136 problems

Gopakumar–Vafa conjecture. The large $N$ limit of $SU(N)$ Chern–Simons gauge theory on $S^3$ is equivalent to the closed topological-string theory on the resolved conifold.

Ooguri–Vafa conjecture. There exists a corresponding Lagrangian submanifold $\mathcal{D}_{\mathcal{K}}$ such that

Let $Z_{BH}$ denote the black hole partition function and let $Z_{\text{top}}$ denote the topological string partition function. OSV conjecture. The black hole partition function e…

Let $X$ be a non-compact Calabi–Yau threefold, let $Z(\epsilon_1,\epsilon_2,t_i)$ be the refined topological-string partition function, and let $\mathbf r$ be a representation. Def…

GHM spectral determinant conjecture.

Let $\mathfrak{X}$ be a toric Calabi–Yau threefold and let $\hat{\mathfrak{X}}$ be its mirror, whose mirror curve $\mathcal{S}_{\hat{\mathfrak{X}}}$ is defined by … Let…

Quantum foam conjecture. The statistical model of random plane partitions is nothing but quantum Kähler gravity.

LMOV conjecture. For every $\mu\in\mathcal{P}_+$, the function $F_{\mu}^{(X,\mathcal{D})}$ has the integral expansion given by the right-hand side of the open multiple-covering for…

In a two-dimensional conformal field theory with Virasoro symmetry, consider its conformal blocks, and in four-dimensional gauge theory consider the instanton partition functions c…

Let $\rho$ be the operator whose Fredholm determinant is denoted by $\Xi(\kappa)$, with $\kappa$ identified with a complex modulus of the mirror curve. Let $\mu$ satisfy the corres…

Let $X$ be a Calabi–Yau threefold, and consider type IIA string theory compactified on $X$. Fix the magnetic D6- and D4-brane charges $P_{6}$ and $P_{4}$, and sum over electric D2-…

Let $F_g(t)$ be the generating function of genus-$g$ open topological-string amplitudes with boundary weights $t$. Open/closed string duality. There is a family $\tilde M(t)$ of Ca…

Let a local Calabi–Yau background have geometry reducing to a complex curve, and let its B-model holomorphic topological-string amplitudes be described by genus-$g$ free energies a…

Let $W(x)$ and $f(x)$ be polynomials of degrees $d+1$ and $d-1$, respectively, and consider the local Calabi–Yau background … Its associated hyperelliptic spectral curve is … which…

Let a noncompact Calabi–Yau manifold be a local background whose geometry reduces to a complex curve, and let $F^{(g)}(t)$ denote the holomorphic genus-$g$ free energies of the ass…

Let $\Psi$ be the B-model wave function in holomorphic polarization, and let ${\rm Fr}\Psi$ denote its $p$-adic Frobenius transform. Let $\zeta$ be an expression whose integrality…

Let $t_c$ be the critical value of the Kähler parameter, and let … be a natural coordinate that measures the distance to the conifold point, so that $\mu=0$ at the conifold. Let…

Let $K$ be a knot and let $\mathcal{H}^{\bullet}_N\{+1\}-\mathcal{H}^{\bullet}_N\{-1\}$ denote the doubly graded complex obtained from the $sl(N)$ homology by shifting the $q$-grad…

Consider a topological string theory obtained by twisting a unitary $N=2$ $d=2$ field theory and coupling it to topological gravity. Let the category of topological D-branes encode…

Consider a noncompact Calabi–Yau threefold whose orientifolded geometry undergoes a local geometric transition: two contracted copies of $\mathbb{P}^1$ and one contracted copy of…

Second Ooguri–Vafa conjecture. The two free energies are identical:

The Chern–Simons system considered in this section has a free energy whose perturbative expansion is compared with the open-string amplitudes, with the local $dP_2$ model denoting…

Global Stokes transformation–BPS spectrum-generator conjecture. The global Stokes transformation can be identified with the spectrum generator of BPS invariants.

Stokes–BPS correspondence. The Stokes constants of the topological string free energy can be identified with the BPS invariants of the associated four-dimensional theory.

For a genus-one fibered Calabi–Yau threefold, let the topological string partition function be decomposed by base degree, and let the topological string coupling be denoted by the…