Classical/quantum integrability in AdS/CFT

By V.A.Kazakov, A.Marshakov, J.A.Minahan, K.Zarembo. 2004

473 citations recorded by the source corpus.

We discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar ﬁelds in N = 4 super Yang-Mills and the energy of their dual semiclassical string states in AdS 5 × S 5 . The anomalous dimensions can be computed using a set of Bethe equations, which for “long” operators reduces to a Riemann-Hilbert problem. We develop a uniﬁed approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU (2) sector and present a general solution, governed by complex curves endowed with meromorphic diﬀerentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions.

Subfields: Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/5f44fe602693259e7d7ece1ea443863e20233996

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Citations in the corpus

Semiclassical Strings and AdS/CFT (2004, methodology): “At the same time, the Bethe ansatz approach [41, 42, 43, 47] is determining the exact eigenvalues of the dilatation operator.”
Tailoring three-point functions and integrability (2010, methodology): “A systematic description of the spectrum in the classical limit is achieved through the finite gap method of KMMZ [27].”
Bethe ansatz for quantum strings (2004, methodology): “Furthermore, it proved possible to exploit the classical integrability of the string sigma model and to derive a Bethe equation describing its semi-classical solutions for the simplest case of strings moving in R × S(3) in full generality [14].”
Four loop twist two, BFKL, wrapping and strings (2008, background): “The Bethe Ansatz Equations have been derived [12, 13, 14, 15, 16, 17, 18, 19] giving the spectrum of states with large quantum numbers.”
Long-range psu(2,2|4) Bethe Ansatze for gauge theory and strings (2005, methodology): “Our spectral parameter x resembles the one of string sigma models, see [9, 29].”
Holographic three-point functions of semiclassical states (2010, unknown): “The periodic solutions of the O(6) sigma-model can be constructed in full generality with the help of integrability methods [25].”
Semiclassical relativistic strings in S 5 and long coherent operators in = 4 SYM theory (2004, methodology): “10) and the corresponding set of Bethe ansatz eigenstates, generalizing the one found in the SU(2) case in [17].”
Five loop Konishi from AdS/CFT (2009, background): “Historically, the developments which led to it concentrated on the Bethe Ansatz part mostly on the gauge theory side [11, 12, 13, 14, 15] as well as for classical string solutions in AdS5 × S5 [16, 17].”
On string S-matrix, bound states and TBA (2007, background): “The corresponding Bethe equations are based on the knowledge of the S-matrix which describes the scattering of world-sheet excitations of the gauge-fixed string sigma-model, or alternatively, the excitations of a certain spin chain in the dual gauge theory [4]-[9].”
Small deformations of supersymmetric Wilson loops and open spin-chains (2006, background): “For closed strings a general classification of classical solutions carrying two charges was provided in [73] which completely agrees with possible solutions of the SU(2) Bethe ansatz in the thermodynamic limit.”
Finite-size effects from giant magnons (2006, methodology): “8 Let us note that the non-periodicity of φ (up to a winding number) is the reason why the giant magnon solution cannot be found by using the KMMZ equations [22].”
Planar N = 4 gauge theory and the Hubbard model (2005, background): “The dual string theory is classically integrable [44], which leads to a complete solution of classical string motions [45] in terms of an algebraic curve [46].”
The all-loop integrable spin-chain for strings on AdS3 × S3 × T4: the massive sector (2013, background): “In this limit the Bethe roots condense on the cuts that appear in the finite gap equations [35].”
Extended Y-system for the AdS5/CFT4 correspondence (2010, background): “In a parallel way, integrability in superstring theory was discovered at classical level [9], extended to semiclassical and partially to quantum level in [10].”
The AdS 5 ×S 5 superstring worldsheet S matrix and crossing symmetry (2006, background): “One of the most fascinating discoveries in recent years was the unravelling of integrable structures in planar N = 4 SYM [1, 2, 3, 4] and in superstring theory on AdS5 × S(5) [5, 6, 7, 8, 9, 10, 11].”
Large spin limit of AdS5×S5 string theory and low energy expansion of ferromagnetic spin chains (2004, methodology): “This agreement was extended to the “two-loop” level using integrable model/Bethe ansatz techniques [20, 17].”
T-systems and Y-systems in integrable systems (2010, background): “We do not intend to cover them here but refer to the literatures that will be cited in the next subsection and [179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192] for example and references therein.”