32 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…

Let $R^{\mathrm{trig}}$ be the trigonometric quantum $R$-matrix, and let $R^{\mathrm{rat}}$ denote its $q\to1$ limit. A quantum integrable model has a $2\times2$ quantum Lax matrix…

Exact-solvability conjecture. Exactly solvability should hold quite widely for super-integrable systems.

Scalar-product conjecture.

Let the Hill determinant be the determinant associated with the spectral problem for the quantum periodic Toda chain, whose zeros determine the relevant spectral values. Simplicity…

Let ${\bfit\gamma}=(\gamma_1,\ldots,\gamma_{N-1})$ be real parameters, let $\psi_{\bfit\gamma}(\bfit x)$ be the Whittaker functions of the quantum periodic Toda chain, and let…

Let $A$ be a Poisson algebra with center $Z$, and let $M$ be the maximal Poisson-commutative subalgebra generated by the spectral invariants of a Lax operator $L(z)$. Let $A_q$ be…

Existence conjecture. There exists an element $K\in {\cal F}\otimes {\cal A}$ satisfying the relations specified subsequently in the source, namely the coaction/reflection relation…

Let a correlation function be any correlation function of the Heisenberg XXX spin-chain model, and let the Riemann zeta-function be denoted by $\zeta$. Rational-coefficient conject…

Quantum complete integrability conjecture. There exists $\kappa=\kappa(N,d)>0$ such that

Let $a_N$ be typically an affine function of $N$, let ${\mathcal{F}}_N({\boldsymbol{\lambda}}\mid{\boldsymbol{x}})$ be an explicit integrand, and let $\Gamma$ be a simple smooth cu…

Let $R$, $a_3$, and $b_1$ be the integral and parameters appearing in the superintegrable system, and let $\bar{\psi}_n$ be the states generated from the ground state by … For each…

Quantum DELL eigenvalue conjecture. The instanton partition function is a simultaneous eigenfunction of the quantum DELL Hamiltonians:

Fix $-3<k<-2$ and $\bar{r}^1,\bar{r}^2\in\mathbb{C}$, and let $s$ be an $\mathfrak{s}$-valued function entering an oper…

The quantum Boussinesq model is a two-dimensional conformal field theory with $\mathcal{W}_3$ symmetry, and its level $N$ states are described by $2N$ complex variables…

The wreath Macdonald polynomials $H_\lambda$ form a basis of the representation ring of $\mathbb{Z}/\ell\mathbb{Z}\wr\Sigma_n$, and the commuting algebra of operators that diagonal…

Frenkel–Reshetikhin's conjecture. For more general quantum integrable systems constructed via finite-dimensional representations of $\mathcal{U}_q(\hat{\mathfrak{g}})$, the spectra…

Consider an integrable quantum chain with transfer matrix and Sklyanin's $B$-operator, in the setting where the $B$-operator is diagonalizable and its eigenvalues have the stated f…

Consider the quantum chain and the operators, twists, inhomogeneities, and separation-of-variables bases appearing in Properties 1 and 2, where Property 1 concerns the diagonalizab…

Higher Hamiltonians conjecture. There exist nonzero meromorphic functions $\boldsymbol{\tau}_i(z)$, for $iboldsymbol{rho} obreakboldsymbol{E}$, such that each has degree $i+1$; the…

Litvinov's deformation conjecture. The algebra of local Hamiltonians can be naturally deformed by adding an extra Heisenberg algebra.

Gaudin Bethe ansatz conjecture. The spectrum of the non-local integrals of motion is described by solutions of Gaudin Bethe ansatz equations associated to affine…

Exact quantization conjecture. The spectra of these Hamiltonians can be obtained from exact quantization conditions, analogous to those for the Toda lattice and involving topologic…

Higher-Hamiltonian eigenvalue conjecture.

Let $\vec{\lambda}$ and $\vec{\mu}$ be $N$-tuples of Young diagrams, and let $c^{(\pm)}_{\vec{\lambda},\vec{\mu}}(q,t\mid u_1,\ldots,u_N)$ be the coefficients in the action … For a…