A. V. Litvinov
In these notes we study the duality between sigma-models and Toda QFT's. We claim that $\mathfrak{gl}(n|n)$ affine Toda field theory behaves in the strong coupling limit as $η-$deformed $\mathbb{CP}(n-1)$ sigma-model plus a free field.
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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 …
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.
In our previous papers we described the structure of trajectories of the symmetric Toda system on normal real forms of various Lie algebras and showed that it was totally determined by the Hasse diagram of the Bruhat order on the corresponding Weil group. This note deals with the simplest non-split real Lie algebra, $\mathfrak{so}(2,4)$. It turns out, that the phase diagram in this case is also closely related with the Bruhat order of the relative Weyl group, but a bit more information is necessary to describe the dimensions of the trajectory spaces.
In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements $w$, $w'$ in the Weyl group $W(\mathfrak g)$, the corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell $Y_{w'}$ iff $w\prec w'$ in the Bruhat order on $W(\mathfrak g)$. Here $\mathfrak g$ is a normal real form of a semisimple complex Lie algebra $\mathfrak g_\mathbb C$. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations…
In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric $\mathfrak{sl}_n$ Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation group. In the present paper we extend this result to the case when some eigenvalues of the Lax matrix coincide. In that case the trajectories are described in terms of the projection to a partial flag space where the induced dynamical system verifies the same properties as before: we show …
We consider the two-dimensional $\mathfrak{sl}_n$ quantum Toda field theory with an imaginary background charge. This conformal field theory has a higher spin symmetry ($W_n$ algebra), a central charge $c \leq n-1$ and a continuous spectrum. Using the conformal bootstrap, we compute structure constants involving two arbitrary scalar fields and a semi-degenerate field of Wyllard type. The solution obtained is not the analytic continuation of the usual Toda three-point function. Non-scalar primary fields and their three-point functions are also discussed. Non-scalar primary fields are classified…
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…
Unitary representations of $U_{q}(\mathfrak{sl}(2,\RR))$, the modular double, and the multiparticle q-deformed Toda chains
本文研究了量子q变形Toda链的解析理论，所用的方法结合了表示论和量子逆散射方法。关键的现象，这是在审查的作用，模对偶概念（首先发现的L.法捷耶夫）在表示理论的非紧半单量子群。给出了用双正弦函数表示的Whittaker矢量的显式表达式，并将N粒子q变形开Toda链的波函数作为Mellin-Barnes型重积分给出.对于周期链，导出了两个对偶Baxter方程。
The Yang-Baxter $σ$-model is an integrable deformation of the principal chiral model on a Lie group $G$. The deformation breaks the $G \times G$ symmetry to $U(1)^{\textrm{rank}(G)} \times G$. It is known that there exist non-local conserved charges which, together with the unbroken $U(1)^{\textrm{rank}(G)}$ local charges, form a Poisson algebra $\mathscr U_q(\mathfrak{g})$, which is the semiclassical limit of the quantum group $U_q(\mathfrak{g})$, with $\mathfrak{g}$ the Lie algebra of $G$. For a general Lie group $G$ with rank$(G)>1$, we extend the previous result by constructing local and n…
A $(q,t)$-deformation of the 2d Toda integrable hierarchy
本研究通过将对称代数提升为量子环面$\u27e8gl(1)$代数，构造出二维Toda可积层级的$(q,t)$形变，推导其双线性结构与多tau函数方程，定义普适细化tau函数并得到类$RLL$的二次方程，丰富了可积系统的形变理论。
We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra $\mathfrak{g}$ leads naturally to the appearance of the "generalized tangent bundle" $\mathbb{T}M \equiv TM \oplus T^*M$ by means of composite fields. Gauge transformations of the composite fields follow the Courant bracket, closing upon the choice of a Dirac structure $D \subset \mathbb{T}M$ (or, more generally, the choide of a "small Dirac-Rinehart sheaf" $\cal{D}$), in which the fields as well as the symmetry parameters are to take values. In these new variables, the gauge …
We elaborate on the class of deformed T-dual (DTD) models obtained by first adding a topological term to the action of a supercoset sigma model and then performing (non-abelian) T-duality on a subalgebra $\tilde{\mathfrak{g}}$ of the superisometry algebra. These models inherit the classical integrability of the parent one, and they include as special cases the so-called homogeneous Yang-Baxter sigma models as well as their non-abelian T-duals. Many properties of DTD models have simple algebraic interpretations. For example we show that their (non-abelian) T-duals---including certain deformatio…
Superconformal Selfdual Sigma-Models
一系列玻色子模型可以表示为（有时是广义的）$σ$ -模型，其运动方程来自自对偶约束。我们表明，在D=2，这是很容易扩展到超对称的情况下，在超空间的方法。特别地，我们发现满足某种自对偶约束的超共形$\mathfrak{G}/\mathfrak{H}$陪集模型的场的组态是该模型运动方程的自动解。最后，我们证明了对称空间$σ$ -模型可以看作是由自对偶方程约束的无限维$\tfG/\tfH$模型，$\tfG$是$\mathfrak{G}$的循环扩展，$\tfH$是极大子群。它确保这些模型具有隐藏的全局$\tfG$对称性以及局部$\tfH$规范对称性。
Higher Toda Mechanics and Spectral Curves
对于每一个李代数$\mathfrak{gl}_{n}$和$\widetilde {\mathfrak{gl}}_{n}$，我们构造了一族由两个整数$m_{+}$和$m_{-}$刻画的Toda链的可积推广.明确给出了Lax矩阵和运动方程，运动积分可根据Lax矩阵$L$的幂迹计算。对于$m_{+}=m_{-}$的情形，我们发现了每个广义Toda链的对称约化，并给出了约化系统初值问题的解。我们还研究了周期性$(m_{+},m_{-})$-Toda链的光谱曲线，发现对于不同的$m_{+}$和$m_{-}$对，其光谱曲线有很大的不同.最后我们还得到了$(m_{+},m_{-})$-Toda链的显式非交换推广.
Exact four dimensional string solutions and Toda-like sigma models from `null-gauged' WZNW theories
通过度量秩为2的非紧群$G$的WZNW模型的具有“null”生成元的独立左、右幂零子群，构造了一类新的具有四维目标空间特征度量的弦精确解（$-,+,+,+$）.生成器（${\rm Tr }(N_n N_m)=0$）的“空”属性意味着测量的一致性和没有$\a'$ -从测量的WZNW模型获得的半经典背景的校正。在极大非紧群（$G= SL(3), SO(2,2), SO(2,3), G_2$）的情况下，构造对应于对高斯分解中幂零“步”算子所生成的某些子群进行度量。秩2的情况下是一个特殊的例子，一个一般的建设，导致共形背景与一个类时方向。通过积分出规范场而得到的共形理论可以看作是Toda模型的西格玛模型类似物（它们的经典运动方程与Toda模型方程等价）。“零测量”的程序也适用于其他非紧群。
Auxiliary Field Sigma Models and Yang-Baxter Deformations
本研究结合Yang-Baxter、双Yang-Baxter形变与高自旋辅助场形变，构造了半单李群上主手征模型的多参数可积形变族，并通过Lax表示证明其弱经典可积性。
Integrable $σ$-models, such as the principal chiral model, ${\mathbb{Z}}_T$-coset models for $T \in {\mathbb{Z}}_{\geq 2}$ and their various integrable deformations, are examples of non-ultralocal integrable field theories described by (cyclotomic) $r/s$-systems with twist function. In this general setting, and when the Lie algebra ${\mathfrak{g}}$ underlying the $r/s$-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space $σ…
Integrable sigma model with generalized $\mathcal{F}$ structure, Yang-Baxter sigma model with generalized complex structure and multi-Yang-Baxter sigma model
本文构造带广义 $\mathcal{F}$ 结构的可积 sigma 模型，并基于 Lie 代数算子关系建立带广义复结构的 Yang-Baxter sigma 模型及含多个相容 Nijenhuis 结构的多 Yang-Baxter 模型，且给出实例。
In this paper, we propose using the nonlinear sigma model (NLSM) with the Wess-Zumino-Witten (WZW) term as a general description of deconfined quantum critical points that separate two spontaneously symmetry-breaking (SSB) phases in arbitrary dimensions. In particular, we discuss the suitable choice of the target space of the NLSM, which is in general the homogeneous space $G/K$, where $G$ is the UV symmetry and $K$ is generated by $\mathfrak{k}=\mathfrak{h}_1\cap \mathfrak{h}_2$, and $\mathfrak{h}_i$ is the Lie algebra of the unbroken symmetry in each SSB phase. With this specific target spac…
We investigate certain properties of $\mathfrak{su}(N)$-valued two-dimensional soliton surfaces associated with the integrable $\mathbb{C}P^{N-1}$ sigma models constructed by the orthogonal rank-one Hermitian projectors, which are defined on the two-dimensional Riemann sphere with finite action functional. Several new properties of the projectors mapping onto one-dimensional subspaces as well as their relations with three mutually different immersion formulas, namely, the generalized Weierstrass, Sym-Tafel and Fokas-Gel'fand have been discussed in detail. Explicit connections among these three…
This paper investigates the relations between the Toda conformal field theories, quantum group theory and the quantisation of moduli spaces of flat connections. We use the free field representation of the $\mathcal{W}$-algebras to define natural bases for spaces of conformal blocks of the Toda conformal field theory associated to the Lie algebra ${\mathfrak s}{\mathfrak l}_3$ on the three-punctured sphere with representations of generic type associated to the three punctures. The operator-valued monodromies of degenerate fields can be used to describe the quantisation of the moduli spaces of f…
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.
We study the dual description of the $η$-deformed $OSP(N|2m)$ sigma model in the asymptotically free regime ($N>2m+2$). Compared to the case of classical Lie groups, for supergroups there are inequivalent $η$-deformations corresponding to different choices of simple roots. For a class of such deformations we propose the system of screening charges depending on a continuous parameter $b$, which defines the $η$-deformed $OSP(N|2m)$ sigma model in the limit $b\rightarrow\infty$ and a certain Toda QFT as $b\rightarrow0$. In the sigma model regime we show that the leading UV asymptotic of the $η$-d…
Supersymmetric Toda Field Theories
我们提出了共形Toda和$A^{(1)}_N$仿射Toda场论的新的超对称扩展。这些新的理论是使用类似的方法，已开发的二维玻色子西格玛模型的标量势的超对称扩展。特别地，我们证明了共形Toda场论允许（1，1）-超对称扩张，而$A^{(1)}_N$仿射Toda场论允许（1，0）-超对称扩张.
On affine Toda field theories related to ${\bf D}_r$ algebras and their real Hamiltonian forms
本文研究与${\bf D}_r$型单李代数相关的二维仿射Toda场论，证明复化后限制为实哈密顿形式的过程与李代数外自同构可交换，并通过两个Dynkin图外自同构的实例加以验证。
A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The Sinh-Gordon model, the affine sl(3) Toda mo…
Integrability of Supergravity Billiards and the generalized Toda lattice equation
本文证明了超引力场方程在纯含时背景下可化为一维西格玛模型的场方程，且具有拉克斯对表示，是完全可积的。在有效的西格玛模型的情况下，是一个最大分裂的非紧陪集U/H（最大超引力或分区较低的超对称超引力），我们也能够构建一个完全明确的解析积分算法，适应的方法介绍了Kodama等人在最近的一篇论文。一般积分的性质特别具有启发性。初始数据由一对C_0，h_0表示，其中C_0在U的李代数的CSA中，而H/W中的h_0在由U的Weyl群模化的紧子群H中。在渐近早和渐近晚的时候，Lax算子总是在Cartan子代数中，并且由于等谱性质，这两个极限仅由Weyl群的某些元素的作用而不同。因此，整个宇宙的演化可以看作是一个台球散射与量子化的角度定义的外尔群。求解算法实现了从H/W到W的映射。
Integrable Sigma-models and Drinfeld-Sokolov Hierarchies
以经典李群为目标流形的西格玛模型中的局部交换荷与Drinfeld-Sokolov（广义mKdV）族中的守恒量有关.相反地，德林费尔德-索科洛夫构造可以用来推导这些模型和更广泛的西格玛模型中交换荷的存在性，包括那些目标流形是例外群或对称空间的模型。这建立了可积西格玛模型和仿射户田场论中的交换量之间的直接联系。
Full symmetric Toda system and vector fields on the group $SO_n(\R)$
本文将全对称Toda系统的首次积分与正交群上的向量场联系起来，并由此给出$\mathfrak{sl}_n(\R)$对偶空间上$B^+(\R)$-不变函数李代数的向量场表示。
A generating function of the single Hurwitz numbers of the Riemann sphere $\mathbb{CP}^1$ is a tau function of the lattice KP hierarchy. The associated Lax operator $L$ turns out to be expressed as $L = e^{\mathfrak{L}}$, where $\mathfrak{L}$ is a difference-differential operator of the form $\mathfrak{L} = \partial_s - ve^{-\partial_s}$. $\mathfrak{L}$ satisfies a set of Lax equations that form a continuum version of the Bogoyavlensky-Itoh (aka hungry Lotka-Volterra) hierarchies. Emergence of this underlying integrable structure is further explained in the language of generalized string equat…
We propose new strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten's cigar model described by the $\mathfrak{sl}(2)/\mathfrak{u}(1)$ coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from $\mathfrak{sl}(2)$ Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type $\mathfrak{sl}(N+1)/(\mathfrak{sl}(N) \times \mathfrak{u…
Generalized Toda mechanics associated with classical Lie algebras and their reductions
对于任何经典李代数$g$，我们构造了一族Toda力学的可积推广，它们被标记为一对有序整数$(m,n)$ .给出了Lax对的普遍形式、运动方程、哈密顿量和Poisson括号，并给出了$\mathfrak{g}=B_{r},C_{r},D_{r}$与$m,n\leq3$的显式例子.对于所有的$m,n$，它表明，$(m,n-1)$ -和$(m-1,n)$ -Toda链的动力学是自然减少的$(m,n)$ -链，和$m=n$，也有一个家庭的对称减少Toda系统，$(m,m)_{\mathrm{Sym}}$ -Toda系统，这也是可积的。在量子情形下，所有具有$m>1$或$n>1$的$(m,n)$ -Toda系统都描述了标准Toda变量与非对易变量耦合的动力学。除了对称约化的情形外，所有$(m,n)$ -Toda系统的可积性在量子化后仍然存在.
Twists of trigonometric sigma models
本文提出$\mathbb{Z}_N$扭曲的三角sigma模型，通过在4d Chern-Simons理论中引入分支割线并施加自同构构造出四个双重形变模型，并证明特定扭曲会产生具有不同对称性的新可积模型。
Non-Linear Sigma Models on a Half Plane
在有边界可积场论的背景下，讨论了半平面上二维可积非线性西格玛模型，如$O(N)$模型、主手征模型、${\rm CP}^{N-1}$模型和复格拉斯曼西格玛模型.与著名的sine-Gordon、非线性薛定谔和仿射Toda场论的情况相反，这些二维非线性西格玛模型如果被限制在半平面上则不是经典可积的。结果表明，在自由（Neumann）边界条件下，表征整个平面上可积性的非局域电荷的无限集合是不守恒的。如果我们要求这些非定域电荷守恒，那么解就变得微不足道了。
Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
本文利用李对代数构造有限维可积系统在余伴随轨道上的拉格朗日多形式，其欧拉-拉格朗日方程给出Lax形式相容方程，并推广至一般轨道，以开Toda链和Gaudin模型为例验证。
Higher-Spin Currents and Flows in Auxiliary Field Sigma Models
本研究证明辅助场西格玛模型中存在任意自旋的守恒流，给出构造自旋-2n流的统一方法，并求解自旋-3流的微扰解，扩展了可积形变的适用范围。
Bispectral duality and separation of variables from surface defect transition
本文通过二维规范线性 sigma 模型耦合的表面缺陷跃迁，建立了 $\mathfrak{gl}(2)$ XXX 自旋链与 $\mathfrak{sl}(N)$ Gaudin 模型谱问题之间的精确对偶性，并给出 KZ/BPZ 对应的形变与高秩推广以及量子变量分离。
Higher-Spin Poisson Sigma Models and Holographic Duality for SYK Models
本文针对具有高自旋对称性的SYK模型，构建了高自旋规范场与标量物质场的相互作用，提出两种模型，为高自旋全息提供互补视角。
Free field realisation and the chiral universal centraliser
本文通过构造从Kostant-Toda格子覆盖到泛中心化子的开辛嵌入，为任意单群提出了手征泛中心化子的自由场实现，并将其应用于亏格为零、最多六个穿孔的类$\mathcal{S}$理论，使广义$S$-对偶性完全显式化。
We show that the four-dimensional Chern-Simons theory studied by Costello, Witten and Yamazaki, is, with Nahm pole-type boundary conditions, dual to a boundary theory that is a three-dimensional analogue of Toda theory with a novel 3d W-algebra symmetry. By embedding four-dimensional Chern-Simons theory in a partial twist of the five-dimensional maximally supersymmetric Yang-Mills theory on a manifold with corners, we argue that this three-dimensional Toda theory is dual to a two-dimensional topological sigma model with A-branes on the moduli space of solutions to the Bogomolny equations. This…
Scattering in the dual regime of Yang-Baxter deformed O(2N) sigma models
本文确认了 Yang-Baxter 变形 O(2N) sigma 模型的对偶区间对应于扭曲的 D_N^{(2)} 三角 R 矩阵，并求得了幺正化函数及热力学性质。
In this paper we study a suitable limit of integrable QFT with the aim to identify continuous non-relativistic integrable models with local interactions. This limit amounts to sending to infinity the speed of light c but simultaneously adjusting the coupling constant g of the quantum field theories in such a way to keep finite the energies of the various excitations. The QFT considered here are Toda Field Theories and the O(N) non-linear sigma model. In both cases the resulting non-relativistic integrable models consist only of Lieb-Liniger models, which are fully decoupled for the Toda theori…
Extended Complex Trigonometry in Relation to Integrable 2D-Quantum Field Theories and Duality
n阶的多复数有一个相关的三角函数（具有（n-1）个参数的多正弦函数），这导致了正弦-戈登模型的自然扩展。参数的约束从本地电流守恒的要求。在二维n < 6已知的可积模型（变形Toda和非线性sigma，纯仿射Toda.）与对偶对应物以这种方式从多复空间MC本身和从自然嵌入$\MC_n \subset \MMC_m, n < m$获得。对于$ n \ge 6$，由相互作用Lagragien中的一阶电流守恒得到了参数空间上的一般约束。
Pohlmeyer reduction of AdS_5 x S^5 superstring sigma model
出于对AdS_5 × S^5超弦世界片理论在物理自由度方面的有用的二维洛伦兹不变重构的渴望，我们构造了相应的西格玛模型的Pohlmeyer约化版本。Pohlmeyer还原过程涉及几个步骤。从共形规范中的陪集空间弦西格玛模型开始，用电流来表示经典方程，可以修正剩余的共形对称性和κ对称性，并引入一组新的变量（与电流局部相关，但与原始弦坐标场非局部相关），从而自动满足维拉索罗约束。由此产生的规范固定方程可以从一个拉格朗日的非阿贝尔户田型：一个规范WZW模型与可积的潜力也耦合到一组2D费米子场。通过对规范WZW模型的二维规范场进行积分，可以得到规范固定形式的Pohlmeyer约化理论。它的小涨落谱包含8个玻色子自由度和8个费米子自由度，质量相等。我们推测，简化模型具有世界片超对称性和紫外有限性。我们证明了在AdS_2 × S^2超弦模型的特殊情况下，约化理论确实是超对称的：它等价于sine-Gordon模型的N=2超对称扩展。
We address the novel structures arising in quantum and string integrable theories, as well as construct methods to obtain them and provide further analysis. Specifically, we implement the automorphic symmetries on periodic lattice systems to obtain integrable hierarchies, whose commutativity and integrable transformations induce a generating structure of integrable classes. This prescription is first applied to 2-dim and 4-dim setups, where we find the new $ \mathfrak{sl}_{2} $ sector, $ \mathfrak{su}(2) \oplus \mathfrak{su}(2) $ with superconductive modes, Generalised Hubbard type classes and…
Four-Dimensional Chern-Simons and Gauged Sigma Models
本文提出从四维陈-西蒙斯规范理论构造规范σ模型的新方法，基于双耦合4d CS场与新型规范缺陷给出统一作用量，推测模型可积性并给出两类实例。
Integrable Discretizations of Chiral Models
手征模型（广义sigma模型在二维时空连续体上使用微分形式的守恒律的构造是扩展的，它也包括相应的离散版本。这是通过变形的常微分。特别地，非线性Toda晶格以这种方式从线性（连续）波动方程得到。该方法被应用到几个进一步的例子。我们还构造拉克斯对和贝克隆变换的类模型在这项工作中考虑。
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 derive correspondences of correlation functions among dual conformal field theories in two dimensions by developing a "first order formulation" of coset models. We examine several examples, and the most fundamental one may be a conjectural equivalence between a coset $(SL(n)_k \otimes SL(n)_{-1})/SL(n)_{k-1}$ and $\mathfrak{sl}(n)$ Toda field theory with generic level $k$. Among others, we also complete the derivation of higher rank FZZ-duality involving a coset $SL(n+1)_k /(SL(n)_{k} \otimes U(1))$, which could be done only for $n=2,3$ in our previous paper. One obstacle in the previous wo…
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