the conformal gauge ghost contribution. 32 Let us mention that if we perform the sum over n first, then using the residue theorem in the integral over ω we arrive back at the same sum as below. 33 The IR singular contribution goes away if one separates it before doing the integral over ω. Then we get for the large κ0 behaviour of P n Cn: 2 P ∞ n=1 Cn = κ0(ln κ0 − 6 ln 2 − 3 2 + γE) + O((κ0) 0 ). 34 To justify the expansion in (B.11) we need to omit the turning point contribution. That can be done by shifting the upper limit of the integration over σ in (3.22)-(3.24) to π 2 − ǫ, ǫ → +0. Then the coefficient of Cn will become sinh[(π − 2ǫ)κ0] ∼ η −1+ 2 π ǫ and thus is subleading compared to the contribution of order O(η 0 ).
Computing the remaining n A n contribution in (3.25), (3.26) using the Euler-MacLaurin formula The contribution of this term in Γ can be easily computed as a product over integer n of a matrix determinant (after ∂ 1 → in).
Since detP ω = −det 3 (−∂ 2 1 + ω 2 ) we may write the relevant contribution from (3.13) as Since ln det(−∂ 2 1 + ω 2 + κ 2 ) = ∞ n=−∞ ln(n 2 + ω 2 + κ 2 ), doing the integral over ω we finally obtain the 1-loop correction to the string energy to order O(η) as This turns out to be a direct generalization of the leading-order result of [10] where κ should be replaced by κ 0 in the fluctuation mass terms (but not in the overall 1 κ factor due to t = κτ ). Using again the Euler-MacLaurin formula to transform the sum into an integral we find 35 This is, of course, in agreement with the result of [10] and also, for the subleading term, with ref. [44]. 36 35 Note that the sum of (3.26) is minus the derivative over κ0 of the sum in (3.33), which explains why the corresponding coefficients are closely related. 36 Ref. [44] considered, following [10], the formal sum (3.32) with κ0 → κ and with the n = 0 term omitted (this term was omitted in [10] since there it was subleading in the infinite κ limit). As a result, the expression in [44] contained an extra (minus "zero mode") term 3 − √ 2. Note that the coefficient in the exponent of the leading e −2πκ 0 exponential correction [44] is determined by the mass of the lightest mode -in the present case of the fermionic mode. The exponential term in the square bracket has also a prefactor of √ κ0.
As we shall argue in Appendix C, an additional contribution that may come from near turning point regions can be parametrized as follows: where c is an undetermined constant (we included factor of π for convenience).
Inverting the relation between S and η in (B.9) to order O(η) we get (3.36) which, plugged into (3.7), gives 37 ln S S 2 should be subleading and should be ignored in the approximation we considered above where we dropped terms of higher order in η at earlier stages; these terms are expected to cancel out in a more systematic treatment. 38 Thus we find that the 1-loop correction to the folded string energy to order O( ln 2 S S 2 ) can be written in the same form as the classical energy (2.3), i.e. as in (1.2),(1.3) The contribution from E Thus finally we obtain for the full 1-loop coefficients Note that in the expression for the energy in (3.5) we need to keep κ to order O(η) to get the correction in E1 also to order η ∼ 1 S . 38 The role of these subleading terms and their possible resummation remains to be studied in more detail.
The functional and reciprocity relations in (1.18), (1.22) at strong coupling are (see discussion in Appendix E) They imply that the coefficients in (3.38) should obey (see (E.6),(E.7)) Recalling the values of the leading coefficients at the classical level in (2.3)

Note Added
As we have learned (N. Gromov, private communication) an independent way of evaluating the 1-loop correction to the folded string energy based on the algebraic curve approach to extracting fluctuation frequencies [53] leads to the value c = 6 ln 2 + π for the undetermined coefficient in (1.6),(1.9),(1.11),(3.35), (3.41), (3.42). This c contribution changes the 1-loop coefficient in (1.6) from 3 ln 2 to −3 ln 2− π. It is interesting to note that for c = 6 ln 2+d, where d is a constant not involving ln 2, the coupling redefinition √ λ → √ λ + 3 ln 2 (suggested to us by G. Korchemsky) removes all ln 2 terms from the leading 1-loop coefficients in (1.5)-(1.11), just as it did in the cusp anomaly coefficient in [37]. Namely, then we get

Appendix A: Comments on conformal algebra realizations
Starting with a conformal theory in R 1,3 with the standard SO(2, 4) conformal group generators P m , M mn , K m , D (m, n = 0, 1, 2, 3) one may define the collinear SL(2, R) subgroup as generated by the following light-cone components [30]: Then the standard spin is S = Σ 12 = M 12 , the conformal spin is S ′ = Σ 34 = 1 2 (K 3 − P 3 ), and the conformal energy is the rotation generator in the 05 plane, i.e. the global AdS 5 energy, E = Σ 05 = 1 2 (K 0 + P 0 ). In general, the energy E of a string state in global AdS 5 space with boundary R × S 3 should be equal to the energy of the corresponding SYM state on R × S 3 . Through radial quantization (and analytic continuation) this state may be associated to a local operator in R 1,3 that creates it. The AdS 5 energy E = Σ 05 or conformal Hamiltonian generates an SO(2) subgroup while the dilatation operator D = Σ 54 generates an SO(1, 1) subgroup of SO(2, 4). 39 After the 39 Their eigenvalues happen to be the same since the two representations (the unitary one classified by SO(4) × SO(2) and the one classified by SO(4) × SO(1, 1)) are related by a global SO(2, 4) similarity transformation (see, e.g., [46]).
Euclidean continuation of the embedding coordinate Y 0 → iY 0E (to allow for the mapping from R × S 3 to R 4 ) one may exchange Y 0E with Y 4 which exchanges the generator Σ 54 = D with E = Σ 05 = 1 2 (P 0 + K 0 ). To relate the SO(1, 2) subgroup of SO(2, 2) which is a symmetry of global AdS 3 subspace of AdS 5 where the folded spinning string is moving to the collinear SL(2, R) subgroup classifying the operators like tr(ΦD S + Φ) one is also to perform an additional analytic continuation that interchanges the euclidean (12) plane with the hyberbolic (+−) plane. Since different choices are formally related via SO(2, 4) transformations and a re-identification of the generators one may expect that the two representations should be equivalent. 40 Still, the representations of SO(2, 2) or string states in AdS 3 are naturally labeled by (E, S), and the relation to SO(1, 2) labels 1 2 (E+S) does not appear to be natural, unless one is interested in the large spin expansion (see in this connection [47,55]). That relation may possibly be made more explicit by choosing a different set of coordinates in global AdS 5 in which the boundary is not R × S 3 but AdS 3 × S 1 (see [48] where such coordinates in the boundary theory where used to explain the leading E ∼ ln S behaviour).
Let us mention also that the relation (1.23) or E − S = f(E + S) is reminiscent of a light-cone gauge expression, where f would be a light-cone Hamiltonian (cf. [49,50,47]).

Appendix B: Review of folded string solution with J = 0
In this Appendix we review the folded spinning string solution in AdS 3 [7,6] and consider its large spin expansion (see also [10]).
The condition satisfied at the turning point ρ 0 at σ = π 2 is ρ ′ ( π 2 ) = 0. To construct the full (2π periodic) folded closed string solution one should glue together four such functions on π 2 intervals to cover the full 0 ≤ σ ≤ 2π interval; e.g., for π 2 < σ < π we have The expressions for the parameter κ, the energy and the spin in terms of η are In this paper we are interested in the large spin or long string limit, i.e. small η expansion.
Since in section 3 we compute 1-loop correction only to order 1 S we will only need expansions to order O(η). Expanding κ, E, S in small η we obtain 41 Expanding the solution (B.4) in small η we obtain, for 0 < σ < π 2 , sinh ρ = sinh(κσ) − To leading order when η → 0, κ 0 → ∞ the string touches the boundary of AdS 5 (ρ 0 = ∞) and the solution can be approximated (away from the turning points) by simply ρ = κ 0 σ. This limiting case proved to be a useful framework for computing 1-and 2-loop string corrections [10,36,38,39]. At the next order in small η expansion the "ends" (turning points) of the string are close to the boundary but no longer touch it.
We should add a word of caution about the use of the formal expansion in (B.10) or (B.11).
Notice that it goes in powers of η with coefficients containing ln η and is not, strictly speaking, valid close enough to the turning points. Indeed, for σ = π 2 we have sinh(2κ 0 σ) = sinh(πκ 0 ) ≈ 1 2 e πκ 0 ∼ η −1 and similarly cosh(κ 0 σ) = sinh( π 2 κ 0 ) ∼ η −1/2 . Hence at the turning point the order 41 Let us note that these expansions were found using pre-Mathematica 6 versions of Mathematica (Mathematica 6 apparently has some bug leading to inconsistent expansions for some elliptic and hypergeometric functions).
If σ is slightly away from the turning point the subleading terms are smaller than the leading term but then the expansion and the contributions to the energy need to be resummed. The resummation at the level of the string profile ρ(σ) is completely equivalent to its expansion near the turning point σ = π 2 , as will be discussed in Appendix C. In section 3 we ignored the regions near the turning points and thus uses the formal expansion (B.11) to compute it. Similar assumption was made in [10] in the computation of the 1-loop shift of the coefficient of the ln S term in the energy (where it was indeed justified). A reason behind this assumption is that the masses of string fluctuations in (3.2) depend on ρ ′2 which is small near the turning points where ρ ′ = 0. Thus the correction to the leading result of [10] should mainly come from the "internal" parts of the σ-interval, where the expansion (B.11) is justified.
We shall comment more on this point in Appendix C. In section 3 we included a possible term which we may thus miss with an aribitrary coefficient yet to be deterined.
For the computation of the 1-loop correction in section 3 we will need the following expansions Notice again that these expansions are formally invalid at the turning point (but are justified away from it). This is evident, e.g., in (B.12) where the leading term does not vanish at the turning point where one should have ρ ′ = 0. What happens is that at σ = π 2 the leading term κ 2 0 gets cancelled agianst the sum of subleading terms which all are of the same order, i.e. are proportional to κ 2 0 (see Appendix C). The masses appearing in the fluctuation Lagrangian in section 3 are then expanded as follows Appendix C: Resummation of "long string" expansion near the turning points Continuing the discussion of the previous Appendix B here we will show that it is possible to resum systematically the terms ∼ e 2 n κ 0 σ appearing in the formal η expansion of ρ(σ) in (B.10). These terms are potentially dangerous since they scale at the turning point σ = π/2 as e n π κ 0 = (16/η) n and spoil the perturbative expansion.
In the first "quarter-string" interval 0 ≤ σ ≤ π 2 , the function ρ(σ) obeys the differential equation (B.2) with plus sign and ρ(0) = 0. Its formal expansion in powers of η (treating κ 0 in (B.7) as a constant parameter) reads (cf. (B.10)) Since here, in fact, κ 0 = − 1 π ln η 16 the hyperbolic functions potentially reduce the true order in the η expansion. The dangerous terms can be easily identified by setting and neglecting exponentially suppressed terms in the above expansion. The next-to-leading (NLO) result which is correct at O(η) can be written All terms (η t 2 ) k are O(1) at σ = π 2 and the above infinite series need resummation. This can be accomplished as follows. Introducing h(t) ≡ r(t) − ln t we get Taking the large t limit we arrive at the following equation for the leading order term h LO which are the leading terms in r(t).
The NLO approximation h NLO (t) is simply obtained by including an extra piece in the square root and taking into account that the ratio κ/κ 0 has a non trivial expansion in η, Integrating this equation, substituting the necessary terms in the expansion of κ/κ 0 , and neglecting all NNLO terms we then get The expansion of (C.9) in powers of η gives which agrees indeed with the expansion of r NLO in (C.3).
In terms of the string profile ρ(σ) = r(e κ 0 σ ) = h(e κ 0 σ ) − κ 0 σ this result can be written as This expression resums at NLO order the contributions near the boundary point σ = π 2 . This expression is not, of course, expected to be correct near σ = 0, but it must reproduce the exact value of ρ( π 2 ) with order O(η) included. This is true since is in agreement with the exact value of ρ( π 2 ) which is Also, as an additional check, we immediately reproduce that ρ ′ NLO ( π 2 ) = 0.
In order to obtain the resummation in a systematic way and to show that it comes from the behaviour of the string profile around the turning point we can work out the expansion of the differential equation for ρ(σ) around σ = π 2 . To this aim, let us define and solve the corresponding equation (cf. (B.2)) Here the valueρ(0) was left unexpanded. Expanding it consistently we get the final result If we now use the definition of x = κ π 2 − σ in this expression, we get precisely the NLO resummation in Eq. (C.11) plus a new η 2 term which is beyond the order of accuracy of (C.11).
One may wonder if this systematic resummation of ρ(σ) can be used to resum the associated contributions in the 1-loop correction to string energy discussed in section 3. This is not, however, immediately clear. Plugging the expansion of ρ around σ = π 2 in the Q ω operator in (3.13) and denoting the resulting terms with label "fold" to indicate the expansion point, we In the large κ 0 limit, we can make the following replacements ( ∞ 0 dz δ + (z) = 1) After this substitution we can write ω is the same operator (3.12) we found in section 3 in the expansion valid near σ = 0, while the new piece is This is a O(κ 0 ) perturbation over Q (0) ω whose matrix elements are O(κ 2 0 ) (since n, ω ∼ κ 0 in the combined sum and integral like in (3.24)). Unfortunately, higher order terms coming from this term can be estimated to have the same order of magnitude and thus must be resummed. In principle, the contribution from Q (0) ω,fold must be treated exactly and separately, a task which we leave for the future.
Still, it is encouraging to note that a possible non-zero contribution from the near-turningpoint region is expected to change the one-loop energy by a term proportional to η + · · · = 1 + 1 2 π S + · · · . (C. 24) This means that the induced modification of the coefficients in appearing in (3.38) must obey δb 0 = δb 11 = 0 , δb 10 = 1 2 π δb c . (C. 25) Remarkably, this is precisely what is required by the reciprocity conditions in (3.45),(3.46).
Appendix D: Details of large spin expansion for folded (S, J) spinning string In this section we collect some details on large spin expansions used in Section 2.2.
Let us mention also that in the case of the m-folded string the interval 0 ≤ σ < 2π is split into 4m segments: for 0 < σ < π 2m the function ρ(σ) increases reaching its maximal value ρ 0 , then decreases to zero for π 2m ≤ σ ≤ π m , etc. This implies the condition 2π = There is, however, a subtlety in formulating the reciprocity condition in the context of large spin expansion at strong coupling as defined by string semiclassical perturbation theory where all non-zero charges are automatically large at large λ. For example, the case of finite twist J = 2, 3, ... can not be distinguished from the formal case of J = 0. It is usually assumed that the folded string in AdS 5 with zero angular momentum in S 5 describes an operator of small twist, but that can be J = 2 or J = 3, etc. To establish a relation to the definition of reciprocity in weakly coupled gauge theory expansion with finite twist one would need to consider the case of semiclassical (S, J) string and then resum the series for its energy (both in J and in √ λ) so that the limit of finite J would make sense.
Here we shall assume that in checking the reciprocity ( We have verified the validity of these relations for b 11 and b 10 in section 3.
Appendix F: Large S expansions for twist 2 and twist 3 anomalous dimensions at weak coupling Here we shall collect the coefficients of large spin expansion of anomalous dimensions for planar SYM operators of twist 2 and 3, up to four loops in the gauge coupling and up to 1/S 3 order. 43 They are derived from the closed expressions in terms of the harmonic sums that were obtained (mainly exploiting the maximum transcendentality principle and the asymptotic Bethe ansatz), respectively, in [17] (at four loops) for the twist two scalar sector, in [51,17] for the twist three scalar sector and in [33] (at three loops) and [18] (at four loops) for the "gauge" sector.
All these expansions were proven to satisfy the reciprocity property, for a review see [18].
The expansions are indeed of the generic form (1.16) where the coefficients satisfy the relations (1.21) once J (and the flavor index ℓ, see footnote 9 in the Introduction) are fixed accordingly. 43 In the case of twist 3 operators, the anomalous dimensions we will consider here are the minimal in the band.
The coefficients of the leading ln m S/S m terms below are manifestly universal in twist and flavor. 44 As far as these leading terms are concerned, there is no need to explicitly write down the results for the twist two gaugino and gauge sectors, and the twist three gaugino sector.
Indeed, the closed formulas for their anomalous dimensions can be deduced from the one for the twist two scalar case by just shifting the argument of the harmonic sums 45 but such shifts do not affect the coefficients of the leading ln m S/S m terms. It is worth stressing again that this universality, a well-known feature of the leading ln S coefficient (or cusp anomaly), is a nontrivial consequence of the functional relation (1.14), as was noticed in [33] and emphasized in [4]. whereS = e γ E S and the coefficients will be power series inλ = λ 16π 2 . Then one finds: