Generalized Gribov-Lipatov Reciprocity and AdS/CFT

Abstract

Planar 𝒩=4 SYM theory and QCD share the gluon sector, suggesting the investigation of Gribov-Lipatov reciprocity in the supersymmetric theory. Since the AdS/CFT correspondence links 𝒩=4 SYM and superstring dynamics on AdS₅×𝑆⁵, reciprocity is also expected to show up in the quantum corrected energies of certain classical string configurations dual to gauge theory twist-operators. We review recent results confirming this picture and revisiting the old idea of Gribov-Lipatov reciprocity as a modern theoretical tool useful for the study of open problems in AdS/CFT.

1. Introduction and Overview

An intense activity in the study of the duality between the planar, large 𝑁 limit of the 𝒩=4 super-Yang-Mills (SYM) theory with 𝑆𝑈(𝑁) gauge group and the free type IIB superstrings in AdS₅×𝑆⁵ is based on the development of analytic tools that exploit the classical integrability of the string side [1], as well as an internal integrability of the superconformal theory [2, 3]. In the latter case, the scale dependence of renormalized composite operators is governed, even at higher loops, by a local, integrable, super spin chain Hamiltonian whose interaction range increases with the loop order [4, 5]. This fact has firstly set the long range asymptotic Bethe equations of [5] as a natural tool for calculating anomalous dimensions of the gauge single traces operators of the theory. Although the relevant two-particle scattering matrix [6] was determined in a gauge theory framework [7], its tensor structure agrees with perturbative calculations in the gauge-fixed world-sheet theory [8]. Its form is determined by the global symmetry of the two theories, 𝔭𝔰𝔲(2,2|4), up to a phase (dressing factor) for which a crossing-like equation has been proposed [9]. For its solution [10], based also on 1-loop string data [11], an analytically continued weak-coupling expansion has been formulated [12], whose effects on the anomalous dimensions of the twist-two operators remarkably agree with the direct calculation of the four-loop cusp anomalous dimension [13]. As a result, from the asymptotic Bethe equations (ABA) an integral equation for such cusp anomaly (or universal scaling function) has been derived, on which in fact is based one of the most nontrivial tests of the structure of the AdS/CFT correspondence. Its strong coupling solution [14] (see also [15–20]) has been in fact shown to perfectly match the expression for the cusp anomaly up to 2-loops term as computed directly from the quantum superstring [21, 22].

Due to their asymptotic nature, the Bethe equations furnish predictions for the anomalous dimensions that, for “short” operators [23], need to be corrected by wrapping effects [24]. To this aim, a clever generalization of the Lüscher formulas [25] has successfully given the correct finite-size correction in [26, 27] to the asymptotic anomalous dimension derived from the Bethe Ansatz [23], which has been confirmed by a purely field-theoretical calculation [28]. For the complete spectral equations of 𝒩=4 SYM, however, it is believed that thermodynamic Bethe Ansatz (TBA) methods ought to be applied, as has been initiated conjecturing a Y-system, which should yield anomalous dimensions of arbitrary local operators of planar 𝒩=4 SYM [29–31], and TBA equations for string and gauge theory [32, 33]. Relevant tests of these proposals have been already carried on [34–36], which however, in the case of short operators anomalous dimensions at strong coupling [35], still have to find a full numerical agreement with purely string theoretical computations [37, 38]¹ and might need further elaboration [39, 40].

To the purpose of furnishing closed formulas for anomalous dimensions which might check the TBA proposals at high orders of perturbation theory, the asymptotic Bethe equations, corrected with generalized Lüscher formulas and further inputs, still stand as a powerful tool for multiloop calculations [34, 41]. The class of operators mostly relevant in this framework are the twist operators, also named quasipartonic [42–44]. These are single-trace operators constructed with an arbitrary number of light-cone derivatives acting on the fundamental fields (scalars, gauginos, or gauge fields). Their anomalous dimension depends on their spin (total number of derivatives), and their interest relies on their similarities with the QCD twist operators arising in the analysis of deep inelastic scattering [45].

It is a general fact that, while 𝒩=4 SYM and QCD are in many details different, a compared analysis of their properties has been crucial for a deeper understanding of both of them. Integrability itself appeared for the first time in four-dimensional gauge field theories in a QCD context, in the high-energy Regge behavior of scattering amplitudes, and in the scale dependence of composite operators [46–49]. About conformal symmetry, unbroken in QCD at one loop, it does not appear to be a necessary condition for integrability, as discussed in [50–53], but it certainly plays an important role by imposing selection rules and multiplet structures. A notable common issue between 𝒩=4 SYM and QCD is their infrared structure [54, 55], and it is believed that QCD would benefit a lot from an ultimate all-loop solution of its superconformal version, since this would provide a representation for the “dominant” part of the perturbative gluon dynamics [56].

A remarkable example of such an interplay between 𝒩=4 SYM theory and QCD in the framework of integrability is the maximum transcendentality principle [57–61], according to which the anomalous dimension of twist-two operators at 𝑛 loops is a linear combination of generalized harmonic sums of transcendentality 2𝑛−1. The principle has been the key via which closed multiloop expressions for the anomalous dimension of special twist operators have been derived [6, 23, 34, 41, 62–66] and has been independently confirmed in a space-time framework [67] as well as exploiting the Baxter approach [68]. A second crucial connection is the relationship to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach [69–77] for describing high-energy scattering amplitudes in gauge theory, which furnishes a prediction for the pole structure of the analytically continued anomalous dimensions of twist operators. The (supersymmetric generalization of the) BFKL equation appears to be a testing device for any conjecture on the exact higher-loop spectrum of anomalous dimensions in the 𝒩=4 model, and in fact it was determinant to state both the failure of Bethe equations in describing the spectrum of short operators [23] as well as the correctness of the full result including the wrapping correction [78].

In this Review, we will report on another fascinating and as yet not fully explained link between QCD, 𝒩=4 SYM, and string theory. This is centered on the so-called reciprocity and consists in a surprising pattern that emerges in studying all the available anomalous dimensions of twist-two operators in QCD, their analogue in 𝒩=4 SYM, together with the energies of their dual string configurations. The reciprocity condition is a constraint on the large spin behavior of a transform of the anomalous dimension, which should run in even negative powers of the Casimir of the collinear group SL(2;ℝ). This constraint, arising in the QCD context, has been presented in [79, 80] as a special (space-time symmetric) reformulation of the parton distribution function evolution equations, while in [81] it has been approached from the point of view of the large spin expansion and generalized to operators of arbitrary twist. Reciprocity has been checked in various multiloop calculations of weakly coupled 𝒩=4 gauge theory [62, 64, 66, 82–84].

A natural tool to investigate the presence of reciprocity relations at strong coupling is the AdS/CFT correspondence, according to which such an organized structure of subleading terms in the large spin expansion should be visible also in the energies of the semiclassical string states corresponding to twist operators. This strong coupling analysis, initiated in [81] for a particular solution at the classical level, has been extended in [85] to more general configurations and beyond the classical result. Given the complicated form of the relevant solutions, however, the large spin expansion for corrections to the leading string energy is a nontrivial task. Remarkably, although not as manifestly as in the weak coupling case, also here the underlying integrable structure of the AdS/CFT system plays a crucial role in making feasible the analysis of reciprocity. The recent findings of [86], demonstrating that the semiclassical fluctuation problem is governed by simple finite-gap operators, have provided us with analytic expressions for the fluctuation determinants that permit to carry out well-defined expansions in the large spin limit. As a notable outcome, the large spin expansion of the string energy happens to have exactly the same structure as that of the anomalous dimension in the perturbative gauge theory, respecting reciprocity relations up to one-loop in string perturbation theory. Interesting generalizations of this analysis at strong coupling are the study [87] of reciprocity for the first commuting charges defined in [88], as well as the generalized reciprocity [89] present in the 𝒩=6 superconformal Chern-Simons theory in three dimensions [90].

We must stress that reciprocity is not a rigorous prediction, in that it is still missing a first-principles derivation. Instead, it is based on sound physical arguments and always needs to be verified, both at weak and at strong couplings. However, its persistent validity is an intriguing empirical observation which can be at the moment qualified as a kind of hidden symmetry for the spectrum of the AdS/CFT system. While the analysis of this review is focussed on the planar limit of the latter, where the emergence of integrability opens the way for a variety of tools to be used in the study of its spectrum, we emphasize that the reciprocity relation is not tied to the planar limit or to the integrability of the theories. Indeed, it holds in QCD for an arbitrary number of colors in the sectors in which integrability is not present [81]. In the AdS/CFT system, the powerful predictive power of reciprocity on the spectrum of the theories has been already successfully employed to formulate a five-loop analytic formula for the anomalous dimension of twist-three operators [34], which has been confirmed by a purely field-theoretical calculation [91].²

The plan of this Review is the following. In Section 2 we recall the original Gribov-Lipatov formulation of the reciprocity property in QCD and sketch a modern reinterpretation of it as in [79–81]. In Section 3 we present its generalized definition to the supersymmetric case of 𝒩=4 SYM theory. In Sections 4.1 and 4.2, after a short introduction on the outcomes of integrability-based techniques, we collect the information on the relevant multiloop results for the anomalous dimensions of quasipartonic operators at weak coupling. We proceed then in Section 4.3 illustrating with specific examples how reciprocity has been checked on those anomalous dimensions, explaining then in Section 4.4 the way reciprocity can be used to produce new higher-order formulas. Section 4.5 summarizes the weak coupling analysis. In Section 5, we present the strong coupling analysis of reciprocity, based on the perturbative (in the sigma model loop expansion) energies of folded and spiky string solutions in AdS₅×𝑆⁵. The final Section 6 is devoted to a short list of open problems related to the subject of this Review. Three Appendices follow, in which we recall the basic properties of harmonic sums (Appendix A) and briefly illustrate the checks of reciprocity in the first commuting charges of the 𝔰𝔩(2) sector (Appendix B) as well as the generalized reciprocity of the so-called ABJM [90] theory (Appendix C).

2. Generalized Gribov-Lipatov Reciprocity in QCD

The anomalous dimensions 𝛾(𝑆) of the twist-two operators with spin 𝑆 emerging in the QCD analysis of deep inelastic scattering (DIS) [45, 92] are expected to contain important information encoded in their dependence on 𝑆. Connecting the total spin 𝑆 to its dual in Mellin space, the Bjorken variable 𝑥, two opposite regimes emerge in a natural way. The first is small 𝑥→0 and is captured by the BFKL equation. It can be analyzed by considering the Regge poles of 𝛾(𝑆) analytically continued to negative (unphysical) values of the spin.

Here, we will be interested in the properties of the second quasielastic regime which is 𝑥→1, that is, large 𝑆. From the large 𝑆 behavior of the known three-loop twist-two QCD results as well as from general results valid at higher twist [93], the following general features can be inferred. The leading large 𝑆 behavior of the anomalous dimensions 𝛾(𝑆) is logarithmic𝛾(𝑆)=𝑓(𝜆)log𝑆+𝒪(𝑆⁰),𝑆→∞, (2.1) where 𝑓(𝜆) is a universal function of the coupling related to soft gluon emission [93–96]. It appears as (twice) the cusp anomalous dimension governing the renormalization of a light-cone Wilson loop describing soft-emission processes as quasiclassical charge motion. About the subleading ∼log^𝑝𝑆/𝑆^𝑞 corrections, they are found to obey special relations first investigated by Moch et al. in [97, 98] (see also, at two loops, [99]) and known as MVV relations. Roughly speaking, they predict the three-loop 1/𝑆 contributions in terms of the 𝑆⁰ two-loop ones. The MVV relations have received a relatively recent intriguing explanation in terms of a nontrivial generalization of the one-loop Gribov-Lipatov reciprocity [100, 101] which is the subject of the next sections.

2.1. Old Gribov-Lipatov Reciprocity: A Review

In the QCD context, the idea of reciprocity arises from the attempt to symmetrically treat deep inelastic scattering (DIS) and its crossed version, that is, 𝑒⁺𝑒⁻ annihilation into hadrons. In DIS, the nonperturbative information is contained in the space-like (𝑆) splitting functions 𝑃_𝑆(𝑥), governed by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution [100–104] and related to the anomalous dimensions 𝛾_𝑆(𝑆) [105] of the twist-two operators via a Mellin transform. Instead, the crossed process involves the nonperturbative fragmentation functions, whose scale evolution is related to the time-like splitting functions 𝑃_𝑇(𝑥); in this case the Mellin transform defines a time-like anomalous dimensions 𝛾_𝑇(𝑆).

The two types of splitting functions were related by analytic continuation through the singular point 𝑥=1 in the relation worked out by Drell et al. [73, 106–111]Drell-Levy-Yan:𝑃_𝑇1(𝑥)=−𝑥𝑃_𝑆(1𝑥). (2.2) A second relation has been proposed by Gribov and Lipatov [100, 101], stating an identical parton evolution for the two processesGribov-Lipatov:𝑃_𝑇(𝑥)=𝑃_𝑆(𝑥)≡𝑃(𝑥). (2.3) Combining the two relations above, one can deduce a “reciprocity property” of the common function 𝒫(𝑥)1Gribov-Lipatovreciprocity:𝑃(𝑥)=−𝑥𝑃(𝑥). (2.4) In Mellin space,𝛾(𝑆)≡𝒫(𝑆)=¹₀𝑑𝑥𝑥𝑥^𝑆𝑃(𝑥), (2.5) and it can be shown [81, 82] that this means³ (in the sense of asymptotic expansions at large 𝑆) 𝛾(𝑆)=𝑓(𝐶²),𝐶²=𝑆(𝑆+1),𝑆→∞. (2.6) Gribov-Lipatov reciprocity holds at one-loop, but fails at two loops [99, 112]. For instance, the explicit violation for the case of the nonsinglet twist-two quark operator can be written as⁴12[𝑃^(2loops)_{𝑇,𝑞𝑞}(𝑥)−𝑃^(2loops)_{𝑆,𝑞𝑞}]=¹₀𝑑𝑧𝑧{𝑃^(1loop)_𝑞𝑞(𝑥𝑧)}₊𝑃^(1loop)_𝑞𝑞(𝑧)log𝑧. (2.7)

2.2. Reciprocity Respecting Evolution Equations

The evolution equations for the parton distributions or fragmentation functions 𝐷_𝜎(𝑥,𝑄²) (𝜎=𝑆,𝑇) take the standard convolution form 𝜕_𝜏𝐷_𝜎(𝑥,𝑄²)=¹₀𝑑𝑧𝑧𝑃_𝜎(𝑧,𝛼_𝑠(𝑄²))𝐷_𝜎(𝑥𝑧,𝑄²), (2.8) where 𝑃_𝜎 are the space or time-like splitting functions, 𝛼_𝑠(𝑄²) is the QCD running coupling constant, and 𝜏=log𝑄². By Mellin transforming, this reads 𝜕_𝜏𝐷_𝜎(𝑆,𝑄²1)=−2𝛾_𝜎(𝑆,𝛼_𝑠(𝑄²))𝐷_𝜎(𝑆,𝑄²), (2.9) where 𝐷_𝜎(𝑆,𝑄²)=¹₀𝑑𝑥𝑥𝑥^𝑆𝐷_𝜎(𝑥,𝑄²),𝛾_𝜎(𝑆,𝑄²1)=−2¹₀𝑑𝑥𝑥𝑥^𝑆𝑃_𝜎(𝑥,𝛼_𝑠(𝑄²)). (2.10) Based on several deep physical ideas, it has been proposed to rewrite the evolution equation in a way that aims at treating the DIS and 𝑒⁺𝑒⁻ channels more symmetrically, in the spirit of Gribov-Lipatov reciprocity [80, 113, 114]. This proposal takes the form⁵𝜕_𝜏𝐷_𝜎(𝑥,𝑄²)=¹₀𝑑𝑧𝑧𝒫(𝑧,𝛼_𝑠(𝑧⁻¹𝑄²))𝐷_𝜎(𝑥𝑧,𝑧^𝜎𝑄²), (2.11) where 𝜎=−1,1 for the space-like and time-like channels, respectively. The crucial point is that the evolution kernel 𝒫(𝑧) is the same in both channels. As an immediate two-loop check, one recovers for the nonsinglet quark evolution the Curci-Furmansky-Petronzio relation (2.7). Other features related to the Low, Burnett, Kroll theorems [115, 116] (LBK) as well as to the inheritance idea are further discussed in [80, 113, 114]. A three-loop check of the above proposal to reproduce the explicit QCD anomalous dimensions requires, however, the addition of a term proportional to the 𝛽-function, as mentioned after (2.14) below.

2.3. Moch-Vermaseren-Moch Relations and Reciprocity of the Kernel 𝒫

The previous formulation of reciprocity is in 𝑥-space, but has important consequences in the large spin expansion of the anomalous dimensions. This point of view is adopted in Basso and Korchemsky [81] who propose a very simple way of testing (2.11).

Neglecting effects due to the running couplings (we are going to discuss 𝒩=4 SYM which is ultraviolet finite), one immediately derives from (2.11) the nonlinear relation (after a rescaling of 𝒫)𝛾_𝜎1(𝑆)=𝒫(𝑆−2𝜎𝛾_𝜎(𝑆)). (2.12) In the spirit of the derivation of the reciprocity respecting evolution equation (2.11), it is natural to guess that the Mellin transform of the kernel 𝒫 in (2.12) obeys the Gribov-Lipatov reciprocity relation (2.4).

As an immediate corollary, the following general parametrization of the large 𝑆 expansion of 𝛾_𝜎 (we define 𝑆=𝑆𝑒^𝛾E and 𝐴=𝑓(𝜆)) 𝛾_𝜎(𝑆)=𝐴log𝑆+𝐵+𝐶_𝜎log𝑆𝑆+(𝐷_𝜎+121𝐴)𝑆+⋯, (2.13) must satisfy the constraints 𝐶_𝜎1=−2𝜎𝐴²,𝐷_𝜎1=−2𝜎𝐴𝐵, (2.14) which are highly nontrivial since 𝐴,𝐵,𝐶, and 𝐷 are functions of the gauge coupling. The first relation in (2.14) is indeed verified at three loops by the explicit evaluation of 𝛾_𝜎, being part of the above-mentioned MVV relations. Most importantly, as discussed in [81], the second (subleading) relation requires, in QCD, a correction in the relation (2.12) that is related to the nonzero value of the 𝛽 function. For twist-two operators in the finite 𝒩=4 SYM theory, it is correct as it stands.

Thus, the two MVV relations in (2.14) strongly suggest that, when formulated for the Mellin transform of the kernel 𝒫 defined in (2.12), the reciprocity relation (2.4) holds. In 𝑆-space, it is equivalent to the claim that 𝒫(𝑆) has a large 𝑆 expansion in integer powers of 𝐶² of the form 𝒫(𝑆)=_𝑛𝑎_𝑛(log𝐶)𝐶^2𝑛, (2.15) where 𝐶²=𝑆(𝑆+1), and 𝑎_𝑛 are polynomials which can be computed in perturbation theory as series in 𝛼_𝑠. The expansion (2.15) can be read as a parity invariance under 𝑆→−𝑆−1, although this must be considered only as an analytic continuation around 𝑆=∞ and not at any 𝑆 in strict sense because of the Regge poles at negative 𝑆.

The property (2.15), or its equivalent form (2.4), has indeed been checked at three loops in [81] for several classes of twist-two operators in QCD. It generates an infinite set of MVV-like relations for all the subleading terms in the large 𝑆 expansion of the anomalous dimensions. The previous relations (2.14) are just the first cases.

3. Generalized Reciprocity in 𝒩=4 SYM

Reciprocity has been discussed in QCD, a theory which shares the gluon sector with 𝒩=4 SYM. This suggests to explore its validity in the latter, highly symmetric theory where one can exploit integrability to compute multiloop anomalous dimensions.

Since the leading order evolution kernel of 𝒩 = 4 SYM theory is purely classical in the LBK sense [117], there is hope to derive one day all-loop expressions for the anomalous dimensions of the operators of the theory within a simple description, that is, in which higher-order terms are dynamically inherited from the first loop. QCD would greatly benefit from such a result, and in general from investigations in which 𝒩=4 SYM is studied with the aim of putting under full theoretical control the dominant part of the perturbative QCD gluon dynamics.

The conformal invariance of the 𝒩=4 SYM theory allows one to extend the results of the previous section to the anomalous dimensions of the so-called quasipartonic operators of arbitrary twist 𝐽. The definition of the quasipartonic operators [42–44, 118] goes back to the conformal limit of the QCD and is in fact unrelated to the presence of supersymmetry.

In the conformal limit, the light-cone ray is left invariant by an SL(2,ℝ) collinear subgroup of the conformal group, generated by translations and dilatations along the ray, and rotations in the 𝑥⁰±𝑥¹ plane [119]. In light cone gauge, one can identify preferred components (SL(2,ℝ) primary fields) of the elementary scalars (in supersymmetric theories), Weyl fermions and field strength with minimal collinear twist.⁶ Composite operators built with this set and an arbitrary number of covariant derivatives correspond to physical degrees of freedom, as it is clear in light-cone gauge and are called quasipartonic operators.

We will then write a general quasipartonic single-trace gauge invariant operator as 𝒪(𝑧₁,…,𝑧_𝐽)=Tr{𝑋(𝑧₁𝑛)⋯𝑋(𝑧_𝐽𝑛)}, (3.1) where 𝑧𝑛^𝜇 is the light-like ray and 𝑋 can be a (suitable) 𝒩=4 scalar field 𝜑, gaugino component 𝜆, or holomorphic combination 𝐴 of the physical gauge field 𝐴^𝜇_⟂ [119]. The number of the constituent fields 𝐽 is the twist (classical dimension minus spin) of the operator.

At one-loop, these operators have simple transformation properties with respect to the collinear group; they transform as [ℓ]^⊗𝐿 where [ℓ] is the infinite-dimensional 𝔰𝔩(2) representation with conformal spin, respectively, [119]1ℓ(𝜑)=23,ℓ(𝜆)=1,ℓ(𝐴)=2. (3.2)

A suitable generalization of the analysis of reciprocity in [80, 81, 113, 114] to the case of 𝒩=4 SYM assumes that 𝛾(𝑆) obeys at all orders the nonlinear equation⁷1𝛾(𝑆)=𝒫(𝑆+2𝛾(𝑆)), (3.3) and the reciprocity relation takes the form𝒫(𝑆)=_𝑛𝑎_𝑛(log𝐶)𝐶^2𝑛, (3.4) where 𝑎_𝑛(log𝐶) are suitable polynomials, and 𝐶 is obtained by replacing 𝑆(𝑆+1) with the Casimir of the collinear conformal subgroup SL(2,ℝ)⊂SO(4,2)𝐶²=𝑠(𝑠−1)≡(𝑆+𝐽ℓ−1)(𝑆+𝐽ℓ). (3.5) Here, 𝑠=(𝑆+Δ₀)/2=𝑆+𝐽ℓ is the “bare” conformal spin 𝑠 of the operator (with Δ₀ being the canonical dimension of the operator) defined in terms of the conformal spin ℓ of the fields (3.2) out of which the operator is built. The constraint (3.4) is simply a parity invariance under (large) 𝐶→−𝐶.

This generalization is related to the proposal by [81] of tracing back the origin of the nonlinear relation (3.3) to the conformal symmetry of the theory (for the same reason, and as mentioned above, in gauge theories with nonvanishing beta-function, like QCD, the anomalous dimensions receive conformal symmetry breaking contributions). Quasipartonic operators can be in fact classified according to representations of the collinear SL(2,ℝ) subgroup of the SO(2,4) conformal group which are labeled by the conformal spin of the operator [119], whose general definition 𝑠=(𝑆+Δ)/2 involves in fact the scaling dimension of the operator. Since this get renormalized, receiving anomalous contribution 𝛾 at higher orders, one may argue that the anomalous dimension itself should be a function of 𝑆 only through its dependence on the “renormalized” conformal spin redefined in terms of Δ=𝑆+𝐽+𝛾(𝑆,𝐽). This then leads to the nonlinear functional relation for 𝛾⁸1𝛾(𝑆,𝐽)=𝑓(𝑠;𝐽)≡𝑓(𝑆+21𝐽+2𝛾(𝑆,𝐽);𝐽). (3.6) Suppressing the dependence on 𝐽 in 𝛾 and 𝑓, one may write such functional relation simply as (3.3).

One can notice that without further information, (3.3) 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 anomalous dimension 𝛾(𝑆,𝐽). The nontrivial information is in fact contained in the parity invariance (3.4), from which an infinite set of constraints can be derived between subleading coefficients in a general large spin expansion of the anomalous dimension, exactly as it happens in (2.13) and (2.14) above.

3.1. A Strong Form of Reciprocity from the Simplicity of 𝒫

We conclude this section with some interesting observation about the large spin expansion of the function 𝒫. Its leading logarithmic behavior, as follows from the structure of (3.3), coincides with the leading behavior of 𝛾 in (2.1), where the coupling-dependent scaling function 𝑓(𝜆) (cusp anomaly) is expected to be universal in both twist and flavour [93, 120]. Concerning the subleading terms, as remarked in [79, 81], the function 𝒫(𝑆) obeys up to three loops a powerful additional simplicity constraint, in that it does not contain logarithmically enhanced terms ∼log^𝑛(𝑆)/𝑆^𝑚 with 𝑛≥𝑚. This immediately implies that the leading logarithmic functional relation1𝛾(𝑆)=𝑓(𝜆)log(𝑆+2𝑓(𝜆)log𝑆+⋯)+⋯ (3.7) predicts correctly the maximal logarithmic terms log^𝑚𝑆/𝑆^𝑚𝑓𝛾(𝑆)∼𝑓log𝑆+²2log𝑆𝑆−𝑓³8log²𝑆𝑆²+⋯ (3.8) whose coefficients are simply proportional to 𝑓^𝑚+1 [66, 85, 121].

Notice that the fact that the cusp anomaly is known at all orders in the coupling via the results of [12, 14] would in principle imply (under the “simplicity” assumption for 𝒫) a proper prediction for such maximal logarithmic terms at all orders in the coupling constant, and in particular for those appearing in the large spin expansion of the energies of certain semiclassical string configurations (dual to the operators of interest). As we will report in the sections dedicated to the strong coupling checks of reciprocity, such “inheritance” has indeed been checked in [85] up to one loop in the sigma model semiclassical expansion, as well as in [122] at the classical level. An independent strong coupling confirmation of (3.8) up to order 1/𝑆 has recently been given for twist-two operators in [123].

However, the asymptotic part of the four-loop anomalous dimension for twist-two operators and of the five-loop anomalous dimension for twist-three operators reveals an exception to this “rule”, being the term log²𝑆/𝑆² not given only in terms of the cusp anomaly.⁹ This seems to indicate that, at least for twist-two and twist-three operators in the 𝔰𝔩(2) sector and at critical wrapping order, the 𝒫-function ceases to be “simple” in the meaning of [79], thus preventing the tower of subleading logarithmic singularities log^𝑚𝑆/𝑆^𝑚 to be simply inherited from the cusp anomaly. In order to clarify how the observed difference in the simplicity of the 𝒫 at weak and strong coupling works, further orders in the semiclassical sigma model expansion would be needed.

4. Reciprocity Tests at Weak Coupling in 𝒩=4 SYM

Given our interest in testing reciprocity in 𝒩=4 SYM, the next step is to exploit integrability in this theory to achieve a closed form for 𝛾(𝑆) of specific classes of operators at many loops.

4.1. Multiloop Calculation of Anomalous Dimensions via Integrability

The calculation of the anomalous dimensions in the planar limit of 𝒩=4 SYM theory is in fact dramatically simplified by its integrability properties. The gauge theory composite operators can be mapped to states of a PSU(2,2|4) invariant integrable spin chain, which for quasipartonic operators coincides at one loop with the XXX_−ℓ chain [124–129]. The energy of the spin chain is the image of the gauge theory dilatation operator. Thus, the calculation of the coupling-dependent energy levels of the spin chain provides the multiloop anomalous dimension of specific gauge theory composite operators.

We can illustrate the general strategy with a specific example which will be relevant in the following discussion. We consider the subsector 𝔰𝔩(2)⊂𝔭𝔰𝔲(2,2|4) which is perturbatively closed at all orders under renormalization. This sector contains composite operators which can be written schematically as 𝒪_𝐽,𝑆=𝜑^𝐽−1𝒟^𝑆𝜑, where 𝜑 is a scalar field and 𝒟 a certain projected covariant derivative.

The integrable structure of the spin chain, the conformal spin (3.2) being here ℓ=1/2, leads to the following Bethe equations at one-loop:(𝑢_𝑘+𝑖/2𝑢_𝑘)−𝑖/2^𝐽=^𝑆_{𝑗=1,𝑗≠𝑘}𝑢_𝑘−𝑢_𝑗−𝑖𝑢_𝑘−𝑢_𝑗,+𝑖 (4.1) where 𝑢_𝑖 are the Bethe roots, in terms of which is written the one-loop anomalous dimension𝛾₁=^𝑆_𝑘=11𝑢²_𝑘.+1/4 (4.2) The same equations can be conveniently reformulated in the language of the Baxter operator [130]. In this simple context and for the physical solutions, its eigenvalue is a degree 𝑆 polynomial in the spectral parameter 𝑢, 𝑄(𝑢)=∏^𝑆_𝑖=1(𝑢−𝑢_𝑖) which obeys the equation𝑖(𝑢+2)^𝐽𝑖𝑄(𝑢+𝑖)+(𝑢−2)^𝐽𝑄(𝑢−𝑖)=𝑡(𝑢)𝑄(𝑢), (4.3) where 𝑡(𝑢) is the transfer matrix of the integrable chain. In terms of 𝑄(𝑢), the one-loop anomalous dimension reads𝛾₁=𝑖(log𝑄(𝑢))^′|^𝑢=+𝑖/2_{𝑢=−𝑖/2}. (4.4) In the simplest case of twist 𝐽=2, the transfer matrix is a second-order polynomial 𝑡(𝑢)=2𝑢²−(𝑆²+𝑆+1/2), and the solution is easily identified with the Hahn function 𝑄(𝑢)=₃𝐹₂(−𝑆,𝑆+1,1/2−𝑖𝑢;1,1;1). Thus, the anomalous dimension at the 1-loop order, or 𝑔²=𝑔²_YM𝑁/16𝜋², is 𝛾(𝑆)=2𝑔²(𝑄^′(𝑖2)−𝑄^′𝑖(−2))=8𝑔²𝑆₁(𝑆),𝑆₁(𝑆)=^𝑆_𝑛=11𝑛. (4.5) This construction can be extended to all loops both in terms of Bethe Ansatz equations [5] as well as with the Baxter formalism [131–134].

In principle, the Baxter method is superior to the other, since it provides an analytical expression to the anomalous dimension as a function of the number 𝑆 of Bethe roots. Nevertheless, this approach has not been pursued in full detail for higher-rank subsectors of the theory and a practical alternative is the maximal transcendentality principle [57–61].

This QCD-inspired idea¹⁰ predicts that at each order 𝑛 the solution can be entirely expressed in terms of certain combinations of generalized harmonic sums of order 2𝑛−1  or in terms of products of harmonic sums 𝑆_𝐚 and zeta functions 𝜁(𝑏_𝑖) in such a way that the sum of their transcendentalities |𝐚| and 𝑏_𝑖 (see Appendix A for definitions) is again equal to 2𝑛−1. One can then use the maximal transcendentality principle to write the anomalous dimensions as a combination of harmonic sum of fixed order with coefficient to be determined. The rational coefficients can be then computed by fitting numerically with high precision the perturbative expansions of the Bethe equation at fixed 𝑆.

A crucial point is that the derivation of the Bethe equations, or equivalently of the Baxter equation, is based on the assumption that the length of the composite operator, that is, the spin chain length, is sufficiently large to avoid finite size effects related to interactions which wrap around the chain. The additional wrapping contributions which occur for short chains were for the first time correctly evaluated in [27] via a clever generalization of the Lüscher formulas [25] previously proposed for the AdS₅×𝑆⁵ sigma model in [26]. Such finite size effects are the object of recent investigations exploiting thermodynamical Bethe Ansatz methods and relying on the AdS/CFT duality with the superstring dynamics on AdS₅×𝑆⁵ [29–33]. The general statement is then that the full anomalous dimension must be written as𝛾(𝑔)=𝛾^ABA(𝑔)+𝛾^wrapping(𝑔), (4.6) where 𝛾^ABA(𝑔) is captured by the asymptotic Bethe Ansatz equations of [5] and 𝛾^wrapping(𝑔) is the wrapping contribution that can be evaluated with the tools mentioned above.

From the point of view of this Review, it is expected that reciprocity holds for the full anomalous dimension (4.6), since the above splitting has a more technical than physical nature. In all the explored examples to be discussed in the next section, reciprocity holds in fact for both the asymptotic and the wrapping part. It is however remarkable that this happens separately for the individual contributions.

4.2. Applications to Quasipartonic Composite Operators

We collect here the information on the relevant multiloop results for the anomalous dimensions of a class of quasipartonic operators in 𝒩=4 SYM. The discussion about the reciprocity properties of these results will follow in the next section.

As mentioned in Section 4.1, the emergence of integrability in the planar limit allows one to construct (at least at the one-loop level) a dictionary of correspondences between quasipartonic operators and generalized spin chains. In the spin-chain language, quasipartonic operators correspond to fixed length states and the anomalous dimensions are the hamiltonian eigenvalues of the relevant XXX_−ℓ chain [124–129]. At the one loop level, these sets are closed under perturbative renormalization, while at higher loops only the operators built out of scalar fields and gauginos continue to scale autonomously. In fact, the 𝒩=4  𝔰𝔩(2) subsector is closed at all orders, and even though operator with gauginos spans the 𝔰𝔩(2|1) subsector where there is mixing between scalars and fermions, this is not true in the quasipartonic set of operators built out of suitably projected components of gaugino fields [135]. Finally, in the case of gauge operators ¹¹ [136], mixing effects start immediately beyond one-loop (see the discussion in [64]).

4.2.1. Scalar Operators

The most studied and simplest sector is the 𝔰𝔩(2) subsector of the theory, whose representative operators 𝒪_𝐽,𝑆=𝜑^𝐽−1𝒟^𝑆𝜑, built out of scalar fields 𝜑 and covariant derivatives acting on them, were introduced in Section 4.1. In the chain language, each covariant derivative is thought as an “excitation” of the vacuum state Tr𝜑^𝐽. The number of these excitations 𝑆=∑𝑛_𝑖, the total spin, is not limited, being the −1/2 representation of 𝔰𝔩(2) infinite-dimensional.

The relevance of this bosonic subsector is due to the fact that, in the important case of twist-two operators, it is exhaustive of the whole theory. All twist-two operators fall in fact in a single supermultiplet [44, 137, 138] and their anomalous dimension is expressed in terms of a universal function 𝛾_univ with shifted arguments𝛾^𝜑_𝐽=2(𝑆)=𝛾_univ(𝑆),𝛾^𝜓_𝐽=2(𝑆)=𝛾_univ(𝑆+1),𝛾^𝐴_𝐽=2(𝑆)=𝛾_univ(𝑆+2). (4.7)

For the twist-two anomalous dimensions, closed expressions at two loops are known from explicit field-theory calculations [139] and at three loops from a conjecture inspired from the maximum transcendentality principle [57–61] applied to the QCD splitting functions [97, 98]. Up to three loops, the same formulas can also be computed by the asymptotic Bethe ansatz [6] for fixed values of 𝑆. It is only recently that the three-loop conjecture has been proved via the Baxter approach method [68]. In [23, 78] the ABA and wrapping part for the four-loop anomalous dimensions for twist-two scalar operators in the 𝔰𝔩(2) have been computed, with the techniques explained in the previous section. This result has been confirmed by a field-theoretical calculation [28, 140]. With similar ABA techniques and in absence of wrapping corrections, closed (in 𝑆) expressions for the anomalous dimensions of twist-three operators were derived in [23, 62].

Exploiting an Ansatz based on reciprocity (see next section), a five-loop formula for the anomalous dimensions was proposed in [34] for the twist-three operators and, in a similar fashion, in [41] for the case of twist-two. While in the first 𝐽=3 case, the formula involves a leading order (generalized) Lüscher correction, in the case of 𝐽=2 a nontrivial next-to-leading order wrapping contribution (together with a modification of the quantization condition) comes into play. This is due to the general fact that, in the 𝔰𝔩(2) sector, for twist 𝐽 operators, the wrapping effect starts at order 𝑔^2𝐽+4, delayed by superconformal invariance. The twist-three five-loop formula has been later confirmed by a purely field-theoretical calculation [91], while the correctness of the recent five-loop twist-two proposal is strongly supported by the fact that it respects the correct weak-coupling constraints deriving from a BFKL analysis and double-logarithmic behavior.

The same techniques used for the anomalous dimensions work in the case of the higher conserved charges of the chain model [88], something discussed so far only for the first few charges in the scalar sector [87] and reviewed in Appendix B.

4.2.2. Fermion Operators

These operators are built out of helicity +1/2 component of the gaugino fields 𝜆_𝛼, and covariant derivatives acting on them, defined in [52], where twist-three representatives have been studied at two loops in 𝒩=1,2,4 SYM by direct computation of the dilatation operator. The high level of symmetry of the 𝒩=4 theory results in a number of degeneracies in the spectrum of anomalous dimensions, with unexpected relations between composite operators of different twist [5]. The Bethe Ansatz reflects of course such remarkable structural properties related to supersymmetry.

An excellent example of this fact is precisely the case of twist-three operators built out of gauginos whose anomalous dimension was first proved in [63] to be related to the “universal” twist-two anomalous dimension (4.7) as𝛾^𝜓_𝐽=3(𝑆)=𝛾^𝜑_𝐽=2(𝑆+2). (4.8) This statement has been rigorously proved at three loops and attributed to a hidden 𝔭𝔰𝔲(1|1) invariance of the 𝔰𝔲(2|1) subsector of the theory.

4.2.3. Gauge Operators

These quasipartonic operators have as constituents gauge fields 𝐴 on which an arbitrary number of covariant derivatives act, where 𝐴 stands for the holomorphic combination of the physical gauge degrees of freedom 𝐴^𝜇_⟂ (suitable projected components of the field strength) defined in [119]. Twist-three gauge operators were considered in [64] at three loops, and in [82] at four loops and without wrapping effects.

At one-loop, this sector is described by a noncompact 𝑋𝑋𝑋_−3/2 spin chain with 𝐽 sites, and the anomalous dimension is known as an exacts solution of the Baxter equation. Beyond this order, no simple spin-chain correspondence exist and mixing effects come into play. In order to find a closed formula for the anomalous dimension, one can then hope to make use of the full 𝔭𝔰𝔲(2,2|4) Bethe equations in which the quantum numbers belonging to the correct superconformal primary that describes this sector have to appear. This can be done exploiting the superconformal properties of the (maximally symmetric) tensorial product of three singletons [141]. As usual, using as an input the one-loop solution𝛾^𝐴_𝐽=3(𝑆)=4𝑆₁(𝑆24+1)−5+,𝑆+4 (4.9) one can solve numerically the Bethe equations order by order in perturbation theory and fit the coefficient in an appropriate Ansatz. However, in this case the latter cannot be inspired by the standard maximum transcendentality principle, which is violated already at one loop as shown explicitly from the formula above. The latter is fully consistent with the QCD analysis of maximal helicity 3-gluon operators [142], where the dilatation operator can be decomposed as an integrable piece ℋ₀ plus a perturbation and the lowest eigenvalue is𝜀=4𝑆₁(𝑆24+1)+𝑆+4+4. (4.10) Inspired by a similar QCD calculation [143], the following Ansatz can be made which generalizes the one-loop result at 𝑘 loops:𝛾_𝑘(𝑛)=^2𝑘−1_𝜏=0𝛾^(𝜏)(𝑛),𝛾^(𝜏)(𝑛)=_{𝑝+ℓ=𝜏}ℋ_𝜏,ℓ(𝑛)(𝑛+1)^𝑝𝑆,𝑛=2+1, (4.11) where ℋ_𝜏,ℓ(𝑛) is a combination of harmonic sums with homogeneous fixed transcendentality ℓ. The terms with 𝑝=0 have maximum transcendentality; all the others have subleading transcendentality. Making use of this Ansatz and in the usual way, a three-loop [64] and a four-loop formula [82] were derived for the anomalous dimension of these twist-three gauge operators.

4.3. Proof of Reciprocity in Closed Form

Reciprocity is checked on the function 𝒫 which is obtained inverting (3.3) as𝒫(𝑆)=^∞_𝑘=111𝑘!(−2𝜕_𝑆)^𝑘−1[𝛾(𝑆)]^𝑘1=𝛾−4(𝛾²)^′+124(𝛾³)^′′−1192(𝛾⁴)^′′′+⋯; (4.12) inheriting thus the perturbative expansion of the anomalous dimension𝛾=^∞_𝑘=1𝑔^2𝑘𝛾_𝑘,𝒫=^∞_𝑘=1𝑔^2𝑘𝒫_𝑘. (4.13) One way to operate is checking directly the parity invariance (3.4). One should perform the large 𝑆 expansion of (4.12), rewrite it as a large 𝐶 expansion inverting (3.5), and check the absence of odd inverse powers of 𝐶. Three-loop tests of reciprocity for QCD and for the universal twist-two supermultiplet in 𝒩=4 SYM were discussed this way in [81], and it is also the procedure adopted up to now in the strong coupling analysis of reciprocity (see Section 5). At weak coupling, however, there is a much more elegant and powerful way to proceed. Considering that each term of the perturbative expansion of 𝒫 is a linear combination of products of harmonic sums, the idea is to find a new basis for the harmonic sums with definite properties under the (large-)𝐶 parity 𝐶→−𝐶.

This has been done in [84], where the map 𝜔_𝑎, 𝑎∈ℕ has been introduced, which acts linearly on linear combinations of harmonic sums as follows (we omit, in the following, the dependence of the harmonic sums on the spin 𝑆): 𝜔_𝑎(𝑆_𝑏,𝐜)=𝑆_{𝑎,𝑏,𝐜}−12𝑆_{𝑎∧𝑏,𝐜}, (4.14) where, for 𝑛,𝑚∈ℤ⧵{0}, the wedge-product is defined as 𝑛∧𝑚=sign(𝑛)sign(𝑚)(|𝑛|+|𝑚|). (4.15) One can also consider a complementary map 𝜔_𝑎 acting in a similar way on complementary sums defined in Appendix A.

Following [66, 79], the combinations of (complementary) harmonic sums can be introduced¹²Ω_𝑎=𝑆_𝑎,Ω_𝑎=𝑆_𝑎=𝑆_𝑎,Ω_𝑎,𝐛=𝜔_𝑎(Ω_𝐛),Ω_𝑎,𝐛=𝜔_𝑎(Ω_𝐛), (4.16) for which the following two theorems hold [84].

Theorem 4.1. ¹³The subtracted complementary combination Ω_𝐚, 𝐚=(𝑎₁,…,𝑎_𝑑), has definite parity 𝒫_𝐚 under the (large-) 𝐶 transformation 𝐶→−𝐶 and 𝒫_𝐚=(−1)^{|𝑎1|+⋯+|𝑎𝑑|}(−1)^𝑑𝑑_𝑖=1𝜀_𝑎𝑖. (4.17)

Theorem 4.2. The combination Ω_𝐚, 𝐚=(𝑎₁,…,𝑎_𝑑) with odd positive 𝑎_𝑖 and even negative 𝑎_𝑖 has positive parity 𝒫=1.

The strategy to prove the reciprocity property of the kernel 𝒫 is then the following. At each perturbative order ℓ, one starts from the expression of the kernel 𝒫_ℓ written in the canonical basis, something that can always be done using the shuffle algebra (A.2), and isolate in this expression the sums with maximum depth. Each of them, say 𝑆_𝐚, appears uniquely as the maximum depth term in Ω_𝐚. One then subtracts all the Ω's required to cancel these terms, keeping track of this subtraction and repeating the procedure decreasing the depth by one. If one ends the algorithm with a zero remainder and the full subtraction is composed by Ω's with the right parities (see Theorem 4.2), one can conclude that the kernel 𝒫 is parity respecting at the investigated order.

For example, the four-loop wrapping contribution from twist-two anomalous dimension calculated in [78]𝛾^wrapping₄(𝑆)=256(𝑆₋₅−𝑆₅+2𝑆_−2,−3−2𝑆_3,−2+2𝑆_4,1−4𝑆_−2,−2,1)𝑆²₁+−640𝜁₅𝑆²₁−512𝑆₋₂𝜁₃𝑆²₁ (4.18) can be conveniently rewritten only in terms of allowed Ω's𝒫^wrapping₄=−128Ω²₁(5𝜁₅+4𝜁₃Ω₋₂+8Ω_−2,−2,1+4Ω_3,−2). (4.19) This way reciprocity was proven at four-loops for the whole (ABA part included) anomalous dimension of twist-two operators. In a totally similar way, four loop reciprocity tests have been performed for twist-three operators in the scalar [66] and in the gauge sector [82].

4.4. Reciprocity-Based Ansatz

Based on the exceptional number of checks done for a variety of operators and reversing the usual logic, reciprocity can be simply assumed, and used as a tool to reduce the number of unknown coefficients in the standard Ansatz based on the maximum transcendentality principle to be solved via Bethe equations.

To see how this procedure can be used in practice, let us consider an illustrative example, the two-loop anomalous dimension for twist-three scalar operators. One starts with the following Ansatz of transcendentality 𝜏=3 made of harmonic sums with positive indices and argument 𝑆/2 (as is the case for twist-three operators made of scalars)𝛾₂=𝑎₁𝑆₃+𝑎₂𝑆_1,2+𝑎₃𝑆_2,1+𝑎₄𝑆_1,1,1. (4.20) The corresponding kernel has the following form in the canonical basis: 𝒫₂=𝛾₄−14𝛾₂𝛾^′₂≡(𝑎₁−16)𝑆₃+(𝑎₂+16)𝑆_1,2+(𝑎₃+16)𝑆_2,1−16𝜁₂𝑆₁+𝑎₄𝑆_1,1,1, (4.21) and when rewritten in terms of the Ω basis, the result is𝒫₄=𝑐₁Ω₁+𝑐₃Ω₃+𝑐_1,2Ω_1,2+𝑐_2,1Ω_2,1+𝑐_1,1,1Ω_1,1,1+const, (4.22) where the 𝑐_𝑖 are linear combinations of the coefficients 𝑎_𝑖. The combinations Ω₁,Ω₃,Ω_1,1,1 are all reciprocity respecting, according to the above theorem. Imposing reciprocity on 𝑃₂ implies the vanishing of the coefficients of those Ω with wrong parity, namely,𝑐_1,2=𝑎₂𝑎+16+⁴2=0,𝑐_2,1=𝑎₃𝑎+16+⁴2=0. (4.23) This leads to the conditions 𝑎₃=𝑎₂ and 𝑎₄=−2(16+𝑎₂), that are indeed satisfied by the known two-loop expression for the anomalous dimension [23, 62]. Thus, reciprocity has determined 2 of the 4 unknown coefficients in the initial Ansatz for the anomalous dimension.¹⁴ This procedure was used in [34] to deduce the five-loop asymptotic part of the anomalous dimension for twist-three scalar operators. At this loop order, starting with a linear combination of harmonic sums of transcendentality 𝜏=2𝑛−1=9, one finds in principle 256 terms which potentially contribute to the anomalous dimension. Fitting numerically all the coefficients, that should come out in exact (rational) form, is rather hard due to computational limitation. Imposing reciprocity, one obtains instead an overdetermined set of linear equations, which is solvable.¹⁵ In the same paper, the leading wrapping correction has been computed, which turns out to be separately reciprocity respecting. We recall that the result based on this assumption has been later confirmed by a purely field-theoretical calculation [91].

A similar reciprocity-based Ansatz was used and was also adopted in [41] to derive the five-loop calculations for the anomalous dimensions of twist-two operators (see Section 4.1 point 1. above).

4.5. Summary of Weak-Coupling Reciprocity Tests

The successful application of the methods that we have just illustrated proves that the reciprocity property of 𝒩=4 SYM has a wider range of validity than expected. It is confirmed at higher loops for the twist-2 universal multiplet and is also valid for twist-3 operators built with elementary fields of any conformal spin. Table 1 summarizes the present status of weak coupling tests.

Table 1 

Status of weak coupling reciprocity on minimal dimensions for twist operators.

The results about the universal twist-two supermultiplet (first row in the table) are a consequence of the four-loop check (ABA and wrapping contributions) in the scalar sector [84], of the five-loop result of [41], and of the fact that the constant shift in the spin that relates the anomalous dimensions in the supermultiplet as in (4.7) does not affect their large spin expansion properties, which are at the basis of the reciprocity. With the same motivation and due to (4.8), reciprocity holds with the same features for twist-thee operators built out of gauginos (third row in the table). For the twist-three scalar sector (second row in the table), reciprocity has been proved up to four loops in [66] and is present separately both in the asymptotic (trivially) and in the wrapping contribution of the five-loop result of [34]. Reciprocity for twist-three gauge operators has been proved at three [64] and at four loops [82] for the asymptotic part of the anomalous dimension (last row in the table).

Let us note that anomalous dimensions of operators with twist higher than two occupy a band [96], the lower bound of which is the minimal dimension for given 𝑆 and 𝐽. Every successful check of reciprocity has been performed at weak coupling only for minimal anomalous dimensions, while in fact anomalous dimensions of operators with twist higher than two with trajectories close to the upper boundary of the band do not respect reciprocity, as seen in the twist-three case at weak coupling in [121]. However, it is interesting that a relation like (3.8) also holds for such excited trajectories [121]. (This is also what we will see at strong coupling on the example of the spiky strings.)

A brief discussion of further results concerning reciprocity properties of higher conserved charges is contained in Appendix B. The extension of the analysis to ABJM models [90] has also been investigated and is illustrated in Appendix C.

5. Reciprocity at Strong Coupling: Semiclassical Strings in AdS₅×𝑆⁵

The analysis of the reciprocity property in the strong coupling regime of 𝒩=4 SYM is performed by making use of the AdS/CFT correspondence, namely, considering energies of the “semiclassical string states" which are believed to be dual to the quasipartonic operators [144]. The string states, one referring to, are solitonic solutions of the string equations of motion carrying a finite 2d energy that can be expressed in terms of other charges (spins), and the standard semiclassical expansion refers to the energies of strings in AdS₅×𝑆⁵ having large quantum numbers and thus dual to “long” SYM operators with large canonical dimensions.

In the following, we will study reciprocity at the level of the energy in the two cases of folded string and spiky strings, extending the analysis at one loop in the semiclassical expansion for the folded string solution. We will then discuss a generalization of reciprocity at the level of the eigenvalues of the first few commuting charges defined in [88].

It is of interest to recall that in such analysis, neither we will explicitly refer to the classical integrability of the string sigma model [1] nor to the semiclassical approach directly relying on such classical general finite gap description [145–148]. Interestingly enough, however, integrability will come up again at the one-loop level via the connection with the integrable, finite-gap, Lamé equation [86].

5.1. Classical Folded String in AdS₃×𝑆¹

The first and most important example in this sense is the nontrivial rigid string solution of [149] describing a folded spinning string rotating in the (𝜌,𝜙) plane of AdS₅ and moving along the 𝜑-circle of 𝑆⁵. For this configuration, the integrals of motion are the space-time energy 𝐸=√𝜆ℰ and the two spins 𝑆=√𝜆𝒮 and 𝐽=√𝜆𝒥 (conserved momenta conjugate to 𝑡 and to 𝜙, 𝜑, resp.). In the full quantum theory 𝑆 and 𝐽 should take quantized values. In the semiclassical approximation we will consider, however, that their values are assumed to be very large, in such a way that 𝒮 and 𝒥 are finite for √𝜆≫1.

The expressions for the “semiclassical” energy and spins can be found [150] in terms of the elliptic functions 𝔼 and 𝕂 of an auxiliary variable 𝜂𝜅ℰ=𝜅+𝜔𝜔𝒮,²−𝒥²𝜅²−𝒥²√≡1+𝜂,𝒮=2𝜋𝜔𝜂𝜅²−𝒥²1[𝔼(−𝜂1)−𝕂(−𝜂)],𝜅²−𝒥²=2𝜋√𝜂1𝕂(−𝜂). (5.1)(5.2) Here 𝜅 and 𝜔 (or 𝜂) are parameters of the classical solution which we should have eliminated to find ℰ as a function of 𝒮 and 𝒥.

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 AdS₅). For such long string one has 𝜂→0.

In the limit in which the 𝑆⁵ momentum 𝐽 of the string state can be ignored, solving for 𝒮 in (5.2) for small 𝜂 and substituting it into the first of (5.2), one finds for ℰ as a function of 𝒮 the expansionℰ=𝒮+log𝒮−1𝜋+log𝒮−12𝜋²𝒮−2log²𝒮−9log𝒮+516𝜋³𝒮²+2log³𝒮−18log²𝒮+33log𝒮−1448𝜋⁴𝒮³+⋯,𝒮≡8𝜋𝒮. (5.3)

In the case in which the 𝑆⁵ angular momentum of the string is not negligible compared to 𝑆, that is, when the string state is dual to an operator with large spin 𝑆 and large twist 𝐽, one can work out analogous expansions. We will be interested in large 𝒮 expansion with 𝒮≫𝒥 since only in this case the expansions like (2.13), that is, going in the inverse powers of 𝒮 with the coefficients being polynomials in log𝒮, will apply (see also [81, 150]).

In the large 𝒮≫𝒥 or long string limit, when 𝜂≪1, one should distinguish between “small” or “large” 𝒥 cases [150, 151]. In the “slow long string” approximation (corresponding to taking 𝒮 to be large with ℓ≡𝒥/log𝒮 fixed and then expanding in powers of ℓ), the leading terms in the semiclassical energy read (cf. (5.3))1ℰ−𝒮−𝒥≈𝜋(log𝒮−1)+𝜋𝒥²2log𝒮−𝜋³𝒥⁴8log³𝒮1(1−log𝒮+4)+⋯𝒮[1𝜋(log𝒮−1)+𝜋𝒥²2log²𝒮−3𝜋³𝒥⁴4log⁴𝒮2(1−3log𝒮−4)+⋯]𝒮²[1𝜋(2log²𝒮−9log𝒮+5)+𝜋𝒥²3(1+2log𝒮−1log²𝒮−2log³𝒮)+⋯], (5.4) where 𝒮≡8𝜋𝒮 and dots stand for higher-order corrections depending on 𝒥. In the case of “fast long string”, when log𝒮≪𝒥≪𝒮, the corrections to the energy read1ℰ−𝒮−𝒥≈𝜋²𝒥[12log²𝒮𝒮−log𝒮+4log𝒮+4𝒮²3(−2log𝒮+1+𝒮+2loglog²𝒮+1+⋯)+⋯]𝜋⁴𝒥³[−log⁴𝒮8−2𝒮(3log²1𝒮+log𝒮+1+𝒮+1loglog²𝒮−2+⋯)𝒮²(2log³𝒮−19log²𝒮+11log𝒮+13+13𝒮+log11log²𝒮+⋯)+⋯], (5.5) where 𝒮≡8𝑆/𝐽=8𝒮/𝒥≫1. Dots in the square brackets indicate corrections in 1/𝒮, corrections in 1/log𝒮 can be added in the round brackets, and terms like log(log𝒮) have been neglected.

With the large spin expansions (5.3)–(5.5) at hand, we first observe a general agreement in the structure of the large 𝑆 expansion as found in perturbative string theory and in perturbative gauge theory; see (2.13). This agreement is nontrivial since 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 log𝑆 term) the reason for the validity of the functional relation (3.3) and, moreover, of the reciprocity property (3.4) is obscure on the semiclassical string theory side.

We can furthermore study the compatibility of the expansions found with the functional relation (3.3). In particular, the coefficients of the leading (log𝒮/𝒮)^𝑚 terms in (5.3) happen, indeed, to be consistent with (3.8), with the leading term in the function f being simply the logarithm√𝐸−𝑆=𝜆𝜋1log[𝑆+2√𝜆𝜋log𝑆+⋯]+⋯ (5.6) The same is true for the expression (5.4), where the leading terms in the expression of (5.3) dominate in the limit when 𝒥²/log𝒮≪log𝒮/𝒮. In the case of the expansion (5.5), the leading terms can be summed up as [96]ℰ−𝒮=𝒥²+1𝜋²log²8𝒮𝒥+⋯, (5.7) where log𝒮/𝒥≪1 plays the role of an expansion parameter. Notice that in contrast to the slow long string case where the expansion (5.4) has the same structure as in (2.13), in the fast long string case (5.5), one gets higher powers of log𝒮 not suppressed by 𝒮. (For this kind of discrepancy with the weak-coupling behavior, one would in general need a resummation of the type discussed at the end of this section.) Nevertheless, the reciprocity property can be successfully checked as we explain below.

It is then possible to proceed as follows with the analysis of reciprocity. If one identifies the energy 𝐸, the angular momenta 𝑆 and 𝐽 of a string rotating in a plane in global AdS₅ with dimension, Lorentz spin and twist of the gauge theory quasipartonic operators, the functional relation (3.3) would then imply that the anomalous dimension should be a function (that we rename as f in this strong coupling context) of itself as in1𝛾=𝐸−𝑆−𝐽=f(𝑆+2𝛾). (5.8) To take into account the peculiarity of the string semiclassical perturbation theory, where all nonzero charges are automatically large at large 𝜆, we will use the semiclassical analogs 𝛾=𝛾/√𝜆, f̃=f/√𝜆 of the function appearing in (5.8), checking therefore whether the function f̃ defined in𝛾=f(̃𝒮+12𝛾)asf̃=^∞_𝑘=111𝑘!(−2𝑑)𝑑𝒮^𝑘−1[𝛾]^𝑘 (5.9) admits an expansion in even negative powers of the semiclassical analog 𝒞=𝐶/√𝜆 of the Casimir in (3.5). This will be 𝒞≡𝒮 in the case of a folded string rotating only in AdS, and 𝒞≡𝒮+(1/2)𝒥 in the case of the folded string rotating in AdS₅ with nonzero angular momentum in 𝑆⁵¹⁶.

Specifically, for the AdS folded string, the large 𝒮 expansion of the function f̃ (its leading term in the strong-coupling limit) is much simpler than that of the anomalous dimension 𝐸−𝑆 in (5.3) and contains only even powers of 𝐶⁻¹∼𝒮⁻¹f(̃𝒮)=1𝜋[log𝒮−1+log𝒮+116𝜋²𝒮²1+𝒪(𝒮⁴1)]+𝒪(√𝜆). (5.10)

A more systematic analysis of the reciprocity (parity invariance) property of the function f̃ is possible with the help of an integral representation for it. Using that (5.9) implies f(̃𝒮^′)=𝛾(𝒮^′−(1/2)f(̃𝒮^′)), where 𝒮^′=𝒮+(1/2)𝛾(𝒮), 𝛾(𝒮)=ℰ−𝒮, and renaming 𝒮^′→𝒮, we havef(̃𝒮)=12𝜋𝑖_Γ𝛾𝑑𝜔𝛾(𝜔)1+(1/2)^′(𝜔),𝜔−𝒮+(1/2)𝛾(𝜔) (5.11) where the contour Γ encircles the pole of the integrand and prime stands for derivative.¹⁷ It is natural to replace the variable 𝜔 in (5.11) with the expression (5.2) for the semiclassical spin 𝒮(𝜂)f(̃𝒮)=12𝜋𝑖_Γ𝑠̃𝑑𝜂𝛾(𝜂)^′(𝜂)̃,𝑠(𝜂)−𝒮 (5.12) where 𝑠(̃𝜂)≡𝒮(𝜂)+(1/2)𝛾(𝜂)=(1/2)(ℰ+𝒮) is the renormalized “conformal spin”, see formula (3.6), expressed in terms of the semiclassical quantities. The integral then gives the function 𝛾 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 spin 𝑠̃.

To verify the reciprocity property of the function f(̃𝒮) in (5.12), it is useful to redefine the variable 𝜂 as¹⁸𝜂→−1+16𝜂+1+256𝜂² and examine the large 𝒮 or small 𝜂 limit of the expressions. One finds that 𝛾(𝜂) is a series in even powers of 𝜂𝛾(𝜂)=−1+log𝜂𝜋+4(log𝜂+12)𝜋𝜂²−6(62log𝜂+777)𝜋𝜂⁴+⋯, (5.13) while the expression for the conformal spin runs in odd powers of 𝜂̃1𝑠(𝜂)=+8𝜋𝜂11+2log𝜂2𝜋𝜂−877+92log𝜂𝜂2𝜋³+⋯ (5.14) From the equation for the pole of the integrand in (5.12), 𝑠̃−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 𝒮. As a result, f(̃𝒮), which is same as 𝛾(𝜂) evaluated at the pole, should also run only in even negative powers 𝒞≡𝒮.

Coming to the case of the folded AdS₅ string with nonzero angular momentum in 𝑆⁵, one may again make use of the integral representation for the functional relation as in (5.11). The discussion will apply to both the “slow” and the “fast” long string limits. Here the renormalized “conformal spin” is 𝑠̃=(1/2)(𝒮+ℰ)=𝒮+(1/2)𝒥+(1/2)𝛾, and we anticipated that the semiclassical value of the Casimir operator is 𝒞≡𝒮+(1/2)𝒥. Then the integral in (5.12) can be written asf(̃𝒞)=12𝜋𝑖_Γ𝑠̃𝑑𝜂𝛾(𝜂)^′(𝜂)̃,̃1𝑠(𝜂)−𝒞𝑠(𝜂)=𝒮(𝜂)+2𝛾(𝜂). (5.15) After a redefinition of 𝜂, one can then show that the expansion of f̃ in large 𝒞 runs only in even negative powers of 𝒞 (see [85, Appendix D]). In the kinematic region of “fast” long strings, with 1≪log𝒮≪𝒥≪𝒮, this parity invariance property was already demonstrated in a closely related way in [81].

Notice that 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. This is due to the subtlety of semiclassical string expansion, again because all nonzero charges are automatically large at large 𝜆 and, for example, the case of finite twist 𝐽=2,3,… cannot be distinguished from the formal case of 𝐽=0. It is usually assumed that the folded string in AdS₅ with zero angular momentum in 𝑆⁵ describes an operator of small twist, but that can be 𝐽=2 or 𝐽=3, and so forth.

5.2. Spiky Strings in AdS₅ and Classical Violation of Reciprocity

It is interesting to mention a relevant example in which reciprocity is violated already at classical level. This is the case of the spiky spinning string in AdS₅ [152], the integrals of motion are the energy, the spin (angular momentum in AdS₅), and the difference between the position of the spike and of the middle of the valley between the two spikes, Δ𝜃=𝜋/𝑛, expressed in terms of the number of the spikes 𝑛. Also in this case it is possible to perform a large spin expansion, corresponding to the ends of the spikes approaching the boundary of AdS₅, which reads [85]𝑛ℰ−𝒮=2𝜋[log𝒮+𝑝₁+4𝒮(log𝒮+𝑝₂4)−𝒮²(2log²𝒮+𝑝₃log𝒮+𝑝₄)+323𝒮³(2log³𝒮+𝑝₅log²𝒮+𝑝₆log𝒮+𝑝₇)+⋯], (5.16) where 𝒮=(16𝜋/𝑛)𝒮 and𝑝₁𝜋=−1+logsin𝑛,𝑝₂𝜋=−1+logsin𝑛+𝜋(𝑛−2)𝜋2𝑛cot𝑛,𝑝₃=−10+2𝜋(𝑛−2)𝑛𝜋cot𝑛−2cot²𝜋𝑛𝜋−4logcsc𝑛+csc²𝜋𝑛,𝑝₄=6−csc²𝜋𝑛+𝜋²(𝑛−2)²2𝑛²−4𝜋(𝑛−2)𝑛𝜋cot𝑛+cot²𝜋𝑛[𝜋²(𝑛−2)²𝑛²𝜋+1]+logcsc𝑛[2cot²𝜋𝑛−2𝜋(𝑛−2)𝑛𝜋cot𝑛−csc²𝜋𝑛𝜋+2logcsc𝑛𝑝+10],₅=−18+𝒪(𝑛−2),𝑝₆=33+𝒪(𝑛−2),𝑝₇=−14+𝒪(𝑛−2). (5.17) It is easy to check that (5.16) coincides with the energy (5.3) for the folded string in AdS₅ when 𝑛=2. Retaining in (5.16) only the dominant contributions at each order of the above expansion, we obtain𝑛ℰ−𝒮=𝑛2𝜋log𝒮+²8𝜋²𝒮𝑛log𝒮−³64𝜋³𝒮²log²𝑛𝒮+⁴384𝜋⁴𝒮³log³𝒮+⋯. (5.18) This may be rewritten as√𝐸−𝑆=𝜆𝑛12𝜋log[𝑆+2√𝜆𝑛2𝜋log𝑆]+⋯, (5.19) implying that the functional relation is satisfied (cf. (3.8)).

Evaluating now the analog of the function f(̃𝒮) in (5.10), one finds the following expansion:f(̃𝒮)=𝑛2𝜋[log𝒮+𝑞₁+𝑞₂𝒮+1𝒮²(𝑞₃log𝒮+𝑞₄1)+𝒮³(𝑞₅log𝒮+𝑞₆)⋯]+⋯, (5.20) where𝑞₁𝜋=−1+logsin𝑛,𝑞₂=2𝜋(𝑛−2)𝑛𝜋cot𝑛,𝑞₃=4csc²𝜋𝑛,𝑞₄=4+2𝜋²(𝑛−2𝑛)²(1−2csc²𝜋𝑛𝜋)+4logsin𝑛csc²𝜋𝑛,𝑞₅=𝒪(𝑛−2),𝑞₆=𝒪(𝑛−2), (5.21) with 𝑞₅,𝑞₆ are nonzero for 𝑛≠2. The expansion (5.20), even if considerably simpler compared to the energy (5.16), is not parity invariant under 𝒮→−𝒮. The parity invariance is restored in the case of the folded string when 𝑛=2, where indeed (5.20) coincides with (5.10).

This breakdown of parity invariance for a string with 𝑛>2 spikes is actually not only nonsurprising, but expected. In fact, such spiky string should correspond to an operator with nonminimal anomalous dimension for a given spin, while the reciprocity was checked at weak coupling only for the minimal anomalous dimensions. Indeed and as already mentioned, anomalous dimensions of operators of twist higher than two with trajectories close to the upper boundary of the band present features completely analog to the one seen here, in that they satisfy (3.8) while violating reciprocity [121].¹⁹

5.3. Reciprocity in String Perturbation Theory

The observation that reciprocity holds at 1-loop in string semiclassical expansion, first made in [85], has been confirmed and extended in [86] in the case of a folded string rotating in AdS. The standard string semiclassical approximation is based on expanding the energy 𝐸 in large √𝜆 with 𝒮=𝑆/√𝜆 kept fixed,𝑆𝐸=𝐸(√𝜆,√𝜆)=√𝜆ℰ₀(𝒮)+ℰ₁1(𝒮)+√𝜆ℰ₂(𝒮)+⋯, (5.22) where ℰ₀, the classical energy, coincides with (5.1), and ℰ₁, ℰ₂ are the 1-loop and 2-loop energies translates into an analog semiclassical expansion within the relation (5.9). Namely, the “anomalous dimension” can be written as𝛾=𝛾₀+1√𝜆𝛾₁+⋯,where𝛾₀=ℰ₀(𝒮)−𝒮,𝛾₁=ℰ₁(𝒮) (5.23) from which the function f̃ defined by (5.9) can be determined as inf̃=f₀̃+1√𝜆f₁̃+⋯, (5.24) withf₀̃=^∞_𝑘=111𝑘!(−2𝑑)𝑑𝒮^𝑘−1[𝛾₀]^𝑘,f₁̃=^∞_𝑘=111𝑘!(−2𝑑)𝑑𝒮^𝑘−1[𝑘𝛾₀^𝑘−1𝛾₁]. (5.25) Due to the observation that the semiclassical fluctuation problem is governed by standard single-gap Lamé operators, the possibility to write down an analytic exact expression for the relevant functional determinants it is a recent achievement [86]. From the exact one-loop energy ℰ₁≡𝛾₁ that can be written in terms of them, it has been possible to extract the following expression for its large spin (small 𝜂) expansion (see also the comment at the end of this section): 𝛾₁=𝜅₀𝜅[(𝑐₀₁𝜅₀+𝑐₀₀+𝑐_0,−1𝜅₀)+(𝑐₁₁𝜅₀+𝑐₁₀+𝑐_1,−1𝜅₀)𝜂+(𝑐₂₁𝜅₀+𝑐₂₀+𝑐_2,−1𝜅₀)𝜂²+(𝑐₃₁𝜅₀+𝑐₃₀+𝑐_3,−1𝜅₀)𝜂³+𝒪(𝜂⁴)], (5.26) where 𝜅₀=(1/𝜋)log(16/𝜂) and the explicit values for the coefficients are𝑐₀₁=−3log2,𝑐₀₀6=1+𝜋log2,𝑐_0,−15=−,𝑐12₁₁=0,𝑐₁₀3=−𝜋log2,𝑐_1,−1=1+2𝜋3log2𝜋²,𝑐₂₁𝜋=−−33232log2,𝑐₂₀=1+1639log232𝜋,𝑐_2,−1=−13−64𝜋63log232𝜋²,𝑐₃₁=𝜋+33232log2,𝑐₃₀3=−−3213log216𝜋,𝑐_3,−1=29+192𝜋85log264𝜋². (5.27) Solving for the parameter 𝜂 explicitly in terms of 𝒮, the first few terms in (5.26) read𝛾₁=−3log2𝜋log𝒮+𝜋+6log2𝜋−5𝜋12log𝒮1+−𝒮[24log2𝜋log𝒮−4𝜋+36log2𝜋+5𝜋3log²𝒮1]+𝒪(𝒮²), (5.28) with 𝒮=8𝜋𝒮. Working out f₁̃ and looking at all terms which are odd under 𝒮→−𝒮, we find that they vanish if the following reciprocity constraints hold 𝑐₁₀=1𝜋𝑐₀₁,𝑐_1,−1=1𝑐2𝜋⁰⁰,𝑐₃₁=−𝑐₂₁,𝑐₃₀=−𝑐₂₀−1𝑐6𝜋⁰¹+1𝜋𝑐₂₁,𝑐_3,−1=−𝑐_2,−1+14𝜋²𝑐₀₁−1𝑐12𝜋⁰⁰+1𝑐2𝜋²⁰. (5.29) With the list of explicit coefficients above (5.27), these relations are indeed satisfied [86].

As we remarked, the expression of the one-loop energy derived in [86] is exact. However, its expansion at large spin is quite nontrivial. It contains a part which can be computed analytically in closed form and a reminder, starting at order 𝒪(𝜂²), which is known (as yet) only in implicit form. It is the large spin expansion of the first contribution, namely formula (5.26) above, which turns out to be separately reciprocity respecting.²⁰

6. Open Problems and Perspectives

From the point of view of AdS/CFT, it is quite important to look for common structures shared by the two sides of the correspondence. Integrability is certainly one of them. The reciprocity property discussed in this Review is another example. Hence, we believe that it is important to pursue its investigation and for this reason we list in this final section some related open problems.

First of all, as remarked in the introduction, there is no rigorous proof of reciprocity neither at weak nor at strong coupling. It would be nice to establish the validity of this (discrete) hidden symmetry by a solid physical argument or, possibly, as a mathematical feature of the integrable structures of AdS/CFT, that is, Bethe Ansatz equations, Baxter formalism, or exact 𝑆-matrix. In fact, as emphasized above, reciprocity is not tied to the planar limit or to the integrability of the theories. In the case of 𝒩=4 SYM, this suggests that nonplanar corrections to the dilatation operator should also verify the reciprocity relation.

We furthermore observe that, in gauge theory, the reciprocity relation relates to each other time-like and space-like anomalous dimensions. In analyzing the spectrum of 𝒩=4 SYM, we considered the space-like case, the one in which anomalous dimensions are directly connected with composite local gauge theory operators and, via AdS/CFT, with the anomalous dimensions of the folded string configurations their dual. Referring to [81] for a detailed discussion of the time-like case, we add that it remains an open problem, a possible physical interpretation of the same time-like/space-like relation in string theory, namely, what is the meaning of the state reciprocal to a folded spinning string in AdS.

As commented in Section 4.5, in the case of operators with twist higher than two, the reciprocity relation only holds for the lower band of the band formed from their anomalous dimensions, while it is broken for the excited states. This happens with a surprisingly similar mechanism also at strong coupling in the case of spiky strings (Section 5.2). It would be nice to clarify the reasons for such a breakdown of reciprocity, as well as verify the possibility to restore it.

Another issue is the connection between reciprocity and so-called wrapping corrections. The latter ones are under intense study and are expected to clarify several interesting facets of a very non trivial pair of integrable models. From this point of view, the observation that reciprocity is separately satisfied by the asymptotic Bethe Ansatz predictions as well as from the wrapping corrections is an unsolved puzzle. As a related problem, reciprocity deserves of course further study in larger (with rank greater than one) sectors of the theory.

Our final comment concerns the strong coupling regime of the gauge theory, which is string perturbation theory. There are currently two apparently alternative formalisms to work out quantum corrections for string configurations in AdS₅×𝑆⁵. The first is standard field-theoretical analysis of the string world-sheet 𝜎-model. This approach, certainly boosted by integrability, is a priori independent on it. The second method is based on of the algebraic spectral curve which, instead, imposes and exploits integrability from scratch. Currently, it is not totally clear how to relate the two approaches. The signals of reciprocity that we have illustrated in the world-sheet calculations are, in our opinion, a very interesting check and a challenge for the spectral curve method.

Appendices

A. Harmonic Sums

The nested harmonic sums 𝑆_{𝑎1,…,𝑎ℓ} are defined recursively as 𝑆_𝑎(𝑆)=^𝑆_𝑛=1𝜀^𝑛_𝑎𝑛^|𝑎|,𝑆_𝑎,𝐛(𝑆)=^𝑆_𝑛=1𝜀^𝑛_𝑎𝑛^|𝑎|𝑆_𝐛(𝑛), (A.1) where 𝜀_𝑎=+1(−1) if 𝑎≥0 (𝑎<0). The depth of a given sum 𝑆_𝐚=𝑆_{𝑎1,…,𝑎ℓ} is defined by the integer ℓ, while its transcendentality is the sum |𝐚|=|𝑎₁|+⋯+|𝑎_𝑛|. The product between harmonic sums can be reduced to linear combinations of single sums iteratively using the so-called shuffle algebra [153] 𝑆_{𝑎1,…,𝑎ℓ}(𝑆)𝑆_{𝑏1,…,𝑏𝑘}(𝑆)=^𝑆_𝑝=1𝜀^𝑝_𝑎1𝑝^|𝑎1|𝑆_{𝑎2,…,𝑎ℓ}(𝑝)𝑆_{𝑏1,…,𝑏𝑘}(𝑝)+^𝑆_𝑝=1𝜀^𝑝_𝑏1𝑝^|𝑏1|𝑆_{𝑎1,…,𝑎ℓ}(𝑝)𝑆_{𝑏2,…,𝑏𝑘}−(𝑝)_𝑆𝑝=1𝜀^𝑝_𝑎1𝜀^𝑝_𝑏1𝑝^{|𝑎1|+|𝑏1|}𝑆_{𝑎2,…,𝑎ℓ}(𝑝)𝑆_{𝑏2,…,𝑏𝑘}(𝑝). (A.2)

A.1. Complementary and Subtracted Sums

Let 𝐚=(𝑎₁,…,𝑎_ℓ) be a multi-index. For 𝑎₁≠1, it is convenient to adopt the concise notation 𝑆_𝐚(∞)≡𝑆^∗_𝐚. (A.3) We define the complementary harmonic sums recursively by 𝑆_𝑎=𝑆_𝑎 and 𝑆_𝐚=𝑆_𝐚−^ℓ−1_𝑘=1𝑆_{𝑎1,…,𝑎𝑘}𝑆^∗_{𝑎𝑘+1,…,𝑎ℓ}. (A.4) Note that the definition is ill when 𝐚 has some rightmost 1 indices; in this case, we will treat 𝑆^∗₁ as a formal object in the above definition and will set it to zero in the end. Since 𝑆_𝐚^∗<∞ in all remaining cases, it is meaningful to introduce the subtracted complementary sums, defined as follows:𝑆_𝐚=𝑆_𝐚−𝑆^∗_𝐚. (A.5) The explicit form of the above definition is𝑆_𝐚(𝑆)=(−1)^ℓ∞_𝑛1=𝑆+1𝜀^𝑛1_𝑎1𝑛^|𝑎1|₁^∞_{𝑛2=𝑛1+1}𝜀^𝑛2_𝑎2𝑛^|𝑎2|₂⋯^∞_{𝑛ℓ=𝑛ℓ−1+1}𝜀^𝑛ℓ_𝑎ℓ𝑛^|𝑎ℓ|_ℓ. (A.6)

B. Reciprocity of Higher Conserved Charges

To the notion of integrability for the spin chains corresponding to 𝒩=4 SYM composite operators is associated the existence of an infinite tower of commuting charges, in standard notation {𝑞_𝑟}_𝑟≥2. The first of them 𝑞₂ is identified with the Hamiltonian of the chain and one refers to a hierarchy of conserved charges. Actually, in our context all the 𝑞_𝑟 are on the same footing and is then natural to extend the analysis of the reciprocity properties to the full set of conserved charges. An attempt in this direction is the paper [87] where the reader can find more details. Here, we just summarize the main outcomes of that analysis.

In [87], a few higher charges in the 𝔰𝔩(2) subsector are studied. In the weak coupling regime, the first two non trivial charges 𝑞_4,6 have been computed at three and two loops, respectively.

The result of the analysis is that reciprocity is indeed at work. The definition of the kernel 𝒫_𝑟 (see (3.3)) can be consistently generalized to the full tower of charges according to𝑞_𝑟(𝑆)=𝒫_𝑟1(𝑆+2𝑞₂(𝑆)). (B.1) Notice that this definition involves the renormalized conformal spin 𝑆+(1/2)𝑞₂(𝑆) as argument of the kernel, in agreement with light-cone quantization. The naive argument 𝑆+(1/2)𝑞_𝑟(𝑆) implicitly defines a nonreciprocity-respecting kernel.

The strong coupling regime can be explored at the classical level considering the first higher charges of the sigma model, which can be derived from those of the 𝔰𝔲(2) sector [88] by analytic continuation and then analyzed following the same strategy adopted for the energy case. At this leading order, the parity invariance is satisfied by all the examined charges.

As a final comment, we remark that the wrapping corrections for the higher charges have not been computed yet, even at the leading order. It would be very nice to include them in the TBA treatment.

C. Reciprocity and ABJM Theory

In this Review, we considered 𝒩=4 SYM duality with string propagation on AdS₅×𝑆⁵. Actually, integrability appears in other instances of the AdS/CFT correspondence. In particular, the correspondence between the so-called ABJM theory [90] and IIA string on AdS₄×ℂℙ³ has been recently widely studied.

Again, the string model is classically integrable [154–156]. The dual gauge theory is an 𝒩=6 superconformal theory in three dimensions, with 𝑈(𝑁)×𝑈(𝑁) gauge group and Chern-Simon action with opposite levels +𝑘, −𝑘, emerging in the low energy limit of a theory of 𝑁 branes at a ℂ⁴/ℤ_𝑘 singularity.

In [157, 158], it has been shown that the dilatation operator for single-trace operators built with the scalars of the theory leads to an 𝑆𝑈(4) integrable spin chain, and soon the set of all-loop Bethe-Ansatz equations for the full 𝔬𝔰𝔭(2,2|6) theory has been proposed. Despite that the 𝒩=4 SYM and the ABJM theory present a very different structure, one can identify a 𝔰𝔩(2) [159, 160] sector in the ABJM theory, and the relative all-loop conjectured Bethe equations show strong similarities with the SYM case. Thus, it is an interesting task to try to investigate to which extent one can recover the QCD-inspired reciprocity properties in such an exotic gauge theory. Some breaking of reciprocity is expected since now the gauge structure is rather far from the QCD one and the physical arguments supporting reciprocity are missing or at least much weaker.

The analysis of [89] shows that twist-one operators obey a four-loop parity invariance closely related to the reciprocity discussed in this Review. This four-loop result for the twist-one operators includes the leading-order wrapping correction, computed using the Y-system formalism [29–31]. In the twist-two case, parity invariance is badly broken, although some remnants can still be seen in the fine structure of the kernel 𝒫.

Acknowledgments

The authors would like to thank A. V. Belitsky, Yu. L. Dokshitzer, G. V. Dunne, A.V. Kotikov, T. Lukowski, G. Marchesini, M. Pawellek, A. Tirziu, A. A. Tseytlin, and S. Zieme for the nice collaboration on some of the papers on which this paper is based. V. Forini is supported by the Alexander von Humboldt foundation.

Endnotes

1. The purely field-theoretical predictions in [37, 38] on the strong coupling expansion of the anomalous dimension for the Konishi operator differ both from [35] as well as from each other.
2. With a similar reciprocity-based Ansatz, a five-loop formula for the twist-two anomalous dimension was worked out in [41].
3. If there are several operators mixing among themselves, the reciprocity property is expected to hold for the eigenvalues of the mixing matrix anomalous dimensions and not for the separate matrix elements. See [81] for a detailed analysis of the case of QCD flavour singlet quark and gluon twist-2 operators.
4. The “+” distribution is defined for an arbitrary function 𝑓(𝑥) in the standard way ∫¹₀1𝑑𝑥𝑓(𝑥)(1−𝑥)₊=∫¹₀𝑑𝑥𝑓(𝑥)−𝑓(1).1−𝑥
5. An evolution equation analogue to (2.11) has been first discussed for time-like anomalous dimensions in the small 𝑥 limit in [161].
6. SL(2,ℝ) primary fields Φ have definite scaling dimension 𝑑 and collinear spin 𝑐 is defined by𝐷Φ=𝑑Φ,Σ₊₋Φ=𝑐Φ,where 𝐷 and Σ_𝜇𝜈 are the dilatation and Lorentz spin generators. The collinear twist (collinear dimension minus collinear spin) is minimal for 𝑡=𝑑−𝑐=1.
7. Since by 𝛾(𝑆) one means the anomalous dimension of a gauge invariant operator in 𝒩=4 SYM theory, it is quite natural to adopt for such generalization the case of 𝜎=−1 in the nonlinear QCD relation (2.12), corresponding to the space-like case. In fact, the QCD time-like anomalous dimensions are not related to composite local gauge operators, due to the general fact that fragmentation functions do not admit the operator product expansion [81].
8. The relation between the notation used in [81] and ours is 𝑁→𝑆, 𝐿→𝐽, 𝐽→𝐶 and 𝑗→𝑠.
9. Interestingly enough, the large spin expansion of the wrapping contribution of [78] and of [34], which correctly does not change the leading asymptotic behavior (cusp anomaly), first contributes at the same order, but not in such a way that the total log²𝑆/𝑆² coefficient results in (−𝑓³/8) as required from (3.8).
10. Inspired by the structure of the two-loop anomalous dimension of 𝒩=4 twist-two operators in the 𝔰𝔩(2) sector, it has been proposed [57–61] that the three-loop answer could be extracted by simply picking up the “most transcendental terms” from the three-loop non-singlet QCD anomalous dimension derived in [97, 98]. The conjectured three-loop formula has been then independently confirmed in the framework of the Bethe ansatz equations [6] as well as within a space-time approach [67].
11. The name stems from the one-loop description of a class of scaling operators. Beyond one-loop, additional fields mix.
12. A different basis for harmonic sums with well-defined reciprocity-respecting properties has been recently proposed in [41].
13. A special case of Theorem 4.1 appeared in [79]. A general proof of Theorem 4.1 in the restricted case 𝐚=(𝑎₁,…,𝑎_ℓ) with positive 𝑎_𝑖>0 and rightmost indices 𝑎_ℓ≠1 can be found in [66].
14. The coefficient 𝑎₄ has only been kept to show the exact number of constraints coming from reciprocity. It could have been set to zero from the beginning because at large 𝑀 the term 𝑆_1,1,1∼log³𝑀 is not compatible with the universal leading logarithmic behavior (cusp anomaly).
15. We should stress, however, that reciprocity as an assumptions only acts as a computational tool. As usual in such kind of conjectures, there is a powerful numerical test that can be applied to any guesswork, and the closed formulas presented in [34] have been always double checked numerically as solutions of the Bethe equations.
16. The choice for this case of ℓ=1/2 in the semiclassical version of (3.5) follows from the fact that the nonzero 𝑅-charge for classical bosonic solutions automatically selects the sl(2) sector identified in fact by ℓ=1/2.
17. The expression that multiplies 𝛾 in the integrand has residue 1, so that the integral is 𝛾 evaluated at the pole 𝜔=𝒮−(1/2)𝛾. Then defining 𝑥=𝒮−(1/2)f(̃𝒮), we have 2𝒮−2𝑥=𝛾 which coincides with the equation for the pole with 𝑥=𝜔.
18. This choice is not unique. An analogous transformation was used in [81].
19. It is interesting that our strong-coupling result (5.19), (5.20) has close similarity with weak-coupling one found for 𝑛=3 in [121]: the functional relation (3.8) is still satisfied, and the parity invariance is broken at level 1/𝑆.
20. This situation is for certain aspects similar to the ABA + wrapping splitting discussed at weak coupling.