18 problems

Ooguri–Vafa integrality conjecture. The wavefunction has the product expansion

Let $L$ be a Lagrangian with boundary $S_\Gamma$, let $A$ be a choice of phase and framing, and let $\Psi_\Gamma^{(A)}$ be the corresponding algebraic wavefunction. For coordinates…

Let $L$ be a Lagrangian with boundary $S_\Gamma$, let $A$ be a choice of phase and framing, and let $W_\Gamma^{(A)}$ be a primitive function whose graph of the differential gives t…

Let $X$ be a symplectic manifold and let $L\subset X$ be a null-homologous Lagrangian. Let $HC^-_*(Fuk(X),L)$ denote the relative cyclic homology associated to the object $L$, and…

Let $S$ be a moduli specification, let $\vec{a}$ be a vector of nonnegative integers, and let $\vec{\gamma}$ be a vector of cohomology classes in $H^*_T(\mathbb{P}^1)$. The combina…

Let $(X,D)$ be a nef Looijenga pair admitting a deformation to a pair $(X',D')$ with $X'$ toric, $D'_i$ prime toric divisors for $i<l$, and $D'_l$ nef. Let $(Y,L)$ be the correspon…

Let $(X,D)$ be a nef log Calabi–Yau surface pair with $D=D_1+\cdots+D_l$, and let $(Y,L)$ be the associated toric Calabi–Yau threefold with framed Aganagic–Vafa branes…

Let $Y(D)$ be a log Calabi–Yau surface with maximal boundary, let $d$ be a $D$-convex curve class, and let $Y^{\rm op}(D)=(Y^{(l)}\setminus D^{(l)},L)$ be the associated open geome…

Consider the topological A-model on a local toric Calabi–Yau 3-fold $Y_{\bm{l}}$ with a special Lagrangian submanifold $L$. Let $F_n^{(g)}(\mathsf{x}_1,\ldots,\mathsf{x}_n)$ be the…

Ooguri–Vafa integrality conjecture. One has

Let $X=X_\Delta^3$ be a Calabi–Yau 3-fold and let $L=L_{AV}$ be an Aganagic–Vafa A-brane. Let $X^\vee\subset(\mathbb{C}^\times)^2\times\mathbb{C}^2$ be its mirror, and let $W$ be t…

The SYZ-map conjecture. The SYZ map $\phi$ is inverse to the mirror map $\psi$. This is an equivalent formulation of the equality between the Hori–Vafa and Lagrangian Floer superpo…

Let $X$ be a semi-Fano toric manifold, meaning that its anti-canonical divisor $-K_X$ is nef. Let $W^{\text{HV}}$ and $W^{\text{LF}}$ denote its Hori–Vafa and Lagrangian Floer supe…

Let $\mathcal{X}\rightarrow X\leftarrow Y$ be toric Calabi--Yau threefolds satisfying the closed crepant resolution conjecture, let…

Let $W_H$ be the B-model prepotential obtained by integrating $\log y/q_0$ with respect to $q_0$, and write … where $W_{H,\mathrm{inst}}$ is the instanton part. Let…

Let $N$ and $k$ be the parameters of the projective hypersurface, let $P_f$ be the set of partitions $\sigma_f=(f_1,\ldots,f_{l(\sigma_f)})$ of $f$, and let … Here…

Let $M_k^k$ be the degree-$k$ projective hypersurface, let $\tau_k(x)$ and $F_o^k(x)$ be the functions introduced above, and define the mirror map … Write $x(t)$ for its inverse an…

BKMP conjecture. These integrated correlators are equal to the A-model framed open Gromov–Witten potentials of $(X,L)$, where $L$ is the mirror brane to $\Gamma_X\subset\widehat X$…