194 problems

Let $K$ be a hyperbolic knot in $S^3$. A realization assigns functions … for positive integers $N$, where $\mathsf{X}(K)$ parametrizes hyperbolic structures on the complement of…

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.

Let $K$ be a knot in $S^3$, let $J_N(K,q)$ denote its colored Jones polynomial, and consider the $\operatorname{SL}(2,\mathbb{C})$ Chern-Simons knot theory on $S^3\setminus K$. Let…

For the Deligne–Mumford moduli space of stable curves $\overline{{\mathcal M}}_{g,h}$, let $\Lambda_g^\vee(u)$ denote the total Chern polynomial of the Hodge bundle and let…

Let $SU(N)_k$ Chern–Simons theory have a bulk fusion category of anyonic topological sectors, and let the boundary CFT have local operators and topological defect lines. Bulk–bound…

Coercivity conjecture. The condition above holds for every $g>1$.

For an oriented compact three-manifold $X$, let $G$ be a semisimple, simply connected Lie group, and let $\mathcal{Z}_G^{(K)}(X)$ denote the Reshetikhin–Turaev TQFT invariant at le…

Let $L(p,q)$ be a lens space, let $G$ be a simply connected compact simple Lie group with complexified Lie algebra $\mathfrak g$, and let $\Lambda^{R}$ be the coroot lattice. The C…

Let $M$ be a closed 3-manifold, let $\mathcal{M}$ be the moduli space of flat connections on $M$, and let $\mathcal{M}'$ denote its smooth irreducible stratum. The Chern–Simons fun…

Let $Y$ be a closed oriented $3$-manifold. For each $\operatorname{SU}(2)$ Chern–Simons action $S\in\operatorname{CS}(Y)$, there exists a Puiseux series … such that the WRT invaria…

Let $\mathcal{K}$ be a hyperbolic knot. Let $N,k\rightarrow\infty$ with $u=\frac{N}{k}$ fixed, and consider the colored Jones polynomial…

Let $\mathcal{K}_{\tau}$ be a knot in $S^3$ with framing $\tau\in\mathbb{Z}$, and let $Z_{CS}^{(S^3,\mathcal{K}_{\tau})}(q,a,\mathbf{x})$ be its Chern–Simons partition function. De…

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…

Assume the asymptotic expansion conjecture holds for a closed oriented $3$-manifold $M$. Let $\mathcal{M}(M)$ be the moduli space of flat $\operatorname{SU}(N)$-connections, and le…

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

Labastida–Mariño integrality conjecture. The reformulated invariants satisfy

Moduli algebra conjecture. The moduli algebra is a quantum algebra of observables of Hamiltonian Chern–Simons theory.

Let $H$ be a finite-dimensional Hopf algebra whose quantum double is factorizable and unimodular but is often not ribbon. Consider the construction of framed-manifold invariants fr…

Quantum $SO_3$ spin-Chern-Simons conjecture. The quantum $SO_3$ spin-Chern-Simons theory is, in fact, a TQFT and at the $1$ (mod $2$)-valued $SO_3$ level $2m-1$, it is isomorphic t…

Let $X$ be a closed oriented $3$-manifold and let $G=\operatorname{SU}(2)$. Let $\mathcal M_j$ be the union of the connected components of the moduli space of flat $G$-connections…

Let $X$ be a closed oriented $3$-manifold, and let $\tau_r(X)$ denote its Reshetikhin–Turaev invariant associated to ${\mathfrak{sl}}_2(\mathbb{C})$ at integer level $r\geq 2$. Let…

Let $E$ be a unitary vector bundle with a self-dual connection, let $\widetilde{E}$ be the corresponding holomorphic bundle on the twistor space $M$, and let $n$ be the rank of…

Mariño–Vafa conjecture. The generating series of these triple Hodge integrals, for all $g$ and all $h$, can be expressed by closed finite formulas in terms of representations of sy…

Let $Z_2(K)$ be the perturbative configuration-space-integral invariant of a framed knot $K$, and let $Z^{\text{anomaly}}$ denote the framing-dependent renormalization term needed…

Let $C\subset M$ be a special Lagrangian cycle in a Calabi–Yau threefold $M$, and let $W$ be its mirror Calabi–Yau threefold. The ordinary Chern–Simons theory is defined on $C$, wh…