19 problems

Let $a,b,c,d$ be non-zero vertex weights satisfying … For each $n\geqslant 0$, consider the transfer matrix of the periodic eight-vertex model with an odd number $L=2n+1$ of vertic…

Let $L\geqslant 1$, let $u_1,\dots,u_L$ be inhomogeneity parameters, and define the inhomogeneous open-boundary transfer matrix … where $K^-(u)=K(u)$, $K^+(u)=K(u+2\eta)$, … … Set…

Let $s_n(z)$ be the polynomial defined from the Bazhanov–Mangazeev polynomial by … Let $p_n$ be the polynomial introduced by Mangazeev and Bazhanov, indexed by $n\in\mathbb Z$. Man…

Consider the higher-spin eight-vertex model with spin parameter $\ell$, and let the fused transfer matrix corresponding to the spin chain with local interaction determine the physi…

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…

Elementary scalar particle conjecture. The operator $t(\zeta)$ satisfies

Polynomiality conjecture. (a) The functions ${\mathcal P}_n(x,z)$ are polynomials in $x,z$ of degree $n$ in $x$; each $r_i^{(n)}(z)$ is a polynomial in $z$ satisfying

Let ${\cal A}_{q,p}(\widehat{sl}_2)$ denote the elliptic algebra associated with the quantum affine algebra $\widehat{sl}_2$, with parameters $q$ and $p$. The elliptic-algebra symm…

Let the vertex operator algebras be the algebras introduced in the paper for the eight-vertex model and its associated representations. Let the bosonization procedure mean the cons…

Let $\Lambda(u_0)^{m-2k}_m{\,}^{n-2l}_n$ denote the operator appearing in the free-field representation, and let $X(u)$ and $\tilde Y(u)$ be the vertex operators used in that repre…

Let $n\geqslant 0$, let $\zeta$ be the parameter used to define the vectors $|\phi_n^{\mathrm{BM}}\rangle$, and let $\bar s_n$ denote the polynomial appearing in the Bazhanov–Manga…

Let $L=2n+1$ with $n\geqslant 0$, let $eta=pi/3$, and let $V$ be the two-dimensional spin space. For inhomogeneity parameters $u1,dots,u{2n+1}$, consider the eigenvalue problem ……

Let $a,b,c,d$ be non-zero vertex weights satisfying … and let $\mathcal T$ be the transfer matrix of the resulting supersymmetric eight-vertex model. Let $H_{\text{XYZ}}$ be the XY…

Let $L=2n+1$, let $T_L(u|x_1,\ldots,x_L)$ be the inhomogeneous eight-vertex transfer matrix, and let $\Psi_L(x_1,\ldots,x_L)$ satisfy the eigenvector equations with eigenvalue…

Let $L=2n+1$ be odd, let $\eta=\pi/3$, and let $T_L(u|x_1,\ldots,x_L)$ be the inhomogeneous eight-vertex transfer matrix with spectral parameters $u,x_1,\ldots,x_L$. Let $a(x)$ and…

Vanishing conjecture. The naive string operator satisfies

Let $K$ and $\eta$ be the elliptic parameters, let $N$ be an odd number of sites with periodic boundary conditions, and let $a(v)$ and $b(v)$ denote the eight-vertex Boltzmann weig…

Similarity conjecture. In general,

Functional-equation conjecture. The matrix $Q^{(1)}_{72}(v)$ satisfies a functional equation not involving $T(v)$.