62 problems

Let $\mathcal{M}$ be the complex symplectic manifold arising from the three-dimensional moduli space of a four-dimensional $\mathcal{N}=2$ supersymmetric gauge theory of class $S$…

Nekrasov–Rosly–Shatashvili conjecture. The superpotential $\mathcal{W}^{\mathrm{eff}}$ may be obtained as a generating function of opers in terms of these Darboux coordinates.

Real-monodromy oper conjecture. The Langlands parameters should be in correspondence with the set of $^LG_\mathbb C$-opers on $X$ having real monodromy, after restricting to single…

Simple-representative conjecture. The simple representative $\Pi^{\mathrm{simp}}(z,\Lambda)$ is a generating function for the eigenvalues $\mathcal{I}^{(2p-1)}_{\ell,\{t_i\}}$.

Affine geometric Langlands conjecture. The Geometric Langlands correspondence generalises to affine Gaudin models, so that their spectra are encoded in affine opers.

Consider generic holomorphic $\operatorname{SL}(N,\mathbb{C})$-connections on $C_{0,2}$ with irregular singularities of the mildest possible type at $0$ and $\infty$. Let…

Let $(J,\tphi)$ be a loop-group $K$-type arising from the toral Yu-data construction, and let $\nabla_{\tphi}$ be its conjectural formal $\LG$-connection. A minimal oper canonical…

Generic-oper contractibility conjecture. The space of generic oper structures on a given irreducible local system is homologically contractible.

Let $G$ be a reductive group, let $\breve{\rho}$ denote the half-sum of positive coroots, and let $\buildrel\circ\over D$ be the formal punctured disc. For…

Let $W_r$ be the polynomial defined from the unique exponential-Gaudin solution indexed by $r$, and consider the monodromy system in the variables $w_i$ displayed in the source.…

Let $L_0$ be the displayed BLZ-type oper, with distinct points $t_j\notin\{0,1\}$, satisfying the condition $b_0+b_1+\sum_iq_i=0$ and having monodromy-free singularities at $x=1$ a…

Let $L_1$ be the displayed scalar oper with distinct nonzero points $s_i$, integer $r$, and parameters $k,l,n_1$ as in the source. Gaudin no-monodromy conjecture. For generic $k,l$…

Let $\mathfrak{g}$ be the affine Lie algebra associated with the finite-dimensional Lie algebra $\mathfrak{g}'$, and consider the relation between quantum $\mathfrak{g}'$-Drinfeld–…

Let $X$ be a smooth projective irreducible curve over a local field $F$, let $G$ be a connected reductive algebraic group over $F$, and let $S$ be a finite set of $F$-points in $X$…

Torelli conjecture for opers. The following conditions are equivalent:

Generic transversality conjecture. The two real submanifolds $\mathrm{Op}_P(\mathscr{X},G)$ and $\mathrm{Rep}_P^0(\Gamma,G(\mathbb{R}))$ intersect transversally at every point. Equ…

Superoper–Gaudin conjecture. Under certain nondegeneracy conditions, $Z$-twisted Miura–Plücker opers are $Z$-twisted Miura opers for certain choices of ${L_i(z)}_{i\in I_g}$. Moreo…

Nekrasov's conjecture. The generating function is the effective twisted superpotential, more precisely,

Beilinson–Drinfeld's conjecture. The Hecke eigensheaf corresponding to the local system represented by $\rho$ is the $\mathcal{D}_{\mathfrak{L}}$-module $\Delta_\rho$.

Fortuna's conjecture. The functor $\Psi_I$ is exact, and its degree-zero functor

Let $(\mathfrak{g},p)$ be one of the examples under consideration, and let $\mathcal{V}^{\infty}$ denote the limiting vertex algebra, with zero fibre…

Let $\mathfrak{g}$ be a non-simply-laced simple Lie algebra, let $\mathfrak{g}^{\sigma}$ denote the corresponding twisted affine data, and let $K_0(\mathcal{O})$ be the Grothendiec…

Let $G$ be a connected complex semi-simple group with real structure, let $\mathcal C$ be a compact complex Riemann surface with antiholomorphic involution $\tau$ and no real point…

Let $\mathcal C$ be a compact complex curve, let $G^{\vee}$ be the Langlands dual group, and let $o$ be a real $G^{\vee}$-oper. For a dominant coweight $\lambda$, let $V_\lambda$ b…

Let $\operatorname{Oper}_{G^{\vee}}(\mathcal C)^{\mathbb R}$ denote the real locus of $G^{\vee}$-opers on a compact complex curve $\mathcal C$, and let $L^2(\operatorname{Bun})_o$…