17 problems

Dorey–Dunning–Tateo conjecture. The spectrum of ${\cal H}_{N,l}$ is real and positive for $N\ge 2$ and $l$ small. This is a further ${\cal PT}$-symmetric generalization of the Bess…

Let $\alpha$, $p$, and $j\in\mathbb{Z}_{\ge 0}$ be as above, and let $\mathcal{Q}_{\pm,j}^{(\alpha)}(E;p)$ denote the eigenvalues of the graded operators $\hat{\mathbf{Q}}_\pm(E)$…

Vacuum-eigenvalue conjecture. For general $r$, the vacuum eigenvalue of the sixth integral of motion is

Fix $d_0$ and $k>-2$, and let $r$ and the Gaudin and BLZ systems be as in the source. Define $Q_{+,r}(\lambda)=\lambda^rQ_+(\lambda)$ for the Gaudin spectral determinants, and use…

Let $L^G$ be obtained from a solution of the Gaudin Bethe Ansatz equations with $d_0-d_1=r$, and let $Q_\pm(\lambda)$ be the associated $Q$-functions. Order conjecture. The functio…

Let $L^G$ be a Schrödinger operator obtained from a solution of the $chat{\mathfrak{sl}}_2$ Gaudin Bethe Ansatz equations, and let $Q_\pm(\lambda)$ be its connection coefficients,…

Perturbative BLZ conjecture. For every partition $\nu$ of $N$, there exists a neighbourhood $D$ of $0$ and a unique algebraic function

Weak BLZ conjecture. For every $\alpha$ and $L$, the number of solutions of the BLZ system is at most $p(N)N!$. For generic values of $\alpha$ and $L$, the number of solutions is…

Consider an integrable massive quantum field theory associated with the $A_{d-1}$ Lie algebra, with $d-1$ massive particle species and two-particle S-matrix elements…

Let $G$ be a compact simple Lie group with rank $r_G$, Coxeter number $h_G$, and level $k$. The associated parafermionic coset conformal field theory $G_k/U(1)^{r_G}$ has central c…

Let $M>1/2$, let $k$ be any value, and let $\psi^{(1)}_{[k]}(z,\overline{z};\theta)$ denote the Symanzik-rescaled solution of the differential equations for $\psi^{(1)}$ and…

Let $L\in\mathbb{Z}_{\geq 0}$, and consider the level-$L$ subspace of an irreducible module over the Virasoro algebra. Let the quantum KdV Hamiltonians act on this subspace, and le…

Consider the modified affine Toda field equations associated with simply-laced or twisted affine Lie algebras. The massive ODE/IM correspondence is the proposed correspondence betw…

Let $\hat{\mathfrak{g}}^{\vee}$ be a modified affine Toda field equation, and let $\Psi^{(a)}$ denote its subdominant solutions. The massive $\psi$-system is the set of functional…

Let $\hat{\mathfrak{g}}^{\vee}$ be an affine Toda algebra, and let $Q^{(a)}_i(\lambda,g)$ be the radial spectral determinants obtained by expanding the subdominant solutions of the…

Bazhanov–Lukyanov–Zamolodchikov symmetric-polynomial conjecture. The eigenvalues of the operators $\mathbf{I}_{2n-1}^{\scriptscriptstyle{\textrm{BLZ}}}$ are symmetric polynomials o…

Bazhanov–Lukyanov–Zamolodchikov conjecture. Descendant fields could be found through relatively simple generalisations of the initial differential equation.