179 problems

Let a Gaudin model have a corresponding system of Bethe ansatz equations, with parameters in a generic regime, and let the model act on a representation of dimension $d$. Bethe ans…

Affine Bethe-ansatz completeness conjecture. For generic $(k,\ell)$, the Bethe ansatz provides all eigenstates of the integrable structure in every space…

Fabricius–McCoy conjecture. There exist zero-energy excitations in the form of string operators, manifested in the Bethe ansatz equations as singular solutions.

Let a Bethe ansatz state be specified by parameters satisfying a corresponding system of on-shell equations, and let its norm be the scalar product of the state with itself. The Ja…

Let $M$ be a finite-dimensional irreducible $\mathrm{Y}(\mathfrak{gl}_{m|n})$-module of highest $\ell$-weight $\boldsymbol\zeta=(\zeta_i(u))_{i\in\bar I}^s$. Let…

Let $\operatorname{BAE}(\ell)$ be the Bethe ansatz equations, and call a solution physical if it is either regular or a physical singular solution satisfying the Nepomechie–Wang co…

Let ${\mathfrak g}$ be a simple Lie algebra, and let ${{\rm KR}}_{{\alpha},i}(z)$ be Kirillov–Reshetikhin Yangian modules with pairwise distinct spectral parameters. For a tensor p…

Let $\mathfrak g$ be a Kac–Moody algebra, let $\Lambda_1,\dots,\Lambda_n$ be dominant integral weights, let $z_1,\dots,z_n$ be points, and let $\boldsymbol b$ be symmetric paramete…

Let $L$ be the chain length and let $N=1,\dots,L/2$. In the sector ${\cal H}_{L,L/2-N}$ with fixed magnetization $\mathfrak{m}=1/2-N/L$, solutions of the Bethe Ansatz equations are…

Low-$T$ Bethe root pattern conjecture. All excitations of the quantum transfer matrix at low $T$ and sufficiently large $N$ can be parametrised in this way, and, up to corrections…

Let $n$ be the fixed number of Bethe roots, let $H$ be the parameter in the transfer matrix, and let $I_j$ denote the corresponding Bethe quantum numbers. Bethe quantum-number conj…

The closed XXX spin-$\frac{1}{2}$ chain with generic complex twist parameters $\{\kappa,\widetilde{\kappa},\kappa_+,\kappa_-\}$ has quantum space…

Let $\mathfrak{g}$ be the Lie algebra under consideration, let $\mathcal H_{\mathbf n}=\prod_{\alpha,m}{{\rm KR}}_{\alpha,m}^{\otimes n_m^{(\alpha)}}$ be the Hilbert space of the g…

Let $(\nu, \{\lambda_1, \dots, \lambda_M\})$ be a solution of the Bethe equations, and let $\nu”$ and $\nu'$ be the parities of the Bethe vector…

Let $V_1,V_2$ be representations of a simple Lie algebra, let $X$ be a Bethe vector obtained from a critical point $t^0$ of the corresponding phase function $\Phi$, and let $B$ be…

Let $\mathfrak{g}$ be a simple Lie algebra, let $V_1,V_2$ be representations of $\mathfrak{g}$, and let $\Phi$ be the corresponding phase function in variables assigned to the simp…

Let $\mathfrak{g}$ be a simple Lie algebra with simple roots $\alpha_i$, let $V_1,V_2$ be representations of $\mathfrak{g}$ with highest weights $\lambda_1,\lambda_2$, and let…

Conjecture. The constructions give the same result for any irreducible finite-dimensional representation of $U_q(\widehat{\mathfrak{gl}}_N)$.

Let $L$ be a tensor product of irreducible finite-dimensional representations of $sl_{N+1}(\mathbb{C})$, marked by complex parameters $z=(z_1,\ldots,z_n)$. A Bethe vector is a vect…

Non-vanishing conjecture. In the $sl_{N+1}({\mathbb C})$ Gaudin model, every Bethe vector is non-zero, for any $z$. For some values of $z$ the number of Bethe vectors, equivalently…

Let $V_{\boldsymbol{\Lambda}}=V_{\boldsymbol{\Lambda}^1}\otimes\dots\otimes V_{\boldsymbol{\Lambda}^k}$ be the tensor product of the relevant $\mathfrak g$-modules. Let $z^0$ have…

Let the master function associated with the inhomogeneous Gaudin model have critical points, considered modulo the relevant symmetry so that they form orbits, and let the correspon…

Let $P^+$ be the set of dominant weights, let $R$ be the root system, and let $\tilde{\xi}=\sum_{i=1}^{N-1}m_i\Lambda_i=\sum_{i=1}^{N}m'_i\epsilon_i\in P^+$. Let $F_{N,l}$ be a com…

Let $m=(m^{(a)}_j)$ be a content such that … where … Let $\Omega(m)$ be the Bethe-ansatz multiplicity, let $W$ be the Weyl group of $A_n$, and let $\lambda(m)$ be the associated do…

Let $p\in P(m)$ be a state with soliton content $m$, and let $(T^{(r)}_l)^t(p)\neq0$ for every $t\geq0$. Denote by ${\mathcal P}^{(r)}_l$ the least common multiple defined earlier…