S. Belliard, N. A. Slavnov
We consider $\mathfrak{gl}_2$-invariant quantum integrable models solvable by the algebraic Bethe ansatz. We show that the form of on-shell Bethe vectors is preserved under certain twist transformations of the monodromy matrix. We also derive the actions of the twisted monodromy matrix entries onto twisted off-shell Bethe vectors.
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Supersymmetric composite generalized quantum integrable models solvable by the algebraic Bethe ansatz are studied. Using a coproduct in the bialgebra of monodromy matrix elements and their action on Bethe vectors, formulas for Bethe vectors in the composite models with supersymmetry based on the super-Yangians $Y[\mathfrak{gl}(2|1)]$ and $Y[\mathfrak{gl}(1|2)]$ are derived.
Multiple actions of the monodromy matrix elements onto off-shell Bethe vectors in the $\mathfrak{gl}(m|n)$-invariant quantum integrable models are calculated. These actions are used to describe recursions for the highest coefficients in the sum formula for the scalar product. For simplicity, detailed proofs are given for the $\mathfrak{gl}(m)$ case. The results for the supersymmetric case can be obtained similarly and are formulated without proofs.
We study Bethe vectors of integrable models based on the super-Yangian $Y(\mathfrak{gl}(m|n))$. Starting from the super-trace formula, we exhibit recursion relations for these vectors in the case of $Y(\mathfrak{gl}(2|1))$ and $Y(\mathfrak{gl}(1|2))$. These recursion relations allow to get explicit expressions for the Bethe vectors. Using an antimorphism of the super-Yangian $Y(\mathfrak{gl}(m|n))$, we also construct a super-trace formula for dual Bethe vectors, and, for $Y(\mathfrak{gl}(2|1))$ and $Y(\mathfrak{gl}(1|2))$ super-Yangians, show recursion relations for them. Again, the latter all…
We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}_3$-invariant $R$-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.
We study integrable models with $\mathfrak{gl}(2|1)$ symmetry and solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for scalar products of Bethe vectors, when the Bethe parameters obey some relations weaker than the Bethe equations. This representation allows us to find the norms of on-shell Bethe vectors and obtain determinant formulas for form factors of the diagonal entries of the monodromy matrix.
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 …
A class of $\mathfrak{o}_{2n+1}$-invariant quantum integrable models is investigated in the framework of algebraic Bethe ansatz method. A construction of the $\mathfrak{o}_{2n+1}$-invariant Bethe vector is proposed in terms of the Drinfeld currents for the double of Yangian $\mathcal{D}Y(\mathfrak{o}_{2n + 1})$. Action of the monodromy matrix entries onto off-shell Bethe vectors for these models is calculated. Recursion relations for these vectors were obtained. The action formulas can be used to investigate structure of the scalar products of Bethe vectors in $\mathfrak{o}_{2n+1}$-invariant m…
We study scalar products of Bethe vectors in integrable models solvable by nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(2|1)$ symmetry. Using explicit formulas of the monodromy matrix entries multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors i…
We present a nested algebraic Bethe ansatz for a one dimensional open spin chain whose boundary quantum spaces are irreducible $\mathfrak{so}_{2n}$- or $\mathfrak{sp}_{2n}$-representations and the monodromy matrix satisfies the defining relations of the Olshanskii twisted Yangian $Y^\pm(\mathfrak{gl}_{2n})$. We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a $\mathfrak{so}_{2n}$- or $\mathfrak{sp}_{2n}$-symmetric open spin chain to that of a $\mathfrak{gl}_{n}$-symmetric periodic spin chain. We explicitly derive th…
We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(N)$-invariant $R$-matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of the Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products…
We study scalar products of Bethe vectors in the models solvable by the nested algebraic Bethe ansatz and described by $\mathfrak{gl}(m|n)$ superalgebra. Using coproduct properties of the Bethe vectors we obtain a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of Bethe parameters. We also obtain recursions for the Bethe vectors. This allows us to find recursions for the highest coefficient of the scalar product.
We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians $Y(\mathfrak{gl}(m|n)$ in terms of the current generators of the Yangian double $DY(\mathfrak{gl}(m|n))$. More specifically, we use the method of projections onto intersections of different type Borel subalgebras in this infinite dimensional algebra to construct the Bethe vectors. Calculating these projection the supersymmetric Bethe vectors can be expressed through matrix elements of the universal monodromy matrix elements. Using two different but isomorphic current realizations of the Yangian doub…
We consider quantum integrable models associated with $\mathfrak{so}_3$ algebra. We describe Bethe vectors of these models in terms of the current generators of the $\mathcal{D}Y(\mathfrak{so}_3)$ algebra. To implement this approach we use isomorphism between $R$-matrix and Drinfeld current realizations of the Yangians and their doubles for classical types $B$, $C$, and $D$ series algebras. Using these results we derive the actions of the monodromy matrix elements on off-shell Bethe vectors. We show that these action formulas lead to recursions for off-shell Bethe vectors and Bethe equations f…
We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$. We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, whose factors are characterized by two highest coefficients. We provide different recursions for these highest coefficients. In addition, we show that when the Bethe vectors are on-shell, their norm takes the form of a Gaudin…
We study quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(m|n)$-invariant $R$-matrix. We compute the norm of the Hamiltonian eigenstates. Using the notion of a generalized model we show that the square of the norm obeys a number of properties that uniquely fix it. We also show that a Jacobian of the system of Bethe equations obeys the same properties. In this way we prove a generalized Gaudin hypothesis for the norm of the Hamiltonian eigenstates.
We study $\mathfrak{gl}(2|1)$ symmetric integrable models solvable by the nested algebraic Bethe ansatz. Using explicit formulas for the Bethe vectors we derive the actions of the monodromy matrix entries onto these vectors. We show that the result of these actions is a finite linear combination of Bethe vectors. The obtained formulas open a way for studying scalar products of Bethe vectors.
Scalar products and norm of Bethe vectors in $\mathfrak{o}_{2n+1}$ invariant integrable models
本文研究$o_{2n+1}$对称可积模型中离壳Bethe向量的标量积，给出基于最高系数的划分求和公式，证明其递推关系、留数定理及与$gl_n$模型约化的一致性，并将在壳范数表示为Gaudin行列式。
We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian $\mathrm Y(\mathfrak{gl}_{m|n})$. To a solution we associate a rational difference operator $\mathcal D$ and a superspace of rational functions $W$. We show that the set of complete factorizations of $\mathcal D$ is in canonical bijection with the variety of superflags in $W$ and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
Bethe vectors and recurrence relations for twisted Yangian based models
本文用代数 Bethe ansatz 研究 Olshanski 扭曲 Yangian 开放自旋链的奇情形，显式构造 Bethe 向量并利用复合模型方法导出偶、奇两种情形下的递推关系。
Bethe Ansatz without Nesting
研究有理$\mathfrak{gl}_\ell$自旋链，从量子谱曲线等出发推导无嵌套贝塞耳方程组，推广到任意秩，识别递归层次结构，给出转移矩阵本征值表达式，揭示相关联系并研究准经典极限。
On the nested algebraic Bethe ansatz for spin chains with simple $\mathfrak{g}$-symmetry
本文为具有任意简单李代数对称性的闭合有理自旋链提出了新的嵌套代数 Bethe ansatz 框架，通过移除单根并利用剩余 U(1) 荷与 R-矩阵分块 Gauss 分解，重新推导出 Yangian 代数嵌套和 AB 对易关系等标准结果。
We describe a reproduction procedure which, given a solution of the $\mathfrak{gl}_{M|N}$ Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family $P$ of other solutions called the population. To a population we associate a rational pseudodifferential operator $R$ and a superspace $W$ of rational functions. We show that if at least one module is typical then the population $P$ is canonically identified with the set of minimal factorizations of $R$ and with the space of full superflags in $W$. We conjecture that the singular eigenvectors (up to rescal…
We construct exact eigenvectors and eigenvalues for $U_q(\mathfrak{sp}_{2n})$- and $U_q(\mathfrak{so}_{2n})$-symmetric closed spin chains by means of a nested algebraic Bethe ansatz method. We use a fusion procedure to construct higher-dimensional Lax operators. Our approach generalises and extends the results obtained by Reshetikhin and De Vega-Karowski. We also present a generalisation of Tarasov-Varchenko trace formula for nested Bethe vectors.
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…
Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry
本文利用含边缘无限维 Verma 模的新型 $U_{\mathfrak{q}}\mathfrak{sl}_2$ 不变链及 Schur-Weyl 对偶，克服非对角边界开放 XXZ 链缺失参考态的问题，导出 Bethe 方程并解释 Nepomechie 条件。
We present a supermatrix realisation of q-deformed spinor-spinor and spinor-vector R-matrices. These R-matrices are then used to construct transfer matrices for $U_{q^2}(\mathfrak{so}_{2n+1})$- and $U_{q}(\mathfrak{so}_{2n+2})$-symmetric closed spin chains. Their eigenvectors and eigenvalues are computed.
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.
The Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra $U(\mathfrak{g}[t])$ and the coordinate ring of the first congruence subgroup $\mathcal{O}(G_1[[t^{-1}]])$. Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Yangian. Bethe subalgebras form a natural family of commutative subalgebras depending on a group element $C$ of the adjoint group $G$. The images of these algebras in tensor products of fundament…
We reformulate nested relations between off-shell $U_q(\widehat{\mathfrak{gl}}_N)$ Bethe vectors as a certain equation on generating series of strings of the composed $U_q(\widehat{\mathfrak{gl}}_N)$ currents. Using inversion of the generating series we find a new type of hierarchical relations between universal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et al., Ann. Henri Poincaré 7 (2006), 1217-1268, math-ph/0512037] for the parameters of universal …
An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_3)$ is given. This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms of the total currents of a "new" realization of the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_3)$.
In these notes we study integrable structures of conformal field theory with $\textrm{BCD}$ symmetry. We realise these integrable structures as affine Yangian $\mathfrak{gl}(1)$ "spin chains" with boundaries. We provide three solutions of Sklyanin $\textrm{KRKR}$ equation compatible with affine Yangian $R$-matrix and derive Bethe ansatz equations for the spectrum.
Bethe algebras for unitarizable modules over classical Lie (super)algebras and a duality
研究经典李（超）代数单位化模块的Bethe代数作用，证明在特定条件下该代数在张量积模块上可对角化，并建立一般线性李（超）代数的Bethe代数对偶性。
On Bethe eigenvectors and higher transfer matrices for supersymmetric spin chains
本文研究了$\mathfrak{gl}_{m|n}$ XXX超对称自旋链，证明了在壳Bethe向量是高阶转移矩阵的特征向量并计算出对应特征值，从而证实了先前文献的猜想并将已有结果扩展至超对称情形及其经典极限下的Gaudin模型。
We give combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for tensor products of irreducible evaluation modules over the Yangian $Y({\mathfrak{gl}}_N)$ and the quantum affine algebra $U_q(\widetilde{\mathfrak{gl}_N})$. The results of the paper were obtained in 1998 and were used in math.QA/9905137, math.QA/0302148, math.QA/0610517.
In this paper we compare two constructions of weight functions (off-shell Bethe vectors) for the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$. The first construction comes from the algebraic nested Bethe ansatz. The second one is defined in terms of certain projections of products of Drinfeld currents. We show that two constructions give the same result in tensor products of vector representations of $U_q(\hat{\mathfrak{gl}}_N)$.
We give a detailed description of the nested algebraic Bethe ansatz. We consider integrable models with a $\mathfrak{gl}_3$-invariant $R$-matrix as the basic example, however, we also describe possible generalizations. We give recursions and explicit formulas for the Bethe vectors. We also give a representation for the Bethe vectors in the form of a trace formula.
The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz
本文研究一般线性李超代数Gaudin模型，证明奇异空间上的Gaudin代数是循环模和Frobenius代数，并对一般参数证明其可对角化且具有单谱，从而给出Bethe ansatz完备性的新解释。
Bethe ansatz equations for quantum $\mathcal{N}=2$ KdV systems
本研究基于对$\widehat{\mathfrak{gl}}(1|1)$仿射Yangian的前期研究，提出了适用于$\mathcal{N}=2$量子KdV系统能谱的Bethe ansatz方程，为该量子可积系统的能谱求解提供了新方法。
q-Opers, QQ-Systems, and Bethe Ansatz
本文研究q-Opers与Bethe方程的对应关系，提出QQ系统在量子可积模型与经典几何对象间的联系，揭示了简单李代数与非简单李代数下的不同可积模型。
We study quantum integrable models with GL(3) trigonometric $R$-matrix and solvable by the nested algebraic Bethe ansatz. Using the presentation of the universal Bethe vectors in terms of projections of products of the currents of the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_3)$ onto intersections of different types of Borel subalgebras, we prove that the set of the nested Bethe vectors is closed under the action of the elements of the monodromy matrix.
The Gaudin model and the super Bethe ansatz for unitarizable modules over the general linear Lie (super)algebra
本文研究一般线性李（超）代数 Gaudin 代数在酉模奇异权空间上的作用，证明其为循环模且 Gaudin 代数为 Frobenius 代数，并建立超 Bethe 拟设。
Multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis
本文研究了量子齐次代数$U_q(\widehat{\mathfrak{gl}}_N)$的多重交换关系，推导了向量表示的Gelfand-Tsetlin基构造方法，并探讨了其与Bethe向量的关系。
Black Holes, the Bethe Ansatz, and Elliptic Calogero--Moser Systems
本文定义了四维N=4超杨-米尔斯BAE解到对应椭圆卡洛杰罗-莫塞系统势极值与极点的映射，证明其关联两系统对称性，提出该映射为双射的猜想，用秩-2代数的实例为猜想提供证据。
Bethe ansatz equations for orthosymplectic Lie superalgebras and self-dual superspaces
本文研究正交辛李超代数 Bethe ansatz 方程的解，通过再生程序构造解的种群，并证明种群与自对偶超空间中各向同性完全超旗及对称完全分解之间存在典范双射。
We compute an universal weight function (off-shell Bethe vectors) in any representation with a weight singular vector of the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$ applying the method of projections of Drinfeld currents developed in arXiv:math/0610398.
Fusion approach for quantum integrable system associated with the $\mathfrak{gl}(1|1)$ Lie superalgebra
本文提出了一种基于$\mathfrak{gl}(1|1)$李超代数的量子可积系统精确解法，通过融合技术推导出算子恒等式，确定能谱和Bethe方程，并探讨了Bethe态的推导方法。
We introduce and study a category $\text{Fin}$ of modules of the Borel subalgebra of a quantum affine algebra $U_q\mathfrak{g}$, where the commutative algebra of Drinfeld generators $h_{i,r}$, corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional $U_q\mathfrak{g}$ modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in $\text{Fin}$. Among them we find the Baxter $…
The zero modes method is applied in order to get action of the monodromy matrix entries onto off-shell Bethe vectors in quantum integrable models associated with $U_q(\mathfrak{gl}_N)$-invariant $R$-matrices. The action formulas allow to get recurrence relations for off-shell Bethe vectors and for highest coefficients of the Bethe vectors scalar product.
New Combinatorial Formulae for Nested Bethe Vectors
研究为杨式代数\(Y(\mathfrak{gl}_4)\)上评估模的向量值权重函数给出新组合公式，任意\(n\)时\(Y(\mathfrak{gl}_n)\)情况在相关文献中被考虑。
We study the Gaudin models associated with $\mathfrak{gl}(1|1)$. We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation $\mathfrak{gl}(1|1)[t]$-modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. Therefore, we conf…
We propose a way to separate variables in a rational integrable $\mathfrak{gl}(n)$ spin chain with an arbitrary finite-dimensional irreducible representation at each site and with generic twisted periodic boundary conditions. Firstly, we construct a basis that diagonalises a higher-rank version of the Sklyanin B-operator; the construction is based on recursive usage of an embedding of a $\mathfrak{gl}(k)$ spin chain into a $\mathfrak{gl}(k+1)$ spin chain which is induced from a Yangian homomorphism and controlled by dual diagonals of Gelfand-Tsetlin patterns. Then, we show that the same basis …
We study integrable models solvable by the nested algebraic Bethe ansatz and described by $\mathfrak{gl}(2|1)$ or $\mathfrak{gl}(1|2)$ superalgebras. We obtain explicit determinant representations for form factors of the monodromy matrix entries. We show that all form factors are related to each other at special limits of the Bethe parameters. Our results allow one to obtain determinant formulas for form factors of local operators in the supersymmetric t-J model.
We consider quantum integrable models solvable by the algebraic Bethe ansatz and possessing $\mathfrak{gl}(2)$-invariant $R$-matrix. We study the models of both periodic boundary conditions and boundary conditions based on reflection algebra. We present a new method to calculate the scalar products based on formula of an action of transfer matrix of a model onto Bethe vector. Using this method we identify some coefficient in the multiple action of the transfer matrix with the scalar product between on-shell and off-shell Bethe vectors. This allows us to find determinant representation of the s…
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.
We consider quantum integrable models with $\mathfrak{gl}(2|1)$ symmetry. We derive a set of multiple commutation relations between the monodromy matrix entries. These multiple commutation relations allow us to obtain different representations for Bethe vectors.
We study the ODE/IM correspondence for all states of the quantum $\widehat{\mathfrak{g}}$-KdV model, where $\widehat{\mathfrak{g}}$ is the affinization of a simply-laced simple Lie algebra $\mathfrak{g}$. We construct quantum $\widehat{\mathfrak{g}}$-KdV opers as an explicit realization of the class of opers introduced by Feigin and Frenkel, which are defined by fixing the singularity structure at $0$ and $\infty$, and by allowing a finite number of additional singular terms with trivial monodromy. We prove that the generalized monodromy data of the quantum $\widehat{\mathfrak{g}}$-KdV opers s…
Derivations for the MPS overlap formulas of rational spin chains
本文利用代数 Bethe Ansatz 和 KT-关系，推导出 gl_N 自旋链中可积 MPS 与 Bethe 本征态重叠的普适公式，并将其推广到 so_N 和 sp_N 情形，拓宽了量子淬火与缺陷 CFT 中可精确处理的初态范围。
A note on odd reflections of super Yangian and Bethe ansatz
本文从新视角重新推导超 Yangian 的奇反射转换规则，建立其与 XXX 型 Bethe ansatz 费米子再生程序的联系，并给出 q-特征标变化算法及斜表示的显式计算。
We consider relations between Gauss coordinates of $T$-operators for the Yangian doubles of the classical types corresponding to the algebras $\mathfrak{g}$ of $A$, $B$, $C$ and $D$ series and the current generators of these algebras. These relations are important for the applications in the quantum integrable models related to $\mathfrak{g}$-invariant $R$-matrices and construction of the Bethe vectors in these models.
In this note we propose a compatible set of equations which commutes with the Knizhnik-Zamolodchikov equations based on the $\mathfrak{gl}(n\vert m)$ symmetry algebra and establish the Matsuo-Cherednik correspondence in this context.
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