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Exact duality transformations for sigma models and gauge theories. (English) Zbl 1062.82009
Summary: We present an exact duality transformation in the framework of statistical mechanics for various lattice models with non-Abelian global or local symmetries. The transformation applies to sigma models with variables in a compact Lie group $$G$$ with global $$G\times G$$-symmetry (the chiral model) and with variables in coset spaces $$G/H$$ and a global $$G$$-symmetry [for example, the nonlinear $$\text{O}(N)$$ or $$\mathbb {RP}^N$$ models] in any dimension $$d \geq 1$$. It is also available for lattice gauge theories with local gauge symmetry in dimensions $$d\geq 2$$ and for the models obtained from minimally coupling a sigma model of the type mentioned above to a gauge theory. The duality transformation maps the strong coupling regime of the original model to the weak coupling regime of the dual model. Transformations are available for the partition function, for expectation values of fundamental variables (correlators and generalized Wilson loops) and for expectation values in the dual model which correspond in the original formulation to certain ratios of partition functions (free energies of dislocations, vortices or monopoles). Whereas the original models are formulated in terms of compact Lie groups $$G$$ and $$H$$, coset spaces $$G/H$$ and integrals over them, the configurations of the dual model are given in terms of representations and intertwiners of $$G$$ and $$H$$. They are spin networks and spin foams. The partition function of the dual model describes the group -theoretic aspects of the strong coupling expansion in a closed form.
##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 81T25 Quantum field theory on lattices 81T10 Model quantum field theories 81T13 Yang-Mills and other gauge theories in quantum field theory 82B10 Quantum equilibrium statistical mechanics (general)
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