26 problems

Self-dual critical-point conjecture. In these systems, the critical point is conjectured to be

Let $\r=(\varepsilon,\mu)$ and \r'=(\varepsilon',\mu') be finite-support electric and magnetic charge distributions, with total charges q_\r and q_{\r'}. Let \pi_\r and…

Lattice-model equivalence conjecture. The lattice gauge model presented in the paper indeed coincides with the Hamiltonian Chern–Simons theory.

Let the lattice gauge models with plaquette action $\left(\frac{1+L(U_P)}{2}\right)^p$ be considered in dimension $d$, where $L(U_P)=\operatorname{Tr}(U_P+U_P^*)$ and $U_P^*$ denot…

Let $\nu=\nu(\beta,q,d)$ be Potts lattice gauge theory on $\mathbb{Z}^d$. Let $\sigma_N=[0,1]^2\times\{N\}\times\{0\}$ be a plaquette, and for a plaquette $\sigma$ let…

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…

Let $a$ and $\hat a$ be ${\mathbb Z}_n$-valued 1-cochains. Consider the Euclidean theory with action … It has a ${\mathbb Z}_n\times{\mathbb Z}_n$ 1-form symmetry and a…

Let $a$ be a ${\mathbb Z}_2$-valued 1-cochain on a triangulated 3-manifold, with Euclidean action … This theory has a global 1-form symmetry $a\mapsto a+\alpha$ for a 1-cocycle…

Plaquette invaded-cluster convergence conjecture. The stationary distribution of the plaquette invaded-cluster algorithm converges weakly to Potts lattice gauge theory in the infin…

Let electrons hop on a planar, bipartite graph at half-filling, and consider magnetic fluxes through its lattice plaquettes. Flux-phase conjecture. The optimum, energy-minimizing m…

Restriction bijectivity conjecture. The function $Res$ is a bijection.

Fix integers $q\geq 2$ and $0\leq i<d$. For hyperrectangular $(i-1)$-boundaries $\gamma$ in $\mathbb{Z}^d$, let $\operatorname{Area}(\gamma)$ and $\operatorname{Per}(\gamma)$ denot…

Let $q$-state Potts lattice gauge theory be defined on $\mathbb{Z}^d$, and let $W_\gamma$ denote the Wilson loop variable associated with a loop $\gamma$. A perimeter law means exp…

The two-dimensional Villain model is the lattice gauge model with compact phase variables and Villain interaction at inverse temperature $\beta$. At large scales, its field is expe…

Low-energy taming-background conjecture. The only taming backgrounds that need to be analyzed at low energies are those that minimize the potential terms in $H$.

Relevant-background conjecture. These are all the backgrounds that are relevant at low energies.

Maxwell crossover conjecture. Intermediate couplings should show a crossover between the topological and confined phases.

Let $G$ be a compact subset of the unit octonions, and consider the lattice gauge theory and its prelinearized version on a polygon triangulated by regular triangles. For $G=SU(2)$…

In Hamiltonian lattice gauge theory, let the configuration space be decomposed into gauge orbits with a stratification into orbit-type strata, and let the quantum Hamiltonian have…

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 $l$ 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…

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 system of $48$ polynomial equations in the parameters $\{\phi_{\boldsymbol{j},\mu}\}$, which has $11664$ complex solutions for generic parameter values. Restrict the p…

Consider a lattice Yang–Mills model with microscopic gauge group $G$, whose ’t Hooft flux takes values in a subgroup $\Pi_2$ of $\pi_1(G)$ and whose discrete theta-angles are label…

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…