Patrick Dorey, Clare Dunning, Ferdinando Gliozzi, Roberto Tateo
Within the framework of the ODE/IM correspondence, we show that the minimal conformal field theories with c<1 emerge naturally from the monodromy properties of certain families of ordinary differential equations.
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We study the linear differential system associated with the supersymmetric affine Toda field equations for affine Lie superalgebras, which has a purely odd simple root system. For an affine Lie algebra, the linear problem modified by conformal transformation leads to an ordinary differential equation (ODE) that provides the functional relations in the integrable models. This is known as the ODE/IM correspondence. For the affine Lie superalgebras, the linear equations modified by a superconformal transformation are shown to reduce to a couple of ODEs for each bosonic subalgebra. In particular, …
An integrable system is introduced, which is a generalization of the $\mathfrak{sl}(2)$ quantum affine Gaudin model. Among other things, the Hamiltonians are constructed and their spectrum is calculated within the ODE/IQFT approach. The model fits within the framework of Yang-Baxter integrability. This opens a way for the systematic quantization of a large class of integrable non-linear sigma models. There may also be some interest in terms of Condensed Matter applications, as the theory can be thought of as a multiparametric generalization of the Kondo model.
A connection between integrable quantum field theory and the spectral theory of ordinary differential equations is reviewed, with particular emphasis being given to its relevance to certain problems in PT-symmetric quantum mechanics.
We study the two-dimensional affine Toda field equations for affine Lie algebra $\hat{\mathfrak{g}}$ modified by a conformal transformation and the associated linear equations. In the conformal limit, the associated linear problem reduces to a (pseudo-)differential equation. For classical affine Lie algebra $\hat{\mathfrak{g}}$, we obtain a (pseudo-)differential equation corresponding to the Bethe equations for the Langlands dual of the Lie algebra $\mathfrak{g}$, which were found by Dorey et al. in study of the ODE/IM correspondence.
The massive ODE/IM correspondence is a relation between the linear problem associated with modified affine Toda field equations and two-dimensional massive integrable models. We study the massive ODE/IM correspondence for the $A_r^{(1)}$-type modified affine Toda field equations. Based on the $ψ$-system satisfied by the solutions of the linear problem, we derive the Bethe ansatz equations and determine the asymptotic behavior of the Q-functions for large value of the spectral parameter. We derive the non-linear integral equations for the Q-functions from the Bethe ansatz equations. We compute …
Anisotropic Kondo line defect and ODE/IM correspondence
该研究针对手征SU(2) WZW模型中的各向异性近藤线缺陷，借助三角情形四维陈-西蒙斯理论提出对应ODE/IM对应关系，经紫外微扰与红外精确WKB分析验证，并推导了这类线缺陷的红外动力学。
We study the massive ODE/IM correspondence for modified $B_2^{(1)}$ affine Toda field equation. Based on the $ψ$-system for the solutions of the associated linear problem, we obtain the Bethe ansatz equations. We also discuss the T-Q relations, the T-system and the Y-system, which are shown to be related to those of the $A_3/{\bf Z}_2$ integrable system. We consider the case that the solution of the linear problem has a monodromy around the origin, which imposes nontrivial boundary conditions for the T-/Y-system. The high-temperature limit of the T- and Y-system and their monodromy dependence …
Integrals of motion in conformal field theory with W-symmetry and the ODE/IM correspondence
本文通过计算 Pochhammer 围道上的 WKB 周期积分并与圆柱上 $WA_r$/$WD_r$ 代数真空本征值比较至六阶，将 ODE/IM 对应推广到高阶常微分方程。
We study the quantum spectral curve of the Argyres-Douglas theories in the Nekrasov-Sahashvili limit of the Omega-background. Using the ODE/IM correspondence we investigate the quantum integrable model corresponding to the quantum spectral curve. We show that the models for the $A_{2N}$-type theories are non-unitary coset models $(A_1)_1\times (A_1)_{L}/(A_1)_{L+1}$ at the fractional level $L=\frac{2}{2N+1}-2$, which appear in the study of the 4d/2d correspondence of ${\cal N}=2$ superconformal field theories. Based on the WKB analysis, we clarify the relation between the Y-functions and the q…
We review a surprising correspondence between certain two-dimensional integrable models and the spectral theory of ordinary differential equations. Particular emphasis is given to the relevance of this correspondence to certain problems in PT-symmetric quantum mechanics.
Deforming the ODE/IM correspondence with $\mathrm{T}\bar{\mathrm{T}}$
本文证明在对应两侧引入 $\mathrm{T}\bar{\mathrm{T}}$ 扰动后，ODE/IM 对应依然成立，sinh-Gordon 模型的形变 Lax 对导出了与量子谱流相同的 Burgers 型方程，且该结论可推广至已知 ODE/IM 实例。
The Fateev model is somewhat special among two-dimensional quantum field theories. For different values of the parameters,it can be reduced to a variety of integrable systems. An incomplete list of the reductions includes O(3) and O(4) non-linear sigma models and their continuous deformations (2D and 3D sausages, anisotropic principal chiral field), the Bukhvostov-Lipatov model, the N=2 supersymmetric sine-Gordon model, as well as the integrable perturbed SU_2(n)\otimes SU_2(p-2)/SU_2(n+p-2) coset CFT. The model possesses a mysterious symmetry structure of the exceptional quantum superalgebras…
This article reviews a recently-discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.
Integrals of motion in $WE_6$ CFT and the ODE/IM correspondence
本文研究了与仿射李代数 $E_6^{(1)}$ 相关的常微分方程的 ODE/IM 对应关系，通过对角化方法计算了 WKB 系数沿 Pochhammer 轨道的周期积分，并展示了二维共形场论中与 $E_6^{(1)}$ 关联的 W 对称性守恒积分的特征值与周期积分在六阶内一致。
We study numerically the ODE/IM correspondence for untwisted affine Lie algebras associated with simple Lie algebras including exceptional type. We consider the linear problem obtained from the massless limit of that of the modified affine Toda field equation. We found that the Q-functions in integrable models are expressed as the inner product of the solution of the dual linear problem and the subdominant solution of the linear problem. Using Cheng's algorithm to obtain the solution of the linear problem, we can determine efficiently the zeros of the Q-function, which is known to provide the …
The ODE/IM Correspondence between $C(2)^{(2)}$-type Linear Problems and 2d $\mathcal{N}=1$ SCFT
研究$C(2)^{(2)}$型线性问题与二维$\mathcal{N}=1$ SCFT之间的ODE/IM对应关系，通过边界条件和Lax算子对角化得到WKB展开，验证WKB周期与局部守恒量的一致性。
We study the linear problem associated with modified affine Toda field equation for the Langlands dual $\hat{\mathfrak{g}}^\vee$, where $\hat{\mathfrak{g}}$ is an untwisted affine Lie algebra. The connection coefficients for the asymptotic solutions of the linear problem are found to correspond to the $Q$-functions for $\mathfrak{g}$-type quantum integrable models. The $ψ$-system for the solutions associated with the fundamental representations of $\mathfrak{g}$ leads to Bethe ansatz equations associated with the affine Lie algebra $\hat{\mathfrak{g}}$. We also study the $A^{(2)}_{2r}$ affine …
This review describes a link between Lax operators, embedded surfaces and Thermodynamic Bethe Ansatz equations for integrable quantum field theories. This surprising connection between classical and quantum models is undoubtedly one of the most striking discoveries that emerged from the off-critical generalisation of the ODE/IM correspondence, which initially involved only conformal invariant quantum field theories. We will mainly focus of the KdV and sinh-Gordon models. However, various aspects of other interesting systems, such as affine Toda field theories and non-linear sigma models, will …
On W-algebras and ODE/IM correspondence
本文发展了基于形式WKB展开的系统化算法，通过Bethe根计算具有Virasoro和W_N对称性的二维共形场论中局域积分运动的本征值，并揭示了镜像曲线的几何结构及其与谱数据的联系，在Argyres-Douglas极小模型中验证了方法的正确性。
A primer of the complex WKB method, with application to the ODE/IM correspondence
本讲义以ODE/IM对应中的非谐振子为例，介绍复WKB方法的核心对象，并推导证明该类振子谱在两种渐近情形下的行为。
ODE/IM Correspondence at the Free-Fermion Point. Laguerre Wronskians, Shifted Symmetric Functions, and Quantum KdV
本文在Virasoro中心荷c=-2（自由费米子点）处证明了ODE/IM对应的完备性，通过Crum-Darboux变换和拉盖尔朗斯基行列式将Bethe方程的解与量子谐振子的有理扩展谱对应，并在IM侧显式对角化量子KdV的前三个哈密顿算子，其特征值为移位对称函数。
The first part of this work consists of a study of the ODE/IM correspondence for simply-laced affine Toda field theories. It is a first step towards a full generalisation of the results of Lukyanov and Zamolodchikov on $\hat{\mathfrak a}_1$ to a general affine Lie-Kač-Moody algebra $\hat{\mathfrak g}$. In order to achieve our goal, we investigate the structure of evaluation representations of $\hat{\mathfrak g}$ and show how their tensor products are related by what we call projected isomorphisms. These isomorphisms are used to construct a set of quadratic functional relations, called $ψ$-syst…
Integrable Structure of Higher Spin CFT and the ODE/IM Correspondence
本文通过 ODE/IM 对应和 Zhu 递推关系，计算了 ${\mathcal W}_3$ 代数高自旋共形场论中量子 Boussinesq 荷的本征值与热关联子，推导出守恒荷流密度并证明其热期望值为作用于特征标的准模微分算子。
Exact Sum Rules and Zeta Generating Formulas from the ODE/IM correspondence
本文基于ODE/IM对应关系，通过融合关系和谱zeta函数，推导出PT对称势和厄米特势的精确求和公式和zeta生成公式，揭示了代数信息损失现象，并为无质量ABS猜想提供了谱理论的证明方法。
Generalised Gibbs Ensembles from the ODE/IM Correspondence: the Lee--Yang Model
本文提出一种系统方案构建广义吉布斯系综的ODE/IM对应物，将其应用于李-杨共形场论手征部分，推导自旋5系综，验证其满足李-杨Y系统，结果与共形场论计算匹配，方案可推广至其他可积模型。
We study the ODE/IM correspondence for ODE associated to $\hat{\mathfrak g}$-valued connections, for a simply-laced Lie algebra $\mathfrak g$. We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called $Ψ$-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.
We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $Ψ$-system among subdominant solutions. We then use the $Ψ$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explici…
Free-Field Construction of Heterotic String Compactified on Calabi-Yau Orbifolds via Correspondence with $\mathcal{N}{=}2$ SCFT Minimal Models
建立了四维异质弦在 Berglund-Hübsch 型 Calabi-Yau 流形及其轨形上紧致化的 Calabi-Yau 扇区的自由场构造与最小模型构造之间的对应关系，并验证了模不变性。
We study the large momentum limit of the monster potentials of Bazhanov-Lukyanov-Zamolodchikov, which -- according to the ODE/IM correspondence -- should correspond to excited states of the Quantum KdV model. We prove that the poles of these potentials asymptotically condensate about the complex equilibria of the ground state potential, and we express the leading correction to such asymptotics in terms of the roots of Wronskians of Hermite polynomials. This allows us to associate to each partition of $N$ a unique monster potential with $N$ roots, of which we compute the spectrum. As a conseque…
In analogy to the chiral-linear multiplet correspondence we establish a relationship between the 3-form (or gaugino condensate) multiplet and a coupled non-minimal (0,1/2) multiplet, illustrated by a simple explicit example.
We study the AGT correspondence between four-dimensional supersymmetric gauge field theory and two-dimensional conformal field theories in the context of W_N minimal models. The origin of the AGT correspondence is in a special integrable structure which appears in the properly extended conformal theory. One of the basic manifestations of this integrability is the special orthogonal basis which arises in the extended theory. We propose modification of the AGT representation for the W_N conformal blocks in the minimal models. The necessary modification is related to the reduction of the orthogon…
We consider the quantization of the massless minimally coupled scalar field in de Sitter spacetime. The no-boundary Euclidean prescription naturally picks out the de Sitter invariant vacuum state of Kirsten and Garriga. We extend Strominger's dS/CFT correspondence to this case which allows us to interpret the massless field in terms of a Euclidean CFT. The extension is non-trivial and requires careful treatment of the zero mode. Since the graviton is massless, this work may also be considered a step towards a theory of gravity in de Sitter space.
We study the ODE/IM correspondence for all the states of the quantum Boussinesq model. We consider a particular class of third order linear ordinary differential operators and show that the generalised monodromy data of such operators provide solutions to the Bethe Ansatz equations of the Quantum Boussinesq model.
On the origin of the correspondence between classical and quantum integrable theories
本文从定义量子可积理论的泛函关系出发，经线性积分方程与 Marchenko 型方程推导经典 Lax 对，阐明 ODE/IM 对应及经典—量子可积理论联系的起源。
We present a complete derivation of absorption cross-section and Hawking radiation of minimal and fixed scalars from the Strominger-Vafa model of five-dimensional black hole, starting right from the moduli space of the D1-D5 brane system. We determine the precise coupling of this moduli space to bulk modes by using the AdS/CFT correspondence. Our methods resolve a long-standing problem regarding emission of fixed scalars. We calculate three-point correlators of operators coupling to the minimal scalars from supergravity and from SCFT, and show that both vanish. We make some observations about …
We address the problem of computing scattering amplitudes related to the correlation function of two Wilson lines and/or loops elongated along light-cone directions in strongly coupled gauge theories. Using the AdS/CFT correspondence in the classical approximation, the amplitudes are shown to be related to minimal surfaces generalizing the {\em helicoid} in various $AdS_5$ backgrounds. Infra-red divergences appearing for Wilson lines can be factorized out or can be cured by considering the IR finite case of correlation functions of two Wilson loops. In non-conformal cases related to confining …
3D bulk field theories for 2D non-unitary N=1 supersymmetric minimal models
本文提出与2D非幺正N=1超对称极小模型通过非幺正体-边界对应相关的3D N=4 rank-0超共形场论，并通过验证非平凡字典测试该提议。
We prove a tensor equivalence between full subcategories of a) graded matrix factorisations of the potential x^d-y^d and b) representations of the N=2 minimal super vertex operator algebra at central charge 3-6/d, where d is odd. The subcategories are given by a) permutation-type matrix factorisations with consecutive index sets, and b) Neveu-Schwarz-type representations. The physical motivation for this result is the Landau-Ginzburg / conformal field theory correspondence, where it amounts to the equivalence of a subset of defects on both sides of the correspondence. Our work builds on result…
Black Hole-Tower Correspondence: Species backreaction in minimal black hole limits
该研究拓展黑洞-弦对应至黑洞-塔对应，分析含弦振荡器与卡鲁扎-克莱因模式的塔，纳入引力反作用，发现杂化弦中两相变平滑、II型受阻，证实匹配在自相互作用下鲁棒。
Conformal blocks and braiding matrices for RCFTs I: Virasoro minimal models
本文提出一种系统方法，通过Shapovalov形式直接计算共形块并固定Fuchsian ODE的附加参数，从而数值与精确地求解Virasoro最小模型中的编织F矩阵，并猜想其元素平方位于有理数的分圆扩张中。
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