16 problems

Weak-coupling Yang–Mills mass gap conjecture. For any bare coupling constant $g\in[0,g_0[$, the spectrum of $H^g$ contains $0$ as a simple eigenvalue for the vacuum eigenstate and…

Let $G$ be the gauge group and let the nontrivial Yang–Mills theory on Minkowskian spacetime $\mathbf{R}^{1,3}$ be the theory whose existence and quantization are asserted by the Y…

Yang–Mills mass gap conjecture. For any compact simple Lie group $G$ there exists a nontrivial Yang–Mills theory on the Minkowskian $\mathbf{R}^{1,3}$ whose quantization satisfies…

Trivial LGBs are associated with pre-classical theories. There exists a $\lambda \in \Omega^1(M;\mathcal{g})$ such that $\widetilde{\nabla^{\mathrm{YM}}}^{\lambda}$ is flat.

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 $G$ be a Lie group satisfying the condition given at the beginning of Section 3.1. Let $\mathfrak{g}$ be its Lie algebra, and let $\{\mathbf{A}^n\}_{n\geq 1}$ be a sequence of…

Let $G$ be a Lie group satisfying the condition given at the beginning of Section 3.1, and let $\mathcal{X}$ be the state space of Yang–Mills heat-flow trajectories. For $t>0$, wri…

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 ${\sf HF}$ be a regular Markovian holonomy field, and let an admissible Lévy process determine a Yang–Mills field through the Yang–Mills-fields construction. Yang–Mills charact…

Global existence and convergence conjecture. There is a smooth solution $A(t)$ for $t\in[0,\infty)$ to this flow for the Yang–Mills energy functional with respect to the $L^2$ Riem…

Lagrangianity conjecture. The submanifold $L_M$ is Lagrangian for non-Abelian Yang–Mills theory.

Let $G$ be a compact Lie group and let $P$ be a principal $G$-bundle over a Riemann surface $\Sigma$. Write $\mathcal A(P)$ for the space of connections on $P$, $\mathcal G_0(P)$ f…

Let $(X,L)$ be a smooth compact polarized variety and let $E$ be a holomorphic vector bundle over $X$ of rank $r$ and degree zero. For positive coupling constants…

Multiple-solutions conjecture. For a rather general family of boundary values, multiple solutions to $(\mathcal{D}_{\epsilon})$ should exist in each component $\mathcal{A}_k(A_0)$,…