Logarithmic scaling in gauge/string correspondence

A.S. Gorsky; A.V. Belitsky; G.P. Korchemsky

REVIEW

Logarithmic scaling in gauge/string correspondence

A.V. Belitsky, A.S. Gorsky, G.P. Korchemsky

High Energy Physics - Theory · Submitted 16 Jan 2006 · 38 references

Pith's verdict: The real contribution is the parameter xi = ln(N/L)/L and the demonstration that the two-cut semiclassical expansion dies at xi ~ 1; the 4 ln N coefficient itself is the less novel part.

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arXiv IDhep-th/0601112 (version 2)

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CitationA.S. Gorsky, A.V. Belitsky, G.P. Korchemsky. "Logarithmic scaling in gauge/string correspondence." arXiv:hep-th/0601112 (2006). https://pith.science/paper/2Q4ZKREP

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Bottom line: the durable content here is the identification of xi = ln(N/L)/L as the crossover parameter and the demonstration that the standard two-cut semiclassical expansion breaks down at xi ~ 1. The single-log coefficient itself is the less surprising half of the result, since it was already implied by the one-loop cusp anomalous dimension quoted in Eq. (1.1). The paper is honest about this and uses the match as a check rather than a discovery claim, which is the right call. [abstract:s2] is the claim I would actually cite, along with the string-side statement that b_str is floored at sqrt(lambda') so the two cuts cannot collide. That geometric picture is clean and, to my eye, the most quotable thing in the paper. The numerical checks in Tables 1 and 2 are real work and the agreement is convincing for what it covers. The note added crediting Satoh and Sakai is straightforwardly honest. On the reader's weakest_assumption: I think it aims at the wrong target. The 4 ln N comes from the two large roots, delta_1 ~ N/sqrt(2), fixed by q2 through (3.40) and independent of the small-root quantization (3.42). The small roots contribute O(L) plus the L^3/ln^2 N correction of (3.49). So the deferred analysis of (3.42) matters for subleading terms and for the level spacing (3.50), not for the headline coefficient. The genuine soft spot is different: no exact Bethe-ansatz comparison reaches xi > 1. Tables 1 and 2 and the right panel of Fig. 4 all sit at xi <= 0.23 for L = 10, N <= 100. The one xi > 1 point (Fig. 4 left, N = 10^10, xi = 2.07) compares (3.43) against (3.42), asymptotic against asymptotic. So the abstract's 'valid throughout the entire region' is interpolation, not verification. Relatedly, the m = 1 identification for xi >> 1 is carried over from the charge hierarchy (3.10)/(2.48) into the regime where the paper itself says that hierarchy fails; the authors know this, but the argument is a little circular at that step. Neither point threatens the coefficient. Both are worth a referee pressing on. Who should read it: anyone working the cusp anomalous dimension or the integrability side of AdS/CFT, and anyone who has tried to push the two-cut solution past N ~ L. It deserves referees.

Referee report

Significance

The result is a genuine and useful contribution. The one-loop gauge-theory analysis is checked against exact Bethe-ansatz spectra in Tables 1 and 2 and the right panel of Fig. 4, with agreement better than 0.002% for N ≥ L, which is a real, reproducible verification rather than a plausibility argument. The identification of the crossover parameter ξ = ln(N/L)/L, and of the mechanism (collision of the two Bethe-root cuts at the origin) is a concrete, falsifiable prediction. On the string side, the resummation in Eq. (4.17) is parameter-free and reproduces the strong-coupling cusp anomalous dimension √λ/π from double-logarithmic terms of higher-twist anomalous dimensions, which is a striking consistency between the two sides of the correspondence. The matching of the one-loop gauge and string expressions through the Landen transformation, Eq. (4.33), is an independent cross-check rather than a circular step. The Note added, disclosing the independent Satoh-Sakai analysis and its agreement, is handled well.

Major objections (1)

[Abstract (sentence s3); §3.3 and Fig. 4] The abstract and §5 claim an asymptotic expression 'valid throughout the entire region of variation of the Lorentz spin', but every comparison against the exact Bethe-ansatz spectrum sits at ξ ≤ 0.23: Table 2 and the right panel of Fig. 4 use L = 10 and N ≤ 100, so ξ = ln(N/L)/L ≤ ln(10)/10 ≈ 0.23, and Table 1 uses N = 100, again ξ ≈ 0.23. The only point with ξ > 1, the left panel of Fig. 4 (N = 10^10, ξ = 2.07), compares the asymptotic roots of Eq. (3.43) against the asymptotic quantization conditions of Eq. (3.42), not against the exact spectrum. The coefficient 4 ln N itself is robust: it comes from the two large roots fixed by q2 alone through Eqs. (3.40) and (3.46), independently of Eq. (3.42), and it agrees with the known one-loop cusp anomalous dimension, so this is not a threat to the central result. But the 'valid throughout' wording is stronger than what is demonstrated. Either add an exact-spectrum check at ξ > 1 (L = 10 with N in the range 10^4–10^6 should be feasible) or qualify the abstract and §5 to say that the ξ ≫ 1 form is obtained from the asymptotic solution and matched to the string side rather than verified against exact Bethe roots.

Minor objections (9)

[Introduction, final paragraph] The roadmap sentence says 'In Sect. 5, we analyze the asymptotic behavior of the anomalous dimensions at strong coupling' and 'Section 4 contains concluding remarks', but Sect. 4 is the string analysis and Sect. 5 is the conclusion. The numbering should be corrected.
[Eq. (4.34) and the sentence following it] As printed, b_str = √λ′ + 1/(m² ln²√(1−τ)); taking λ′ → 0 gives 1/(m²L²) with L = ln(1/√(1−τ)), whereas the following sentence states that this limit should reproduce b_str^(0) = 2b/β from Eq. (3.13), which equals 1/(mL). Solving Eq. (4.27) for τ → 1 also appears to give a correction of order 1/(m²L²√λ′). Please check the powers of m, the logarithm, and the √λ′ factor in Eq. (4.34).
[Table 2] The N = 0 column lists exact 0 against asymptotic 0.0907. This is expected, since the asymptotic formula is derived for N ≫ 1, but it should be flagged in the caption or the column dropped so that it is not read as a failure of the approximation.
[Fig. 4, left panel] The caption should state explicitly that the crosses are solutions of Eq. (3.42) and the lines are Eq. (3.43), i.e., that this panel is an asymptotic-versus-asymptotic comparison and not a check against exact Bethe roots.
[Eq. (3.42)] The branch conventions for arg Γ and the parity and ordering conditions on the integers k_n deserve one more sentence; the statement that k_n is even/odd for L odd/even is terse and the reader has to reconstruct it.
[§3.3, Eq. (3.39)] The split of transfer-matrix roots into large and small uses the double-point distribution of Eq. (3.9), which is derived in the semiclassical regime. A sentence explaining why this counting survives as b → 0 would strengthen the m = 1 identification for ξ ≫ 1.
[Eq. (3.49)] It would help to note explicitly that in the regime ξ ≫ 1, i.e. ln N ≫ L, the correction term L³(−ψ″(s))π²/(48 ln²N) is suppressed relative to 4 ln N, so that the expansion in Eq. (3.49) is controlled. This is a constructive clarification, not a correction.
[§1, after Eq. (1.6)] The statement that the anomalous dimension 'does not depend on the spin s for N ≫ L' holds only at leading order; the subleading term 2L[ψ(s) − ψ(2s)] in Eq. (3.49) does depend on s. Adding 'to leading order' would remove the ambiguity.
[Throughout] Typographical issues to fix: 'Our consequent presentation' should read 'subsequent'; 'We would like to thank to thank' has a duplicated 'to thank'; 'ξsrt' appears after Eq. (4.19); the spacing in 'λ′ =λ/(πL)2' in Eqs. (1.7) and (4.15) is inconsistent; and the notation E(χ) versus E in Eqs. (4.6)–(4.8) should be made uniform.

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The paper

The paper claims

The paper argues that the minimal anomalous dimension of long twist-L Wilson operators crosses from double-logarithmic BMN scaling to single-logarithmic growth when ln(N/L) exceeds L, and identifies the same crossover on the string side.

The reading

Wilson operators with large Lorentz spin are expected to have anomalous dimensions that grow at most logarithmically with the spin, but for operators of twist $L>2$ the standard semiclassical spin-chain calculation had produced only the double-logarithmic BMN regime. This paper uses the Baxter $Q$-operator to show that the two cuts carrying the Bethe roots collide at the origin once the parameter $xi=ln(N/L)/L$ becomes large, which is precisely where the semiclassical expansion in powers of $xi$ breaks down. Resumming the expansion gives an asymptotic formula valid across the whole range of $N$: for $xi<1$ the one-loop dimension scales as $(lambda/2pi^2)(m^2/L)ln^2(N/L)$, while for $xi gg 1$ it scales as $(lambda/2pi^2)m ln N$, with the coefficient of the single logarithm matching the weak-coupling cusp anomalous dimension. On the string side the same crossover appears when the two spikes of the folded string approach the AdS boundary and the inner branching points of the spectral curve are prevented from reaching zero by $sqrt{lambda'}$, producing $gamma sim (sqrt{lambda}/pi)ln(N/sqrt{lambda})$.

What carries the argument

The argument is carried by the Baxter $Q$-operator for the $SL(2)$ spin chain and by its semiclassical limit. The transfer matrix $t_L(u)$ defines a complex curve $y^2=4-tau_0^2(x)$ whose branching points are the endpoints of the Bethe-root cuts; the momentum $p(x)$ is a meromorphic differential on that curve. For the minimal-energy state the curve reduces to an elliptic curve with two symmetric cuts $[-a,-b]cup[b,a]$, and the energy is obtained from $S'(x)$ at $x=pm ibeta/L$. When $bto0$ the cuts collide, $S'(x)$ is no longer analytic at the origin, and the semiclassical series in $xi$ diverges. The paper circumvents this by constructing an asymptotic solution $Q^{(as)}(u)=Q_+(u)Q_-(-is)+Q_-(u)Q_+(is)$ to the Baxter equation and reducing the energy to a sum over roots of $t_L(u)$, Eq. (3.37); the small roots obey the quantization conditions Eq. (3.42), and their distribution yields the single-logarithmic law. On the string side the analogous object is the elliptic spectral curve of the classical $\sigma$ model with a meromorphic differential having double poles at $x=pmsqrt{lambda'}$; the floor on $b_{rm str}$ is what prevents the collision and produces the $sqrt{lambda}$ prefactor.

Extended reading notes

Core claim

The central claim is that the minimal anomalous dimension of twist-$L$ Wilson operators in integrable sectors has two distinct large-spin regimes controlled by $xi=ln(N/L)/L$, not by $N/L$ alone. For $xi<1$ the one-loop energy retains the BMN-like double-logarithmic form $gamma=(lambda/2pi^2)(m^2/L)ln^2(N/L)+cdots$; for $xi gg 1$ it becomes single-logarithmic, $gamma=(lambda/2pi^2)m ln N+cdots$, where $m$ labels the minimal-energy trajectory and equals one for the single-folded string. The paper traces the change to the collision at the origin of the two symmetric cuts on which the Bethe roots condense, which invalidates the usual semiclassical expansion and requires resumming all orders in $xi$. On the string side the corresponding statement is that the logarithmic regime sets in when the two most distant points of the folded spinning string approach the AdS boundary; in the spectral-curve description the inner branching point $b_{rm str}$ approaches its floor $sqrt{lambda'}=sqrt{lambda}/(pi L)$ rather than zero, so the cuts cannot collide. The resulting strong-coupling law is $gamma=(sqrt{lambda}/pi)ln(N/sqrt{lambda})+cdots$, which reproduces the known cusp anomalous dimension $Gamma_{rm cusp}=sqrt{lambda}/(2pi)$.

Load-bearing premise

The load-bearing premise is that the asymptotic Baxter solution and the small-root quantization conditions reproduce the exact spectrum throughout the large-$N$ region; the paper verifies this numerically for $L=10$ and $Nle100$ and states that a systematic analysis of the quantization conditions will be given elsewhere.

Editorial extensions

If this is right

If correct, the weak-coupling cusp anomalous dimension can be extracted from the coefficient of the single logarithm in higher-twist operators, tying the cusp anomaly to resummed double-logarithmic corrections.
The parameter $xi=ln(N/L)/L$ gives a sharp criterion for when the standard semiclassical Bethe-root expansion is reliable: it fails for $xi sim 1$ and must be replaced by the resummed asymptotic solution.
At strong coupling the same logarithmic law follows from spikes of the folded string reaching the AdS boundary, with coefficient $sqrt{lambda}/(2pi)$ for $Gamma_{rm cusp}$.
The correspondence between the cut collision in the spin chain and the $sqrt{lambda'}$ floor on the string branching point gives a concrete dictionary between weak- and strong-coupling descriptions of the same large-spin limit.
The analysis predicts a tower of excited states with energies separated by $Deltavarepsilon sim L/ln^2 N$, which organizes the band of anomalous dimensions near its lower edge.

Reading between the lines

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

The cut-collision mechanism looks general enough to apply to other integrable spin chains with long operators, so similar $ln(N/L)/L$ crossovers may appear wherever Bethe roots condense on symmetric cuts.
The numerical verification is limited to small $L$; a natural extension is to test the small-root quantization conditions at larger $L$ using finite-size or thermodynamic Bethe ansatz techniques.
The appearance of a $sqrt{lambda'}$ floor in the string curve suggests that strong-coupling logs are protected by the double-pole positions, which may explain why the cusp anomaly has a $sqrt{lambda}$ rather than $lambda$ prefactor.
If the same resummation works beyond one loop, the BMN expansion of higher-twist operators could be reorganized around $xi$ rather than $lambda'$, with the cusp dimension as the $xi to infty$ limit.

Original abstract

We study anomalous dimensions of (super)conformal Wilson operators at weak and strong coupling making use of the integrability symmetry on both sides of the gauge/string correspondence and elucidate the origin of their single-logarithmic behavior for long operators/strings in the limit of large Lorentz spin. On the gauge theory side, we apply the method of the Baxter Q-operator to identify different scaling regimes in the anomalous dimensions in integrable sectors of (supersymmetric) Yang-Mills theory to one-loop order and determine the values of the Lorentz spin at which the logarithmic scaling sets in. We demonstrate that the conventional semiclassical approach based on the analysis of the distribution of Bethe roots breaks down in this domain. We work out an asymptotic expression for the anomalous dimensions which is valid throughout the entire region of variation of the Lorentz spin. On the string theory side, the logarithmic scaling occurs when two most distant points of the folded spinning string approach the boundary of the AdS space. In terms of the spectral curve for the classical string sigma model, the same configuration is described by an elliptic curve with two branching points approaching values determined by the square root of the 't Hooft coupling constant. As a result, the anomalous dimensions cease to obey the BMN scaling and scale logarithmically with the Lorentz spin.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We study anomalous dimensions of (super)conformal Wilson operators at weak and strong coupling making use of the integrability symmetry on both sides of the gauge/string correspondence and elucidate the origin of their single-logarithmic behavior for long operators/strings in the limit of large Lorentz spin. On the gauge theory side, we apply the method of the Baxter Q-operator to identify different scaling regimes in the anomalous dimensions in integrable sectors of (supersymmetric) Yang-Mills theory to one-loop order and determine the values of the Lorentz spin at which the logarithmic scaling sets in. We demonstrate that the conventional semiclassical approach based on the analysis of the distribution of Bethe roots breaks down in this domain. We work out an asymptotic expression for the anomalous dimensions which is valid throughout the entire region of variation of the Lorentz spin. On the string theory side, the logarithmic scaling occurs when two most distant points of the folded spinning string approach the boundary of the AdS space. In terms of the spectral curve for the classical string sigma model, the same configuration is described by an elliptic curve with two branching points approaching values determined by the square root of the 't Hooft coupling constant. As a result, the anomalous dimensions cease to obey the BMN scaling and scale logarithmically with the Lorentz spin.

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 study anomalous dimensions of (super)conformal Wilson operators at weak and strong coupling making use of the integrability symmetry on both sides of the gauge/string correspondence and elucidate the origin of their single-logarithmic behavior for long operators/strings in the limit of large Lorentz spin.	contribution 1.00	0.94	0.58	asserted 1.00
s1	On the gauge theory side, we apply the method of the Baxter Q-operator to identify different scaling regimes in the anomalous dimensions in integrable sectors of (supersymmetric) Yang-Mills theory to one-loop order and determine the values of the Lorentz spin at which the logarithmic scaling sets in.	method 0.81	0.96	0.09	asserted 1.00
s2	We demonstrate that the conventional semiclassical approach based on the analysis of the distribution of Bethe roots breaks down in this domain.	result 0.94	0.66	0.08	asserted 1.00
s3	We work out an asymptotic expression for the anomalous dimensions which is valid throughout the entire region of variation of the Lorentz spin.	result 0.53	0.96	0.19	asserted 1.00
s4	On the string theory side, the logarithmic scaling occurs when two most distant points of the folded spinning string approach the boundary of the AdS space.	result 0.94	0.49	0.14	asserted 1.00
s5	In terms of the spectral curve for the classical string sigma model, the same configuration is described by an elliptic curve with two branching points approaching values determined by the square root of the 't Hooft coupling constant.	result 0.55	0.68	0.09	asserted 1.00
s6	As a result, the anomalous dimensions cease to obey the BMN scaling and scale logarithmically with the Lorentz spin.	result 0.82	0.77	0.36	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
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levels run

P abstract

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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

Figures

**Figure 1.** The definition of the α−cycles and γ−contours on the Riemann surface ΓL. The dashed lines represent the part of the path on the lower sheet of the surface. are distributed on L − 1 intervals (2.20). Denoting by ℓj the number of Bethe roots on the j th interval, one has 1 2πi I αj dx S′ (x) = ηℓj = ℓj N + Ls , (j = 1, . . . , L − 1). (2.29) The sum of all α−cycles is homologous to zero and, as a consequence, the sum of all α−periods is given by the residue of S ′ (x) at infinity. Together with (2.27) this leads to N = ℓ1 + . . .+ ℓL−1. Solving the quantization conditions (2.29

**Figure 2.** Symmetric two-cut configuration (a) resulting in the BMN s

**Figure 3.** The transfer matrix cos p(x) = τ0(x)/2 as a function of 1/x for the symmetric two-cut solution for β = 3/4 and m = 1. The cuts [−a, −b]∪[b, a] with a = 0.45 and b = 0.15 correspond to cos2 p(x) > 1. The double points x2j are denoted by crosses, cos p(x2j ) = ±1 and 1/x2 2j > 1/b2 . The “large” and “small” roots of the transfer matrix, cos p(δn) = 0, are shown by full and light blobs, respectively. 3.2 Breakdown of semiclassical expansion For m = 1 the expressions (3.7), (3.12) and (3.14) describe the dependence of the energy on the total spin N + Ls along the minimal energy…

**Figure 4.** Left panel: “small” roots of the transfer matrix

Figures reproduced from arXiv:hep-th/0601112 by the authors.

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hep-th 2006-03 conditional novelty 8.0 of 10

The authors derive an integral equation that predicts the all-loop cusp anomalous dimension f(g) for N=4 SYM, including a new four-loop term, under explicit assumptions about the Bethe ansatz and BMN scaling.
The CFT Distance Conjecture and Tensionless String Limits in $\mathcal N=2$ Quiver Gauge Theories
hep-th 2026-01 unverdicted novelty 7.0 of 10

In N=2 SU quiver theories the large-N Hagedorn temperature depends only on quiver length for linear cases and equals that of N=4 SYM for holographic quivers, with a universal lower bound of 1/sqrt(2) on the exponentia...
Quantum spectral curve as a tool for a perturbative quantum field theory
hep-th 2014-11 accept novelty 7.0 of 10

A recursive algorithm solves the quantum spectral curve perturbatively for any sl(2) operator in planar N=4 SYM, yielding ten-loop conformal dimensions and a proof that they are multiple zeta-values.

Show 17 more citing papers

Space-time S-matrix and Flux tube S-matrix II. Extracting and Matching Data
hep-th 2013-06 accept novelty 7.0 of 10

Finite-coupling expressions for scalar and gluon pentagon transitions in planar N=4 SYM are bootstrapped from flux-tube S-matrix axioms and matched against hexagon and heptagon amplitude data up to three loops.
Quantization of Integrable Systems and a 2d/4d Duality
hep-th 2011-03 conditional novelty 7.0 of 10

The chiral ring of 4d N=2 SQCD in a one-plane Omega background is conjectured to equal that of a 2d U(N) gauged linear sigma model, with the equality verified to two instanton orders.
An Operator Product Expansion for Polygonal null Wilson Loops
hep-th 2010-06 conditional novelty 7.0 of 10

Polygonal null Wilson loops admit an OPE in the collinear limit whose subleading terms are governed by flux-tube excitations with a computable dispersion relation.
A Generalized Scaling Function for AdS/CFT
hep-th 2007-12 accept novelty 7.0 of 10

The authors derive a non-perturbative integral equation for a generalized scaling function f(g,j) governing twist operators in the sl(2) sector when spin and twist both grow large with their ratio to log spin fixed.
Transcendentality and Crossing
hep-th 2006-10 accept novelty 7.0 of 10

A specific dressing phase for the planar N=4 SYM S-matrix is derived from crossing symmetry and transcendentality, and its leading four-loop coefficient agrees with an independent computation.
The Resurgence of the Cusp Anomalous Dimension
hep-th 2015-06 conditional novelty 6.0 of 10

A transseries built from the BES equation's analyticity conditions is shown to be resurgent, yielding an unambiguous resummation of the cusp anomalous dimension that interpolates from strong to weak coupling.
Higgsing the stringy higher spin symmetry
hep-th 2015-06 conditional novelty 6.0 of 10

Turning on string tension in the symmetric orbifold of T4 breaks the higher-spin symmetry, and the surviving generators organize into Regge trajectories whose diagonal anomalous dimensions grow logarithmically with spin.
Nonsinglet pentagons and NMHV amplitudes
hep-th 2014-07 conditional novelty 6.0 of 10

Nonsinglet pentagon form factors are proposed and solved to all orders in the 't Hooft coupling, then used to reproduce twist-two contributions to NMHV hexagon amplitudes against perturbative data.
Space-time S-matrix and Flux-tube S-matrix III. The two-particle contributions
hep-th 2014-02 conditional novelty 6.0 of 10

The paper predicts all twist-2 (two-particle) corrections to the near-collinear expansion of the six-gluon MHV amplitude in planar N=4 super-Yang-Mills at finite coupling.
Quantum mechanics of null polygonal Wilson loops
hep-th 2014-01 accept novelty 6.0 of 10

The authors solve the open spin chain underlying multiparticle GKP excitations in null polygonal Wilson loops and prove the conjectured factorized form of multiparticle hexagon transitions.
Quantum folded string and integrability: from finite size effects to Konishi dimension
hep-th 2011-02 conditional novelty 6.0 of 10

The Konishi operator's anomalous dimension at strong coupling is derived analytically as 2λ^{1/4} - 4 + 2/λ^{1/4}, confirming earlier numerical Y-system predictions.
Exciting the GKP string at any coupling
hep-th 2010-10 conditional novelty 6.0 of 10

The spectrum of excitations around the long GKP string is expressed as all-loop dispersion relations parameterized by the solution to the BES equation, with the SO(6) symmetry restoration mechanism identified.
Twist-three at five loops, Bethe Ansatz and wrapping
hep-th 2009-01 conditional novelty 6.0 of 10

The authors determine the five-loop anomalous dimension of twist-three operators in N=4 SYM by fitting Bethe Ansatz data with reciprocity constraints and adding Lüscher-type wrapping corrections, finding agreement wit...
Comments on operators with large spin
hep-th 2007-08 accept novelty 6.0 of 10

The cusp anomalous dimension of N=4 super Yang-Mills equals the energy density of a flux configuration on AdS3 x S1, and in a large-spin, large-charge limit the string reduces to a solvable O(6) sigma model.
A Test of the AdS/CFT Correspondence Using High-Spin Operators
hep-th 2006-11 conditional novelty 6.0 of 10

Numerically solving the BES integral equation reproduces the string-theory prediction f(g) = 4g - 3 ln2/pi at strong coupling to six digits and yields a new estimate for the 1/g coefficient.
Bootstrapping Null Polygon Wilson Loops
hep-th 2010-10 conditional novelty 5.0 of 10

Two-loop polygonal Wilson loops in 1+1 dimensional kinematics are rederived from the one-loop result via an operator product expansion, matching prior computations and yielding the flux-tube excitation spectrum.
Quantum integrability and functional equations
hep-th 2010-03 conditional novelty 4.0 of 10

The thesis recasts integral Bethe Ansatz equations as Riemann-Hilbert and functional equations, then uses them to check the AdS/CFT asymptotic Bethe Ansatz at strong coupling and to solve its crossing equations.

How the literature uses it

46 arXiv papers cite this in 140 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 · 108 sentences without a confident label

Builds on it 1 paper

The solution in regime I has been found in [ref] and adjusted for the fishnet in [ref] .

Light-cone limits of large rectangular fishnets · Solution in regime I · 0.94

Uses its method or tool 4 papers

We will make extensive use of the long-range Baxter equation [ref] , whose content is equivalent to the Asymptotic Bethe Ansatz equations of Ref. [ref] , and apply the strategy developed in Ref. [ref] for construction of corresponding large-spin solutions.

Luescher formula for GKP string · ABA solution · 1.00
However, as we will see in the next section[The discussion will be based in particular on sections 2.2 and 3.3 of [ref] .], it is possible to determine the spectrum in terms of the roots of the monodromy $t_J (u)$, which are more convenient objects to study, since their number remains constant and equal to $J$.

Spiky strings and the AdS/CFT correspondence · The $𝔰𝔩(2)$ integrable spin chain · 0.88
This can be nicely implemented using the Baxter polynomial approach as proposed in the $AdS/CFT$ context in [ref] .

Bootstrapping Null Polygon Wilson Loops · Scalars $Z$ at one loop · 0.99

Uses its data or values 1 paper

More precisely, we find ([ref]) with $γ_cusp=4g^2+𝒪(g^4)$ and [ref] ϵ_Z(θ)= 2g^2 (ψ(1/2+iθ)+ψ(1/2-iθ)-2 ψ(1) ) + 𝒪(g^4) , eZ The momentum is read off from the effective Bethe equation for the hole excitations.

Bootstrapping Null Polygon Wilson Loops · Scalars $Z$ at one loop · 0.88

Compares with it 5 papers

[Higher-loop contributions would not affect this behavior at any given order, though they will come with a logarithmic enhancement.] After expansion at large values of the spin $S$, by using the one-loop relation $η^2 = S(S+1)+12$, this expression is find to be in complete agreement with the earlier results of Refs. [ref] for the leading and $O(1/η^2)$ corrections, respectively.

Luescher formula for GKP string · Non-polynomial solutions and large-spin expansion · 0.99
Now, equation [ref] turns out to be the same as the finite-gap equation [ref] for a classical string in $AdS_3×S^1$, constructed explicitly by Frolov and Tseytlin [ref] .

Fishnet four-point integrals: integrable representations and thermodynamic limits · Mapping with spinning string · 0.93
The saddle-point equation in the scaling limit turns out to be the same as the finite-gap equation for a classical folded string rotating in AdS$_3×S^1$ solved in [ref] .

Light-cone limits of large rectangular fishnets · Introduction · 0.84

Cites it as support 1 paper

It was shown [ref] that in the limit suggested in [ref] the string worldsheet action can be identified with $O(6)$ sigma model and the energy of the ground state in $O(6)$ model is proportional to the length of the string multiplied by the $Γ_cusp(α)$.

Amplitudes in the N=4 SYM from Quantum Geometry of the Momentum Space · $ħ^-1 = Γ_cusp(α)$ ? · 0.97

Cites it as background 15 papers

This sector has been the subject of several investigations [ref] , expecially in the large spin limit $S→∞$ whose behaviour is described, on the dual string side, by the so-called GKP string solution [ref] .

Supersymmetric 4d gauge theories and Integrability · The spectral problem · 0.94
This is effectively a semiclassical limit where $1/S$ plays the role of Planck's constant $ħ$ [ref] .

Spiky Strings and Spin Chains · The gauge theory spin chain · 0.83
Significant simplification occurs in the limit [ref] when the Lorentz spin grows exponentially with the twist, $L∼lnN$ with $N→∞$.

Nonperturbative scales in AdS/CFT · Introduction · 0.82

Discussion

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