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...gs on AdS in a more direct fashion? Note that this possibility entirely hinges on whether integrability is really exact in either theory. It does not necessarily depend on whether the AdS/CFT duality =-=[39]-=- is actually valid dynamically. We feel that the present work demonstrates once again that it is surely useful. 2. One-Loop Scattering in Planar N = 4 2.1 The su(2) Bosonic Sector This sector consists...

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...us be quantized exactly [25]. Much intuition about the nature of the elementary excitations in this limit comes from subsequently comparing to gauge theory, i.e. from considering the famous BMN limit =-=[26]-=-. Recent studies of the near-BMN limit [27, 28, 29, 30, 31] yield information on how these excitations begin to interact. As we will show below, these interactions are fully consistent with the idea o...

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...ous λ 1 4 strong coupling behavior. First evidence that this might happen was found in [10]. It would also be very interesting to reproduce other semiclassical limits, such as the large spin limit of =-=[52]-=-, from the S-matrices. Acknowledgments I thank the Kavli Institute for Theoretical Physics for hospitality, and the organizers of the program QCD and String Theory for an inspiring workshop. I also th...

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...e distinguished from creative speculation. The first, crucial hint that conformal N = 4 gauge theory might be integrable in the planar limit was discovered in a beautiful paper by Minahan and Zarembo =-=[1]-=-. There it was shown that the spectrum of N = 4 conformal operators may be obtained by diagonalizing an integrable quantum spin chain. The observation was initially restricted to the subset of scalar ...

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...l(2) sector. A second important source of spectral information comes from the plane wave limit of strings on AdS5 ×S 5 , where the worldsheet theory becomes free, and may be thus be quantized exactly =-=[25]-=-. Much intuition about the nature of the elementary excitations in this limit comes from subsequently comparing to gauge theory, i.e. from considering the famous BMN limit [26]. Recent studies of the ...

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...miltonian, spectral studies for small operators in [14] are consistent with integrability in that certain tell-tale degeneracies (the socalled planar pairs argued to be a hallmark of integrability in =-=[2]-=-) indeed reappear. Further, compelling, evidence will come from the success of the computations below, as they establish that the spectrum may indeed be obtained from the principle of factorized scatt...

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... g2 YMN 8π 2 , (2.13) where in the present case the classical scaling dimension is ∆0 = L. Finally, one commonly expresses the su(2) Bethe equations in a form familiar from the algebraic Bethe ansatz =-=[41]-=-, as in [1, 3]. Changing variables by introducing the so-called Bethe roots uk = 1 2 cot ( ) pk , (2.11) becomes, after rewriting the S-matrix 2 (2.9), ( uk + i 2 uk − i )L = 2 M∏ j=1 j̸=k uk − uj + i...

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...ation operator was then extended to five loops, assuming integrability as well as BMN scaling. A fiveloop Bethe ansatz was found experimentally, and an all-loop (asymptotic) Bethe ansatz was proposed =-=[16]-=-. For a closely related system, see [17]. Note that the three-loop predictions of the three-loop dilatation operator in the su(2) sector were recently spectacularly confirmed, for two states, by a ful...

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...r were recently spectacularly confirmed, for two states, by a full-fledged, rigorous field theory computation [18]. For a detailed review on most aspects concerning the N = 4 dilatation operator, see =-=[19]-=-. One thing to note about the higher loop dilatation operators [2, 14, 16] is that they are very complicated, and appear to increase exponentially in complexity as the loop order increases (see e.g. a...

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...p2x1+ip1x2 if x1 ≪ x2 . (3.2) This is Sutherland’s “asymptotic” Bethe ansatz [45]. It was used by Inozemtsev to find the Bethe ansatz for the hyperbolic version of his spin chain [46], and adapted in =-=[15]-=- to diagonalize the three loop dilatation operator in the bosonic su(2) sector. If the ansatz (3.2) is true, the Schrödinger equation will be satisfied in the asymptotic region x1 ≪ x2 with the energy...

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..., x2) + +2 Ψ(x1, x2) − Ψ(x1, x2 − 1) − Ψ(x1, x2 + 1) , for x2 = x1 + 1 : (2.6) E0 Ψ(x1, x2) = 2 Ψ(x1, x2) − Ψ(x1 − 1, x2) − Ψ(x1, x2 + 1) . This difference equation is easily solved by Bethe’s ansatz =-=[42]-=- for the position space wave function Ψ(x1, x2) which reads Ψ(x1, x2) = e ip1x1+ip2x2 + S(p2, p1) e ip2x1+ip1x2 . (2.7) It is based on the intuition that the partons should freely evolve down the trac...

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... of scalar operators at one loop. However, shortly after, strong evidence was found that integrability extends to higher loops [2], and, – 1 –at least at one loop, to the full set of N = 4 operators =-=[3]-=-. Integrable structures have appeared before in planar QCD, starting with the pioneering work of Lipatov [4] (for a comprehensive review, see [5]). There it was always considered to be due to approxim...

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...ry’s dilatation operator and its Bethe ansatz. The latter may be constructed once the S-matrix is known. At one loop, the dilatation operator was derived for the complete set of psu(2, 2|4) operators =-=[13]-=-. It was shown to be integrable in [3], and the associated Bethe ansatz was constructed. Beyond one loop, we have Beisert’s two- and three loop dilatation operator in the “maximally compact” subsector...

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...ed in the case of two-impurity states, where the spectrum is known from superconformal symmetry [33]. More excitingly, we may compare to recent results of Kotikov, Lipatov, Onishchenko and Velizhanin =-=[35]-=-, who proposed three-loop exact anomalous dimensions for N = 4 twist-two operators. These dimensions were first found at one-loop from the OPE of four-point functions in [37], and the first few, at tw...

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...rable classical mechanics systems of Neumann type [7]. The reduction may be undone by the Bäcklund transformation, and the infinite tower of commuting charges of the original sigma model is recovered =-=[8]-=-. Subsequently, in an important paper by Kazakov, Marshakov, Minahan and Zarembo, it was demonstrated how the spectrum of finite-gap solutions of the classical sigma model may be obtained by the class...

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...n about the nature of the elementary excitations in this limit comes from subsequently comparing to gauge theory, i.e. from considering the famous BMN limit [26]. Recent studies of the near-BMN limit =-=[27, 28, 29, 30, 31]-=- yield information on how these excitations begin to interact. As we will show below, these interactions are fully consistent with the idea of an underlying factorized S-matrix. In fact, we will use t...

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...shown to be integrable in [3], and the associated Bethe ansatz was constructed. Beyond one loop, we have Beisert’s two- and three loop dilatation operator in the “maximally compact” subsector su(2|3) =-=[14]-=-. This includes the su(2) (here it was initially found in [2]) and su(1|1) sectors, but not the sl(2) sector. No two- and three loop Bethe ansatz is known, except for the su(2) sector, where it was co...

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...nishchenko and Velizhanin [35], who proposed three-loop exact anomalous dimensions for N = 4 twist-two operators. These dimensions were first found at one-loop from the OPE of four-point functions in =-=[37]-=-, and the first few, at two loops, in [38]. This author does not understand the derivation of Kotikov et.al., but it is clear that the crucial input is a computationally highly intensive three-loop QC...

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... = L∑ (1 − Px,x+1) = x=1 L∑ x=1 1 2 (1 − ⃗σx · ⃗σx+1) , (2.3) and may be expressed with the help of the permutation operator Px,x+1 which exchanges the partons at sites x and x+1, as first noticed in =-=[1, 40]-=-. It may alternatively be interpreted as an su(2) nearest neighbor spin chain [1] if we say that Z is a spin up with 2-spinor (1, 0) and φ is a spin down with 2-spinor (0, 1). Then the spin operator a...

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... already mentioned [9], where an equation describing the classical spectrum in the su(2) sector (in gauge theory connotation) was derived. Recently, the procedure was also applied to the sl(2) sector =-=[21]-=-. It is found that the most general classical finite-gap solution may be described by an algebraic curve. But does it contain information on the S-matrix, which is an inherently quantum concept? Here ...

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...l scattering equation (4.13) with the following dependence of the excitation momenta on the spectral parameter: p(ϕ) = 1 √ . (4.21) ϕ2 − 4 ω2 9 In much of the literature on the sigma model, including =-=[10]-=-, the total string energy is 1+2ω 2 E. Here we prefer to continue to use spin chain terminology where the energy E is the eigenvalue of the Hamiltonian, i.e. the anomalous part of the dilatation opera...

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...n both the near-BMN and FT situation. There are indeed indications that the discrepancies might worsen when higher corrections (i.e. 1/J 2 corrections to BMN and 1/J corrections to FT) are considered =-=[47, 48]-=-. This is entirely consistent with our suspicion that the S-matrix is to be blamed. It presumably changes as one goes from weak to strong coupling. 8 There are some subtleties, not important for the p...

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...ed three-loop exact anomalous dimensions for N = 4 twist-two operators. These dimensions were first found at one-loop from the OPE of four-point functions in [37], and the first few, at two loops, in =-=[38]-=-. This author does not understand the derivation of Kotikov et.al., but it is clear that the crucial input is a computationally highly intensive three-loop QCD field theory computation recently comple...

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...e that the three-loop predictions of the three-loop dilatation operator in the su(2) sector were recently spectacularly confirmed, for two states, by a full-fledged, rigorous field theory computation =-=[18]-=-. For a detailed review on most aspects concerning the N = 4 dilatation operator, see [19]. One thing to note about the higher loop dilatation operators [2, 14, 16] is that they are very complicated, ...

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...n both the near-BMN and FT situation. There are indeed indications that the discrepancies might worsen when higher corrections (i.e. 1/J 2 corrections to BMN and 1/J corrections to FT) are considered =-=[47, 48]-=-. This is entirely consistent with our suspicion that the S-matrix is to be blamed. It presumably changes as one goes from weak to strong coupling. 8 There are some subtleties, not important for the p...

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...ting from three loops, not identical [15]) to the gauge theory when the associated operators become very “long”. On the gauge side one finds rather similar equations in this “thermodynamic” situation =-=[23, 24]-=-, [15]. Since the latter, which are derived from the discrete Bethe equations, do have a scattering interpretation, the detailed comparison allows one to also bring the string equations into “scatteri...

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...gh we do not know the higher loop dilatation operator. The resulting spectrum is then successfully checked in the case of two-impurity states, where the spectrum is known from superconformal symmetry =-=[33]-=-. More excitingly, we may compare to recent results of Kotikov, Lipatov, Onishchenko and Velizhanin [35], who proposed three-loop exact anomalous dimensions for N = 4 twist-two operators. These dimens...

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...n about the nature of the elementary excitations in this limit comes from subsequently comparing to gauge theory, i.e. from considering the famous BMN limit [26]. Recent studies of the near-BMN limit =-=[27, 28, 29, 30, 31]-=- yield information on how these excitations begin to interact. As we will show below, these interactions are fully consistent with the idea of an underlying factorized S-matrix. In fact, we will use t...

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...gly important to test these ideas in refined situations such as 1/J2 corrections to the near-BMN limit, and 1/L corrections to spinning strings. For first steps in these directions see, respectively, =-=[51]-=- and [47, 48]. A dressing factor as in (6.9) should then also explain the famous λ 1 4 strong coupling behavior. First evidence that this might happen was found in [10]. It would also be very interest...

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...ip2x2 + S(p2, p1) e ip2x1+ip1x2 if x1 ≪ x2 . (3.2) This is Sutherland’s “asymptotic” Bethe ansatz [45]. It was used by Inozemtsev to find the Bethe ansatz for the hyperbolic version of his spin chain =-=[46]-=-, and adapted in [15] to diagonalize the three loop dilatation operator in the bosonic su(2) sector. If the ansatz (3.2) is true, the Schrödinger equation will be satisfied in the asymptotic region x1...

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...re √ λ is the string tension. This is simply a lattice version of the famous BMN energy formula [26]; in fact, one may show that this is the only possible lattice discretization of the BMN expression =-=[2, 34]-=-. Furthermore, a study of the dispersion laws for the higher charges in the su(2) sector, up to five loops, suggests [16] that these are likely given by the expressions qr(pk) = 2 sin ( r−1 2 pk ) ⎛√ ...

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...n about the nature of the elementary excitations in this limit comes from subsequently comparing to gauge theory, i.e. from considering the famous BMN limit [26]. Recent studies of the near-BMN limit =-=[27, 28, 29, 30, 31]-=- yield information on how these excitations begin to interact. As we will show below, these interactions are fully consistent with the idea of an underlying factorized S-matrix. In fact, we will use t...

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...ed than those theories where S-matrices have already been found. Still, one could try to look for similar models in order to derive some hints, which is an approach that has recently been followed in =-=[20]-=-. Alternatively, one can attempt to derive useful clues from available information on the sigma model’s spectrum. It comes from two sources. We already mentioned [9], where an equation describing the ...

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...assical coset sigma model is integrable [6]. Later it was shown in detail how finite-dimensional reductions of the classical sigma model lead to integrable classical mechanics systems of Neumann type =-=[7]-=-. The reduction may be undone by the Bäcklund transformation, and the infinite tower of commuting charges of the original sigma model is recovered [8]. Subsequently, in an important paper by Kazakov, ...

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... interaction, which is, in our case, the considered order of perturbation theory: Ψ(x1, x2) ∼ e ip1x1+ip2x2 + S(p2, p1) e ip2x1+ip1x2 if x1 ≪ x2 . (3.2) This is Sutherland’s “asymptotic” Bethe ansatz =-=[45]-=-. It was used by Inozemtsev to find the Bethe ansatz for the hyperbolic version of his spin chain [46], and adapted in [15] to diagonalize the three loop dilatation operator in the bosonic su(2) secto...

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... e 2iθr(pk,pj) , (6.5) su(1|1) (pk, pj) ≃ −e −iθ1(pk,pj) , (6.6) ∞∏ e −2iθr(pk,pj) . (6.7) S string sl(2) (pk, pj) ≃ ϕ(pk) − ϕ(pj) − i ϕ(pk) − ϕ(pj) + i r=2 r=1 – 30 –Finally, it was demonstrated in =-=[50]-=- that to at least five-loop order a spin chain exists whose S-matrix ˜ Ssu(2) is exactly given by the right hand side of (6.5). Interestingly this model is part of the general three-loop su(2|3) spin ...