Review of AdS/CFT Integrability, Chapter I.2: The spectrum from perturbative gauge theory
C. Sieg
备注：
24 pages, see also overview article arXiv:1012.3982, v2: references to other chapters updated, v3: formulations improved, reference added, final version
We review the constructions and tests of the dilatation operator and of the spectrum of composite operators in the flavour SU(2) subsector of N=4 SYM in the planar limit by explicit Feynman graph calculations with emphasis on analyses beyond one loop. From four loops on, the dilatation operator determines the spectrum only in the asymptotic regime, i.e. to a loop order which is strictly smaller than the number of elementary fields of the composite operators. We review also the calculations which take a first step beyond this limitation by including the leading wrapping corrections.
The D3-D5 probe-brane system is holographically dual to a defect CFT which is known to be integrable. The evidence comes mainly from the study of correlation functions at weak coupling. In the present work we shed light on the emergence of integrability on the string theory side. We do so by constructing the double row transfer matrix which is conserved when the appropriate boundary conditions are imposed. The corresponding reflection matrix turns out to be dynamical and depends both on the spectral parameter and the string embedding coordinates.
We establish a framework for calculating the Hagedorn temperature of AdS5/CFT4 via integrability. Concretely, we derive the thermodynamic Bethe ansatz equations that yield the Hagedorn temperature of planar N=4 super Yang-Mills theory at any value of the 't Hooft coupling. We solve these equations perturbatively at weak coupling via the associated Y-system, confirming the known results at tree-level and one-loop order as well as deriving the previously unknown two-loop Hagedorn temperature. Finally, we comment on solving the equations at finite coupling.
In this letter we study how the exact non-perturbative integrability methods in 4D N=4 Super-Yang-Mills can work efficiently together with the numerical conformal bootstrap techniques to go beyond the spectral observables and access previously unreachable quantities such as correlation functions at finite coupling. In the setup of 1D defect CFT living on a Maldacena-Wilson line, we managed to compute with good precision a non-supersymmetric structure constant for a wide range of the `t Hooft coupling. Our result is particularly precise at strong coupling and matches well with the recent analyt…
We discuss the properties of scattering amplitudes in a conformal bi-scalar fishnet theory that previously appeared in the study of integrable deformations of $\mathcal N=4$ SYM. In distinction with the latter theory, the scattering amplitudes in the conformal fishnet theory do not suffer from infrared divergences and possess enhanced symmetries. We demonstrate that for the simplest four-particle scattering the single- and double-trace partial amplitudes can be computed for arbitrary coupling in the leading color approximation. We also review the invariance of single-trace components of multi-…
We study $T\bar T$ deformations of 2d CFTs with periodic boundary conditions. We relate these systems to string models on $\mathbb{R}\times {S}^1\times{\cal M}$, where $\cal M$ is the target space of a 2d CFT. The string model in the light cone gauge is identified with the corresponding 2d CFT and in the static gauge it reproduces its $T\bar T$ deformed system. This relates the deformed system and the initial one by a worldsheet coordinate transformation, which becomes a time dependent canonical map in the Hamiltonian treatment. The deformed Hamiltonian defines the string energy and we express…
Integrability, spin-chains and the AdS3/CFT2 correspondence
在arXiv：0912.1723的基础上，本文使用可积性技术研究了AdS 3/CFT 2对应关系。我们给出了AdS_3 x S^3 x S^3 x S^1上弦的全循环Bethe Answer（BA），对称性为D（2，1;alpha）^2，对所有alpha值都有效。这种构造依赖于一种新颖的、依赖于α的Zhukovsky映射的概括。我们研究了AdS_3 x S^3 x T^4上弦的这个BA和全圈BA的弱耦合极限。我们构造了对应于这些弱耦合BA的可积短程自旋链和哈密顿量。自旋链分别是交替的和均匀的。交替自旋链可以被看作是对未知的关于AdS_3 x S^3 x S^3 x S^1的弦论的CFT 2对偶的第一个提示。我们表明，在α到1的限制，D（2，1; α）模型的可积结构是非奇异的，并保持跟踪不仅有质量，但也无质量的模式。这提供了一种将无质量模式纳入AdS 3/CFT 2对应的可积性机制的方法。
We compute a set of correlation functions of operator insertions on the 1/8 BPS Wilson loop in $\mathcal{N}=4$ SYM by employing supersymmetric localization, OPE and the Gram-Schmidt orthogonalization. These correlators exhibit a simple determinant structure, are position-independent and form a topological subsector, but depend nontrivially on the 't Hooft coupling and the rank of the gauge group. When applied to the 1/2 BPS circular (or straight) Wilson loop, our results provide an infinite family of exact defect CFT data, including the structure constants of protected defect primaries of arbi…
Protected string spectrum in AdS3/CFT2 from worldsheet integrability
我们推导出在弦张力和三种形式的通量的任意值下，具有16个超荷的AdS 3/CFT 2对偶对的受保护闭弦谱。这些紧接着可积世界片理论的谱的全圈贝特方程。此外，代表底层的可积系统作为自旋链，我们发现，它们的动态涉及长度变化的相互作用和保护状态对应于上面的Berenstein-Maldacena-Nastase真空的无隙激发。在AdS 3 x S3 x T4的情况下，这些算子的简并度与对偶CFT 2和超重力谱的简并度精确匹配。另一方面，我们发现，对于AdS 3 x S3 x S3 x S1，受保护的状态比以前的超引力计算所建议的要少。特别地，受保护状态对于两个三球具有相同的su（2）电荷。
We construct the all loop effective action representing, for small couplings, simultaneously self and mutually interacting current algebra CFTs realized by WZW models. This non-trivially generalizes our previous works where such interactions were, at the linear level, not simultaneously present. For the two coupling case we prove integrability and calculate the coupled RG flow equations. We also consider non-Abelian T-duality type limits. Our models provide concrete realisations of integrable flows between exact CFTs and exhibit several new features which we discuss in detail.
The present thesis is based in the material previously published in [arXiv:1501.04252, arXiv:1511.08759, arXiv:1606.07870] and it is organized in the following way. The second chapter is shown as the isometric group of the space time AdS is exactly (isomorphic) match to the conformal group of the dual CFT theory. We present the precise formulation of the duality AdS/CFT and we describe the behavior of a massive scalar field in AdS space-time. In chapter 3 we present solutions of black holes widely known in the literature, which we will use in some later calculations to gain intuition, we expla…
Integrability and the spectrum of two-dimensional fishnet CFT
King’s College London（伦敦国王学院）；Uppsala University（乌普萨拉大学）；Centre for Mathematical Science, City St George’s, University of London（伦敦大学城市圣乔治学院数学科学中心）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器研究所）
We show that the CFT with symmetry group $G_{k_1}\times G_{k_2}\times \cdots \times G_{k_n}$ consisting of WZW models based on the same group $G$, but at arbitrary integer levels, admits an integrable deformation depending on $2(n-1)$ continuous parameters. We derive the all-loop effective action of the deformed theory and prove integrability. We also calculate the exact in the deformation parameters RG flow equations which can be put in a particularly simple compact form. This allows a full determination and classification of the fixed points of the RG flow, in particular those that are IR st…
We consider (warped) AdS string backgrounds which allow for a GKP spinning string/null cusp solution. Integrability implies that the worldsheet S-matrix should factorize, which in turn constrains the form of the warp factor as a function of the coordinates of the internal space. This constraint is argued to rule out integrability for all supersymmetric AdS7 and AdS6 backgrounds as well as AdS5 Gaiotto-Maldacena backgrounds and a few highly supersymmetric AdS4 and AdS3 backgrounds.
强耦合下规范理论的描述是理论物理中长期存在的问题之一。强耦合规范理论和弦论之间的关系的想法是由特胡夫特、威尔逊和波利亚科夫开创的。10年前，马尔达西纳通过证明四维共形不变理论（最大超对称杨-米尔斯理论）与AdS 5 x S5背景下的弦论的精确等价性，明确了这一关系。在AdS背景下，共形不变理论和弦论之间的对应关系的其他例子最近也被发现了。对应的两侧的比较需要使用非微扰方法。规范理论和弦论中可积结构的发现导致了这样的猜想，即这两个理论对于任何耦合常数都是可积的，并且它们具有相同的非微扰定义的可积结构。在过去的八年里，在确定这个可解模型和明确解决计算理论的共形维数谱的问题方面取得了显着的进展。这些进展来自于对具有可积自旋链的伸缩算符的识别以及对弦西格玛模型的研究。在这篇论文中，我介绍了可积性的概念的发展框架中的AdS/CFT对应和使用这种方法得到的主要结果。
We give an introduction to the Quantum Spectral Curve in AdS/CFT. This is an integrability-based framework which provides the exact spectrum of planar N = 4 super Yang-Mills theory (and of the dual string model) in terms of a solution of a Riemann-Hilbert problem for a finite set of functions. We review the underlying QQ relations starting from simple spin chain examples, and describe the special features arising for AdS/CFT. We also discuss the recently found links between the Quantum Spectral Curve and the computation of correlation functions. To appear in a special issue of J Phys A based o…
