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The symmetric space sine-Gordon models arise by conformal reduction of ordinary 2-dim σ-models, and they are integrable exhibiting a black-hole type metric in target space. We provide a Lagrangian formulation of these systems by considering a triplet of Lie groups F ⊃ G ⊃ H. We show that for every symmetric space F/G, the generalized sine-Gordon models can be derived from the G/H WZW action, plus a potential term that is algebraically specified. Thus, the symmetric space sine-Gordon models describe certain integrable perturbations of coset conformal field theories at the classical level. We also briefly discuss their vacuum structure, Backlund transformations, and soliton solutions. Of course, ordinary σ-models are not consistent string backgrounds quantum mechanically; this geometrical approach is only used to motivate the definition of classical reduced σ-models, as it was originally done in the literature.

Abstract:We consider the geometric and non-geometric faces of closed string vacua arising by T-duality from principal torus bundles with constant H-flux and pay attention to their double phase space description encompassing all toroidal coordinates, momenta and their dual on equal footing. We construct a star-product algebra on functions in phase space that is manifestly duality invariant and substitutes for canonical quantization. The 3-cocycles of the Abelian group of translations in double phase space are seen to account for non-associativity of the star-product. We also provide alternative cohomological descriptions of non-associativity and draw analogies with the quantization of point-particles in the field of a Dirac monopole or other distributions of magnetic charge. The magnetic field analogue of the R-flux string model is provided by a constant uniform distribution of magnetic charge in space and non-associativity manifests as breaking of angular symmetry. The Poincaré vector comes to rescue angular symmetry as well as associativity and also allow for quantization in terms of operators and Hilbert space only in the case of charged particles moving in the field of a single magnetic monopole.