10 problems

The physical configuration space of a gauge theory may be compact in certain directions, as indicated by the possibility that the averaging integral has infinitely many stationary…

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…

Let $N\geq 3$, let $\Phi(x)\in\sqrt{\beta}\,\mathbf{S}^{N-1}\subset\mathbb{R}^{N}$ for $x\in\mathbb{Z}^{2}$, and consider an infinite-volume limit formally corresponding to … Here…

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 $S^k$ be the target sphere of radius $R$, and let $\tilde{\mathcal{L}}_{N,U}(S^k)$ be the manifold with boundary obtained by removing a tubular neighborhood…

Let the arboreal gas be defined on $\mathbb{Z}^2$, let $\beta>0$, and write $\mathbb{P}_{\beta}[i\leftrightarrow j]$ for the probability that $i$ and $j$ lie in the same tree. Let…

Let $SO(N)$ be the group of $N\times N$ orthogonal matrices with determinant $1$, let $\Lambda\subset\mathbb{Z}^d$ be finite, and let $\mu_{\Lambda,N,\beta}$ be the lattice gauge-t…

Consider the mass gap of the bosonic ${\mathbb C}{\rm P}^{N-1}$ model in the weak-coupling, small-circle regime. In this regime, the mass gap is non-perturbative and proportional t…

In Hamiltonian ${\rm SU}(2)$ gauge theory on a finite rectangular lattice with open boundary conditions, let the ground-state and first-excited-state energies be the two lowest ene…

For a quantum crystal in any dimension, let $\gamma$ denote its mass gap, let $J$ be the coupling parameter, and let $q$ be the deformation parameter. Mass-gap conjecture. The mass…