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A series of solvable lattice models with face interaction are introduced on the basis of the affine Lie algebraX (1) =A (1) ,B (1) ,C (1) ,D (1) . The local states taken on by the fluctuation variables are the dominant integral weights ofX (1) of a fixed level. Adjacent local states are subject to a condition related to the vector representation ofX n . The Boltzmann weights are parametrized by elliptic theta functions and solve the star-triangle relation.

The variational calculus for the Faddeev-Hopf model on a general Riemannian domain, with general Kähler target space, is studied in the strong coupling limit. In this limit, the model has key similarities with pure Yang-Mills theory, namely conformal invariance in dimension 4 and an infinite dimensional symmetry group. The first and second variation formulae are calculated and several examples of stable solutions are obtained. In particular, it is proved that all immersive solutions are stable. Topological lower energy bounds are found in dimensions 2 and 4. An explicit description of the spectral behaviour of the Hopf map $${S^3 \rightarrow S^2}$$ is given, and a conjecture of Ward concerning the stability of this map in the full Faddeev-Hopf model is proved.

This article shows that for a large class of discrete periodic Schrödinger operators, most wavefunctions resemble Bloch states. More precisely, we prove quantum ergodicity for a family of periodic Schrödinger operators H on periodic graphs. This means that most eigenfunctions of H on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our results cover the adjacency matrix on $$\mathbb {Z}^d$$ , the triangular lattice, the honeycomb lattice, Cartesian products, and periodic Schrödinger operators on $$\mathbb {Z}^d$$ . The theorem applies more generally to any periodic Schrödinger operator satisfying an assumption on the Floquet eigenvalues.

A spherically symmetric solution of the “already unified field theory” ofRainich (i.e. of the source-free Maxwell-Einstein equations) is presented which represents a static massless charged particle. It is not equivalent to the Reissner-Nordström solution with zero mass, although both metrics repel uncharged test particles.

The main result of this paper is a rigorous computation of Wilson loop expectations in strongly coupled SO(N) lattice gauge theory in the large N limit, in any dimension. The formula appears as an absolutely convergent sum over trajectories in a kind of string theory on the lattice, demonstrating an explicit gauge-string duality. The generality of the proof technique may allow it to be extended other gauge groups.