50 problems

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…

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…

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 $\mathfrak g$ 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…

Kirillov–Reshetikhin conjecture. For every $(a,m)\in H$,

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…

Let $B=B^{r_1,l_1}\otimes\cdots\otimes B^{r_L,l_L}$ be a tensor product of crystals of type $A_n$, and let $m=(m^{(a)}_j)$ be a soliton content. Define … let $p^{(a)}_j$ be the cor…

Bethe-vector basis conjecture. For every $|\mathbf{x}\rangle\in\mathbb{V}$,

Completeness conjecture. The Bethe Ansatz completely determines the spectrum for all but finitely many values of $\Delta$ with $|\Delta|<\varepsilon$.

Let $r=d_0-d_1$ and use the polynomials $W_r,U_r$ and the Hermite-Wronskian polynomial $V_\mu$ defined in the source. Unequal-degree large-$l$ conjecture. For generic $l,m$, the Ga…

Let the exponential Gaudin Bethe Ansatz system be as in the source, and let $d_0=r^2$, $d_1=r(r-1)$ characterize the Virasoro highest-weight locus. Exponential Gaudin nonexistence…

Let $d_0=d_1$ and let $\operatorname{Part}(d_0)$ denote the number of partitions of $d_0$. Large-$l$ Gaudin counting conjecture. For generic $l,m$, the Gaudin Bethe Ansatz system h…

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…

Completeness conjecture. The operator $\mathscr H(x)$ has simple spectrum on $(L_{\bm\Lambda}(\bm b))^{\mathrm{sing}}$. There is a bijection between the monic divisors $y$ of…

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…

Fix the ratio $n/N$, where $n$ is the number of Bethe roots and $N$ is the system size, and let $C$ denote the contour described in the paper. Thermodynamic Bethe-root contour conj…

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…

Let $M=2s$, and let $P_a^M(U_i,\bar U_i)=0$ be the polynomial Bethe-root equations for parameter choice $a=d$ or $a=g$. A solution is admissible if its coordinates are pairwise dis…

Let a quantum $\widehat{\mathfrak{g}}$-KdV oper be an oper of the form … with the additional regular singularities chosen to have trivial monodromy for every value of $\lambda$. Le…

Let $d$ be a positive exponent with $d>1$, let $\gamma$ be a Pochhammer contour, and let $\Psi^{\mathrm{on}}_{\bm c}(\bm w)$ be an on-shell Bethe state in $H_{\bm\lambda}$. Associa…

Let $\boldsymbol\lambda$ be a sequence of arbitrary $\mathfrak{gl}_{1|1}$ weights, and let $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$ be its singular subspace. Let…

Consider a tensor product $L(\boldsymbol\lambda)$ of typical polynomial $\mathfrak{gl}_{1|1}$ modules, with singular subspace $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$, and let…