83 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 $S$…

Gaiotto–Moore–Neitzke's conjecture. As the radial variable $t\to\infty$ on the Hitchin base, the Hitchin metric on $\mathcal{M}$ is asymptotic to the semi-flat metric constructed f…

Let a gravitational instanton be a complete hyperKähler manifold of ALG or ALG$^*$ type, and let a Hitchin moduli space be a moduli space of solutions to a Hitchin system equipped…

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, let $X$ be an irreducible smooth projective variety over $\mathbb{K}$, and let $G$ be a reductive group ove…

Ooguri–Vafa local model conjecture. Near the discriminant locus of the Hitchin base, the hyperkähler metric on a Hitchin moduli space should be described locally by the Ooguri–Vafa…

Let $({\overline{\partial}}_E,\varphi)$ be a Higgs bundle in the regular locus $\mathcal M'$, and let $g_{L^2}$ and $g_{\mathrm{sf}}$ denote Hitchin's $L^2$-metric and the semiflat…

Let $C$ be the underlying smooth projective curve, let $G$ be a reductive group, and let $\mathcal F_{\mathcal P^\vee}$ be the holonomic $\mathcal D_{\mathcal Bun_G}$-module associ…

The geometric Langlands correspondence. There exists a canonical equivalence of categories

The Hitchin system is an algebraically completely integrable system whose phase space is the cotangent bundle of a moduli space of stable bundles, while the Beauville–Mukai system…

The Hitchin and Beauville–Mukai systems are algebraically completely integrable systems associated with a complex curve $C$ and, in the relevant construction, a spectral transforma…

Let $(M,g)$ be a hyperkähler gravitational instanton with quadratic volume growth. Modularity conjecture. The gravitational instanton arises as a Hitchin moduli space of Higgs bund…

Let $(A,\phi,\psi)\in\mathcal{M}_\alpha^G$ be a solution and let $b=\mathrm{Hit}_\alpha^G(A,\phi,\psi)$. Denote by $\Sigma_b$ the associated spectral curve and by $L$ the correspon…

Let a family of smooth curves be given, and consider the Hitchin connections for a group at an arbitrary level. The Hitchin system has smooth spectral covers whose fibers are Prym…

Kostka-weighted formula. The polynomial $C^f_{m,n}(q,t)$ coincides with the bigraded polynomial arising from either the mixed Hodge filtration or the perverse filtration of the Hit…

Let $G=\operatorname{SO}_{2n}$, let $\mathbf{H}_{\overline{\mathbf{O}}}$ be the Hitchin base, and let $\mathbf{A}_P$ denote the Coulomb Hitchin base associated with a parabolic sub…

Let $C$ be a smooth projective curve, let $M_{r,d}$ be the moduli space of semistable Higgs bundles on $C$, and let $\mathbb G_m$ act by $\lambda\cdot(E,\phi)=(E,\lambda\phi)$. For…

Let $C$ be a nonsingular projective curve of genus $g\ge 2$, and let $M_{r,d}$ be the moduli space of semistable Higgs bundles on $C$ with Hitchin fibration $h:M_{r,d}\to A$. Write…

Let $S$ be a Riemann surface, let $\mathfrak{t}$ be a maximal split toral subalgebra of $\mathfrak{p}\subset\mathfrak{g}$, and let $W$ be the Weyl group acting on $\mathfrak{t}$. A…

Let $V_{g,n}$ be the vertex operator algebra associated with the genus-$g$ theory with $n$ regular punctures, and let $\{L_a\}$ be its simple modules. Let ${\cal M}_{g,n}$ be the m…

Let $V_{g,0}$ be the vertex operator algebra associated with the unpunctured genus-$g$ theory, with simple modules $\{L_a\}$ and conformal dimensions $h(L_a)$. Let ${\cal M}_{g,0}$…

Let $X=G/H$ be an affine homogeneous spherical variety with abelian regular centralizers and no type $N$ roots, and let $S_X$ be a polarizable symplectic dual representation of…

Let $X=G/H$ be an affine homogeneous spherical variety with abelian regular centralizers and no type $N$ roots. Let $\mathfrak D\subset\mathfrak c_{G_X^\vee}$ be the divisor define…

Let $G$ be a reductive group over an algebraically closed field, let $C$ be a smooth curve, and let $\mathcal P_G$ and $\mathcal P_{G^\vee}$ be the abelian group stacks over the co…

Let $X\in\mathfrak{so}(2N)$ be a nilpotent element corresponding to a special D-partition $P=(p_1\geq p_2\geq\dots\geq p_{2m})$ of $2N$, and let $g\in\mathfrak{so}(2N)$. Define … F…

Let $\mathcal M$ be a derived Hitchin stack, let $\operatorname{Sht}_{\mathcal M}$ be its associated shtuka stack, and let $c_1$ be the correspondence map whose derived dimension i…