24 problems

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…

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…

Let $\nu=\nu(\beta,q,d)$ be Potts lattice gauge theory on $\mathbb{Z}^d$ defined as a weak limit of finite-volume measures. For a rectangular boundary $\gamma$ in $\mathbb{Z}^d$, l…

Wilson-loop BPS expansion conjecture. An analogous refined BPS expansion with non-negative expansion coefficients can also be applied to Wilson loops.

Wilson-loop gauge/string duality conjecture. The gauge/string duality should extend beyond the partition function and hold for Wilson loop expectations.

Let $X$ be a non-compact Calabi–Yau threefold, let $B$ be a magnetic flux satisfying the flux quantization condition … and let $Z(\epsilon_1,\epsilon_2,t_i)$ be the refined topolog…

Let $\omega^{(0)}$ be the holomorphic-limit Wilson loop in a fixed frame, and let $\ell {\cal A}$ be a Borel singularity, where ${\cal A}$ is a primitive singularity and…

Let $\mathbf{r}_1,\ldots,\mathbf{r}_l$ be representations, let $\mathbf{r}_{i_1}\otimes\cdots\otimes\mathbf{r}_{i_l}$ be their tensor product, and let…

In M-theory, let $X$ be a Calabi–Yau manifold and let $\mathsf{C}$ be a primitive stationary non-compact holomorphic curve class, with associated non-decomposable representation…

Let $(M,a)$ be a closed, area-weighted topological map, embedded in a closed, connected, orientable surface, and let $G_N=\mathrm{U}(N)$. A loop is included in a disc if its drawin…

Let $\Sigma$ be a closed, connected, orientable, two dimensional, Riemannian manifold, the Euclidean plane $\mathbb{R}^2$, or a disc in $\mathbb{R}^2$. Let $G_N$ be a classical gro…

Let $(\mathcal{P},[n])$ be a Wilson loop diagram, with associated positroid cells of dimension $3|\mathcal{P}|$, and let the Amplituhedron parametrize positroid cells of dimension…

Let $W$ be a Wilson loop diagram with associated positroid cell $\Sigma(W)$, and let $R(W)$ denote the denominator of its differential form $\Omega(W)$. The zeros of $R(W)$ produce…

The Amplituhedron is the geometric object associated with the positive Grassmannian in the scattering-amplitude framework. Orientability conjecture. The Amplituhedron is orientable…

Let $G\subseteq U(N)$ be any compact non-Abelian Lie group for some $N\geq 2$, and consider any infinite-volume limit of four-dimensional lattice gauge theory with gauge group $G$…

Let $\Sigma$, $U$, $C$, $a$, and $c=\operatorname{area}(\Sigma)-a$ be as in the preceding conjecture, with $C$ a simple closed curve in the topological disk $U$. For…

Let $\Sigma$ be a surface of area $\operatorname{area}(\Sigma)$, let $U\subset\Sigma$ be a topological disk, and let $C$ be a simple closed curve in $U$. Assign area $a$ to the int…

Let the gauge group be $SU(N)$), and consider planar scattering amplitudes of maximally supersymmetric Yang–Mills theory in the large-$N$ limit. The trace ordering of the gluons de…

In ${\cal N}=4$ supersymmetric Yang–Mills theory, consider the expectation value of circular Wilson loops. Gaussian matrix-model conjecture. The expectation value of a circular Wil…

Let $n\geq 0$, let $u_1,\ldots,u_{2n}$ be Bethe rapidities, and define … The matrix part $\Pi_{mat}^{(2n)}(u_1,\ldots,u_{2n})$ is independent of the coupling $\lambda$ and depends…

Let $N_c\to\infty$ in planar ${\cal N}=4$ super-Yang–Mills theory, and let $\lambda=N_c g_{YM}^2$ be the ’t Hooft coupling. Consider gluon scattering amplitudes and expectation val…

Let $l$ be a closed loop in the lattice and let $W_l=\operatorname{Tr}(Q_{e_1}Q_{e_2}\cdots Q_{e_n})$ be its Wilson loop variable. Let the minimum surface area enclosed by $l$ mean…

QCD evolution conjecture. In a QCD setting beyond leading order, the evolution equation is conjectured to be