Abstract
Gauge-invariant discrete and Abelian ( X- Y) spin models with compact degrees of freedom are studied in ( D + 1) Euclidean (“space-time”) dimensions ( D ⩽ 3). They are formulated in terms of quantum Hamiltonians on a D-dimensional spatial lattice. In this approach the periodic Gaussian and Abelian models are identical. Duality transformations are obtained which relate the topological excitations of one model to the fundamental quantum spin excitations of another. These motivate the extension of the usual gauge-invariant structure to “gauge invariance of the third kind” which involves degrees of freedom associated with the surfaces of the plaquettes. The nature of the excited states and the behaviour of the order (and disorder) parameters are discussed in various limits of the coupling constants. For both discrete and Abelian theories the behaviour of the order parameter (the Wilson loop) depends crucially on whether the spins in the theory carry the fundamental unit of charge or a multiple of it.
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