182 problems

Let $M_{n,a}$ be a smooth hypersurface of degree $a\le n+1$ in $\mathbb{CP}^{n+1}$, let $D_{n,a,i}$ be a union of $i\le n+2$ generic hyperplanes, and let…

Let $(U,f)$ be a Landau–Ginzburg model, where $U$ is a smooth quasi-projective variety and $f\colon U\to \mathbb{A}^1$ is a regular function with reduced pole divisor. The Landau–G…

Let $X$ be a smooth Fano variety. A toric Landau–Ginzburg model existence conjecture asserts that $X$ has a Laurent polynomial satisfying the period, Calabi–Yau, and toric conditio…

Let $W$ be an invertible polynomial, let $W^T$ be its Berglund–Hübsch transpose, and let $G_W$ be the maximal group of diagonal symmetries of $W$. LG/Saito–Givental mirror equivale…

Let $X$ be a Fano manifold. A Landau–Ginzburg model for $X$ is a one-parameter family $Y\to\mathbb{A}^1$, with associated Fukaya category $Fuk(X)$, Fukaya–Seidel category $FS(Y)$,…

For the singularities under consideration, let the partition functions be those arising from the associated topological gravity coupled with matter. Witten's ADE integrable hierarc…

For an admissible LG pair $(W,G)$, let the Fan–Jarvis–Ruan–Witten, Polishchuk–Vaintrob, and Chang–Li–Li–Kiem–Li constructions define three cohomological field theories. Folklore eq…

Let $Y$ be a smooth Fano variety of index $r$ and $D_Y\thinspace\subseteq\thinspace Y$ a smooth anti-canonical divisor. Let $X$ be the $r$-fold cyclic covering of $Y$ branched alon…

Let $X$ be a derived critical locus, and let $f$ range over potentials whose derived critical locus is $X$. Derived spectrum equivalence conjecture. The derived spectrum is the sam…

Let $X$ be a Fano variety and let $(Y,w)$ be a Landau–Ginzburg model forming a homological mirror pair with $X$. Let $f^{p,q}(Y,w)$ denote the Hodge-theoretic invariants of the Lan…

Singular complete-intersection Knörrer periodicity conjecture. Under appropriate hypotheses on $f_1,\dots,f_k$, there is a quasiequivalence of $A_\infty$-categories

Witten's Landau–Ginzburg/Calabi–Yau correspondence conjecture. The Gromov–Witten invariants of $Q_w$ should be computable by the Landau–Ginzburg model above, whose curve-counting i…

Let $(Y_1,w_1)$ and $(Y_2,w_2)$ be gauged Landau–Ginzburg models with $Y_i$ smooth, a group $G$ acting on each $Y_i$, and $w_i$ a section of $\operatorname{\mathcal O}(\chi)$ for a…

Let $X$ be a Fano manifold and consider the moduli spaces of Landau–Ginzburg models for its mirror. Landau–Ginzburg moduli conjecture. The geometry of these moduli spaces determine…

Let $M$ be the formal moduli space of Landau–Ginzburg models, equipped with its $\mathbb{C}^*$-action, and let $V$ be the local system over $M$ arising from the cohomologies of the…

Consider the Lagrangian cones $\mathcal L_{\operatorname{FJRW}}$ and $\mathcal L_{\operatorname{GW}}$ in the symplectic spaces $\mathcal V_{\operatorname{FJRW}}$ and…

Barannikov–Kontsevich conjecture. The Frobenius structure $(S',\eta_F|_{S'},\Phi_F|_{S'})$ should be a Frobenius submanifold of the Frobenius manifold constructed by Barannikov–Kon…

Let $Y_1$ and $Y_2$ be manifolds with mirror Landau–Ginzburg models $(X_1,W_1)$ and $(X_2,W_2)$. The mirror of $Y_1\times Y_2$ is $(X_1\times X_2,W_1+W_2)$, and let…

Let ${\bf G}$ be a Grassmannian and let ${\bf P}$ denote the projective-space case. A Landau–Ginzburg theory is a theory described by a potential function $W$ and mirror-symmetric…

In a quasi-homogeneous Landau–Ginzburg model, let $W$ be the worldsheet superpotential and let $K(x_i,\bar x_i)$ be a Kähler potential for the chiral fields $x_i$. Conformal Kähler…

Let the A-model contain the anti-diagonal Lagrangian 2-sphere in $\mathbb{C}\mathbb{P}^1\times\mathbb{C}\mathbb{P}^1$, and let the B-model contain the two irreducible D2-branes ass…

Landau–Ginzburg compactification conjecture. $X$ admits a Landau–Ginzburg model with a smooth log Calabi–Yau compactification.

Let $r$ be the spin parameter, let $g\geq 2$, let $F_g^{x^r,o}$ denote the open $r$-spin potential, and let $\phi_g$ denote the corresponding closed $r$-spin potential. The notatio…

Folklore conjecture. For generic coefficients $\mathbf{a}$, the Fukaya–Seidel-type category associated with $W_{\mathbf{a}}$ admits a Bridgeland stability condition whose central c…

Exceptional-collection and Landau–Ginzburg correspondence conjecture. The maximal length of an exceptional collection on $X$ equals the number of fibers with ordinary double points…