The Factorized S-Matrix of CFT/AdS

Matthias Staudacher

REVIEW

The Factorized S-Matrix of CFT/AdS

Matthias Staudacher

High Energy Physics - Theory · Submitted 17 Dec 2004 · 53 references

Pith's verdict: The su(1|1) three-loop S-matrix and the perturbative asymptotic Bethe ansatz are real and well checked; the sl(2) deliverable rests on an assumed relation the paper is upfront about, and the KLV check constrains the assumption jointly with the asymptotic ansatz rather than isolating it.

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arXiv IDhep-th/0412188 (version 1)

Pith ID4BQLM6UQ, Pith's permanent record of this paper and its review

CitationMatthias Staudacher. "The Factorized S-Matrix of CFT/AdS." arXiv:hep-th/0412188 (2004). https://pith.science/paper/4BQLM6UQ

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Editor's note

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Worth your time, with one caveat to hold onto while reading it. What is real here is the perturbative asymptotic Bethe ansatz. The device in (3.5) — coupling-dependent correction factors B2, C2, B4, C4 whose range tracks the interaction range, with the phase shift read off the asymptotics — is a genuine technique, not a relabeling of Inozemtsev's ansatz. Applied to su(1|1) it yields the three-loop phase shift (3.7)/(3.8) and the complete three-loop spectrum (3.11) for all Tr Z^{L-M} psi^M. Table 1 is a stronger check than the reader allows: it is diagonalization of the same Hamiltonian, yes, but it tests factorization at M=4 in L=5, which is a many-body statement, not a two-body one. And the compactness point lands — set (3.8) against the page of spin-chain Hamiltonian it came from. Where it is thin is (5.1). The relation S_sl(2) = S_su(1|1) S_su(2)^{-1} S_su(1|1) is read off approximate large-tension S-matrices and Dynkin intuition, then assumed at weak coupling. The stress-test note is right that the KLV twist-two agreement does not cleanly isolate (5.1): L=2 sits inside the three-site interaction range, and the paper says so itself. The M=2 argument at J=2,3 is guaranteed by superconformal symmetry, so it is not independent evidence either. What the KLV check does establish is that the assumed relation plus the asymptotic ansatz, taken together, survive contact with an external three-loop QCD computation for M=2,4,6,8. That is a real constraint on the pair of assumptions, but a joint test, not a proof of (5.1). One correction to the reader: the abstract does carry the conditional. [abstract:s6] says 'Assuming that the relation also holds in gauge theory, we derive the three-loop S-matrix of the sl(2) sector.' The classifier's 'limitations: none found' is wrong; section 5 is explicit about the asymptotic-ansatz worry. Minor: [43], [44], [49] are unpublished or in preparation, and the three-loop Hamiltonian arrives through Beisert's private code. None of these is load-bearing. Send it out. The su(1|1) part is solid and the framing has legs; read the sl(2) part as a conjecture with a good check attached.

Referee report

Significance

If the program holds, this is an important conceptual reorientation: the S-matrix, not the loop-by-loop dilatation operator, becomes the organizing object, and the paper makes that concrete by extracting a compact three-loop result from an 'algorithmic' Hamiltonian that fills pages. The PABA technique is a genuine methodological contribution, and the explicit all-impurity three-loop spectrum (3.11) is a useful, checkable deliverable. The paper is also commendably falsifiable: the twist-three prediction (5.7) and the general twist-L, spin-M entries of Table 2 are stated as predictions that a conventional Feynman-diagram computation could refute, and the author reports that the two-loop part of (5.7) has since been confirmed. The relation (5.1) between the three S-matrices, if it survives beyond the small-momentum string expansion, would be a strong structural statement about the psu(2,2|4) chain.

Major objections (3)

[§5, Eq. (5.1) and the KLV comparison] The sl(2) three-loop S-matrix (5.2) and the twist-two agreement with (5.6) rest on the assumed relation (5.1), which the manuscript derives only from the approximate large-tension, small-momentum string S-matrices (4.28), (4.33), (4.39) plus Dynkin-diagram intuition, and then assumes at weak coupling. The paper says so plainly, which is to its credit, but the two offered checks do not cleanly isolate (5.1). The M=2 consistency (5.4)-(5.5) is guaranteed by superconformal symmetry [33] and therefore tests the ansatz's compatibility with known two-impurity universality rather than (5.1) itself. The KLV comparison is genuine external evidence, but the twist-two operators have L=2 while the three-loop interaction range is three sites, exactly the regime the manuscript itself flags as outside the asymptotic ansatz's justification. The M=2 argument at J=2,3 does not extend to the M=4,6,8 entries that carry the KLV check. The agreement is therefore evidence for the conjunction of (5.1) with the validity of the asymptotic ansatz at L=2, not for (5.1) alone. I would ask the author to state this conjunction explicitly in §5 and in the abstract, and, if possible, to identify a longer-length sl(2) observable where the asymptotic ansatz is safe and (5.1) can be tested separately.
[§3, Eqs. (3.5)-(3.8) and footnote 5] The extraction of the three-loop phase shift theta(p1,p2) proceeds through the correction factors B2, C2, B4, C4 of the PABA ansatz (3.5), which the manuscript states are gauge dependent and partly undetermined (only a linear relation between B4 and C4 is obtained). The claim that theta itself is gauge independent is asserted in footnote 5 rather than demonstrated. Since the su(1|1) three-loop S-matrix is the paper's most concrete result, a short explicit demonstration would materially strengthen it: for instance, repeating the two-loop extraction in a second allowed gauge choice and exhibiting that theta is unchanged, or giving the argument for why the gauge parameters drop out of the asymptotic phase. As written, a reader cannot distinguish 'gauge independent by construction' from 'gauge independent in the one gauge checked'.
[§4 and abstract, sentence [abstract:s4]] The abstract states that current semiclassical and near-plane-wave string knowledge is 'fully consistent' with a factorized S-matrix, and §4 states that the near-BMN and Frolov-Tseytlin physics is reproduced 'to all orders in g2'. The latter is correct only at leading order in 1/L: the phase shifts (4.29) and the S-matrices (4.28), (4.33), (4.39) are obtained by expanding theta to leading order in small momenta of order O(1/L), as the text around (4.10) and (4.13) makes clear. 'Fully consistent' in the abstract therefore reads more strongly than what is established. I would suggest qualifying the abstract to 'consistent at leading order in 1/L' and noting in §4 that the 1/J^2 and 1/L corrections mentioned in §6 are precisely where the approximate string S-matrices have not been tested.

Minor objections (5)

[§3, footnote 4] The three-loop piece H4 of Beisert's Hamiltonian is not reproduced, on the grounds that it is lengthy and little instructive, and the computation is stated to have used Beisert's original code. Since the three-loop su(1|1) result depends on H4, recording it in an appendix (or pointing to a public location for the code) would make the central computation reproducible by an independent reader.
[§5, Eq. (5.7) and ref. [49]] The reported confirmation of the two-loop part of the twist-three prediction (5.7) rests on ref. [49], listed as 'in preparation'. This should be flagged in the text as unpublished at the time of writing, so that the reader can weigh the check appropriately.
[§2.4, Eqs. (2.37)-(2.41) and ref. [44]] The screening argument that eliminates the fermionic roots w_l is attributed in footnote 3 to an independent observation by K. Zarembo, cited as ref. [44] 'unpublished'. Since the argument is reproduced in full in the text this is not a gap, but the citation should either be updated or the dependence on unpublished communication made explicit in the surrounding sentence.
[§6, Eq. (6.9) and footnote 11] The manuscript correctly notes that the string S-matrices (6.6) and (6.7) are not periodic in the momenta for r=1 and hence cannot literally be spin-chain S-matrices. The dressing-factor proposal (6.9)-(6.10) is then presented as a speculation. It would help to state in one sentence what would falsify the dressing-factor picture, beyond the 1/J^2 and 1/L comparisons already cited.
[Editorial] There are a number of typographical and typesetting defects that should be cleaned up: 'of of' in footnote 1, 'the the sl(2)' in the paragraph following (2.34), 'that that' in footnote 5, 'a spin chain might exists' in §6, and the recurring 'Bethe ans¨ atze' / 'ans¨ atze' rendering. The Dynkin diagrams (2.34), (2.35), (2.41), (2.43) are legible but would benefit from a caption explaining the node labels and the meaning of the arrows.

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The paper

The paper claims

Staudacher argues that the spectrum of the large-N CFT/AdS system is best encoded not in its Hamiltonian but in a factorized two-body S-matrix, and uses that idea to compute three-loop data in two sectors.

The reading

The paper's target is a change of tool. If planar $N=4$ super Yang-Mills and IIB strings on $AdS_5$ are genuinely integrable, the author argues, then the elementary excitations hidden inside the trace of a composite operator scatter without diffraction: many-body scattering factorizes into a sequence of two-body collisions that permute momenta but never change their magnitudes. That makes the two-body S-matrix the natural object to compute, and it is far more compact than the dilatation operator, which the paper notes grows explosively with loop order. To make this concrete, the author introduces the perturbative asymptotic Bethe ansatz (PABA) and extracts the three-loop S-matrix of the fermionic $su(1|1)$ sector from the known three-loop $su(2|3)$ dilatation operator. He then reads the available near-plane-wave and semiclassical string results in the $su(2)$, $su(1|1)$ and $sl(2)$ sectors through the same lens and finds them consistent with factorized scattering, which yields a simple relation among the three S-matrices. Assuming that relation survives at weak coupling, he writes down a three-loop $sl(2)$ Bethe ansatz even though that sector's dilatation operator is unknown beyond one loop, and the resulting spectrum matches the three-loop twist-two anomalous dimensions conjectured from a large QCD computation.

What carries the argument

Two objects carry the argument. The first is the factorized Bethe equation $e^{ip_k L} = ∏_{j≠k} ±S(p_k,p_j)$ with $∑_k p_k = 0$, which turns the whole spectral problem into a phase-shift bookkeeping exercise once the two-body S-matrix is known. The second is the perturbative asymptotic Bethe ansatz, a position-space wavefunction ansatz whose correction terms are organized by powers of the coupling, reflecting the finite range of the interaction at each loop order; matching it to the Schrödinger equation at the nearest, next-nearest and next-to-next-nearest separations fixes the phase shift. The paper also uses the long-range lattice phase function $φ(p) = ½ cot(p/2) √(1 + 8g² sin²(p/2))$ and the lattice dispersion laws $q_r(p)$ to translate between continuum string expressions and discrete gauge-theory ones.

Extended reading notes

Core claim

The paper's central claim is that integrability in the CFT/AdS system is equivalent to diffractionless scattering of the hidden elementary excitations, and that the factorized S-matrix should therefore be treated as the primary object rather than the dilatation operator or the $\sigma$-model Hamiltonian. The concrete realization is a new technique, the perturbative asymptotic Bethe ansatz: one assumes the position-space wavefunction is a superposition of in- and outgoing plane waves away from the collision region, and lets the coupling order control the interaction range, so that amplitude corrections near the collision point can be tuned order by order. Applying this to the $su(1|1)$ sector of planar $N=4$ gauge theory, the author extracts the two-body phase shift $S_{su(1|1)}(p_1,p_2) = -e^{iθ(p_1,p_2)}$ to three-loop order from the three-loop $su(2|3)$ dilatation operator, then solves the resulting Bethe equations to give explicit three-loop anomalous dimensions for all operators of the form $Tr Z^{L-M}ψ^M$. The result is checked against direct diagonalization of the same Hamiltonian for states up to four magnons in a length-five chain. A second strand compares the three basic sectors' string-theory S-matrices at large tension and small momentum and finds the relation $S_{sl(2)} = S_{su(1|1)} S_{su(2)}^{-1} S_{su(1|1)}$; assuming it holds in gauge theory gives a three-loop $sl(2)$ S-matrix whose twist-two spectrum agrees with the Kotikov–Lipatov–Onishchenko–Velizhanin prediction.

Load-bearing premise

The relation between the three S-matrices is inferred from approximate large-tension string expressions and from the intuition that it follows from inverting the Dynkin diagram, and the paper then assumes without proof that the same relation holds at weak coupling in order to write down the three-loop $sl(2)$ S-matrix.

Editorial extensions

If this is right

The three-loop anomalous dimensions of every operator of the form $Tr Z^{L-M}ψ^M$ in planar $N=4$ gauge theory are given explicitly by the closed formulas in (3.11), for arbitrary magnon number $M$.
A three-loop $sl(2)$ Bethe ansatz exists even though the corresponding dilatation operator is not known beyond one loop, and it reproduces the three-loop twist-two anomalous dimensions obtained from the QCD splitting-function computation.
The ansatz predicts concrete three-loop values for operators that have not been computed by Feynman diagrams, for example a twist-three state with $E = 15/2 - 225g²/16 + 3195g⁴/64$.
The discrepancies between gauge theory and string theory in the near-BMN and Frolov–Tseytlin limits are attributed to a coupling-dependent change in the S-matrix, i.e. a dressing factor, rather than to a breakdown of the local dispersion laws.
Agreement between gauge and string theory is shown to persist to two loops and to fail first at three loops in all three two-component sectors, which localizes where any strong-coupling deformation must enter.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the S-matrix is really the compact object, the natural next target is an all-loop expression for the $su(1|1)$ phase $θ(p_k,p_j)$, analogous to the conjectured all-loop form of $φ(p)$; the paper explicitly leaves this open, and finding it would be a sharper test than any fixed-order check.
The relation $S_{sl(2)} = S_{su(1|1)} S_{su(2)}^{-1} S_{su(1|1)}$ reads like a statement about reversing the Dynkin diagram, so one might expect it to hold for the full $psu(2,2|4)$ chain as a statement about crossing or inversion of the underlying algebra rather than as an accident of the small-momentum expansion.
The claim that the discrepancy between weak and strong coupling sits in a multiplicative dressing factor suggests a concrete search: look for a function that is trivial at weak coupling, reproduces the semiclassical kernels at strong coupling, and is periodic in the momenta so that a genuine spin chain exists.
Because the perturbative asymptotic Bethe ansatz only sees interaction ranges up to the loop order computed, its predictions for operators shorter than that range are extrapolations; the twist-two agreement is evidence for, but not a proof of, the absence of wrapping corrections at three loops.

Original abstract

We argue that the recently discovered integrability in the large-N CFT/AdS system is equivalent to diffractionless scattering of the corresponding hidden elementary excitations. This suggests that, perhaps, the key tool for finding the spectrum of this system is neither the gauge theory's dilatation operator nor the string sigma model's quantum Hamiltonian, but instead the respective factorized S-matrix. To illustrate the idea, we focus on the closed fermionic su(1|1) sector of the N=4 gauge theory. We introduce a new technique, the perturbative asymptotic Bethe ansatz, and use it to extract this sector's three-loop S-matrix from Beisert's involved algebraic work on the three-loop su(2|3) sector. We then show that the current knowledge about semiclassical and near-plane-wave quantum strings in the su(2), su(1|1) and sl(2) sectors of AdS_5 x S^5 is fully consistent with the existence of a factorized S-matrix. Analyzing the available information, we find an intriguing relation between the three associated S-matrices. Assuming that the relation also holds in gauge theory, we derive the three-loop S-matrix of the sl(2) sector even though this sector's dilatation operator is not yet known beyond one loop. The resulting Bethe ansatz reproduces the three-loop anomalous dimensions of twist-two operators recently conjectured by Kotikov, Lipatov, Onishchenko and Velizhanin, whose work is based on a highly complex QCD computation of Moch, Vermaseren and Vogt.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.62 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.41 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We argue that the recently discovered integrability in the large-N CFT/AdS system is equivalent to diffractionless scattering of the corresponding hidden elementary excitations. This suggests that, perhaps, the key tool for finding the spectrum of this system is neither the gauge theory's dilatation operator nor the string sigma model's quantum Hamiltonian, but instead the respective factorized S-matrix. To illustrate the idea, we focus on the closed fermionic su(1|1) sector of the N=4 gauge theory. We introduce a new technique, the perturbative asymptotic Bethe ansatz, and use it to extract this sector's three-loop S-matrix from Beisert's involved algebraic work on the three-loop su(2|3) sector. We then show that the current knowledge about semiclassical and near-plane-wave quantum strings in the su(2), su(1|1) and sl(2) sectors of AdS_5 x S^5 is fully consistent with the existence of a factorized S-matrix. Analyzing the available information, we find an intriguing relation between the three associated S-matrices. Assuming that the relation also holds in gauge theory, we derive the three-loop S-matrix of the sl(2) sector even though this sector's dilatation operator is not yet known beyond one loop. The resulting Bethe ansatz reproduces the three-loop anomalous dimensions of twist-two operators recently conjectured by Kotikov, Lipatov, Onishchenko and Velizhanin, whose work is based on a highly complex QCD computation of Moch, Vermaseren and Vogt.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We argue that the recently discovered integrability in the large-N CFT/AdS system is equivalent to diffractionless scattering of the corresponding hidden elementary excitations.	interpretation 0.68	0.65	0.73	asserted 0.62
s1	This suggests that, perhaps, the key tool for finding the spectrum of this system is neither the gauge theory's dilatation operator nor the string sigma model's quantum Hamiltonian, but instead the respective factorized S-matrix.	interpretation 0.96	0.22	0.43	hedged possibility 0.98
s2	To illustrate the idea, we focus on the closed fermionic su(1\|1) sector of the N=4 gauge theory.	contribution 0.56	0.70	0.03	asserted 0.86
s3	We introduce a new technique, the perturbative asymptotic Bethe ansatz, and use it to extract this sector's three-loop S-matrix from Beisert's involved algebraic work on the three-loop su(2\|3) sector.	method 0.70	0.95	0.06	asserted 1.00
s4	We then show that the current knowledge about semiclassical and near-plane-wave quantum strings in the su(2), su(1\|1) and sl(2) sectors of AdS_5 x S^5 is fully consistent with the existence of a factorized S-matrix.	result 0.88	0.58	0.11	asserted 0.77
s5	Analyzing the available information, we find an intriguing relation between the three associated S-matrices.	result 0.92	0.93	0.08	asserted 0.98
s6	Assuming that the relation also holds in gauge theory, we derive the three-loop S-matrix of the sl(2) sector even though this sector's dilatation operator is not yet known beyond one loop.	contribution 0.50	0.94	0.18	hedged possibility 0.42
s7	The resulting Bethe ansatz reproduces the three-loop anomalous dimensions of twist-two operators recently conjectured by Kotikov, Lipatov, Onishchenko and Velizhanin, whose work is based on a highly complex QCD computation of Moch, Vermaseren and Vogt.	result 1.00	0.81	0.22	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.64 · evidence from proof or derivation 0.68.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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M. Staudacher, in preparation

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Forward citations

Cited by 52 Pith papers

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140 later papers cite it

Evidence: substantial. 137 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 204 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 1 used its method or tool and 1 compared its own results with it.

Uses its method or tool 1 of 5 papers with this label

Adopted the expression used by Staudacher in this paper for the internal S-matrix.

On string integrability. A journey through the two-dimensional hidden symmetries in the AdS/CFT dualities (2010) · 0.97 · paraphrased: the licence does not allow quoting

Compares with it 1 of 3 papers with this label

Found agreement between its own solved 2-particle S-matrix for this sector and this paper's known S-matrix first obtained to three loops.

Iterative Structure of the N=4 SYM Spin Chain (2008) · 0.93 · paraphrased: the licence does not allow quoting

How the literature uses it

136 arXiv papers cite this in 200 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 127 sentences without a confident label

Builds on it 1 paper

For $Q>1$ , this formula is a generalisation of the exact [ref] magnon dispersion relation obtained in [ref] (see also [ref] ).

Aspects of Integrability in AdS/CFT Duality · The asymptotic states · 0.92

Uses its method or tool 4 papers

The magnitude $z$ is fixed by the asymptotic Bethe ansatz equation [ref] e^ipLS(z, -z)|_𝔰𝔩(2) = (x^+/x^-)^L+2z-i/2/z+i/2 /σ(z, -z)^2 = 1 , where $L = L_1$ is the length of the excited state, $σ= 1+O(g^6)$ is the BES dressing phase [ref] , and $x^± = x(z±i/2)$, with $x(z) = (z+√(z^2-4g^2))/(2g) = z/g+O(g)$.

Structure constants at wrapping order · Asymptotic data · 0.83
This is the asymptotic Bethe ansatz discussed in [ref] .

Reflecting magnons · Appendix: integrability at two loops · 0.82
On the gauge theory side, it is the same, namely we are dealing with the internal S-matrix, adopting the expression used by Staudacher in [ref] .

On string integrability. A journey through the two-dimensional hidden symmetries in the AdS/CFT dualities · Introduction · 0.97

Compares with it 3 papers

Solving for the coefficients $S$ and $c^(i)$ in the wavefunction, we find agreement with the known 2-particle S-matrix for this sector, first obtained to three-loops in [ref] , using the result of [ref] , S(p_1, p_2) -1 - 2 i g^2 (sin(p_1) - sin(p_1-p_2) - sin(p_2)) + 4 i g^4 sin(p_1/2) sin(p_2/2) (sin(p_1 - 3 p_2/2) - 7sin(p_1 - p_2/2) + sin(3 p_1 - p_2/2) + sin(3 p_1 - p_2/2) - 8 i sin(p_1/2) sin(p_2/2) sin^2( p_1 - p_2/2) ). + 𝒪(g^6).

Iterative Structure of the N=4 SYM Spin Chain · S-matrix checks · 0.93
One can check [ref] and even prove [ref] that they coincide with the corresponding results of the ABE.

Five-Loop Anomalous Dimension of Twist-Two Operators · The structure of the anomalous dimension · 0.96
Finally, using Eq. [ref], we obtain [equation] The planar parts of the above results coincide with those obtained with the help of integrability [ref] .

Anomalous dimensions of twist-two operators in extended N=2 and N=4 super Yang-Mills theories · Four-loop anomalous dimension in $=4$ SYM · 1.00

Cites it as support 2 papers

On the other hand, the perturbative asymptotic integrability for the first few orders of perturbation theory have been confirmed in some subsectors of the full theory, see [ref] and [ref] .

Integrability and the AdS/CFT correspondence · Asymptotic Bethe Ansatz · 0.99
The conjectured three loop formula has been then independently confirmed in the framework of the Bethe ansatz equations [ref] as well as within a space-time approach [ref] .] predicts that at each order $n$ the solution can be entirely expressed in terms of certain combinations of generalized harmonic sums of order $2n - 1$ or in terms of products of harmonic sums $S_𝐚$ and zeta functions $ζ(b_i)$ in such a way that the sum of their transcendentalities $|𝐚|$ and $b_i$ (see Appendix A for definitions) is again equal to $2n -1$.

Generalized Gribov-Lipatov Reciprocity and AdS/CFT · Multi-loop calculation of anomalous dimensions via integrability · 0.97

Cites it as background 58 papers

Going beyond one loop, asymptotic all-order Bethe Ansatz equations have been proposed [ref] .

Large spin expansion of the long-range Baxter equation in the sl(2) sector of N=4 SYM · Introduction · 0.85
Recent developments in the study of planar $N=4$ SUSY Yang-Mills (and the dual string theory on $AdS_5×S^5$) have culminated in a proposal for a set of Asymptotic Bethe Ansatz Equations (ABAE) [ref] .

Quantum Scattering of Giant Magnons · Introduction · 0.96
For the undeformed theory, the all-loop asymptotic Bethe equations [ref] have been derived [ref] from the $AdS_5/CFT_4$ $S$-matrix [ref] - [ref] .

Twisted Bethe equations from a twisted S-matrix · Introduction · 0.87

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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M. Staudacher, in preparation

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[2] [44]

K. Zarembo, unpublished

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[3] [49]

B. Eden, in preparation

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N. Beisert, “BMN operators and superconformal symmetry,” Nucl. Phys. B 659 (2003) 79, hep-th/0211032

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J. A. Minahan and K. Zarembo, “The Bethe-ansatz for N = 4 super Yang-Mills,” JHEP 0303 (2003) 013, hep-th/0212208

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N. Beisert and M. Staudacher, “The N = 4 SYM integrable super spin chain,” Nucl. Phys. B 670 (2003) 439, hep-th/0307042

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L. N. Lipatov, “High-energy asymptotics of multicolor QCD and exactly sol vable lattice models,” JETP Lett. 59 (1994) 596 [Pisma Zh. Eksp. Teor. Fiz. 59 (1994) 571], hep-th/9311037

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A. V. Belitsky, V. M. Braun, A. S. Gorsky and G. P. Korchems ky, “Integrability in QCD and beyond,” Int. J. Mod. Phys. A 19, 4715 (2004), hep-th/0407232

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I. Bena, J. Polchinski and R. Roiban, “Hidden symmetries of the AdS5 × S5 superstring,” Phys. Rev. D 69 (2004) 046002, hep-th/0305116

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