20 problems

Vacuum formula conjecture. The vacua are given by

Particle-structure conjecture. The particle structure of the alternating vertex model should consist of the spin-$0$ and spin-$1/2$ particles created by $\psi^{(0)}(\xi)$ and…

Local–particle equivalence conjecture. There should be an isomorphism between the space in (SEMIPATH) and the space in (TENSORPRO).

Let $F^{(\ell)}_{L;m}(\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_N)$ be the weighted sum over configurations with fixed interior edge variables, let $Z^{(\ell)}_{L;m}$ be the c…

Let $\bT(z)$ be the transfer matrix and let $\tQ(z,\us)$ be the matrix obtained by removing the factor $\delta(q^n)$ from $\bQ(z,\us)$: … For $n=0$, the factor $\delta(q^n)$ is mea…

The generalized $Q$-operator for the 6-vertex model is defined by the matrix elements described above, which can contain the formal factor … When $s_1=s_2=0$, the factor…

Let ${\cal H}=\sum_{a,b}{\cal H}_{a,b}$ with $0\leq a\leq m-n$ and $0\leq b\leq n$, and identify ${\cal F}={\cal H}\otimes{\cal H}^*$ with ${\rm End}({\cal H})$. A vacuum state is…

Whole-range arctic curve conjecture. The arctic curve equation remains valid over the whole range $t\in\mathbb{R}$, rather than only on the SE and SW ranges $t\in[\beta(1),\infty)$…

Consider an $n$-state spin model with horizontal and vertical edge weights $W_h$ and $W_v$, and an $n$-state vertex model with weights $w(i_1,i_2\mid i_3,i_4)$. Let the weights be…

Let an $n$-state spin model and an $n$-state vertex model be related by the proposed mapping of Boltzmann weights, and let $L$ denote the lattice size. For the vertex model, write…

For each $n\geq 1$, let $Z^{20V}_n$ denote the total number of 20-vertex configurations with DWBC3 boundary conditions on the quadrangle ${\mathcal Q}_n$, and let $Z^{DT}_n$ denote…

Let ${\mathcal P}_{n,k}$ be the pentagonal square-lattice domain carrying the 20-vertex model with DWBC3 boundary conditions, and let ${\mathcal T}_{n,k}$ be the corresponding exte…

Three-dimensional quantum crystal melting conjecture. The three-dimensional quantum crystal melting model is integrable and has the same mass gap as its one- and two-dimensional co…

$n$-CB algebra conjecture. The $n$-CB algebra is obeyed for all solvable vertex models with $n$ blocks for which the ansatz in equations (2.18) and (2.19) holds.

For nonnegative integers $a,b,c$, let $N_{a,b,c}$ denote the number of 20V configurations on the rectangular grid of size $(a+b+1)\times(b+c+1)$ with DWBC4-type boundary conditions…

Let $P_{n,k}$ be the pentagonal domain carrying the extended DWBC3 boundary conditions, and let ${\mathcal T}_{n,k}$ be the corresponding extended triangle whose domino tilings are…

Let $B_n$ be the sequence associated with 20V configurations with DWBC3 boundary conditions, and let $b_n=T({\mathcal T}_n)$ denote the number of domino tilings of the triangle…

Let $\mathrm{U}$ be the coupling parameter of the vertex model, and let $\mathrm{U}_c$ denote the limiting value of the finite-size critical point at which a pair of real Bethe roo…

The column and bi-partite staggered free-fermion 8-vertex models are related by a mapping imposed under temperature-independent free-fermion conditions for each model. Temperature-…

The $Z_2$ staggered vertex model has a scattering theory with $S$-matrix denoted by $S$ and particle mass $mu$ as specified in the source. Its thermodynamic Bethe Ansatz (TBA) is d…