20 problems

Let $\widehat{\mathfrak{g}}$ be an affine Lie algebra, and let ${}^{\rm L}\widehat{\mathfrak{g}}$ denote its Langlands dual. Affine Gaudin models have local and non-local commuting…

Let $\{z_1,\dots,z_N\}\subset\mathbb{C}\mathbb{P}^1$ be distinct marked points, let $\mathfrak g$ be an affine Kac–Moody algebra, and consider the associated affine Gaudin model wi…

One-loop AGM RG-flow conjecture. The twist function satisfies

Let $\widehat{\mathfrak{g}}$ be an affine Lie algebra and let … For $\mathfrak{g}=\mathfrak{sl}(2)$, affine opers are represented by functions $v(z)$ in differential operators … Th…

Consider the monodromy matrix constructed from the gauge-fixed Lax matrix of the chiral affine Gaudin model. Yang–Baxter monodromy conjecture. There exists a quantisation of this m…

Let an affine Gaudin model be built from $N$ independent affine Kac–Moody current algebras at levels $\{k_j\}_{j=1}^N$, attached to punctures $\{z_j\}_{j=1}^N\subset\mathbb{CP}^1$.…

Let $P(x)=e^{2x}x^\kappa$ define the one-site affine Gaudin model associated with the integrable Kondo problem in the $SU(2)_\kappa$ WZW model. Let $|l,\kappa\rangle$ be the highes…

The affine Gaudin model has classical currents ${\cal J}^a_i$ organized by a spectral parameter $z$, while its quantum transfer matrices are Kondo line defects coupled to quantum K…

Identification conjecture. The three-parameter deformed model coincides with the model of Delduc et al. without TsT transformations.

Consider the quantum hierarchy associated with an affine Gaudin model with twist function. Its poles determine the integration contours, and affine opers provide the geometric data…

Let $S^d(x)$ be the evaluation at a zero $x$ of the twist function of the degree-$d$ quantum operator, obtained as the saddle-point classical limit of the hypergeometric-integral H…

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 $E_+$ be the set of positive exponents of the underlying Lie algebra, let $\varphi(z)$ be the twist function, and let $h^\vee$ be the dual Coxeter number. The quadratic Hamilto…

Let $\lambda_0$ be a finite zero of the twist function, and let $n,m\in\mathcal{E}_{\lambda_0}$ be admissible degrees. The integrated zero-curvature expression associated with the…

Affine Gaudin eigenvector conjecture. For every exponent $i$ and every permitted contour $\boldsymbol{\beta}$,

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…

Schechtman–Varchenko eigenvector conjecture. The vector $\boldsymbol{\theta}$ should be a simultaneous eigenvector of the operators \boldsymbol{Q}^\boldsymbol{\beta}_r, with eige…

Hypergeometric eigenvalue conjecture. These hypergeometric integrals should give the eigenvalues of the local higher affine Gaudin Hamiltonians.

Integral expression conjecture. The affine Gaudin Hamiltonians themselves should be naturally expressed as such hypergeometric integrals.

Higher Hamiltonians conjecture. Quantum Gaudin models in affine types should admit families of higher Hamiltonians, labelled by the exponents, whose eigenvalues are given by functi…