45 problems

The Chern–Simons partition function is expressed in terms of quantum invariants of a link, with $P_{\overrightarrow{A}}(\mathcal{L};q,t)$ denoting the colored quantum invariant and…

Labastida–Mariño–Ooguri–Vafa conjecture. For any $\vec{A}\in\mathcal{P}^{L}$, there exist $P_{\vec{B}}(q,t)$ for $\vec{B}\in\mathcal{P}^{L}$ such that

String formulation conjecture. Chiral intersecting loop amplitudes are described by the modified construction obtained by allowing $f^{-1}(\amalg\Gamma)$ to be a graph and allowing…

String interpretation conjecture. The chiral intersecting Wilson loop amplitudes are obtained by imposing the boundary condition that $f:\partial\Sigma_W\to\amalg\Gamma$ lies in th…

Finite-area equivalence conjecture. The perturbation is equivalent to two-dimensional Yang–Mills theory at finite area, as seen from the contribution of simple Hurwitz space.

Let $CY_3$ be a Calabi–Yau threefold, and consider the topological A and B models targeting this same $CY_3$. S-duality conjecture. There is an S-duality between the A and B models…

Conjecture. The quantity

Topological vertex conjecture. The expression above is the exact trivalent vertex amplitude. This identifies the representation-theoretic expression derived from the two-point func…

The instanton amplitudes are associated with non-perturbative corrections to the perturbative partition function of topological string theory, and they are functionals of that part…

Costello and Li proposed twists of Type II string theories and topological string theories as related physical theories. Costello–Li conjecture. Certain twists of Type II string th…

Consider $SU(N)$ Chern–Simons gauge theory on $S^3$, and let $M_6=T^{\ast}S^3$ be the cotangent bundle of the three-sphere. Witten's large N duality. This theory is conjecturally e…

Type I twist conjecture. The twist of the type I string is the type I topological string, which is an orientifold of the familiar topological $B$ model.

Type IIB twist conjecture. The holomorphic twist of type IIB on a Calabi–Yau $5$-fold is the topological $B$-model.

Twisted type IIB supergravity is conjectured to be equivalent to BCOV theory, and twisted type IIB superstring theory is expected to admit a corresponding description by the topolo…

BPS interpretation conjecture. The Stokes coefficients $\mathsf{S}^{\mathcal A}_{\omega\omega'}$ appearing in this resurgent structure are interesting invariants counting BPS state…

Let $X$ be a CalabiYau $5$-fold. The type I topological string on $X$ is the unoriented topological $B$-model with its open-string sector, whose gauge theory has structure group…

The type I topological $B$-model has fields given by the $/2$-fixed points of KodairaSpencer theory on a CalabiYau $5$-fold, coupled to an ${\rm SO}(32)$ holomorphic ChernSimons th…

Let $Q=e^{2\pi i t}$, and define … Let $F_{\mathrm{GV}}(\lambda,t)$ and $F_{\mathrm{NS}}(g,t)$ denote the Gopakumar–Vafa and Nekrasov–Shatashvili free energies, respectively. Hatsu…

Let $F_{\rm coni}^{\rm resum}(\lambda,t)$ be the analytic function considered in the source, and let $F_{\mathbb R_{>0}}(\lambda,t)$ denote the Borel summation along the positive r…

Let $F_{\rm GV}(\lambda,t)$ be the Gopakumar–Vafa free energy of the resolved conifold, and let $F_{\rm np}(\lambda,t)$ be the non-perturbative completion defined by … Here…

Consider the asymptotic power series arising in topological string theory from conifold free energies and quantized flat coordinates. A peacock pattern is an infinite tower of sing…

Let $\Xi_{{\mathbb F}_0}(\kappa,\hbar,\xi)$ be the spectral determinant of local ${\mathbb F}_0$, and let $\xi$ denote its mass parameter. A q-deformed Painlevé equation is a diffe…

Let $\rho_{{\mathbb F}_0}$ be the operator associated with local ${\mathbb F}_0$, let $\hbar>0$, and let $Z_1(\hbar)=\operatorname{Tr}\rho_{{\mathbb F}_0}$ be its first fermionic s…

A peacock pattern is a resurgent structure involving infinitely many integer Stokes constants. A quantum curve is the quantum-geometric structure underlying a theory based on a qua…

Matrix models arising from topological string theory include the $U(N)$ Chern–Simons matrix model on $S^3$, whose weak-confinement potential is associated with intermediate statist…