Integrable bootstrap for AdS3/CFT2 correlation functions
PPrepared for submission to JHEP
Integrable bootstrap for AdS3/CFT2 correlationfunctions
Burkhard Eden, a Dennis le Plat, a Alessandro Sfondrini b a Institut für Mathematik und Physik, Humboldt-Universität zu Berlin,Zum großen Windkanal 6, 12489 Berlin, Germany b Dipartimento di Fisica e Astronomia, Università degli Studi di Padova,& Istituto Nazionale di Fisica Nucleare, Sezione di Padova,via Marzolo 8, 35131 Padova, Italy
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We propose an integrable bootstrap framework for the computation of corre-lation functions for superstrings in AdS × S × T backgrounds supported by an arbitrarymixture or Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz fluxes. The frameworkextends the “hexagon tessellation” approach which was originally proposed for AdS × S and for the first time it demonstrates its applicability to other (less supersymmetric) setups.We work out the hexagon form factor for two-particle states, including its dressing factorswhich follow from those of the spectral problem, and we show that it satisfies non-trivialconsistency conditions. We propose a bootstrap principle, slightly different from that ofAdS × S , which allows to extend the form factor to arbitrarily many particles. Finally,we compare its predictions with some correlation functions of protected operators. Possibleapplications of this construction include the study of wrapping corrections, of higher-pointcorrelation functions, and of non-planar corrections. a r X i v : . [ h e p - t h ] F e b ontents × S × T psu (1 | ⊕ centrally extended 92.3.2 Physical values of the central charges 102.3.3 Four irreducible representations of psu (1 | ⊕ c.e. 112.3.4 Parametrisation after crossing 132.4 Particle content of the theory 142.4.1 The left representation 142.4.2 The right representation 152.4.3 The massless representations 162.5 Scattering matrix 172.5.1 Left-left scattering 172.5.2 Right-right scattering 182.5.3 Left-right scattering 182.5.4 Right-left scattering 182.5.5 Massless scattering. 192.5.6 Mixed-mass scattering 192.5.7 Dressing factors 20 A.1 The massive sector 48A.1.1 Left-left scattering 48A.1.2 Right-right scattering 49A.1.3 Left-right scattering 49A.1.4 Right-left scattering 50A.2 The mixed-mass sector 50A.2.1 Left-massless scattering 50A.2.2 Massless-left scattering 51A.2.3 Right-massless scattering 51A.2.4 Massless-right scattering 52A.3 The massless sector 52– 1 –
Introduction
Our understanding of theoretical physics has always been shaped by experimental obser-vations on the one side, and by the construction of a theoretical framework which mayallow us to compute, compare and study relevant observables on the other side. Somequestions having to do with the fundamental behaviour and self-consistency of a physicaltheory are much more easily answered when we can compute as many observables as possi-ble exactly, without resorting to truncations, approximations or simulations. The study ofexactly solvable systems is by now a large branch of physics. In the quantum world, it en-compasses integrable spin-chains, lattice models, two-dimensional conformal field theories(CFTs) and integrable quantum field theories (IQFTs). Most recently, this found applica-tions to string theory (which is indeed defined on a two-dimensional worldhseet) as well astheir holographic duals in the AdS/CFT correspondence.One of the best understood examples of AdS/CFT is the correspondence between su-perstrings on AdS × S and the N = 4 supersymmetric Yang-Mills theory in D = 4 ( N = 4 SYM) [1–3]. This is the AdS/CFT setup with the maximal amount of supersymmery. De-spite that, both the string theory and the gauge theory look like tough nuts to crack: on theworldsheet, the superstring theory is supported by Ramond-Ramond fluxes, which makesit hard to describe it as a CFT; in the dual, the gauge theory appears to be as involved asany four-dimensional Yang-Mills theory. A remarkable simplification appears in the planarlimit [4] (or large- N limit, where N is the number of colours), as it was first noticed byMinahan and Zarembo [5]: the one-loop spectrum of anomalous dimensions of certain op-erators is exactly solvable in terms of a Bethe ansatz [6, 7]. In fact, this can be extendedto the all-loop planar spectrum of all local operators, though a complete understandingof the problem requires to overcome the issue of wrapping corrections. To this end it iscrucial to reformulate the problem on the string worldsheet [8] and to introduce a mirrormodel [9]. Eventually, the whole spectral problem could be encoded in a powerful systemof equations known as the quantum spectral curve [10]. We refer the reader to refs. [11, 12]for reviews of the AdS /CFT spectral problem. Quite remarkably, integrability surviveseven beyond the strict N → ∞ limit. In particular, the hexagon tessellation program wasintroduced in reference [13] as a way to turn the computation of three-point functions ofgeneric (non-protected) operators into the computation of an integrable form factor. Inbroad strokes, the idea is that we may entirely constrain, by symmetry arguments, therelevant form factor for the case of one and two particles, and bootstrap it for arbitrar-ily many particles. The resulting form factor takes a relatively simple form [13] in termsof the S matrix that was found by Beisert [14] for the spectral problem. Accounting forwrapping corrections is more problematic and, up to date, an open issue. It is possible toaccount for Lüscher-type [15, 16] order-by-order corrections to wrapping, see for instancerefs. [13, 17–19], but it seems to be hard to push these computations much further, at leastfor generic functions. On the other hand, it is possible to extend such computations to morecomplicated correlation functions: to higher-point planar correlation functions [20, 21], aswell as to non-planar correlators [22–24]. The main aim of this paper is to extend this verypromising program beyond the case of AdS /CFT .– 2 –nother extremely interesting holographic setup is the AdS /CFT correspondence [1].In terms of superstring backgrounds, there are three families of backgrounds that one mayconsider: AdS × S × T , AdS × S × K3 and AdS × S × S × S . Each of these preservessixteen real supersymmetries, half of the amount of AdS × S . The focus of this paper ison what is arguably the simplest of these setups, AdS × S × T . The case of K3 follows toa large extent from that, at least for orbifold K3s, while the case of AdS × S × S × S isinteresting in its own right, but beyond the scope of our discussion. For these backgroundsthere are effectively two parameters that determine the spectrum in the planar limit: looselyspeaking, this is because the background can be supported by a combination of Ramond-Ramond (RR) and Neveu-Schwarz-Neveu-Schwarz fluxes (NSNS), which affect the spectrumvery differently — see ref. [25] for a discussion of the moduli of this background and theireffect on the spectrum. When there is no NSNS flux, the background is most similar to thecase of AdS × S . When there is no RR flux, the spectrum becomes very degenerate, anda continuum appears corresponding to the so-called long strings. For such a background(and only in this case), a simple worldsheet CFT description exists in the Ramond-Neveu-Schwarz formalism. For the AdS part it can be given in terms of a supersymmetric sl (2 , R ) level- k Wess-Zumino-Witten model [26–28], which can then be coupled to an su (2) WZWmodel and free Bosons to account for the remaining compact spaces. The special case k = 1 requires slightly different worldsheet-CFT techniques [29], but it is very interesting becauseit seems to be the only point of the whole moduli space where one has a firm handle onthe CFT dual [30–32], which should be the symmetric-product orbifold CFT of four freeBosons and as many Fermions, Sym N ( T ) .Remarkably, both the pure-RR and pure-NSNS background, as well as anything inbetween, are classically integrable [33, 34]. Integrability, expressed in terms of factorisedscattering, seems also to hold at the quantum level. Largely by analogy with the case ofAdS × S , it was first understood for pure-RR backgrounds, see ref. [35] for a review, andthen extended to mixed-flux backgrounds [36–38]. Also in this case, wrapping correctionsare not entirely under control [39]. A notable exception is the case of pure-NSNS back-grounds, where the spectrum was computed by means of integrability and showed to matchwith the WZW prediction [40, 41], including at the special value of the level k = 1 [42].It is natural to ask whether we may use integrability to compute three- and higher-point correlation functions of generic operators on AdS × S × T . For anything butpure-NSNS background, this would be a major advance as there is currently no techniqueto do so. Conversely, for pure-NSNS backgrounds, this could yield a nice comparison withthe worldsheet-CFT (or RNS) approach, and possibly shine new light on how the hexagonapproach relates to the CFT machinery. This is particularly interesting because the hexagonapproach is formulated in terms of the target-space Fermions, and hence should be closerto the dual theory. (Worldsheet functions have recently been studied in ref. [43] at level k = 1 precisely to map them to the their holographic counterparts.) We will see belowthat indeed, the hexagon approach can be used also for AdS × S × T , with arbitrarybackground fluxes. We will also work out the hexagon form factor for one- and two-particlestates, based on symmetry, and use a factorisation principle to bootstrap the form factor forarbitrarily-many particles. This lays the basis for a systematic investigation of correlation– 3 –unction in the whole moduli space of AdS × S × T .The article is structured as it follows. We decided to dedicate section 2 to a ratherdetailed review of AdS × S × T integrability, given that we will need many results for thespectral problem which are somewhat scattered over the literature; we take this occasion totry to fix some conventions and correct some minor misprints that are floating in the olderliterature. The main part of the paper is section 3 where we set up the hexagon programfor AdS × S × T . In section 4 we check our construction against a result easily availablein the literature, i.e. the three-point functions of certain protected operators; it should bestressed that here, unlike in AdS × S , the spectrum of protected operators is quite rich,and their correlation functions quite non-trivial. Finally, we discuss our result and outlookin section 5. We also have spelled out the full AdS × S × T S matrix in appendix A forthe readers’ convenience (again, the whole result was not explicitly written in any givenarticle to the best of our knowledge). × S × T The AdS × S × T metric superisometries are given by psu (1 , | ⊕ psu (1 , | , (2.1)where each copy contains eight real supergcharges and a Bosonic subalgebra su (1 , ⊕ su (2) .In total, the Bosonic subalgebra is then [ su (1 , ⊕ su (1 , ⊕ [ su (2) ⊕ su (2)] ∼ = so (2 , ⊕ so (4) .The factorisation of the isometries like in eq. (2.1) is a key feature of the AdS / CFT duality. In particular, each of the two copies of the non-compact subalgebra su (1 , ∼ = sl (2 , R ) correpsonds to the chiral and antichiral part of the global conformal algebra inthe dual CFT . In addition to these isometries, we have a four shift isometries u (1) ⊕ corresponding to the T directions. Finally, the flat manifold T enjoys a local so (4) T rotational symmetry, which for later convenience we also decompose as so (4) T ∼ = su (2) • ⊕ su (2) ◦ . This is not a symmetry of the whole theory (it is broken by the boundary conditionsof the T fields) but it will play an important role nonetheless, for two reasons. First of all,locally the Killing spinors will be charged under so (4) T , which will make this algebra usefulto group the symmetry generators; secondly but importantly, states with no momentum orwinding along T are blind to its global features, and as long as we restrict to those (as wewill do), this rotational isometry will be important.The complete type IIB superstring background features additional fields beyond theAdS × S × T metric. Generically, the background will involve Neveu-Schwarz-Neveu-Schwarz (NSNS) and Ramond-Ramond (RR) fluxes. To be concrete, we will consider abackground with a NSNS three-form flux H = dB proportional to the volume form ofAdS × S and a RR three form flux F also proportional to the same volume form. Sucha background can be thought of as arising from the F1-NS5-D1-D5 system. Its dynamicsis dominated by two parameters: the amount of NSNS fluxes and the amount of RR fluxes(in units where the AdS radius is set to one). Within this two-parameter space, there aretwo interesting limits: the case where RR fluxes are absent, which can be described by asupersymmetric sl (2 , R ) WZW model and corresponds to the F1-NS5 system, and the case– 4 –here H = B = 0 which corresponds to the D1-D5 system. The latter is most similar toAdS × S . The whole two-parameter case is classically integrable [33, 34] and is believed tobe integrable at the quantum level, see [35] for a review. It is worth emphasising that theF1-NS5-D1-D5 system has more moduli than the two we just introduced [44]. However, inthe near horizon limit and when restricting to states with no momentum or winding alongT , only these two modules end up being important, see ref. [25] for a detailed discussion.The classical integrability of strings on AdS × S × T was discussed in [33, 34]. Thestudy of the integrable S matrix was initiated in [37, 45] and completed, for the matrixpart, in refs. [38, 46, 47]. The dressing factors were studied in [48–50] for backgroundswithout NS-NS fluxes. The S matrix (including dressing factors) for the case of NS-NSfluxes only , is also known and was worked out in [40, 41], where it was also shown that theresulting mirror TBA reproduces the WZW spectrum. It is worth emphasising that takingthe pure-NSNS limit in the S matrix is subtle because all excitations become massless then.Besides, the dressing factors for the generic mixed-flux backgrounds are not known (see [51]for work in this direction). Even if most of the integrability construction is reviewed in[35], some details and especially the latest developments are scattered over the literature.Hence we find it useful to review its main features below and collect some formulae in theappendices. The supersymmetry algebra is given by two copies of psu (1 , | as in eq. (2.1). The firstcopy, which we label “left”, is given by eight supercharges Q mαa ( α = ± , a = ± , m = ± ), three su (2) (R-symmetry) generators J α ( α = ± , ) and three su (1 , generators L m ( m = ± , ). Notice that the supercharges carry an index a = ± , due to the fact that theytransform in the fundamental representation of an su (2) automorphism. In fact, it can beseen geometrically [47] that such an automorphism is a subalgebra of so (4) T , and in whatfollows we will label it su (2) • . The second copy, which we call “right” and denote with tildas,is given by (cid:101) Q ˙ m ˙ αa , (cid:101) J ˙ α and (cid:101) L ˙ m . Note however that these charges are charged under the same su (2) • automorphism as the “left” ones. The names “left” and “right” correspond to theinterpretation of these charges as symmetries of the two-dimensional dual superconformalsymmetry with N = (4 , symmetry. As remarked above, the two (R-symmetry) su (2) algebras give the isometries of the three sphere, su (2) ⊕ su (2) ∼ = so (4) , and the two su (1 , give the AdS isometries, su (1 , ⊕ su (1 , ∼ = so (2 , . A convenient matrix realisation of su (1 , | is given by (2 | supermatrices, which can bewritten as blocks M = (cid:32) m θη n (cid:33) , (2.2)where Latin letters are bosonic blocks and Greek ones are fermionic. The matrix M mustsatisfy M † + Σ M Σ = 0 , Σ = diag = (1 , − , , . (2.3)– 5 –e can introduce the following explicit parametrisations for the complexified algebra L = 12 − , L + = , L − = , (2.4) J = 12 − , J + = , J − = , (2.5) Q −− = − , Q ++2 = , (2.6) Q ++1 = , Q −− = − , (2.7) Q + − = , Q − +2 = , (2.8) Q − +1 = − , Q + − = − , (2.9)These satisfy the commutation relations [ L , L ± ] = ± L ± , [ L + , L − ] = 2 L , [ J , J ± ] = ± J ± , [ J + , J − ] = 2 J , [ L , Q ± αA ] = ± Q ± αA , [ L ± , Q ∓ αA ] = Q ± αA , [ J , Q a ± A ] = ± Q a ± A , [ J ± , Q a ∓ A ] = Q a ± A , { Q ± + A , Q ± + B } = ± (cid:15) AB L ± , { Q + ± A , Q −± B } = ∓ (cid:15) AB J ± , (2.10)and finally { Q + ± A , Q −∓ B } = (cid:15) AB (cid:0) − L ± J (cid:1) . (2.11)The Weyl-Cartan basis of the algebra is as it follows: [ h i , h j ] = 0 , [ e i , f j ] = δ ij h j , [ h i , e j ] = A ij e j , [ h i , f j ] = − A ij f j , (2.12)– 6 –ith h = − L − J , e = + Q + − f = + Q − +2 , h = 2 J , e = + J + f = + J − , h = − L − J , e = + Q + − f = − Q − +1 , (2.13)with Cartan matrix A = − − − − . (2.14) In this notation, the BPS condition for the algebra is − L − J ≥ . (2.15)This positive-semidefinite operator can therefore be used to define a “Hamiltonian” for theleft algebra. Indeed it can be shown that this is precisely the contribution of left chargesto the light-cone Hamiltonian [45, 47] H ≡ − L − J . (2.16)The supercharges that commute with H are Q ≡ f = + Q − +2 , Q ≡ f = − Q − +1 , S ≡ e = + Q + − , S ≡ e = + Q + − . (2.17)They form the algebra psu (1 | ⊕ ⊕ u (1) (cid:8) Q A , S B (cid:9) = δ AB H ≥ . (2.18)The charges satisfy the Hermiticity conditions ( Q + − ) † = + Q − +2 , ( Q − +2 ) † = + Q + − , ( Q − +1 ) † = − Q + − , ( Q + − ) † = − Q − +1 , (2.19)or equivalently ( Q A ) † = S A , ( S A ) † = Q A , (2.20) Let us now come to the second copy of psu (1 , | , the “right” algebra. The construction ofthe matrix representation is identical. We will however pick a slightly different Weyl-Cartanbasis. Denoting the generators by tildes, we have ˜ h = + (cid:101) L + (cid:101) J , ˜ e = + (cid:101) Q − +1 ˜ f = + (cid:101) Q + − , ˜ h = − (cid:101) L , ˜ e = + (cid:101) L + ˜ f = − (cid:101) L − , ˜ h = + (cid:101) L + (cid:101) J , ˜ e = + (cid:101) Q − +2 ˜ f = − (cid:101) Q + − , (2.21)– 7 –ith Cartan matrix (cid:101) A = − . (2.22)Running a little ahead of ourselves, let us motivate this choice. Given that the algebra (2.1)is factorised, we can choose the positive roots in either copy of the algebra independently.It will turn out however that, when considering a certain class of “off-shell” observables,the symmetries will be extended by two central charges that couple the left and rightalgebras [48]. In that case, our choice of positive roots will prove convenient. Once again we define (cid:101) H ≡ − (cid:101) L − (cid:101) J ≥ , (2.23)and (cid:101) Q ≡ + e = + (cid:101) Q − +1 , (cid:101) Q ≡ + e = + (cid:101) Q − +2 , (cid:101) S ≡ − f = − (cid:101) Q + − , (cid:101) S ≡ − f = + (cid:101) Q + − , (2.24)so that { (cid:101) Q A , (cid:101) S B } = δ AB (cid:101) H ≥ . (2.25)Like before we have that ( (cid:101) Q A ) † = (cid:101) S A , ( (cid:101) S A ) † = (cid:101) Q A . (2.26) Much like in the case of AdS / CFT [14, 52], the algebra relevant for integrability fea-tures a central extension with respect to the superisometry algebra. This central extensionannihilates all physical states. However, it acts nontrivially on the individual worldsheetexcitations that make up a physical state (or, in spin-chain language, on the magnons thatmake up the Bethe state). We refer the reader to [11, 12] for reviews of the construction inthe AdS / CFT setup.For our purposes, it will be sufficient at this stage to recall [45] how the algebra ofsymmetries commuting with H and (cid:101) H may be extended. In the notation just introduced,the algebra is (cid:110) Q A , S B (cid:111) = H δ AB , (cid:110) (cid:101) Q A , (cid:101) S B (cid:111) = (cid:101) H δ AB , (2.27)This allows for a central extension. It is possible to check semiclassically that the cen-tral extension appears for AdS × S × T backgrounds with Ramond-Ramond flux [47].Introducing two central charges P and K we have (cid:110) Q A , (cid:101) Q B (cid:111) = P δ AB , (cid:110) S A , (cid:101) S B (cid:111) = K δ AB , (2.28)where the reality conditions discussed above imply that for a unitary representation K † = P and P † = K . In presence of this central extension, our choice of simple roots (2.17) and(2.24) appears natural. Since K is central, if Q A is a negative root, then (cid:101) Q A needs to be a positive root, and similarly for (cid:101) S A and S A .– 8 – .2.1 Factorisation of the centrally extended algebra The algebra above is psu (1 | ⊕ centrally extended, which plays a role similar to su (2 | ⊕ in AdS / CFT . In the latter case, the factorisation in su (2 | ⊕ was quite useful in sim-plifying many computations: in particular, it was sufficient to work out a su (2 | -invariantS matrix [14] which served as a building block of the full su (2 | ⊕ -invariant S matrix. Toemphasise the similarity in the factorised structure, we introduce the psu (1 | ⊕ centrallyextended algebra, given by (cid:110) q , s (cid:111) = H , (cid:110)(cid:101) q , (cid:101) s (cid:111) = (cid:101) H , (cid:110) q , (cid:101) q (cid:111) = P , (cid:110) s , (cid:101) s (cid:111) = K , (2.29)The algebra in eq. (2.29) plays the role that su (2 | plays for AdS / CFT . We can thenobtain the larger algebra by setting Q ≡ q ⊗ , Q ≡ Σ ⊗ q , S ≡ s ⊗ , S ≡ Σ ⊗ s , (2.30)where Σ is the graded identity, Σ = δ ij ( − F j and (note the lowered indices) (cid:101) Q ≡ (cid:101) q ⊗ , (cid:101) Q ≡ Σ ⊗ (cid:101) q , (cid:101) S ≡ (cid:101) s ⊗ , (cid:101) S ≡ Σ ⊗ (cid:101) s . (2.31)To show that this give the same psu (1 | ⊕ centrally extended as above, note that on anysupercharge we have e.g. Σ q Σ = − q . (2.32)Indices are raised and lowered with the Levi-Civita symbol with ε = − ε = 1 . Having identified the algebra that commutes with the left and right Hamiltonians H and (cid:101) H ,as well as its central extension, it is time to construct its short representation. Worldsheetexcitations will transform in these representations [47, 48]. It is convenient to start fromthe smaller algebra (2.29). psu (1 | ⊕ centrally extended We are interested in the short representations of the smaller algebra (2.29). Let | φ (cid:105) be ahighest weight state, and let us say that q is a lowering operator. Then it must be s | φ (cid:105) = 0 ,because s is a raising operator. From the commutation relation involving P we see that (cid:101) q must also act as a raising operator on q | φ (cid:105) . The representation is short if we can assumethat (cid:101) q | φ (cid:105) = 0 and (cid:101) s ( q | φ (cid:105) ) = 0 , so that no new states are generated and the representationis two-dimensional. In that case we can write (cid:2)(cid:101) s (cid:101) qq − (cid:101) s (cid:101) qq (cid:3) | φ (cid:105) = (cid:2) ( (cid:101) s (cid:101) q + (cid:101) q (cid:101) s ) q − (cid:101) q (cid:101) sq − (cid:101) s ( (cid:101) qq + q (cid:101) q ) (cid:3) | φ (cid:105) = (cid:2) (cid:101) Hq − (cid:101) q (cid:101) sq − (cid:101) sP (cid:3) | φ (cid:105) = (cid:2) (cid:101) Hq − (cid:101) sP (cid:3) | φ (cid:105) . (2.33)By taking the anti-commutator of this expression with s we can find a condition whichdepends only on the central charges and therefore applies to the whole representation (notonly to | φ (cid:105) ). H (cid:101) H = P K , on the representation. (2.34)– 9 –nterestingly, if P = 0 it must be either H = 0 or (cid:101) H = 0 , i.e. , the representation ischiral. (When H = (cid:101) H = 0 the representation decomposes in two unidimensional singletrepresentations.) We will see that such chiral representations appear for all physical states,as well as for any state in a theory with no RR fluxes. In conclusion, the only shortrepresentations (besides singlets) are two dimensional, they consist of a Boson and oneFermion, and we indicate them as ( | ) .A short representation with highest weigth state | φ (cid:105) is parametrised by the eigenvaluesof the central charges, ( P, K, H, (cid:101) H ) . The shortening condition (2.34) implies that, if therepresentation is unitary, P K ≥ . (2.35)For this reason for unitary representations we will henceforth indicate C ≡ P , C † ≡ K . (2.36)The representation has the form q | φ (cid:105) = a | ϕ (cid:105) , s | ϕ (cid:105) = a ∗ | φ (cid:105) , (cid:101) s | φ (cid:105) = b ∗ | ϕ (cid:105) , (cid:101) q | ϕ (cid:105) = b | φ (cid:105) , (2.37)where a, b ∈ C . Note that on this representation | φ (cid:105) = highest-weight state , | ϕ (cid:105) = lowest-weight state , (2.38)Note that | φ (cid:105) and | ϕ (cid:105) must have opposite statistics. We get two distinct type of represen-tations by setting | φ (cid:105) to be a Boson or a Fermion, φ → φ B ≡ Boson , ϕ → ϕ F ≡ Fermion , (2.39)or viceversa φ → φ F ≡ Fermion , ϕ → ϕ B ≡ Boson . (2.40)Finally the central charges take the form: C = C = ab , C † = C ∗ = ( ab ) ∗ , H = H = | a | , (cid:101) H = (cid:101) H = | b | . (2.41)One can solve for a, b as a functions of ( C, H, (cid:101) H ) . We can parametrise the central charges themselves in terms of the coupling constants andthe momentum p of the magnon [45] C = + i h e ip − e iξ , C ∗ = − i h e − ip − e − iξ , (2.42)where ξ is a representation coefficient related to an automorphism of the algebra. Asexplained in ref. [11], ξ arises from the boundary conditions of the fields, and is importantto establish the coproduct of the algebra. Notice that C = C ∗ = 0 when p = 0 mod π , which– 10 –s the case for physical states. Here h ≥ is a property of the background: the amountof RR background flux. In what follows, we will be interested in the “most-symmetriccoproduct” [45], and we will set C = C ∗ = − h sin( p/ . (2.43)Coming to the remaining central charges, let us consider the combinations E ≡ H − (cid:101) H = − L − (cid:101) L − J − (cid:101) J ≥ , M ≡ H − (cid:101) H = − L + (cid:101) L − J + (cid:101) J , (2.44)For physical states ( p = 0 mod π ), the eigenvalues of M should be quantised in integers.For Bosonsic states this is obvious as the AdS and S spins are integer. For Fermionicstates, both spins are half-integer, so that the total spin in M is integer. It turns out thatit is [37, 38] M = | a | + | b | = k π p + m , m ∈ Z . (2.45)Here k = 1 , , , . . . is a property of the string background and measures tha amount ofNSNS flux, which is quantised. In the special case where h = 0 and k > , then k isprecisely the level of the supersymmetric WZW model describing the worldsheet theory.Before commenting more on m , let us use the shortening condition (2.34) to express thelast central charge E as E = M + | C | , E = (cid:115)(cid:18) m + k π p (cid:19) + h sin p . (2.46)It is clear that m plays the role of a mass in the dispersion relation. Therefore we introducethe following nomenclature:• m = 0 : we call the representation massless. Here we have that E = 0 at p = 0 forany value of h and k .• m = +1 , +2 , . . . : we call these representations “left” because at p = 0 we have that E = m and M = m , which implies H = m > and (cid:101) H = 0 .• m = − , − , . . . : we call these representations “right” because at p = 0 we have that E = − m and M = m , which implies H = 0 and (cid:101) H = − m > .We can further distinguish the case | m | = 1 which corresponds to fundamental particles,from the case of | m | = 2 , , . . . , which corresponds to bound states thereof [37, 48]. It isworth emphasising that, unlike what happens in AdS × S , bound-state modules have thesame dimension as fundamental particle modules — they are two-dimensional. psu (1 | ⊕ c.e. Four irreducible representations will be important in what follows. We denote them by ρ L = ( φ BL | ϕ FL ) , ρ R = ( φ FR | ϕ BR ) , ρ o = ( φ Bo | ϕ Fo ) , ρ (cid:48) o = ( φ Fo | ϕ Bo ) , (2.47)– 11 –here the first state is always the highest-weight state, | φ ∗∗ (cid:105) = highest-weight state , | ϕ ∗∗ (cid:105) = lowest-weight state . (2.48)The representations take the same form up to relabeling the representation coefficients: q | φ BL (cid:105) = a L | ϕ FL (cid:105) , s | ϕ FL (cid:105) = a ∗ L | φ BL (cid:105) , (cid:101) s | φ BL (cid:105) = b ∗ L | ϕ FL (cid:105) , (cid:101) q | ϕ FL (cid:105) = b L | φ BL (cid:105) , q | φ FR (cid:105) = a R | ϕ BR (cid:105) , s | ϕ BR (cid:105) = a ∗ R | φ FR (cid:105) , (cid:101) s | φ FR (cid:105) = b ∗ R | ϕ BR (cid:105) , (cid:101) q | ϕ BR (cid:105) = b R | φ FR (cid:105) , q | φ Bo (cid:105) = a o | ϕ Fo (cid:105) , s | ϕ Fo (cid:105) = a ∗ o | φ Bo (cid:105) , (cid:101) s | φ Bo (cid:105) = b ∗ o | ϕ Fo (cid:105) , (cid:101) q | ϕ Fo (cid:105) = b o | φ Bo (cid:105) , q | φ Fo (cid:105) = a o | ϕ Bo (cid:105) , s | ϕ Bo (cid:105) = a ∗ o | φ Fo (cid:105) , (cid:101) s | φ Fo (cid:105) = b ∗ o | ϕ Bo (cid:105) , (cid:101) q | ϕ Bo (cid:105) = b o | φ Fo (cid:105) . (2.49)The explicit form of the representation coefficients can be given in terms of Zhukovskivariables, much like in AdS5. We will be able to describe all representation parameters byintroducing different sets of Zhukovsky variables: a L = e iξ η L ,p , b L = − e iξ e − ip/ x − L ,p η L ,p , a ∗ L = e − iξ e − ip/ η L ,p , b ∗ L = − e − iξ x + L ,p η L ,p ,b R = e iξ η R ,p , a R = − e iξ e − ip/ x − R ,p η R ,p , b ∗ R = e − iξ e − ip/ η R ,p , a ∗ R = − e − iξ x + R ,p η R ,p ,a o = e iξ η o ,p , b o = − e iξ e − ip/ x − o ,p η o ,p , a ∗ o = e − iξ e − ip/ η o ,p , b ∗ o = − e − iξ x + o ,p η o ,p . (2.50)The η parameter is always η ∗ ,p = e ip/ (cid:114) ih x −∗ ,p − x + ∗ ,p ) , (2.51)where we indicated with “ ∗ ” the symbols L, R, o. The Zhukovsky variables, instead, satisfy x + L ,p + 1 x + L ,p − x − L ,p − x − L ,p = 2 i (cid:0) k π p (cid:1) h ,x + R ,p + 1 x + R ,p − x − R ,p − x − R ,p = 2 i (cid:0) − k π p (cid:1) h ,x + o ,p + 1 x + o ,p − x − o ,p − x − o ,p = 2 i (cid:0) k π p (cid:1) h , (2.52)and can be parametrised as it follows x ± L ,p = e ± ip/ h sin (cid:0) p (cid:1) (cid:32)(cid:0) k π p (cid:1) + (cid:113)(cid:0) k π p (cid:1) + 4 h sin (cid:0) p (cid:1)(cid:33) ,x ± R ,p = e ± ip/ h sin (cid:0) p (cid:1) (cid:32)(cid:0) − k π p (cid:1) + (cid:113)(cid:0) − k π p (cid:1) + 4 h sin (cid:0) p (cid:1)(cid:33) ,x ± o ,p = e ± ip/ h sin (cid:0) p (cid:1) (cid:32)(cid:0) k π p (cid:1) + (cid:113)(cid:0) k π p (cid:1) + 4 h sin (cid:0) p (cid:1)(cid:33) . (2.53)– 12 –t satisfies (2.52) as well as x + ∗ ,p − x + ∗ ,p − x −∗ ,p + 1 x −∗ ,p = 2 i Eh . (2.54)It is worth noting that the left and right representation are not simply related bysending m → − m as one may have naïvely expected. Instead, the paramterisation of therepresentation coefficients and of the Zhukovski variables is genuinely different. This isdone so that | x ±∗ ,p | ≥ for physical particles [53, 54].Notice further that for the massless representation we have defined a o = lim m → a L , b o = lim m → b L , x ± o ,p = lim m → x ± L ,p , (2.55)We could have used the right representation instead. Practically, this amounts to flippingthe sign of p in (2.53) and switching a o ↔ b o in (2.50). This is actually allowed and does notintroduce any new physics because the central charges, and in particular M in eq. (2.45), areunchanged. Hence the two representations obtained in the two limits must be isomorphic.This may be seen through a change of basis, e.g. by rescaling the lowest-weight state (butnot the highest-weight one) as it follows: | ϕ (cid:105) → σ p | ϕ (cid:105) , σ p ≡ (cid:104) a L a R (cid:105) m → = − sgn (cid:104) sin p/ (cid:105) . (2.56)It is also useful to note the following identity lim m → (cid:0) x ± L ( p ) x ∓ R ( p ) (cid:1) = 1 , (2.57)which is valid for any k and generalises the fact, valid at k = 0 , that x + o ( p ) = 1 /x − o ( p ) .Let us finally comment on the h → limit, which corresponds to the WZW model.The Zhukovsky variables are divergent in this limit x ±∗ ,p = e ± ip/ h sin( p/ M + | M | O ( h ) , η p = e ip/ (cid:114) M + | M | O ( h ) , (2.58)The leading order in h of the Zhukovsky variables depends on the sign of (2 πm + kp ) , i.e. on the branch of the dispersion relation E ( p ) = (cid:12)(cid:12)(cid:12) m + k π p (cid:12)(cid:12)(cid:12) . (2.59)Therefore, particles moving in the same or in opposite directions have starkly differentlimits [40]. It will be useful in what follows to consider particles whose momentum analytically iscontinued to the crossed region, i.e. p → − p , E ( p ) → − E ( p ) . (2.60)– 13 –dS Bosons S Bosons T Bosons FermionsLeft, m = +1 Z ( p ) Y ( p ) Ψ A ( p ) Right m = − Z ( p ) ˜ Y ( p ) ˜Ψ A ( p ) Massless m = 0 T A ˙ A ( p ) χ ˙ A ( p ) , ˜ χ ˙ A ( p ) Table 1 . The fundamental particles of AdS × S × T are eight Bosons and eight Fermions. Inthis table we arrange them according to which representation they belong (this depends on the signof the central charge M at momentum p = 0 , m = M | p =0 ) and to their geometrical interpretation. This is the analogue of going from the s - to the t - channel in a relativistic theory. Followingthe notation of ref. [13] we indicate the crossed momentum as p γ . This is justified bythe fact that p γ represents the continuation of momentum to the mirror region [9] whichloosely speaking corresponds to “half crossing”. For a comprehensive discussion of the mirrorand crossed regions we refer the readers to [11] and, in the context of AdS × S × T , to[38, 48, 49]. Under the crossing transformation we have [38] x ± L ( p γ ) = 1 x ± R ( p ) , x ± R ( p γ ) = 1 x ± L ( p ) . (2.61)Hence, the Zhukovsky variables and any rational function thereof map to themselves undera γ -shift. Instead, the functions η L ( p ) and η R ( p ) behave as it follows, η L ( p ± γ ) = ± ix + R ( p ) η R ( p ) , η R ( p ± γ ) = ± ix + L ( p ) η L ( p ) . (2.62)Crossing for massless modes is essentially given by the m → limit of the massless caseand by recalling the identity (2.57). We have x ± o ( p γ ) = x ∓ ( p ) , η o ( p ± γ ) = ∓ iσ p e − ip/ η o ( p ) . (2.63) The fundamental particle content of the theory is summarised in Table 1. They can be ar-ranged in representations constructed out of the ρ ± , ρ , ρ (cid:48) representations discussed above.Let ρ be any short representation of psu (1 | ⊕ c.e., which as we saw is two dimensionaland takes the form ( | ) . We want to use it to construct representations of psu (1 | ⊕ c.e.,which we call (cid:37) . Clearly we can set (cid:37) = ρ ± ⊗ ρ ± , or (cid:37) = ρ ⊗ ρ (or indeed, in this lastformula, swap ρ for ρ (cid:48) , as it will turn out to be the case). However, a representation ofthe form e.g. (cid:37) = ρ ± ⊗ ρ ∓ or (cid:37) = ρ ± ⊗ ρ would not be a representation of the algebra in-troduced in Section 2.2. Indeed to obtain a valid representation it is necessary that the two psu (1 | ⊕ representations appearing in the tensor product have the same central charge. To construct the left representation we consider (cid:37) + = ρ + ⊗ ρ + . (2.64)– 14 –e define the following states | Y ( p ) (cid:105) = | φ BL ( p ) ⊗ φ BL ( p ) (cid:105) , | Ψ ( p ) (cid:105) = | ϕ FL ( p ) ⊗ φ BL ( p ) (cid:105) , | Ψ (cid:105) = | φ BL ( p ) ⊗ ϕ FL ( p ) (cid:105) , | Z ( p ) (cid:105) = | ϕ FL ( p ) ⊗ ϕ FL ( p ) (cid:105) , (2.65)By using this definition we see that the supercharges act is as it follows | Y ( p ) (cid:105)| Ψ ( p ) (cid:105) | Ψ ( p ) (cid:105)| Z ( p ) (cid:105) Q , (cid:101) S Q , (cid:101) S Q , (cid:101) S Q , (cid:101) S (2.66)To avoid cluttering the figure we only indicated the lowering operators, and not the raisingones. By using the definitions of section 2.2.1 we get the following action of the supercharges: Q A | Y ( p ) (cid:105) = a L ( p ) | Ψ A ( p ) (cid:105) , Q A | Ψ B ( p ) (cid:105) = ε AB a L ( p ) | Z ( p ) (cid:105) , S A | Ψ B ( p ) (cid:105) = δ AB a ∗ L ( p ) | Y ( p ) (cid:105) , S A | Z ( p ) (cid:105) = − ε AB a ∗ L ( p ) | Ψ B ( p ) (cid:105) , (cid:101) S A | Y ( p ) (cid:105) = b ∗ L ( p ) | Ψ A ( p ) (cid:105) , (cid:101) S A | Ψ B ( p ) (cid:105) = ε AB b ∗ L ( p ) | Z ( p ) (cid:105) , (cid:101) Q A | Ψ B ( p ) (cid:105) = δ AB b L ( p ) | Y ( p ) (cid:105) , (cid:101) Q A | Z ( p ) (cid:105) = − ε AB b L ( p ) | Ψ B ( p ) (cid:105) , (2.67)where we omitted the vanishing actions and we recall our convention ε = − ε = +1 . For the right representation (cid:37) − = ρ − ⊗ ρ − , (2.68)and we define | ˜ Z ( p ) (cid:105) = | φ FR ( p ) ⊗ φ FR ( p ) (cid:105) , | ˜Ψ ( p ) (cid:105) = | ϕ BR ( p ) ⊗ φ FR ( p ) , | ˜Ψ ( p ) (cid:105) = −| φ FR ( p ) ⊗ ϕ BR ( p ) (cid:105) , | ˜ Y ( p ) (cid:105) = | ϕ BR ( p ) ⊗ ϕ BR ( p ) (cid:105) , (2.69)where the reason for the minus sign is that “right” supercharges are canonically defined withlower su (2) • indices, see section 2.2.1. Arranging the representation in this way we see thatthe lowering operator act in the same fashion as above | ˜ Z ( p ) (cid:105)| ˜Ψ ( p ) (cid:105) | ˜Ψ ( p ) (cid:105)| ˜ Y ( p ) (cid:105) Q , (cid:101) S Q , (cid:101) S Q , (cid:101) S Q , (cid:101) S (2.70)– 15 –here the representation takes the form Q A | ˜ Z ( p ) (cid:105) = b R ( p ) | ˜Ψ A ( p ) (cid:105) , Q A | ˜Ψ B ( p ) (cid:105) = − ε AB b R ( p ) | ˜ Y ( p ) (cid:105) , S A | ˜Ψ B ( p ) (cid:105) = δ AB b ∗ R ( p ) | ˜ Z ( p ) (cid:105) , S A | ˜ Y ( p ) (cid:105) = ε AB b ∗ R ( p ) | ˜Ψ B ( p ) (cid:105) , (cid:101) S A | ˜ Y ( p ) (cid:105) = a ∗ R ( p ) | ˜Ψ A ( p ) (cid:105) , (cid:101) S A | ˜Ψ B ( p ) (cid:105) = − ε AB a ∗ R ( p ) | ˜ Z ( p ) (cid:105) , (cid:101) Q A | ˜Ψ B ( p ) (cid:105) = δ AB a R ( p ) | ˜ Y ( p ) (cid:105) , (cid:101) Q A | ˜ Z ( p ) (cid:105) = ε AB a R ( p ) | ˜Ψ B ( p ) (cid:105) . (2.71)Notice that there is a discrete left-right symmetry [38, 53] when swapping the particles | Y (cid:105) ↔ | ˜ Y (cid:105) , | Ψ A (cid:105) ↔ | ˜Ψ A (cid:105) , | Z (cid:105) ↔ | ˜ Z (cid:105) . (2.72) There are actually two massless representations, which carry a charge under another su (2) algebra, which commutes with all symmetries thus far introduced. This is the su (2) ◦ thatemerged from the decomposition of so (4) T . We write (cid:37) ˙ A = (cid:0) ρ ⊗ ρ (cid:48) (cid:1) ⊕ (cid:0) ρ (cid:48) ⊗ ρ (cid:1) , ˙ A = 1 , , (2.73)with the understanding that the two modules ρ ⊗ ρ (cid:48) and ρ (cid:48) ⊗ ρ must also fit into a doubletof su (2) ◦ . This is not in contradiction with the fact that su (2) ◦ commutes with psu (1 | ⊕ centrally extended because, as psu (1 | ⊕ c.e. representations, ρ ⊗ ρ (cid:48) ∼ = ρ (cid:48) ⊗ ρ . In fact,in reference [46] the same representation ρ ⊗ ρ (cid:48) was used for both ˙ A = 1 and ˙ A = 2 . Thisamounts to a change of basis. We now have eight states | χ ˙1 ( p ) (cid:105) = | φ B ( p ) ⊗ φ F ( p ) (cid:105) , | T ˙11 ( p ) (cid:105) = | ϕ F ( p ) ⊗ φ F ( p ) (cid:105) , | T ˙12 ( p ) (cid:105) = | φ B ( p ) ⊗ ϕ B ( p ) (cid:105) , | ˜ χ ˙1 ( p ) (cid:105) = | ϕ F ( p ) ⊗ ϕ B ( p ) (cid:105) , (2.74)and | χ ˙2 ( p ) (cid:105) = i | φ F ( p ) ⊗ φ B ( p ) (cid:105) , | T ˙21 ( p ) (cid:105) = i | ϕ B ( p ) ⊗ φ B ( p ) (cid:105) , | T ˙22 ( p ) (cid:105) = − i | φ F ( p ) ⊗ ϕ F ( p ) (cid:105) , | ˜ χ ˙2 ( p ) (cid:105) = − i | ϕ B ( p ) ⊗ ϕ F ( p ) (cid:105) , (2.75)Note that we have introduced an overall i in the latter representation. This is a matterof convenience that can be also addressed by introducing suitable normalisations later on.Arranging the representation in this way we see that the lowering operator act in the samefashion as above for either module | χ ˙ A ( p ) (cid:105)| T ˙ A ( p ) (cid:105) | T ˙ A ( p ) (cid:105)| ˜ χ ˙ A ( p ) (cid:105) Q , (cid:101) S Q , (cid:101) S Q , (cid:101) S Q , (cid:101) S (2.76)– 16 –nd regardless of the value of the index ˙ A = 1 , the representation takes the form Q A | χ ˙ A ( p ) (cid:105) = a o ( p ) | T ˙ AA ( p ) (cid:105) , Q A | T ˙ AB ( p ) (cid:105) = ε AB a o ( p ) | ˜ χ ˙ A ( p ) (cid:105) , S A | T ˙ AB ( p ) (cid:105) = δ AB a ∗ o ( p ) | χ ˙ A ( p ) (cid:105) , S A | ˜ χ ˙ A ( p ) (cid:105) = − ε AB a ∗ o ( p ) | T ˙ AB ( p ) (cid:105) , (cid:101) S A | χ ˙ A ( p ) (cid:105) = b ∗ o ( p ) | T ˙ AA ( p ) (cid:105) , (cid:101) S A | T ˙ AB ( p ) (cid:105) = ε AB b ∗ o ( p ) | ˜ χ ˙ A ( p ) (cid:105) , (cid:101) Q A | T ˙ AB ( p ) (cid:105) = δ AB b o ( p ) | χ ˙ A ( p ) (cid:105) , (cid:101) Q A | ˜ χ ( p ) (cid:105) = − ε AB b o ( p ) | T ˙ AB ( p ) (cid:105) . (2.77) Up to the dressing factors, the S matrix of AdS × S × T can be constructed by ten-soring two S matrices of psu (1 | ⊕ c.e., which in turn can be determined by imposingcommutation with the symmetries discussed above [45]. All in all, this closely resembleswhat happens with AdS × S , with two main differences: firstly, rather than dealing withthe algebra su (2 | c.e. we have here psu (1 | ⊕ c.e.; secondly, instead of dealing with asingle irreducble representation here we have four irreducible representations. (Recall thatin AdS × S we have four-dimensional representations of su (2 | leading to = 16 di-mensional representations of su (2 | ⊕ ; here we start from two-dimensional representationsinstead.) As a result of having four irreducible representations, the S matrix will consist ofsixteen blocks with as many dressing factors. Fortunately unitarity and other symmetriesreduce the number of independent dressing factors to four.For the reader’s convenience, we introduce below the scattering matrices between psu (1 | ⊕ c.e. representations that play a role in what follows. Here we report the scattering matrix for particles in the ρ L representation of psu (1 | ⊕ centrally extended, S | φ BL ,p φ BL ,q (cid:105) = A LL pq | φ BL ,q φ BL ,p (cid:105) , S | φ BL ,p ϕ FL ,q (cid:105) = B LL pq | ϕ FL ,q φ BL ,p (cid:105) + C LL pq | φ BL ,q ϕ FL ,p (cid:105) ,S | ϕ FL ,p ϕ FL ,q (cid:105) = F LL pq | ϕ FL ,q ϕ FL ,p (cid:105) , S | ϕ FL ,p φ BL ,q (cid:105) = D LL pq | φ BL ,q ϕ FL ,p (cid:105) + E LL pq | ϕ FL ,q φ BL ,p (cid:105) , (2.78)where the matrix elements are determined up to an overall prefactor Σ LL pq , A LL pq = Σ LL pq , B LL pq = Σ LL pq e − i p x + L ,p − x + L ,q x − L ,p − x + L ,q ,C LL pq = Σ LL pq e − i p e + i q x − L ,q − x + L ,q x − L ,p − x + L ,q η L ,p η L ,q , D LL pq = Σ LL pq e + i q x − L ,p − x − L ,q x − L ,p − x + L ,q ,E LL pq = C pq , F LL pq = − Σ LL pq e − i p e + i q x + L ,p − x − L ,q x − L ,p − x + L ,q . (2.79)Notice that we include a minus sign in F LL pq to account for the Fermion permutation (in otherwords, in the free-theory limit our S matrix reduces to the graded permutation operator).– 17 – .5.2 Right-right scattering Here we have S | ϕ BR ,p ϕ BR ,q (cid:105) = A RR pq | ϕ BR ,q ϕ BR ,p (cid:105) , S | ϕ BR ,p φ FR ,q (cid:105) = B RR pq | φ FR ,q ϕ BR ,p (cid:105) + C RR pq | ϕ BR ,q φ FR ,p (cid:105) ,S | φ FR ,p φ FR ,q (cid:105) = F RR pq | φ FR ,q φ FR ,p (cid:105) , S | φ FR ,p ϕ BR ,q (cid:105) = D RR pq | ϕ BR ,q φ FR ,p (cid:105) + E RR pq | φ FR ,q ϕ BR ,p (cid:105) , (2.80)with A RR pq = Σ RR pq , B RR pq = Σ RR pq e − i p x + R ,p − x + R ,q x − R ,p − x + R ,q ,C RR pq = Σ RR pq e − i p e + i q x − R ,q − x + R ,q x − R ,p − x + R ,q η R ,p η R ,q , D RR pq = Σ RR pq e + i q x − R ,p − x − R ,q x − R ,p − x + R ,q ,E RR pq = C pq , F RR pq = − Σ RR pq e − i p e + i q x + R ,p − x − R ,q x − R ,p − x + R ,q . (2.81) Here we have S | φ BL ,p ϕ BR ,q (cid:105) = A LR pq | ϕ BR ,q φ BL ,p (cid:105) + B LR pq | φ FR ,q ϕ FL ,p (cid:105) , S | φ BL ,p φ FR ,q (cid:105) = C LR pq | φ FR ,q φ BL ,p (cid:105) ,S | ϕ FL ,p φ FR ,q (cid:105) = E LR pq | φ FR ,q ϕ FL ,p (cid:105) + F LR pq | ϕ BR ,q φ BL ,p (cid:105) , S | ϕ FL ,p ϕ BR ,q (cid:105) = D LR pq | ϕ BR ,q ϕ FL ,p (cid:105) , (2.82)with A LR pq = Σ LR pq e − i p − x + L ,p x − R ,q − x − L ,p x − R ,q , B LR pq = Σ LR pq e − i p e − i q ih η L ,p η R ,q − x − L ,p x − R ,q ,C LR pq = Σ LR pq , D LR pq = Σ LR pq e − i p e − i q − x + L ,p x + R ,q − x − L ,p x − R ,q ,E LR pq = − Σ LR pq e − i q − x − L ,p x + R ,q − x − L ,p x − R ,q , F LR pq = − B LR pq . (2.83) The right-left S matrix is related to the left-right one by unitarity. It reads S | ϕ BR ,p φ BL ,q (cid:105) = A RL pq | φ BL ,q ϕ BR ,p (cid:105) + B RL pq | ϕ FL ,q φ FR ,p (cid:105) , S | ϕ BR ,p ϕ FL ,q (cid:105) = C RL pq | ϕ FL ,q ϕ BR ,p (cid:105) ,S | φ FR ,p ϕ FL ,q (cid:105) = E RL pq | ϕ FL ,q φ FR ,p (cid:105) + F RL pq | φ BL ,q ϕ BR ,p (cid:105) , S | φ FR ,p φ BL ,q (cid:105) = D RL pq | φ BL ,q φ FR ,p (cid:105) , (2.84)with A RL pq = Σ RL pq e + i q − x + R ,p x − L ,q − x + R ,p x + L ,q , B RL pq = Σ RL pq ih η R ,p η L ,q − x + R ,p x + L ,q ,C RL pq = Σ RL pq e + i p e + i q − x − R ,p x − L ,q − x + R ,p x + L ,q , D RL pq = Σ RL pq ,E RL pq = − Σ RL pq e + i p − x − R ,p x + L ,q − x + R ,p x + L ,q , F RL pq = − B RL pq . (2.85)– 18 – .5.5 Massless scattering. Because the massless representation coefficients may be obtained either from ρ L or ρ R by taking the m → limit, so can the relevant S-matrix elements (up to the dressingfactor: those do not follow immediately from symmetry, so that a more cautious analysisis required [48, 49]). Here we will choose to obtain the massless S-matrix elements fromthe left-left scattering. The only additional caution in this case is relative to the statisticsof the exictations, since in massless representations we may encounter Fermionic highest-weight states. This leads to different signs, which we spell out here, starting by recallingthe standard scattering matrix. S | φ Bo ,p φ Bo ,q (cid:105) = A LL pq | φ Bo ,q φ Bo ,p (cid:105) , S | φ Bo ,p ϕ Fo ,q (cid:105) = B LL pq | ϕ Fo ,q φ Bo ,p (cid:105) + C LL pq | φ Bo ,q ϕ Fo ,p (cid:105) ,S | ϕ Fo ,p ϕ Fo ,q (cid:105) = F LL pq | ϕ Fo ,q ϕ Fo ,p (cid:105) , S | ϕ Fo ,p φ Bo ,q (cid:105) = D LL pq | φ Bo ,q ϕ Fo ,p (cid:105) + E LL pq | ϕ Fo ,q φ Bo ,p (cid:105) , (2.86)When both particles are in the ˜ ρ L representation we have, instead S | ϕ Bo ,p ϕ Bo ,q (cid:105) = − F LL pq | ϕ Bo ,q ϕ Bo ,p (cid:105) , S | ϕ Bo ,p φ Fo ,q (cid:105) = D LL pq | φ Fo ,q ϕ Bo ,p (cid:105) − E LL pq | ϕ Bo ,q φ Fo ,p (cid:105) ,S | φ Fo ,p φ Fo ,q (cid:105) = − A LL pq | φ Fo ,q φ Fo ,p (cid:105) , S | φ Fo ,p ϕ Bo ,q (cid:105) = B LL pq | ϕ Bo ,q φ Fo ,p (cid:105) − C LL pq | φ Fo ,q ϕ Bo ,p (cid:105) . (2.87)Note that we could have also defined, in analogy with the above, A ˜ L ˜ L ≡ − F LL , B ˜ L ˜ L ≡ D LL , C ˜ L ˜ L ≡ − E LL , D ˜ L ˜ L ≡ B LL , E ˜ L ˜ L ≡ − C LL , and F ˜ L ˜ L ≡ − A LL . Similarly, in the mixed case wehave S | φ Bo ,p ϕ Bo ,q (cid:105) = B LL pq | ϕ Bo ,q φ Bo ,p (cid:105) − C LL pq | φ Fo ,q ϕ Fo ,p (cid:105) , S | φ Bo ,p φ Fo ,q (cid:105) = A LL pq | φ Fo ,q φ Bo ,p (cid:105) ,S | ϕ Fo ,p φ Fo ,q (cid:105) = − D LL pq | φ Fo ,q ϕ Fo ,p (cid:105) + E LL pq | ϕ Bo ,q φ Bo ,p (cid:105) , S | ϕ Fo ,p ϕ Bo ,q (cid:105) = − F LL pq | ϕ Bo ,q ϕ Fo ,p (cid:105) , (2.88)and finally S | ϕ Bo ,p φ Bo ,q (cid:105) = D LL pq | φ Bo ,q ϕ Bo ,p (cid:105) + E LL pq | ϕ Fo ,q φ Fo ,p (cid:105) , S | ϕ Bo ,p ϕ Fo ,q (cid:105) = − F LL pq | ϕ Fo ,q ϕ Bo ,p (cid:105) ,S | φ Fo ,p ϕ Fo ,q (cid:105) = − B LL pq | ϕ Fo ,q φ Fo ,p (cid:105) − C LL pq | φ Bo ,q ϕ Bo ,p (cid:105) , S | φ Fo ,p φ Bo ,q (cid:105) = A LL pq | φ Bo ,q φ Fo ,p (cid:105) . (2.89) In a similar way as the above, we may obtain the mixed-mass S matrix by considering themassless limit of the representation parameters only for one of the variables. Additionally,we have to account for the various signs that may arise due to the grading of the highestweight state. Below we list those related to the ρ (cid:48) ⊗ ρ − and ρ − ⊗ ρ (cid:48) representations, sincethe ones related to ρ (cid:48) ⊗ ρ + and ρ + ⊗ ρ (cid:48) are the same as eqs. (2.89) and (2.88), respectively.We have S | ϕ BR ,p ϕ Bo ,q (cid:105) = + C Ro pq | ϕ Bo ,q ϕ BR ,p (cid:105) , S | ϕ BR ,p φ Fo ,q (cid:105) = + A Ro pq | φ Fo ,q ϕ BR ,p (cid:105) − B Ro pq | ϕ Bo ,q φ FR ,p (cid:105) ,S | φ FR ,p φ Fo ,q (cid:105) = − D Ro pq | φ Fo ,q φ FR ,p (cid:105) , S | φ FR ,p ϕ Bo ,q (cid:105) = − E Ro pq | ϕ Bo ,q φ FR ,p (cid:105) + F Ro pq | φ Fo ,q ϕ BR ,p (cid:105) , (2.90)and (correcting a misprint in [47]) S | ϕ Bo ,p ϕ BR ,q (cid:105) = + D oR pq | ϕ BR ,q ϕ Bo ,p (cid:105) , S | ϕ Bo ,p φ FR ,q (cid:105) = − E oR pq | φ FR ,q ϕ Bo ,p (cid:105) − F oR pq | ϕ BR ,q φ Fo ,p (cid:105) ,S | φ Fo ,p φ FR ,q (cid:105) = − C oR pq | φ FR ,q φ Fo ,p (cid:105) , S | φ Fo ,p ϕ BR ,q (cid:105) = + A oR pq | ϕ BR ,q φ Fo ,p (cid:105) + B oR pq | φ FR ,q ϕ Bo ,p (cid:105) . (2.91)– 19 – .5.7 Dressing factors The pre-factors introduced above must obey crossing and unitarity constraints, besideshaving the correct analytic structure to give a sensible S-matrix for the full psu (1 | ⊕ c.e.S matrix. It is possible to write the solutions in the form [48, 49] (cid:0) Σ LL pq (cid:1) = (cid:0) Σ RR pq (cid:1) = e i ( p − q ) σ ∗∗ ( p, q ) x −∗ ,p − x + ∗ ,q x + ∗ ,p − x −∗ ,q − x −∗ ,p x + ∗ ,q − x + ∗ ,p x −∗ ,q , (cid:0) Σ LR pq (cid:1) = e ip σ LR ( p, q ) − x − L ,p x − R ,q − x + L ,p x + R ,q − x − L ,p x + R ,q − x + L ,p x − R ,q , (cid:0) Σ RL pq (cid:1) = e − iq σ RL ( p, q ) − x + R ,p x + L ,q − x − R ,p x − L ,q − x − R ,p x + L ,q − x + R ,p x − L ,q . (2.92)and for the massless case (cid:0) Σ oo pq (cid:1) = − e i ( p − q ) σ oo ( p, q ) x − o ,p − x + o ,q x + o ,p − x − o ,q . (2.93)Notice that our normalisation differs by an overall minus sign from that of [38]. This doesnot affect the crossing equations and leads to a consistent limit in the near-BMN expansion.For the mixed-mass cases we have Σ •◦ Lo ( p, q ) = e + i p x − L ,p − x + o ,q x + L ,p − x + o ,q ζ ( p, q ) 1 σ •◦ Lo ( p, q ) , Σ ◦• oL ( p, q ) = e − i q x − o ,p − x + L ,q x − o ,p − x − L ,q ζ ( p, q ) 1 σ ◦• oL ( p, q ) , Σ •◦ Ro ( p, q ) = e − i ( p + q ) (1 − x − R ,p x + o ,q )(1 − x + R ,p x + o ,q )(1 − x − R ,p x − o ,q ) ˜ ζ ( p, q ) 1 σ •◦ Ro ( p, q ) , Σ ◦• oR ( p, q ) = e + i ( p + q ) (1 − x − o ,p x + R ,q )(1 − x − o ,p x − R ,q )(1 − x + o ,p x + R ,q ) ˜ ζ ( p, q ) 1 σ ◦• oR ( p, q ) , (2.94)where we introduced the functions ζ ( p, q ) = (cid:115) x −∗ ,p − x −∗ ,q x + ∗ ,p − x −∗ ,q x + ∗ ,p − x + ∗ ,q x −∗ ,p − x + ∗ ,q , ˜ ζ ( p, q ) = (cid:115) − x + ∗ ,p x + ∗ ,q − x + ∗ ,p x −∗ ,q − x −∗ ,p x −∗ ,q − x −∗ ,p x + ∗ ,q . (2.95)All of the above formulae are written in terms of some functions σ ∗∗ which have branchcuts on the Zhukovski plane. The transformations of the Zhukovski variables are given insection 2.3.4, while for the detailed description of the cuts of the dressing factors we referthe reader to the review [35] and the original literature [48, 49]. In what follows we will not– 20 –se the explicit form of the dressing factors, but we will use their properties under crossingand unitarity. We have σ LL ( p +2 γ , q ) σ RL ( p, q ) = g RL ( p, q ) , σ LL ( p, q ) σ RL ( p +2 γ , q ) = ˜ g LL ( p, q ) ,σ RR ( p +2 γ , q ) σ LR ( p, q ) = g LR ( p, q ) , σ RR ( p, q ) σ LR ( p γ , q ) = ˜ g RR ( p, q ) ,σ LL ( p, q − γ ) σ LR ( p, q ) = 1˜ g LL ( q γ , p ) , σ LL ( p, q ) σ LR ( p, q − γ ) = 1 g RL ( q γ , p ) ,σ RR ( p, q − γ ) σ RL ( p, q ) = 1˜ g RR ( q γ , p ) , σ RR ( p, q ) σ RL ( p, q − γ ) = 1 g LR ( q γ , p ) , (2.96)where the rational functions g ( p, q ) and ˜ g ( p, q ) are given by g ∗∗ ( p, q ) = e − iq (cid:16) − x + ∗ ,p x + ∗ ,q (cid:17) (cid:16) − x −∗ ,p x −∗ ,q (cid:17)(cid:16) − x + ∗ ,p x −∗ ,q (cid:17) x −∗ ,p − x + ∗ ,q x + ∗ ,p − x −∗ ,q , ˜ g ∗∗ ( p, q ) = e − iq ( x −∗ ,p − x + ∗ ,q ) ( x + ∗ ,p − x + ∗ ,q )( x −∗ ,p − x −∗ ,q ) 1 − x −∗ ,p x + ∗ ,q − x + ∗ ,p x −∗ ,q . (2.97)Similarly, for the massless phase we have that σ oo ( p γ , q ) σ oo ( p, q ) = x + o ,p − x − o ,q x + o ,p − x + o ,q x − o ,p − x + o ,q x − o ,p − x − o ,q , (2.98)and for the mixed-mass phases, σ •◦ Ro ( p γ , q ) σ •◦ Lo ( p, q ) = x − L ,p − x + o ,q x + L ,p − x + o ,q x + L ,p − x − o ,q x − L ,p − x − o ,q = σ ◦• oL ( q γ , p ) σ ◦• oL ( q, p ) ,σ •◦ Lo ( p γ , q ) σ •◦ Ro ( p, q ) = 1 − x + R ,p x + o ,q − x + R ,p x − o ,q − x − R ,p x − o ,q − x − R ,p x + o ,q = σ ◦• oR ( q γ , p ) σ ◦• oR ( q, p ) . (2.99) It was proposed that, for AdS × S superstrings, three- [13] and higher-point functions [20,21] of generic operators may be constructed using integrability techniques. The setup iseasiest to understand for three-point functions [13] by bearing in mind the approach usedfor the spectral problem. In the spectral problem, one goes from a closed string (a finite-volume worldsheet) to a decompactified worldsheet where the S matrix may be defined [11].For three-point functions, too, one wants to consider a decompactification of the “pairof pants” topology by cutting it open in two hexagonal patches. Without reviewing thisconstruction in full detail (we refer the reader to [13]) it suffices to say that either patchcontains a piece of each of the three closed-string states whose correlator we are interestedin computing, see figure 1. We are interested in representing each hexagonal patch as anordinary worldsheet where a non-local “hexagon” operator has been inserted, see figure 2.What is remarkable is that, in the case of AdS × S , it is possible [13] to bootstrap the– 21 – igure 1 . The main idea of ref. [13] is to “cut open” the three-point function in string theory toget two patches of worldsheet with six distinct edges (hexagon tesellation). This is the analogueof considering the infinite-volume worldsheet theory for the spectral problem ( i.e. , cutting open acylinder into a plane). form factors of these operators starting from the light-cone gauge symmetries that helpeddetermine the S matrix. It is therefore natural to ask whether a similar construction maybe applied to more general setups, and in particular to AdS × S × T . This is what wewill discuss in this section. In the case of the S matrix, the original psu (1 , | ⊕ supersymmetry was broken by gaugefixing. Such a gauge fixing relies on the choice of a 1/2-BPS geodesic and, in the dualCFT, amounts to picking a reference two-point function involving one 1/2-BPS operator O (0) and its conjugate O † ( ∞ ) . In the case of three-point functions we need three-operator,sitting at three distinct points. It is useful to construct such an operator following ref. [55],starting from a reference BPS operator and considering its image under translation. Given a 1/2-BPS operator O (0) at z = 0 , we are interested in constructing translatedoperators O ( z ) . To be concrete, let us say that O (0) is a 1/2-BPS scalar operator which isthe highest-weight state in the representation with − L = J = j , − (cid:101) L = (cid:101) J = j , (3.1)see section 2.1. In terms of the psu (1 , | ⊕ generators, translations are given by T = i L − + i (cid:101) L − . (3.2)We are interested in constructing three such operators in such a way as to preserve as much(super)symmetry as possible. We expect this to break some of the psu (1 | ⊕ centrallyextended symmetry described in section 2.2. It is easy to see that this requires combining thetranslation with an R-symmetry rotation [13, 55]. Hence we introduce the supertranslationgenerator T κ = i L − + i (cid:101) L − + κ J − + κ (cid:101) J − , (3.3)– 22 – igure 2 . We can represent each hexagonal worldsheet patch as an ordinary two-dimensionaltheory with the insertion of a non-local “hexagon operator” which creates an excess angle (the redzig-zag line). This operators may absorb excitations (like particle a in the figure) yielding a non-zeroresult — its form factor. This form factor is what we are interested in determining starting fromthe symmetries preserved by the configuration of three operators O , O and O . where κ ∈ C is some constant to be determined. It is worth noting that, while we mayintroduce a distinct κ and ˜ κ for the left and right part of the algebra, we will be able tocarry out the bootstrap procedure with a single κ = ˜ κ . Using T κ we may construct a oneparameter family of operators starting from O (0) , namely O t,κ = e t T κ O (0) e − t T κ , (3.4)which by construction sits at position t . At the same time, we have that the operatoris t -rotated in R-symmetry space. For instance, taking t = ∞ yields O † ( ∞ ) . A genericconfiguration of images of O (0) sitting at t , t , t , . . . will be jointly annihilated by thestabilizer of T κ in psu (1 | ⊕ centrally extended. By direct inspection, this supertranslationoperator preserves four supercharges in psu (1 | ⊕ , namely Q + − A − iκ Q − + A = S A − iκ (cid:15) AB Q B , (cid:101) Q + − A − iκ (cid:101) Q − + A = − (cid:15) AB (cid:101) S B − iκ (cid:101) Q A . (3.5) It is convenient to introduce the notation Q A = S A − iκ (cid:15) AB Q B , (cid:101) Q A = (cid:101) Q A − iκ (cid:15) AB (cid:101) S B , (3.6)to indicate the four supercharges that commute with the supertranslation generator T κ (3.3).By direct inspection, using the relations of section 2.2, we find {Q A , Q A } = 0 , {Q , Q } = − iκ (cid:16) { S , (cid:15) Q } + { (cid:15) Q , S } (cid:17) = 0 . (3.7)Moreover, we have (cid:8) Q A , (cid:101) Q B (cid:9) = − iκ { S A , (cid:15) BC (cid:101) S C } − iκ { (cid:15) AC Q C , (cid:101) Q B } = − iκ (cid:15) AB (cid:16) P − κ K (cid:17) (3.8)– 23 –ere P and K are the central extensions of the psu (1 | ⊕ algebra which are not in psu (1 , | ⊕ . In fact, for a unitary representation of the psu (1 | ⊕ algebra we shouldtake P and K to be Hermitian conjugate to each other; in fact, as reviewed in section 2.2it is possible and convenient to take them to be real, cf. (2.43). Introducing the centralcharge C ≡ − iκ (cid:16) P − κ K (cid:17) , (3.9)we have that on a multi-excitation state involving momenta p , . . . p N , C | p , . . . p N (cid:105) = ( κ − hiκ sin (cid:18) p + · · · + p N (cid:19) | p , . . . p N (cid:105) . (3.10) Let us now specialise to the case of three-point functions. We therefore want to considerthree images of the 1/2-BPS operator O (0) . For this purpose — without loss of generalityowing to conformal symmetry — we may take the images under superstranslation with t =0 , t (cid:48) = 1 and t (cid:48)(cid:48) = ∞ . The first operator will be precisely O (0) , sitting at z = 0 (and beingthe highest-weight state in its R-symmetry multiplet). The third operator will be O † ( ∞ ) ,sitting at z = ∞ and being the lowest weight state in the R-symmetry multiplet. The secondoperator will be sitting at z = 1 and it will be neither the highest- nor the lowest-weightstate in the R-symmetry multiplet. The symmetry algebra preserved by this configurationis generated by the four supercharges ( Q A , (cid:101) Q A ) . Following Basso, Komatsu and Vieira [13]we shall assume that this is the symmetry preserved by the “hexagon operator”. In otherwords, denoting the hexagon operator by h , [ h , Q A ] = 0 , [ h , (cid:101) Q A ] = 0 . (3.11)Indicating the form factor of h with any state Ψ as (cid:104) h | Ψ (cid:105) , it follows that (cid:104) h |Q A | Ψ (cid:105) = 0 , (cid:104) h | (cid:101) Q A | Ψ (cid:105) = 0 , (cid:104) h |C| Ψ (cid:105) = 0 . (3.12)The equality follows by letting ( Q A , (cid:101) Q A ) — or, for the third equations, a suitable anticom-mutator thereof — act on the state. Vice versa, letting the (super)charges act on the ketwe obtain a set of linear constaints that the hexagon form factor must obey. The bootstrap condition (3.12) takes a particular simple form in the case of the centralcharge C , because this acts diagonally and independently on the particles flavour. We seefrom (3.10) that, whenever the the Ramond-Ramond coupling h (cid:54) = 0 , C only annihilatesphysical states — just as is the case for P and K in the spectral problem — unless κ = 1 .Let us recall that κ is a free parameter in our construction, see (3.3); it is up to us to choosethe value of κ most suitable for the bootstrap procedure. Following the reasoning of [13],we must require κ = 1 , because, if not, the hexagon form factor in (3.12) would annihilateall non-physical states, which would be too strong a requirement. In fact, we want to definean off-shell object which, like the S matrix, may act on just a subset of the excitations thatdefine a physical state. Henceforth we will take κ = 1 . (3.13)– 24 – .2 Bootstrapping the hexagon form factor from symmetry Here we will use the bootstrap principle of eq. (3.12) to fix as much as possible of thehexagon form factor. We will consider in particular the case where | Ψ (cid:105) consists of a singleparticle, and when it consists of two. We will then propose a self-consistent ansatz formulti-particle states. As discussed in section 2.4, we can represent the excitations of the theory, which transformunder psu (1 | ⊕ c.e., in terms of tensor products of excitations in psu (1 | ⊕ c.e. — forinstance, for the left massive representation we have that | Y (cid:105) = | φ BL ⊗ φ BL (cid:105) , | Ψ (cid:105) = | ϕ FL ⊗ φ BL (cid:105) , | Ψ (cid:105) = | φ BL ⊗ ϕ FL (cid:105) and | Z (cid:105) = | ϕ FL ⊗ ϕ FL (cid:105) . It is useful to rewrite the supercharges of eq. (3.12)in terms of the same decomposition, Q = s ⊗ + i Σ ⊗ q , Q = Σ ⊗ s − i q ⊗ , (cid:101) Q = (cid:101) q ⊗ + i Σ ⊗ (cid:101) s , (cid:101) Q = Σ ⊗ (cid:101) q − i (cid:101) s ⊗ , (3.14)where Σ is the Fermion sign operator. Imposing now one instance of the bootstrap equa-tion (3.12) we get (cid:104) h |Q | Y ( p ) (cid:105) = 0 ⇒ (cid:104) h | Ψ ( p ) (cid:105) = 0 . (3.15)Similarly, it is easy to find (as expected from su (2) • symmetry) that (cid:104) h | Ψ ( p ) (cid:105) = 0 . Wenote that, naïvely, we have more bootstrap equations than undetermined one-particle formfactors. However, they all result is one single constraint between (cid:104) h | Y ( p ) (cid:105) = i a L ( p ) a L ( p ) ∗ (cid:104) h | Z ( p ) (cid:105) = i (cid:104) h | Z ( p ) (cid:105) , (3.16)where a L ( p ) is the representation coefficient introduced in section 2.3. Note that, since theequations that we are imposing are linear, we will not be able to fix the overall normalisationof the form factor, but at best only the ratio of different elements. Working on the otherrepresentations, we find analogous results: (cid:104) h | Y p (cid:105) = i (cid:104) h | Z p (cid:105) , (cid:104) h | ˜ Z p (cid:105) = − i (cid:104) h | ˜ Y p (cid:105) , (cid:104) h | χ ˙1 p (cid:105) = i (cid:104) h | ˜ χ ˙1 p (cid:105) , (cid:104) h | χ ˙2 p (cid:105) = − i (cid:104) h | ˜ χ ˙2 p (cid:105) , (3.17)while the remaining form factors vanish, (cid:104) h | Ψ Ap (cid:105) = 0 , (cid:104) h | (cid:101) Ψ Ap (cid:105) = 0 , (cid:104) h | (cid:101) T A ˙ Ap (cid:105) = 0 . (3.18)Without loss of generality, we normalise the form factor so that (cid:104) h | Y p (cid:105) = 1 , (cid:104) h | Z p (cid:105) = − i , (cid:104) h | ˜ Y p (cid:105) = 1 , (cid:104) h | ˜ Z p (cid:105) = − i , (cid:104) h | χ ˙1 p (cid:105) = 1 , (cid:104) h | ˜ χ ˙1 p (cid:105) = − i , (cid:104) h | χ ˙2 p (cid:105) = 1 , (cid:104) h | ˜ χ ˙2 p (cid:105) = − i . (3.19)– 25 – .2.2 Two-particle states We can determine the hexagon form factor for two-particle states by explicitly evaluatingthe eq. (3.12). At this point it is worth observing that the symmetry algebra that we areexploiting is a “diagonal” (in the sense of the tensor product decomposition of section 2.2.1) psu (1 | ⊕ subalgebra in psu (1 | ⊕ . In this sense it is not surprising that the two-particleform-factor may be expressed in terms of the only non-trivial intertwiner of two short psu (1 | ⊕ representations, i.e. the Borsato–Ohlsson-Sax–Sfondrini S matrix [53]. This iscompletely analogous to what happens for the AdS × S hexagon in terms of the BeisertS matrix. A solution of all bootstrap equations for the two-particle form factor may bewritten explicitly in terms of the S-matrix elements of section 2.5. Note that, as expected,we are unable to fix one overall prefactor for each choice of irreducible representations; belowwe shall denote such prefactors as h ( p, q ) and postpone their discussion to section 3.4. Form factor for two massive excitations.
We may distinguish two cases dependingon whether the two exitations are left or right. When they are both left we have (cid:104) h | Y p Y q (cid:105) = + A LL pq , (cid:104) h | Z p Z q (cid:105) = + F LL pq , (cid:104) h | Y p Z q (cid:105) = − iB LL pq , (cid:104) h | Z p Y q (cid:105) = − iD LL pq , (cid:104) h | Ψ p Ψ q (cid:105) = + iC LL pq , (cid:104) h | Ψ p Ψ q (cid:105) = − iC LL pq . (3.20)When both particles are right we get (cid:104) h | ˜ Y p ˜ Y q (cid:105) = + A RR pq , (cid:104) h | ˜ Z p ˜ Z q (cid:105) = + F RR pq , (cid:104) h | ˜ Y p ˜ Z q (cid:105) = − iB RR pq , (cid:104) h | ˜ Z p ˜ Y q (cid:105) = − iD RR pq , (cid:104) h | ˜Ψ p ˜Ψ q (cid:105) = − iC RR pq , (cid:104) h | ˜Ψ p ˜Ψ q (cid:105) = + iC RR p,q . (3.21)In the case of mixed chirality, we distinguish two cases depending on the ordering of theparticles. Firstly, for left–right we have (cid:104) h | Y p ˜ Y q (cid:105) = + A LR pq , (cid:104) h | Ψ p ˜Ψ q (cid:105) = − F LR pq , (cid:104) h | Ψ p ˜Ψ q (cid:105) = − B LR pq , (cid:104) h | Z p ˜ Y q (cid:105) = − iD LR pq , (cid:104) h | Y p ˜ Z q (cid:105) = − iC LR pq , (cid:104) h | Z p ˜ Z q (cid:105) = + E LR pq . (3.22)Finally, for right–left we have (cid:104) h | ˜ Y p Y q (cid:105) = + A RL pq , (cid:104) h | ˜Ψ p Ψ q (cid:105) = − F RL pq , (cid:104) h | ˜Ψ p Ψ q (cid:105) = − B RL pq , (cid:104) h | ˜ Z p Y q (cid:105) = − iD RL pq , (cid:104) h | ˜ Y p Z q (cid:105) = − iC RL pq , (cid:104) h | ˜ Z p Z q (cid:105) = + E RL pq . (3.23) One massless and one massive particle.
In this case we can distinguish excitationson whether they are left or right (for the massive particle) and depending on their su (2) ◦ charge (for the massless particle). Moreover, we can also distinguish their order. It turnsout that we may write more compact formulae by explicitly making use of the one-particle– 26 –orm factor (cid:104) h | χ ˙ A (cid:105) . For instance, in the case of one left-massive particle and one masslessparticle we obtain (cid:104) h | Y p χ ˙ Aq (cid:105) = + A Lo pq , (cid:104) h | Z p ˜ χ ˙ Aq (cid:105) = + F Lo pq , (cid:104) h | Y p ˜ χ ˙ Aq (cid:105) = − iB Lo pq , (cid:104) h | Z p χ ˙ Aq (cid:105) = − iD Lo pq , (cid:104) h | Ψ p T ˙ A q (cid:105) = + iC Lo pq , (cid:104) h | Ψ p T ˙ A q (cid:105) = − iC Lo pq . (3.24)Similarly, for one-right massive particle and one massless particle we have (cid:104) h | ˜ Y p χ ˙ Aq (cid:105) = + A Ro pq , (cid:104) h | ˜Ψ p T ˙ A q (cid:105) = − F Ro pq , (cid:104) h | ˜Ψ p T ˙ A q (cid:105) = − B Ro pq , (cid:104) h | ˜ Z p χ ˙ Aq (cid:105) = − iD Ro pq , (cid:104) h | ˜ Y p ˜ χ ˙ Aq (cid:105) = − iC Ro pq , (cid:104) h | ˜ Z p ˜ χ ˙ Aq (cid:105) = + E Ro pq . (3.25)The possibility of writing formulae in such a compact way is a first sign of an underlyingsymmetry structure of the form factor which we shall investigate in the next section. Toconclude here, we list the mixed-mass form factors when particles are in the reversed order, (cid:104) h | χ ˙ Ap Y q (cid:105) = + A oL pq , (cid:104) h | ˜ χ ˙ Ap Z q (cid:105) = + F oL pq , (cid:104) h | χ ˙ Ap Z q (cid:105) = − iB oL pq , (cid:104) h | ˜ χ ˙ Ap Y q (cid:105) = − iD oL pq , (cid:104) h | T ˙ A p Ψ q (cid:105) = − iC oL pq , (cid:104) h | T ˙ A p Ψ q (cid:105) = + iC oL pq , (3.26)and finally (cid:104) h | χ ˙ Ap ˜ Y q (cid:105) = + A oR pq , (cid:104) h | T ˙ A p ˜Ψ q (cid:105) = + F oR pq , (cid:104) h | T ˙ A p ˜Ψ q (cid:105) = + B oR pq , (cid:104) h | ˜ χ ˙ Ap ˜ Y q (cid:105) = − iD oR pq , (cid:104) h | χ ˙ Ap ˜ Z q (cid:105) = − iC oR pq , (cid:104) h | ˜ χ ˙ Ap ˜ Z q (cid:105) = + E oR pq . (3.27)Note that the form factor is blind to the su (2) ◦ index ˙ A , which is unsurprising as the algebrawhich we are using to constrain it commutes with su (2) ◦ . Two massless particles.
In this case we can compactly write (cid:104) h | χ ˙ Ap χ ˙ Bq (cid:105) = + A oo pq , (cid:104) h | ˜ χ ˙ Ap ˜ χ ˙ Bq (cid:105) = + F oo pq , (cid:104) h | χ ˙ Ap ˜ χ ˙ Bq (cid:105) = − iB oo pq , (cid:104) h | ˜ χ ˙ Ap χ ˙ Bq (cid:105) = − iD oo pq , (cid:104) h | T ˙ A T ˙ B (cid:105) = + iC oo pq , (cid:104) h | T ˙ A T ˙ B (cid:105) = − iC oo pq , (3.28)which again is blind to su (2) ◦ .To conclude the discussion of two-particle form factors, it is important to note thatthe equations (3.12) we have imposed are linear, so that we may obtain new solutions bymultiplying each block (for instance, left–left, or left–right, etc. ) by an arbitrary function.In other words, the prefactors Σ LL p,q , Σ LR p,q , etc. , which appear in the S matrix elements canbe changed with no effect (3.12). We shall see later how they may be further constrained,see section 3.4. – 27 – .2.3 General form of the two-particle hexagon form factor It is possible to summarise the form of the two-particle form factor in a way that encom-passes the various representations encountered thus far. Let us denote a generic psu (1 | ⊕ excitation in the tensor product form of section 2.2.1 as Ξ a ´ a ≡ ξ a ⊗ ´ ξ ´ a , (3.29)where the second entry of the tensor product is distinguished by a “prime”. Here ξ a and ´ ξ ´ a could transform under any of the relevant representations which we encountered, i.e. ρ ± , ρ or ρ (cid:48) ; we absorb the information of the representation into the indices a and ´ a to keepthe notation a little lighter. Then, using this notation, we have that (cid:104) h | Ξ a ´ ap Ξ b ´ bq (cid:105) = K p K q ( − ( F a + F ´ a ) F b (cid:104) | ξ bq ξ ap (cid:105) ⊗ S | ´ ξ ´ ap ´ ξ ´ bq (cid:105) (cid:105) = ( − ( F a + F ´ a ) F b S ´ a ´ b ´ d ´ c ( p, q ) K p K q (cid:104) | ξ bq ξ ap (cid:105) ⊗ | ´ ξ ´ dq ´ ξ ´ cp (cid:105) (cid:105) , (3.30)where we have introduced the “contraction operator” K p ≡ (cid:16) h Y ∂∂ ´ φ BL ( p ) ∂∂φ BL ( p ) + h Z ∂∂ ´ ϕ FL ( p ) ∂∂ϕ FL ( p ) (cid:17) + (cid:16) h ˜ Y ∂∂ ´ ϕ BR ( p ) ∂∂ϕ BR ( p ) + h ˜ Z ∂∂ ´ φ FR ( p ) ∂∂φ FR ( p ) (cid:17) + (cid:16) h χ ∂∂ ´ φ Fo ( p ) ∂∂φ Bo ( p ) + h ˜ χ ∂∂ ´ ϕ Bo ( p ) ∂∂ϕ Fo ( p ) (cid:17) + (cid:16) h χ ∂∂ ´ φ Bo ( p ) ∂∂φ Fo ( p ) + h ˜ χ ∂∂ ´ ϕ Fo ( p ) ∂∂ϕ Bo ( p ) (cid:17) , (3.31)where h Y = (cid:104) h | Y (cid:105) , etc. , are the values of the one-particle hexagon form factors of eq. (3.19).Let us explain what we mean by this notation. We begin to note that K p simply picksout the one-particle states with a non-trivial hexagon form factor and assigns them thevalue thereof, i.e. K p | Ξ a ´ a ( p ) (cid:105) = (cid:104) h | Ξ a ´ a (cid:105) . The reason why we go through the trouble ofintroducing this operator — something not necessary in AdS × S — is that here the one-particle hexagon form factors in the massless representations are nonvanishing for particleswith Fermionic statistics. This creates a potential ambiguity for massless particles wheneverwe want to contract multi-particle states: note that indeed the commutator [ K p , K q ] doesnot vanish for massless particles due to the statistics. Realising the contractions in terms ofthe graded differential operator K p will make it easier to properly account for this statistics.Armed with this operator, let us go back to eq. (3.30). In the first equality we rearrangethe excitations to factor out the pieces of the tensor product related to either factor of thediagonal symmetry algebra (distinguished here by the absence or presence of the prime),picking up Fermion signs as appropriate. To this end we defined F a = (cid:40) if ξ a is a Boson if ξ a is a Fermion and F ´ a = (cid:40) if ´ ξ ´ a is a Boson if ´ ξ ´ a is a Fermion (3.32)– 28 –e then scatter the “primed” particles by using the psu (1 | ⊕ c.e. S matrix in the appropriaterepresentation (for instance, ρ L ⊗ ρ L , ρ L ⊗ ρ (cid:48) o , etc. ). We pick up the relative S matrixelements, which now contain a irrep-dependent prefactor h ´ a ´ b ( p, q ) . Lastly, we act with thecontraction operator, again keeping track of the statistics, perfectly reproducing the resultswhich we listed above. It is worth stressing that this prescription can also be applied toAdS × S , yielding a result identical to ref. [13]. Nothing stops us from imposing eq. (3.12) for three- and higher-particle states. However,while for two-particle states we managed to fix the form factor completely (up to an un-avoidable scalar prefactor for each choice of representations), for higher number of particleswe will only be able to fix relatively few coefficients. A better approach, following [13], isto exploit the fact that the two-particle solution can be written in terms of a factorisedS matrix [46, 53]. Then the Yang-Baxter equation allow us to write down a self-consistentansatz with is guaranteed to satisfy all symmetry requirements. We set (cid:10) h (cid:12)(cid:12) Ξ a ´ a p Ξ a ´ a p . . . Ξ a N ´ a N p N (cid:11) ≡≡ ( − F ··· N K ··· N (cid:104)(cid:12)(cid:12) ξ a N p N . . . ξ a p ξ a p (cid:11) ⊗ S ··· N (cid:12)(cid:12) ´ ξ ´ a p ´ ξ ´ a p . . . ´ ξ ´ a N p N (cid:11)(cid:105) . (3.33)where F ··· N ≡ (cid:88) ≤ i Let us start by reviewing the structure of the half-BPS states, which is substantially richerhere than for AdS × S (because overall here we have less supersymmetry). While in AdS × S we have exactly one BPS operator for each value of the “orbital” R-charge J (with energy J owing the the BPS bound) this is not the case here. First ofall, we have two su (2) (left and right) orbital quantum numbers, which we indicate by ( J, ˜ J ) . These are the eigenvalues of the highest-weight state in the BPS representationunder ( J , (cid:101) J ) , respectively. Recall that the psu (1 , | ⊕ BPS bound gives − L = J and − (cid:101) L = (cid:101) J . Then, for every positive integer value of j we have the following diamond ofBPS multiplets, indicated here by the charge of their highest-weight states: (cid:0) j − , j − (cid:1)(cid:0) j , j − (cid:1) ˙ A (cid:0) j − , j (cid:1) ˙ A (cid:0) j +12 , j − (cid:1) (cid:0) j , j (cid:1) ˙ A ˙ B (cid:0) j − , j +12 (cid:1)(cid:0) j +12 , j (cid:1) ˙ A (cid:0) j , j +12 (cid:1) ˙ A (cid:0) j +12 , j +12 (cid:1) (4.1)for a total of 16 multiplets. The dotted indices indicate that some of these states transformin the or ⊗ representation su (2) ◦ . This structure can be related to the Hodge diamondof T or to a Clifford module generated by four Fermion zero-modes. In particular, lookingat the dual CFT, these multiplets may be identified with those arising from the symmetric-product orbifold CFT of T , Sym N T . Using the notation of ref. [57] (which will beconvenient for what follows), the diamond looks like this: V −− j V ˙ A − j V − ˙ Aj V + − j V ˙ A ˙ Bj V − + j V + ˙ Aj V ˙ A + j V ++ j (4.2)where the subscript index j in V ∗∗ j denotes the length of the permutation cycle of theoperator.In the language of integrability that we have so far used, one state is the BMN vacuum | (cid:105) , featuring no particles at all, while the remaining can be constructed by inserting on topof the vacuum the massless Fermions χ ˙ A ( p ) and ˜ χ ˙ A ( p ) at zero momentum [58]. The zero-modes which we can use have charges under ( J , (cid:101) J ) as in table 2 [41] and, owing to Pauli’sprinciple, yield precisely 16 states. Note that, unlike the zero-modes of massive states, thezero-modes of χ ˙ A ( p ) and ˜ χ ˙ A ( p ) do not yield psu (1 , | ⊕ descendants, but genuinely new– 39 –agnon J (cid:101) J lim p → + | χ ( p ) (cid:105) − lim p → − | χ ( p ) (cid:105) + lim p → + | ˜ χ ( p ) (cid:105) + lim p → − | ˜ χ ( p ) (cid:105) − Table 2 . The su (2) L ⊕ su (2) R charge of massless particles. The Fermions | χ ˙ A ( p ) (cid:105) have su (2) spinunder ( J − (cid:101) J ) equal to − / , while have +1 / . Given that they all have M = 0 , the su (1 , spin ( − L + (cid:101) L ) follows, cf. (2.44). This is also consistent with the fact that | ˜ χ ˙ A ( p ) (cid:105) = Q Q | χ ˙ A ( p ) (cid:105) .However, the particles are chiral depending on the sign of sin( p/ , cf. (2.46). Hence, in differentmomentum regions they will be annihilated by either − L and J or by − (cid:101) L and (cid:101) J . Keeping thatinto account, we propose the following identification of the massless modes. psu (1 , | ⊕ multiplets. Based on table 2, the highest-weight states can be identified as itfollows: | χ ˜ χ (cid:105) (cid:0) | χ ˜ χ ˜ χ (cid:105) , | ˜ χ (cid:105) (cid:1) (cid:0) | χ (cid:105) , | χ χ ˜ χ (cid:105) (cid:1) | ˜ χ ˜ χ (cid:105) | (cid:105) ⊕ (cid:0) | χ ˜ χ (cid:105) , | χ ˜ χ χ ˜ χ (cid:105) , | χ ˜ χ (cid:105) (cid:1) | χ χ (cid:105) (cid:0) | ˜ χ (cid:105) , | χ ˜ χ ˜ χ (cid:105) (cid:1) (cid:0) | χ χ ˜ χ (cid:105) , | χ (cid:105) (cid:1) | χ ˜ χ (cid:105) (4.3)where in the middle of the Hodge diamond we have distinguished the and represen-tation of su (2) ◦ . It should be emphasised that the number of magnons (the “length” ofthe operators) is not a quantum number here, and it is not preserved by the su (2) ◦ action.All various magnons are at zero momentum as in table 2. Despite the nice structure, weshould be careful with identifying multiplets from (4.2) to (4.3). From the above we seethat most of the half-BPS multiplets can mix among themselves when going from the in-tegrability description. In fact, there are several multiplets with the exact same charge.For instance, the states V −− j +1 , ε ˙ A ˙ B V ˙ A ˙ Bj and V ++ j − have the same charges and therefore, wecannot distinguish the relative multiples just by their quantum numbers. All of them couldin principle mix with | (cid:105) , | χ ˜ χ (cid:105) and | χ ˜ χ (cid:105) . Fortunately, the multiplets V + − j and V − + j donot mix — neither among themselves nor with any other half-BPS multiplet — so that itis quite convenient to focus on them. We focus on the three-point functions that may be constructed out of operators in themultiplets of V + − j and V − + j for appropriate values of j . Broadly, speaking, they fall in twocategories: three-point functions involving all operators from the same type of multiplets,and those involving three-point functions with two operators from one type of multipletand the third from the other — the other combinations follow from exchanging the left and– 40 –ight algebra. These correlation functions are well-known in the literature [57, 59, 60]. Wetake a look at the result as written in ref. [57], which has the advantage of being presentedquite compactly and including the Clebsch-Gordan coefficients. Recall from section 3.1 thatin our construction we want one of the operators to be the highest-weight state, one to bethe lowest-weight state, and one to have zero magnetic su (2) quantum number — i.e. , tobe the su (2) descendant “in the middle” of the multiplet.The result of [57] reads, in particular (cid:104) V − + j V − + j V − + j (cid:105) = − √ N D J J J D ˜ J ˜ J ˜ J ( j + j + j − j + j + j + 1) √ j j j , (cid:104) V − + j V − + j V + − j (cid:105) = − √ N D J J J D ˜ J ˜ J ˜ J ( j + j − j − j + j − j + 1) √ j j j , (4.4)where in the first line J k = j k + 1 and ¯ J k = j k − , and in the second line the same holdsexcept for operator 3, for which instead J = j − and ¯ J = j + 1 . The factors D J J J and D ˜ J ˜ J ˜ J depend also on the magnetic su (2) charges, i.e. on the J and (cid:101) J charges ofthe operators, respectively. Recall that operator is a highest-weight state, operator is alowest weight state, and operator has vanishing orbital quantum numbers. All in all, forour configuration of states are simply given by D J J J = ( − J +2 J J ! (cid:112) (2 J )! . (4.5)The prefactor / √ N is an overall normalisation common to all three-point functions — N is the number of copies in the symmetric product orbifold CFT Sym N ( T ) . Note that inpractice, for the three-point function to be non-vanishing, we want to specialise (4.4) to thecase J = J , ˜ J = ˜ J . We will describe here the how to use the formalism which we developed in order to reproducethe result (4.4). It should be stressed that the integrability machinery is suitable to computenon-protected correlation functions — this is just intended as a relatively simple check ofour proposal.The operators of interest are those related to V − + j and V + − j , namely V − + j ∼ lim p → + lim q → − | ˜ χ ( p ) ˜ χ ( q ) (cid:105) , V − + j ∼ lim p → + lim q → − | χ ( p ) χ ( q ) (cid:105) , (4.6)constructed over a vacuum of total R-charge j . The expression above stresses that thezero-momentum magnons described above should be treated with some care — we will seethat indeed singularities may arise from the p, q → limit. This is not surprising, giventhat among other things the dispersion relation is singular at that point, see eq. (2.46). Itturns out that things may be simplified a little, namely we can take the limit on the twomomenta symmetrically, V + − j ∼ lim p → + | ˜ χ ( p ) ˜ χ ( − p ) (cid:105) , V − + j ∼ lim p → + | χ ( p ) χ ( − p ) (cid:105) , (4.7)– 41 – igure 6 . We represent schematically some of the terms contributing to the hexagon computationof the three-point function. We cut the three point functions in two hexagons, one correspondingto the “front” of it and one to the “back” (the cut runs parallel to the surface of the page). Then,we have to sum over all possible ways of distributing each pair of particles over the two patches,for a total of (2 ) = 64 possibilities; in the figure we only write the first = 4 terms relativeto moving around { p , − p } (in blue), and one term relative to moving { p , − p } (in green). Thevarious terms have to be weighted as in eq. (4.10). We are interested in inserting three such operators on the three distinguished edges ofthe hexagon, which we have labeled with γ , γ and γ . Hence we have to consider thefollowing excitations (cid:104) V − + j V − + j V − + j (cid:105) ∼ (cid:16) { χ ( p ) , χ ( − p ) } , { χ ( p ) , χ ( − p ) } , { χ ( p ) , χ ( − p ) } (cid:17) . (4.8)Here and from now on, we leave the p j s generic. We will see later how to take the limit.Similarly, we have (cid:104) V − + j V − + j V + − j (cid:105) ∼ (cid:16) { χ ( p ) , χ ( − p ) } , { χ ( p ) , χ ( − p ) } , { ˜ χ ( p ) , ˜ χ ( − p ) } (cid:17) . (4.9)The hexagon prescription [13] requires us to partition the three sets of excitationsidentified above in all possible ways over the two hexagonal patches of worldsheet, see fig-ure 6. Let us consider the case of (4.8). Then we have three sets X = { χ ( p ) , χ ( − p ) } ,– 42 – = { ˜ χ ( p ) , ˜ χ ( − p ) } and X = { χ ( p ) , χ ( − p ) } . Accordingly, we sum over all parti-tions of the form X j = α j ∪ ¯ α j obtaining (cid:16) (cid:89) j =1 (cid:88) X j = α j ∪ ¯ α j ( − ¯ α j w α j , ¯ α j (cid:17) (cid:104) h | α γ α γ α γ (cid:105) (cid:104) h | ¯ α γ ¯ α γ ¯ α γ (cid:105) , (4.10)where the different ordering of the partitions is due to the different orientation of the twohexagonal patches (which is necessary to glue them back to give a three-point function).Accordingly, we have also indicated how the particles have to be analytically continued onthe various edges. It is worth emphasising that, following the rules of section 3.3, a γ -shiftresults in a flavour change, e.g. χ ( p γ ) = i ˜ χ ( p ) . Finally, the sum is weighted by the factor w α, ¯ α , which takes the form w α , ¯ α = α = { χ ( p ) χ ( − p ) } , ¯ α = ∅ e i ( − p ) (cid:96) α = { χ ( p ) } , ¯ α = { χ ( − p ) } S χχ ( p , − p ) e i ( p ) (cid:96) α = { χ ( − p ) } , ¯ α = { χ ( p ) } e i ( p − p ) (cid:96) = 1 α = ∅ , ¯ α = { χ ( p ) χ ( − p ) } (4.11)The expression further simplifies in the small- p limit because S χχ ( p, − p ) → . Here wehave introduced the “bridge length” [13] (cid:96) ; we have (cid:96) = j + j − j , (cid:96) = j + j − j , (cid:96) = j + j − j . (4.12)Similar formulae hold for the weight factors for the other partitions, up to cycling the indices , and . Furthermore, it is also true that S ˜ χ ˜ χ ( p, − p ) → . Finally, it should be noted thatthere is some confusion in the literature concerning the signs which should be assigned to agiven partition, especially when the permuted particles are Fermionic [13, 61]. In our casewe will impose that the signs satisfy all relevant self-consistency and symmetry conditions,at which point we will be able to obtain the result and match the existing literature. As we have mentioned, the limit p j → will require some care. We can expect two typesof singular behaviour: one arises because of possible singularities at p = 0 , while the otheris due to pair of momenta approaching each other, p j = p k . Recall from the discussionof crossing (section 3.3) that a particle-antiparticle pair results in a pole; this is whatwill happen when, e.g. , p → p in our setup. There is one further complication thatwe should bear in mind: the hexagon formalism should not depend on the details of howwe construct the external states — for instance, it should not depend on the ordering ofthe particles within each state. This is indeed the case, but only as long as the particlesin each state satisfy the Bethe equations. In other words, in order to have a consistentformalism we need to require p , p and p to obey the Bethe equations. These are verysimple in our setup, because we are interested in a limit where particles behave as free, i.e. S χχ ( p, − p ) = S ˜ χ ˜ χ ( p, − p ) = 1 . Still, they do impose three non-trivial conditions, e ip k j k = 1 ⇒ p k = 2 πν k j k , ν k ∈ Z , k = 1 , , . (4.13)– 43 –his discrete structure calls for a little more care in taking the coincident-momenta limit.To this end we introduce a small ε > and three real numbers (cid:15) k and redefine j k → j k ε (cid:15) k , p k → p k (1 + ε (cid:15) k ) , (4.14)which leaves the Bethe equations (4.13) unchanged. In this language, we can get thecoincident-momenta limit by setting p = p = p , ε → . (4.15)This will also provide us with a check of our construction: the limit should be independentfrom (cid:15) , (cid:15) and (cid:15) .In practice, in our computation it will be useful to consider one additional limit. Thestructure constants for the three-point correlation functions of protected operators are them-selves protected [62]. As a result, we may choose any value of h, k that we want. Fromeq. (2.53) we note that, for massless particles, kinematics only depends on the ration h/k (up to an overall factor of k which washes out of all S-matrix elements). Hence it is conve-nient to take the limit h/k → with k arbitrary. The upshot is that, in this way, we mayrewrite all the ingredients necessary for the computation in terms of the new variables y ± ( p ) ≡ e ± i p sin (cid:0) p (cid:1) p , (4.16)which play the role of x ± o . In terms of these, we can easily rewrite the various S-matrixelements necessary for the hexagon computation, including the relevant scalar factor. Forinstance, we have h ◦◦ ( p γ , p γ ) A ( p γ , p γ ) → ˜ h e i ( p + p ) ( y − − y − ) ,h ◦◦ ( p γ , p γ ) B ( p γ , p γ ) → ˜ h e i ( p − p ) ( y − − y +2 ) ,h ◦◦ ( p γ , p γ ) C ( p γ , p γ ) → ˜ h ˜ γ ˜ γ ,h ◦◦ ( p γ , p γ ) D ( p γ , p γ ) → ˜ h e i ( p − p ) ( y +3 − y − ) ,h ◦◦ ( p γ , p γ ) E ( p γ , p γ ) → ˜ h ˜ γ ˜ γ ,h ◦◦ ( p γ , p γ ) F ( p γ , p γ ) → ˜ h e − i ( p + p ) ( y +2 − y +3 ) , (4.17)when both momenta have the same sign. Here ˜ γ j := (cid:113) i ( y − j − y + j ) , ˜ h = sgn ( p − p ) (cid:113) ( y − − y − )( y +2 − y +3 ) . (4.18)The expressions become even simpler when momenta have opposite signs: in that case thereflection part of the S matrix vanishes ( C = E = 0 ) and one is left with a free S matrix,up to frame factors — exactly how it was argued in refs. [41, 63].– 44 – .2.2 Computation of the form factor We now turn to the computation of the hexagon form factors for the two correlators ofinterest (4.8–4.9). For the correlator involving three identical states, we can expect asingularity when any pair of momenta become singular. Hence the most singular partof (4.8) should go like ε (cid:15) − (cid:15) ) ( (cid:15) − (cid:15) ) ( (cid:15) − (cid:15) ) . (4.19)Conversely, for the correlator of eq. (4.9) we expect a pole from the decoupling conditiononly for operators one and two — the third operators being different — so that it will golike ε (cid:15) − (cid:15) ) . (4.20)A first obvious issue to address is how to resolve this mismatch, given that both correlatorsshould eventually give a finite result, possibly up to an overall factor. Let us start from thecompletely symmetric case of eq. (4.8). Among all various way of partitioning the particles,the one yielding the highest O ( ε − ) singularity occurs when the three particles with positivemomenta { p , p , p } sit on one hexagon, {− p , − p , − p } sit on the other, or when theyall sit on the same hexagon. In the former case, when e.g. { p , p , p } are on the the “front”hexagon we pick up a numerator proportional to the following polynomial P in y ± k : P = + y − y − y +1 − y − y − y +1 − y − y − y +2 + y − y − y +2 − y − y +1 y +2 + y − y +1 y +2 − y − y − y +3 + y − y − y +3 − y − y +1 y +3 + y − y +1 y +3 + y − y +2 y +3 − y − y +2 y +3 . (4.21)Conversely, when { p , p , p } are on the the “back” hexagon we pick the complex conju-gate P ∗ , which is obtained from P by swapping y ± k ↔ y ∓ k . Repeating the computationfor {− p , − p , − p } we come to the conclusion that the full result is proportional to PP ∗ ;similarly, when all particles are on the same hexagon we get P or ( P ∗ ) . It is useful tointroduce the quantity ∆ ∓ ij ≡ y ∓ i − y ∓ j , (4.22)in terms of which we can encode the y ± k dependence in all but one variable, say y ± . Thenwe have P = (∆ − ∆ +12 − ∆ − ∆ +13 + ∆ − ∆ +13 ) y − − (∆ − ∆ +12 − ∆ − ∆ +13 + ∆ − ∆ +13 ) y +1 − ∆ − ∆ − ∆ +12 + ∆ − ∆ +12 ∆ +13 , (4.23)and similarly for P ∗ . We see that in the coincidence limit when p → p and p → p , thenumerator goes like O ( ε ) . In conclusion, the term which naïvely would be the most diver-gent (4.19) eventually goes like O ( ε − ) , exactly like eq. (4.20). By way of example, if e.g.the particle with momentum + p is moved from the partition { p , p , p } , {− p , − p , − p } into the other hexagon such as to obtain { p , p } , { p , − p , − p , − p } the back hexagonruns up only a simple pole, while the front one still is maximally singular. On the otherhand, we lose P from the back hexagon, while P ∗ will agan arise on the front one. Again,the result has only a second order pole. – 45 –arrying out the computation to the end, we find that in the coincident-momenta limitthe result for the symmetric correlator (4.8) is proportional to (cid:0) j + j + j (cid:1) ( y − − y + ) ( y − y + ) (1 − y − )(1 − y + ) − (8 d − d − d s + d s − s + 7 d s + 12 s − d s − s + s ) 64 ( y − y + ) (1 − y − ) (1 − y + ) , (4.24)with s = y − + y + , d = y − − y + . Notice that this expression is real. In the limit p → this finally gives (cid:0) j + j + j − (cid:1)(cid:0) j + j + j + 1 (cid:1) p + . . . , (4.25)as expected from eq. (4.4). In a similar way we can compute the hexagon form factor forthe correlator (4.9), where the third operator is different from the other two. In this case,the result is proportional to (cid:0) (cid:96) (cid:1) ( y − − y + ) ( y − y + ) (1 − y − )(1 − y + ) − (8 d − d − d s + d s − s + 10 d s + 12 s − d s − s + s ) 64 ( y − y + ) (1 − y − ) (1 − y + ) . (4.26)The resulting p → limit is − − (cid:96) )(1 − (cid:96) ) p + . . . , (4.27)which matches with (4.4). In particular, if we disregard the overall normalisations, we findthat the ratio of the two families of correlation functions match for arbitrary j , j and j .We have not mentioned a selection rule concerning the results (4.24) and (4.26) andtheir limits: in the whole discussion it was assumed that the J i are such that the l ij areinteger as suggested by perturbative field theory. Further, we have to distinguish the cases (cid:80) ν k ∈ Z for which formulae (4.24), (4.26) are valid, and (cid:80) ν k ∈ Z + 1 for which thecorrelators actually vanish.To conclude this discussion, we note that our result only relied on the “asymptotic”part of the hexagon, without accounting for wrapping effects. This can be done in thisformalism order by order [13, 17–19] by considering Lüscher-type corrections. It is naturalto ask why our result nonetheless matches those in the literature. This is because we aredealing with half-BPS states, or equivalently precisely with states that are composed of zero-momentum excitations only. The argument was first noted in refs. [58, 64] in the contextof the computation of the spectrum for the very same operators. Essentially, the transfermatrix appearing in the computation of (arbitrarily high) wrapping effects only involveszero-momentum excitations. As such it get precisely the same and opposite corrections forFermionic and Bosonic wrapping effect, leading to a complete cancellation of wrapping. In this article we have seen that the hexagon approach for the computation of three-pointfunctions by integrability set out in ref. [13] can also be applied to AdS × S × T . This is– 46 –he first example of an integrable superstring background with this feature other than theoriginal AdS × S . The main aim of this paper was to perform the bootstrap procedurefor the hexagon form factor, check its internal consistency and perform a basic check ofthe resulting machinery. In this regard, we have been successful. There are now manyinteresting directions that should be studied.Our framework can be used to study background with a mixture of NSNS and RRbackground fluxes. In a sense, the case of pure-RR background fluxes seems simplestbecause in that case we know all the dressing factors [48, 49] and much of the intuitionfrom AdS × S may be exploited; this is also the case that naturally corresponds to theD1-D5 brane systems, which is of interest in holography. Conversely, the pure-NSNS casewould also be very interesting to study, as in that case we should be able to make contactwith the computation of correlation functions by worldsheet CFT techniques [28]. Themain obstacle in this case is that we do not know the scalar factors; however, given therelative simplicity of the system at the pure-NSNS point — which is quite apparent whenstudying the spectral problem [40, 41] — it is possible that we could make an educatedguess for them. The most-general case of mixed-flux backgrounds will possibly be the mostchallenging, as once again the scalar factors are unknown and probably highly nontrivial,see also ref. [51].Another interesting point is how to incorporate finite-size (“wrapping”) effects, whichis the bane of most integrability approaches. In AdS × S this can be done order-by-order [13, 17–19]. Here it is likely that things are more complicated, at least in general, dueto the presence of massless modes [39]. However, we expect that in the pure-NSNS case weshould be able to deal quite easily with all wrapping effects, due to the simple structurehighlighted in ref. [41]. In fact, studying wrapping effect in this context may well be atraining ground for incorporating them in more general backgrounds.It is worth emphasising that the hexagon formalism may be used also to constructhigher-point correlation functions [20, 21] as well as non-planar correlators [22–24]. Thisgives another setup in which wrapping may be manageable, namely the correlation functionsof BPS operators. The simplest case is that of a four-point function, which would show avery non-trivial dependence on the conformal and R-symmetry cross-ratios. In AdS × S it is possible to study this in quite some detail at small ’t Hooft coupling, see refs. [21, 65].Can we perform a similar study here? If so, this would undoubtedly shed new lights on thestructure of interactions at generic points of at AdS /CFT moduli space [66].Finally, it is natural to wonder which other backgrounds are amenable to this bootstrapapproach. Two natural candidates from the point of view of integrability are AdS × CP [67]and AdS × S × S × S [68]. The main obstacle which we encounter here is that neither ofthese backgrounds has a factorised symmetry algebra — unlike the case of AdS × S whereone could identify a diagonal su (2 | , and of AdS × S × T where we found a diagonal su (1 | ⊕ . All the same, these backgrounds are all integrable as far as the spectral problemis concerned, and their integrable structure is remarkably similar. It would almost seemunnatural if their correlation functions cannot be bootstrapped. The same goes for thevarious integrable deformations of all these setups that one may consider. Among those, itwould be particularly interesting to consider “quantum” deformations [69], whose geometric– 47 –escription [70, 71] was recently shown to include a consistent string background [72, 73].We hope to return in the near future to some of these intriguing questions. Acknowledgements We thank Riccardo Borsato, Andrea Dei, Sergey Frolov and Fiona K. Seibold for discus-sions. AS thanks Riccardo Borsato, Olof Ohlsson Sax, Bodgan Stefański jr. and AlessandroTorrielli for past collaboration and discussions on many points related to this article. Theauthors are grateful to ETH Zurich for hospitality at various stages of this project. BE andAS acknowledge support from the Swiss National Science Foundation under Spark grantn. 190657. DlP acknowledges support by the Stiftung der Deutschen Wirtschaft. A Explicit form of the full S Matrix In order to check that the bootstrapped two-particle hexagon form factor obeys the Watsonequation, we must use the explicit form of the S-matrix in the different sectors. This is knowin the literature [38, 45, 45] but the explicit expressions are somewhat scattered betweendifferent papers that have slightly different notations. Hence we collect it here. The full psu(1 | S matrix can be obtained by taking the graded tensor product [45] of two copiesof the psu(1 | S matrix of [53], S = S ˆ ⊗ ´ S, (A.1)which can be defined in terms of the matrix elements by ( M ˆ ⊗ ´ M ) I ´ I,J ´ JK ´ K,L ´ L = ( − F ´ K F L + F J F ´ I M IJKL ´ M ´ I ´ J ´ K ´ L . (A.2)We recall our convention for (cid:15) ab here (cid:15) = − (cid:15) = − (cid:15) = (cid:15) = 1 . (A.3) A.1 The massive sectorA.1.1 Left-left scattering S | Y p Y q (cid:105) = + A LL pq A LL pq | Y q Y p (cid:105) , S | Z p Z q (cid:105) = + F LL pq F LL pq | Z q Z p (cid:105) , S | Y p Z q (cid:105) = + C LL pq C LL pq | Y q Z p (cid:105) + B LL pq B LL pq | Z q Y p (cid:105) − B LL pq C LL pq (cid:15) ab | Ψ aq Ψ bp (cid:105) , S | Z p Y q (cid:105) = + D LL pq D LL pq | Y q Z p (cid:105) + E LL pq E LL pq | Z q Y p (cid:105) − D LL pq E LL pq (cid:15) ab | Ψ aq Ψ bp (cid:105) , S | Y p Ψ aq (cid:105) = + A LL pq C LL pq | Y p Ψ aq (cid:105) + A LL pq B LL pq | Ψ aq Y p (cid:105) , S | Z p Ψ aq (cid:105) = + E LL pq F LL pq | Z p Ψ aq (cid:105) − D LL pq F LL pq | Ψ aq Z p (cid:105) , S | Ψ ap Y q (cid:105) = + A LL pq D LL pq | Y q Ψ ap (cid:105) + A LL pq E LL pq | Ψ aq Y p (cid:105) , S | Ψ ap Z q (cid:105) = − B LL pq F LL pq | Z q Ψ ap (cid:105) + C LL pq F LL pq | Ψ aq Z p (cid:105) , S | Ψ ap Ψ bq (cid:105) = δ ab A LL pq F LL pq | Ψ bq Ψ ap (cid:105) + (cid:15) ab ( C LL pq D LL pq | Y q Z p (cid:105) + B LL pq E LL pq | Z q Y p (cid:105) )+ (1 − δ ab )( C LL pq E LL pq | Ψ aq Ψ bp (cid:105) − B LL pq D LL pq | Ψ bq Ψ ap (cid:105) ) . (A.4)The last process can be further simplified by using the identity C LL pq E LL pq − B LL pq D LL pq = A LL pq F LL pq . (A.5)– 48 – .1.2 Right-right scattering S | ˜ Y p ˜ Y q (cid:105) = + A RR pq A RR pq | ˜ Y q ˜ Y p (cid:105) , S | ˜ Z p ˜ Z q (cid:105) = + F RR pq F RR pq | ˜ Z q ˜ Z p (cid:105) , S | ˜ Y p ˜ Z q (cid:105) = + C RR pq C RR pq | ˜ Y q ˜ Z p (cid:105) + B RR pq B RR pq | ˜ Z q ˜ Y p (cid:105) − B RR pq C RR pq (cid:15) ab | ˜Ψ aq ˜Ψ bp (cid:105) , S | ˜ Z p ˜ Y q (cid:105) = + D RR pq D RR pq | ˜ Y q ˜ Z p (cid:105) + E RR pq E RR pq | ˜ Z q ˜ Y p (cid:105) − D RR pq E RR pq (cid:15) ab | ˜Ψ aq ˜Ψ bp (cid:105) , S | ˜ Y p ˜Ψ aq (cid:105) = + A RR pq C RR pq | ˜ Y p ˜Ψ aq (cid:105) + A RR pq B RR pq | ˜Ψ aq ˜ Y p (cid:105) , S | ˜ Z p ˜Ψ aq (cid:105) = + E RR pq F RR pq | ˜ Z p ˜Ψ aq (cid:105) − D RR pq F RR pq | ˜Ψ aq ˜ Z p (cid:105) , S | ˜Ψ ap ˜ Y q (cid:105) = + A RR pq D RR pq | ˜ Y q ˜Ψ ap (cid:105) + A RR pq E RR pq | ˜Ψ aq ˜ Y p (cid:105) , S | ˜Ψ ap ˜ Z q (cid:105) = − B RR pq F RR pq | ˜ Z q ˜Ψ ap (cid:105) + C RR pq F RR pq | ˜Ψ aq ˜ Z p (cid:105) , S | ˜Ψ ap ˜Ψ bq (cid:105) = δ ab A RR pq F RR pq | ˜Ψ bq ˜Ψ ap (cid:105) + (cid:15) ab ( C RR pq D RR pq | ˜ Y q ˜ Z p (cid:105) + B RR pq E RR pq | ˜ Z q ˜ Y p (cid:105) )+ (1 − δ ab )( C RR pq E RR pq | ˜Ψ aq ˜Ψ bp (cid:105) − B RR pq D RR pq | ˜Ψ bq ˜Ψ ap (cid:105) ) . (A.6)The last process can be further simplified by using the identity C RR pq E RR pq − B RR pq D RR pq = A RR pq F RR pq . (A.7) A.1.3 Left-right scattering S | Y p ˜ Y q (cid:105) = + A LR pq A LR pq | ˜ Y q Y p (cid:105) − B LR pq B LR pq | ˜ Z q Z p (cid:105) − A LR pq B LR pq (cid:15) ab | ˜Ψ aq Ψ bp (cid:105) , S | Z p ˜ Z q (cid:105) = − F LR pq F LR pq | ˜ Y q Y p (cid:105) + E LR pq E LR pq | ˜ Z q Z p (cid:105) + E LR pq F LR pq (cid:15) ab | ˜Ψ aq Ψ bp (cid:105) , S | Y p ˜ Z q (cid:105) = + C LR pq C LR pq | ˜ Z q Y p (cid:105) , S | Z p ˜ Y q (cid:105) = + D LR pq D LR pq | ˜ Y q Z p (cid:105) , S | Y p ˜Ψ aq (cid:105) = + A LR pq C LR pq | ˜Ψ aq Y p (cid:105) − B LR pq C LR pq | ˜ Z q Ψ ap (cid:105) , S | Z p ˜Ψ aq (cid:105) = − D LR pq E LR pq | ˜Ψ aq Z p (cid:105) + D LR pq F LR pq | ˜ Y q Ψ ap (cid:105) , S | Ψ ap ˜ Y q (cid:105) = + A LR pq D LR pq | ˜ Y q Ψ ap (cid:105) − B LR pq D LR pq | ˜Ψ aq Z p (cid:105) , S | Ψ ap ˜ Z q (cid:105) = − C LR pq E LR pq | ˜ Z q Ψ ap (cid:105) + C LR pq F LR pq | ˜Ψ aq Y p (cid:105) , S | Ψ ap ˜Ψ bq (cid:105) = − C LR pq D LR pq δ ab | ˜Ψ bq Ψ ap (cid:105) − (cid:15) ab ( A LR pq F LR pq | ˜ Y q Y p (cid:105) − B LR pq E LR pq | ˜ Z q Z p (cid:105) )+ (1 − δ ab )( A LR pq E LR pq | ˜Ψ bq Ψ ap (cid:105) − B LR pq F LR pq | ˜Ψ aq Ψ bp (cid:105) ) . (A.8)The last process can be further simplified by using the identity A LR pq E LR pq − B LR pq F LR pq = − C LR pq D LR pq . (A.9)– 49 – .1.4 Right-left scattering S | ˜ Y p Y q (cid:105) = + A RL pq A RL pq | Y q ˜ Y p (cid:105) − B RL pq B RL pq | Z q ˜ Z p (cid:105) + A RL pq B RL pq (cid:15) ab | Ψ aq ˜Ψ bp (cid:105) , S | ˜ Z p Z q (cid:105) = − F RL pq F RL pq | Y q ˜ Y p (cid:105) + E RL pq E RL pq | Z q ˜ Z p (cid:105) − E RL pq F RL pq (cid:15) ab | Ψ aq ˜Ψ bp (cid:105) , S | ˜ Y p Z q (cid:105) = + C RL pq C RL pq | Z q ˜ Y p (cid:105) , S | ˜ Z p Y q (cid:105) = + D RL pq D RL pq | Y q ˜ Z p (cid:105) , S | ˜ Y p Ψ aq (cid:105) = + A RL pq C RL pq | Ψ aq ˜ Y p (cid:105) + B RL pq C RL pq | Z q ˜Ψ ap (cid:105) , S | ˜ Z p Ψ aq (cid:105) = − D RL pq E RL pq | Ψ aq ˜ Z p (cid:105) − D RL pq F RL pq | Y q ˜Ψ ap (cid:105) , S | ˜Ψ ap Y q (cid:105) = + A RL pq D RL pq | Y q ˜Ψ ap (cid:105) + B RL pq D RL pq | Ψ aq ˜ Z p (cid:105) , S | ˜Ψ ap Z q (cid:105) = − C RL pq E RL pq | Z q ˜Ψ ap (cid:105) − C RL pq F RL pq | Ψ aq ˜ Y p (cid:105) , S | ˜Ψ ap Ψ bq (cid:105) = − C RL pq D RL pq δ ab | Ψ bq ˜Ψ ap (cid:105) − (cid:15) ab ( A RL pq F RL pq | Y q ˜ Y p (cid:105) − B RL pq E RL pq | Z q ˜ Z p (cid:105) )+ (1 − δ ab )( A RL pq E RL pq | Ψ bq ˜Ψ ap (cid:105) − B RL pq F RL pq | Ψ aq ˜Ψ bp (cid:105) ) . (A.10)The last process can be further simplified by using the identity A RL pq E RL pq − B RL pq F RL pq = − C RL pq D RL pq . (A.11) A.2 The mixed-mass sectorA.2.1 Left-massless scattering S | Z p T ˙ Aaq (cid:105) = − D Lo pq F Lo pq | T ˙ Aaq Z p (cid:105) − E Lo pq F Lo pq | ˜ χ ˙ Aq Ψ ap (cid:105) , S | Y p T ˙ Aaq (cid:105) = + A Lo pq B Lo pq | T ˙ Aaq Y p (cid:105) − A Lo pq C Lo pq | χ ˙ Aq Ψ ap (cid:105) , S | Ψ ap ˜ χ ˙ Aq (cid:105) = + B Lo pq F Lo pq | ˜ χ ˙ Aq Ψ ap (cid:105) + C Lo pq F Lo pq | T ˙ Aaq Z p (cid:105) , S | Ψ ap χ ˙ Aq (cid:105) = − A Lo pq D Lo pq | χ ˙ Aq Ψ ap (cid:105) + A Lo pq E Lo pq | T ˙ Aaq Y p (cid:105) , S | Z p ˜ χ ˙ Aq (cid:105) = + F Lo pq F Lo pq | ˜ χ ˙ Aq Z p (cid:105) , S | Y p χ ˙ Aq (cid:105) = + A Lo pq A Lo pq | χ ˙ Aq Y p (cid:105) , S | Z p χ ˙ Aq (cid:105) = + D Lo pq D Lo pq | χ ˙ Aq Z p (cid:105) + E Lo pq E Lo pq | ˜ χ ˙ Aq Y p (cid:105) + D Lo pq E Lo pq (cid:15) ab | T ˙ Aaq Ψ bp (cid:105) , S | Y p ˜ χ ˙ Aq (cid:105) = + B Lo pq B Lo pq | ˜ χ ˙ Aq Y p (cid:105) + C Lo pq C Lo pq | χ ˙ Aq Z p (cid:105) + B Lo pq C Lo pq (cid:15) ab | T ˙ Aaq Ψ bp (cid:105) , S | Ψ ap T ˙ Abq (cid:105) = − δ ab A Lo pq F Lo pq | T ˙ Aaq Ψ bp (cid:105) + (cid:15) ab ( C Lo pq D Lo pq | χ ˙ Aq Z p (cid:105) + B Lo pq E Lo pq | ˜ χ ˙ Aq Y p (cid:105) )+ (1 − δ ab )( B Lo pq D Lo pq | T ˙ Aaq Ψ bp (cid:105) − C Lo pq E Lo pq | T ˙ Abq Ψ ap (cid:105) ) , (A.12)The last process can be further simplified by using the identity C Lo pq E Lo pq − B Lo pq D Lo pq = A Lo pq F Lo pq . (A.13)– 50 – .2.2 Massless-left scattering S | T ˙ Aap Z q (cid:105) = − B oL pq F oL pq | Z q T ˙ Aap (cid:105) + C oL pq F oL pq | Ψ aq ˜ χ ˙ Ap (cid:105) , S | T ˙ Aap Y q (cid:105) = + A oL pq D oL pq | Y q T ˙ Aap (cid:105) + A oL pq E oL pq | Ψ aq χ ˙ Ap (cid:105) , S | ˜ χ ˙ Ap Ψ aq (cid:105) = + D oL pq F oL pq | Ψ aq ˜ χ ˙ Ap (cid:105) − E oL pq F oL pq | Z q T ˙ Aap (cid:105) , S | χ ˙ Ap Ψ aq (cid:105) = − A oL pq B oL pq | Ψ aq χ ˙ Ap (cid:105) − A oL pq C oL pq | Y q T ˙ Aap (cid:105) , S | ˜ χ ˙ Ap Z q (cid:105) = + F oL pq F oL pq | Z q ˜ χ ˙ Ap (cid:105) , S | χ ˙ Ap Y q (cid:105) = + A oL pq A oL pq | Y q χ ˙ Ap (cid:105) , S | χ ˙ Ap Z q (cid:105) = + B oL pq B oL pq | Z q χ ˙ Ap (cid:105) + C oL pq C oL pq | Y q ˜ χ ˙ Ap (cid:105) − B oL pq C oL pq (cid:15) ab | Ψ aq T ˙ Abp (cid:105) , S | ˜ χ ˙ Ap Y q (cid:105) = + D oL pq D oL pq | Y q ˜ χ ˙ Ap (cid:105) + E oL pq E oL pq | Z q χ ˙ Ap (cid:105) − D oL pq E oL pq (cid:15) ab | Ψ aq T ˙ Abp (cid:105) , S | T ˙ Aap Ψ bq (cid:105) = − δ ab A oL pq F oL pq | Ψ bq T ˙ Aap (cid:105) − (cid:15) ab ( B oL pq E oL pq | Z q χ ˙ Ap (cid:105) + C oL pq D oL pq | Y q ˜ χ ˙ Ap (cid:105) )+ (1 − δ ab )( D oL pq B oL pq | Ψ bq T ˙ Aap (cid:105) − E oL pq C oL pq | Ψ aq T ˙ Abp (cid:105) ) , (A.14)The last process can be further simplified by using the identity C oL pq E oL pq − B oL pq D oL pq = A oL pq F oL pq . (A.15) A.2.3 Right-massless scattering S | ˜ Z p T ˙ Aaq (cid:105) = − D Ro pq E Ro pq | T ˙ Aaq ˜ Z p (cid:105) + D Ro pq F Ro pq | χ ˙ Aq ˜Ψ ap (cid:105) , S | ˜ Y p T ˙ Aaq (cid:105) = + A Ro pq C Ro pq | T ˙ Aaq ˜ Y p (cid:105) − B Ro pq C Ro pq | ˜ χ ˙ Aq ˜Ψ ap (cid:105) , S | ˜Ψ ap χ ˙ Aq (cid:105) = − A Ro pq D Ro pq | χ ˙ Aq ˜Ψ ap (cid:105) + B Ro pq D Ro pq | T ˙ Aaq ˜ Z p (cid:105) , S | ˜Ψ ap ˜ χ ˙ Aq (cid:105) = + C Ro pq E Ro pq | ˜ χ ˙ Aq ˜Ψ ap (cid:105) − C Ro pq F Ro pq | T ˙ Aaq ˜ Y p (cid:105) , S | ˜ Z p χ ˙ Aq (cid:105) = + D Ro pq D Ro pq | χ ˙ Aq ˜ Z p (cid:105) , S | ˜ Y p ˜ χ ˙ Aq (cid:105) = + C Ro pq C Ro pq | ˜ χ ˙ Aq ˜ Y p (cid:105) , S | ˜ Z p ˜ χ ˙ Aq (cid:105) = + E Ro pq E Ro pq | ˜ χ ˙ Aq ˜ Z p (cid:105) − F Ro pq F Ro pq | χ ˙ Aq ˜ Y p (cid:105) + F Ro pq E Ro pq (cid:15) ab | T ˙ Aaq ˜Ψ bp (cid:105) , S | ˜ Y p χ ˙ Aq (cid:105) = + A Ro pq A Ro pq | χ ˙ Aq ˜ Y p (cid:105) − B Ro pq B Ro pq | ˜ χ ˙ Aq ˜ Z p (cid:105) − B Ro pq A Ro pq (cid:15) ab | T ˙ Aaq ˜Ψ bp (cid:105) , S | ˜Ψ ap T ˙ Abq (cid:105) = + C Ro pq D Ro pq δ ab | T ˙ Aaq ˜Ψ bp (cid:105) + (cid:15) ab ( A Ro pq F Ro pq | χ ˙ Aq ˜ Y p (cid:105) − B Ro pq E Ro pq | ˜ χ ˙ Aq ˜ Z p (cid:105) )+ (1 − δ ab )( B Ro pq F Ro pq | T ˙ Aaq ˜Ψ bp (cid:105) − A Ro pq E Ro pq | T ˙ Abq ˜Ψ ap (cid:105) ) . (A.16)The last process can be further simplified by using the identity B Ro pq F Ro pq − A Ro pq E Ro pq = C Ro pq D Ro pq . (A.17)– 51 – .2.4 Massless-right scattering S | T ˙ Aap ˜ Z q (cid:105) = − C oR pq E oR pq | ˜ Z q T ˙ Aap (cid:105) + C oR pq F oR pq | ˜Ψ aq χ ˙ Ap (cid:105) , S | T ˙ Aap ˜ Y q (cid:105) = + A oR pq D oR pq | ˜ Y q T ˙ Aap (cid:105) − B oR pq D oR pq | ˜Ψ aq ˜ χ ˙ Ap (cid:105) , S | χ ˙ Ap ˜Ψ aq (cid:105) = − A oR pq C oR pq | ˜Ψ aq χ ˙ Ap (cid:105) + B oR pq C oR pq | ˜ Z q T ˙ Aap (cid:105) , S | ˜ χ ˙ Ap ˜Ψ aq (cid:105) = + D oR pq E oR pq | ˜Ψ aq ˜ χ ˙ Ap (cid:105) − D oR pq F oR pq | ˜ Y q T ˙ Aap (cid:105) , S | χ ˙ Ap ˜ Z q (cid:105) = + C oR pq C oR pq | ˜ Z q χ ˙ Ap (cid:105) , S | ˜ χ ˙ Ap ˜ Y q (cid:105) = + D oR pq D oR pq | ˜ Y q ˜ χ ˙ Ap (cid:105) , S | ˜ χ ˙ Ap ˜ Z q (cid:105) = + E oR pq E oR pq | ˜ Z q ˜ χ ˙ Ap (cid:105) − F oR pq F oR pq | ˜ Y q χ ˙ Ap (cid:105) + F oR pq E oR pq (cid:15) ab | ˜Ψ aq T ˙ Abp (cid:105) , S | χ ˙ Ap ˜ Y q (cid:105) = + A oR pq A oR pq | ˜ Y q χ ˙ Ap (cid:105) − B oR pq B oR pq | ˜ Z q ˜ χ ˙ Ap (cid:105) − A oR pq B oR pq (cid:15) ab | ˜Ψ aq T ˙ Abp (cid:105) , S | T ˙ Aap ˜Ψ bq (cid:105) = + C oR pq D oR pq δ ab | ˜Ψ aq T ˙ Abp (cid:105) + (cid:15) ab ( A oR pq F oR pq | ˜ Y q χ ˙ Ap (cid:105) − B oR pq E oR pq | ˜ Z q ˜ χ ˙ Ap (cid:105) )+ (1 − δ ab )( B oR pq F oR pq | ˜Ψ aq T ˙ Abp (cid:105) − A oR pq E oR pq | ˜Ψ bq T ˙ Aap (cid:105) ) . (A.18)The last process can be further simplified by using the identity B oR pq F oR pq − A oR pq E oR pq = C oR pq D oR pq . (A.19) A.3 The massless sector S | T ˙ Aap T ˙ Bbq (cid:105) = − A oo pq F oo pq δ ab | T ˙ Baq T ˙ Abp (cid:105) + (cid:15) ab ( C oo pq D oo pq | χ ˙ Bq ˜ χ ˙ Ap (cid:105) + B oo pq E oo pq | ˜ χ ˙ Bq χ ˙ Ap (cid:105) ) − (1 − δ ab )( C oo pq E oo pq | T ˙ Baq T ˙ Abp (cid:105) − B oo pq D oo pq | T ˙ Bbq T ˙ Aap (cid:105) ) , S | T ˙ Aap ˜ χ ˙ Bq (cid:105) = − B oo pq F oo pq | ˜ χ ˙ Bq T ˙ Aap (cid:105) − C oo pq F oo pq | T ˙ Baq ˜ χ ˙ Ap (cid:105) , S | ˜ χ ˙ Ap T ˙ Baq (cid:105) = − D oo pq F oo pq | T ˙ Baq ˜ χ ˙ Ap (cid:105) − E oo pq F oo pq | ˜ χ ˙ Bp T ˙ Aap (cid:105) , S | T ˙ Aap χ ˙ Bq (cid:105) = + A oo pq D oo pq | χ ˙ Bq T ˙ Aap (cid:105) − A oo pq E oo pq | T ˙ Baq χ ˙ Ap (cid:105) , S | χ ˙ Ap T ˙ Baq (cid:105) = + A oo pq B oo pq | T ˙ Baq χ ˙ Ap (cid:105) − A oo pq C oo pq | χ ˙ Bp T ˙ Aap (cid:105) , S | ˜ χ ˙ Ap ˜ χ ˙ Bq (cid:105) = − F oo pq F oo pq | ˜ χ ˙ Bq ˜ χ ˙ Ap (cid:105) , S | χ ˙ Ap χ ˙ Bq (cid:105) = − A oo pq A oo pq | χ ˙ Bq χ ˙ Ap (cid:105) , S | ˜ χ ˙ Ap χ ˙ Bq (cid:105) = − D oo pq D oo pq | χ ˙ Bq ˜ χ ˙ Ap (cid:105) − E oo pq E oo pq | ˜ χ ˙ Bq χ ˙ Ap (cid:105) − E oo pq D oo pq (cid:15) ab | T ˙ Baq T ˙ Abp (cid:105) , S | χ ˙ Ap ˜ χ ˙ Bq (cid:105) = − B oo pq B oo pq | ˜ χ ˙ Bq χ ˙ Ap (cid:105) − C oo pq C oo pq | χ ˙ Bq ˜ χ ˙ Ap (cid:105) − B oo pq C oo pq (cid:15) ab | T ˙ Baq T ˙ Abp (cid:105) . (A.20)The first process can be further simplified by using the identity C oo pq E oo pq − B oo pq D oo pq = A oo pq F oo pq . (A.21) References [1] J. M. Maldacena, The large N limit of superconformal field theories and supergravity , Adv.Theor. Math. Phys. (1998) 231 [ hep-th/9711200 ].[2] E. Witten, Anti-de Sitter space and holography , Adv. Theor. Math. Phys. (1998) 253[ hep-th/9802150 ]. – 52 – 3] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from non-criticalstring theory , Phys. Lett. B428 (1998) 105 [ hep-th/9802109 ].[4] G. ’t Hooft, A Planar Diagram Theory for Strong Interactions , Nucl. Phys. B (1974) 461.[5] J. A. Minahan and K. Zarembo, The Bethe-ansatz for N = 4 super Yang-Mills , JHEP (2003) 013 [ hep-th/0212208 ].[6] H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomicchain , Z. Phys. (1931) 205.[7] L. D. Faddeev, How algebraic Bethe ansatz works for integrable model , hep-th/9605187 .[8] J. Ambjørn, R. A. Janik and C. Kristjansen, Wrapping interactions and a new source ofcorrections to the spin-chain/string duality , Nucl. Phys. B736 (2006) 288 [ hep-th/0510171 ].[9] G. Arutyunov and S. Frolov, On string S-matrix, bound states and TBA , JHEP (2007)024 [ ].[10] N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for AdS /CFT , Phys. Rev. Lett. (2014) 011602 [ ].[11] G. Arutyunov and S. Frolov, Foundations of the AdS × S superstring. part I , J. Phys. A A42 (2009) 254003 [ ].[12] N. Beisert, C. Ahn, L. F. Alday, Z. Bajnok, J. M. Drummond, L. Freyhult et al., Review ofAdS/CFT Integrability: An Overview , Lett. Math. Phys. (2012) 3 [ ].[13] B. Basso, S. Komatsu and P. Vieira, Structure constants and integrable bootstrap in planar N = 4 SYM theory , .[14] N. Beisert, The su (2 | dynamic S -matrix , Adv. Theor. Math. Phys. (2008) 945[ hep-th/0511082 ].[15] M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 1.Stable particle states , Commun. Math. Phys. (1986) 177.[16] M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 2.Scattering states , Commun. Math. Phys. (1986) 153.[17] B. Eden and A. Sfondrini, Three-point functions in N = 4 SYM: the hexagon proposal atthree loops , JHEP (2016) 165 [ ].[18] B. Basso, V. Goncalves, S. Komatsu and P. Vieira, Gluing Hexagons at Three Loops , Nucl.Phys. B907 (2016) 695 [ ].[19] B. Basso, V. Goncalves and S. Komatsu, Structure constants at wrapping order , .[20] B. Eden and A. Sfondrini, Tessellating cushions: four-point functions in N = 4 SYM , JHEP (2017) 098 [ ].[21] T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions , JHEP (2017) 130[ ].[22] B. Eden, Y. Jiang, D. le Plat and A. Sfondrini, Colour-dressed hexagon tessellations forcorrelation functions and non-planar corrections , JHEP (2018) 170 [ ].[23] T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling Handles I:Nonplanar Integrability , . – 53 – 24] T. Bargheer, F. Coronado and P. Vieira, Octagons I: Combinatorics and Non-PlanarResummations , JHEP (2019) 162 [ ].[25] O. Ohlsson Sax and B. Stefański, Closed strings and moduli in AdS /CFT , JHEP (2018)101 [ ].[26] J. M. Maldacena and H. Ooguri, Strings in AdS and SL(2 , R ) WZW model. I , J. Math.Phys. (2001) 2929 [ hep-th/0001053 ].[27] J. M. Maldacena, H. Ooguri and J. Son, Strings in AdS and the SL(2 , R ) WZW model. II:Euclidean black hole , J. Math. Phys. (2001) 2961 [ hep-th/0005183 ].[28] J. M. Maldacena and H. Ooguri, Strings in AdS and the SL(2 , R ) WZW model. III:Correlation functions , Phys. Rev. D65 (2002) 106006 [ hep-th/0111180 ].[29] N. Berkovits, C. Vafa and E. Witten, Conformal field theory of AdS background withRamond-Ramond flux , JHEP (1999) 018 [ hep-th/9902098 ].[30] G. Giribet, C. Hull, M. Kleban, M. Porrati and E. Rabinovici, Superstrings on AdS at k = 1 , JHEP (2018) 204 [ ].[31] M. R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS , JHEP (2018)085 [ ].[32] L. Eberhardt, M. R. Gaberdiel and R. Gopakumar, The Worldsheet Dual of the SymmetricProduct CFT , JHEP (2019) 103 [ ].[33] A. Babichenko, B. Stefański, jr. and K. Zarembo, Integrability and the AdS / CFT correspondence , JHEP (2010) 058 [ ].[34] A. Cagnazzo and K. Zarembo, B-field in AdS / CFT correspondence and integrability , JHEP (2012) 133 [ ].[35] A. Sfondrini, Towards integrability for AdS / CFT , J. Phys. A48 (2015) 023001 [ ].[36] B. Hoare and A. A. Tseytlin, On string theory on AdS × S × T with mixed 3-form flux:tree-level S-matrix , Nucl. Phys. B873 (2013) 682 [ ].[37] B. Hoare, A. Stepanchuk and A. Tseytlin, Giant magnon solution and dispersion relation instring theory in AdS × S × T with mixed flux , Nucl. Phys. B879 (2014) 318 [ ].[38] T. Lloyd, O. Ohlsson Sax, A. Sfondrini and B. Stefański, jr., The complete worldsheet Smatrix of superstrings on AdS × S × T with mixed three-form flux , Nucl. Phys. B891 (2015) 570 [ ].[39] M. C. Abbott and I. Aniceto, Massless Lüscher terms and the limitations of the AdS asymptotic Bethe ansatz , Phys. Rev. D93 (2016) 106006 [ ].[40] M. Baggio and A. Sfondrini, Strings on NS-NS Backgrounds as Integrable Deformations , Phys. Rev. D98 (2018) 021902 [ ].[41] A. Dei and A. Sfondrini, Integrable spin chain for stringy Wess-Zumino-Witten models , JHEP (2018) 109 [ ].[42] A. Sfondrini, Long Strings and Symmetric Product Orbifold from the AdS Bethe Equations , .[43] L. Eberhardt, M. R. Gaberdiel and R. Gopakumar, Deriving the AdS /CFT correspondence , JHEP (2020) 136 [ ]. – 54 – 44] F. Larsen and E. J. Martinec, U(1) charges and moduli in the D1-D5 system , JHEP (1999) 019 [ hep-th/9905064 ].[45] R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański, jr. and A. Torrielli, The all-loopintegrable spin-chain for strings on AdS × S × T : the massive sector , JHEP (2013)043 [ ].[46] R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, jr., Towards the all-loopworldsheet S matrix for AdS × S × T , Phys. Rev. Lett. (2014) 131601 [ ].[47] R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, jr, The complete AdS × S × T worldsheet S-matrix , JHEP (2014) 66 [ ].[48] R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański, jr. and A. Torrielli, Dressing phasesof AdS / CFT , Phys. Rev. D88 (2013) 066004 [ ].[49] R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański, jr. and A. Torrielli, On the dressingfactors, Bethe equations and Yangian symmetry of strings on AdS × S × T , J. Phys. A50 (2017) 024004 [ ].[50] A. Fontanella and A. Torrielli, Geometry of Massless Scattering in Integrable Superstring , JHEP (2019) 116 [ ].[51] A. Babichenko, A. Dekel and O. Ohlsson Sax, Finite-gap equations for strings onAdS × S × T with mixed 3-form flux , JHEP (2014) 122 [ ].[52] G. Arutyunov, S. Frolov, J. Plefka and M. Zamaklar, The off-shell symmetry algebra of thelight-cone AdS × S superstring , J. Phys. A40 (2007) 3583 [ hep-th/0609157 ].[53] R. Borsato, O. Ohlsson Sax and A. Sfondrini, A dynamic su (1 | S-matrix for AdS / CFT , JHEP (2013) 113 [ ].[54] R. Borsato, O. Ohlsson Sax and A. Sfondrini, All-loop Bethe ansatz equations forAdS / CFT , JHEP (2013) 116 [ ].[55] N. Drukker and J. Plefka, The Structure of n-point functions of chiral primary operators inN=4 super Yang-Mills at one-loop , JHEP (2009) 001 [ ].[56] M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiralprimary 3-point functions , JHEP (2012) 137 [ ].[57] A. Pakman and A. Sever, Exact N = 4 correlators of AdS / CFT , Phys. Lett. B652 (2007)60 [ ].[58] M. Baggio, O. Ohlsson Sax, A. Sfondrini, B. Stefański, jr. and A. Torrielli, Protected stringspectrum in AdS /CFT from worldsheet integrability , JHEP (2017) 091 [ ].[59] M. R. Gaberdiel and I. Kirsch, Worldsheet correlators in AdS / CFT , JHEP (2007)050 [ hep-th/0703001 ].[60] A. Dabholkar and A. Pakman, Exact chiral ring of AdS / CFT , Adv. Theor. Math. Phys. (2009) 409 [ hep-th/0703022 ].[61] J. Caetano and T. Fleury, Fermionic Correlators from Integrability , JHEP (2016) 010[ ].[62] M. Baggio, V. Niarchos and K. Papadodimas, On exact correlation functions in SU(N) N = 2 superconformal QCD , JHEP (2015) 198 [ ]. – 55 – 63] M. Baggio, A. Sfondrini, G. Tartaglino-Mazzucchelli and H. Walsh, On T ¯ T deformations andsupersymmetry , .[64] R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, On the spectrum ofAdS × S × T strings with Ramond-Ramond flux , J. Phys. A49 (2016) 41LT03[ ].[65] T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions II: Two-ParticleContributions , JHEP (2018) 177 [ ].[66] J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point , JHEP (2017) 013 [ ].[67] T. Klose, Review of AdS/CFT integrability, Chapter IV.3: N = 6 Chern-Simons and stringson AdS × CP , Lett. Math. Phys. (2010) 401 [ ].[68] R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, jr., The AdS × S × S × S worldsheet S matrix , J. Phys. A48 (2015) 415401 [ ].[69] N. Beisert and P. Koroteev, Quantum deformations of the one-dimensional Hubbard model , J. Phys. A41 (2008) 255204 [ ].[70] F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS × S superstringaction , Phys. Rev. Lett. (2014) 051601 [ ].[71] G. Arutyunov, R. Borsato and S. Frolov, S-matrix for strings on η -deformed AdS × S , JHEP (2014) 002 [ ].[72] B. Hoare and F. K. Seibold, Supergravity backgrounds of the η -deformed AdS × S × T andAdS × S superstrings , JHEP (2019) 125 [ ].[73] F. K. Seibold, S. J. van Tongeren and Y. Zimmermann, The twisted story of worldsheetscattering in η -deformed AdS × S , JHEP (2020) 043 [ ].].