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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.

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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.

Four-dimensional string backgrounds with local realizations of N = 4 world-sheet supersymmetry have, in the presence of a rotational Killing symmetry, only one complex structure which is an SO(2) singlet, while the other two form an SO(2) doublet. Although N = 2 world-sheet supersymmetry is always preserved under Abelian T-duality transformations, N = 4 breaks down to N = 2 in the rotational case. A non-local realization of N = 4 supersymmetry emerges, instead, with world-sheet parafermions. For SO(3)-invariant metrics of purely rotational type, like the Taub-NUT and the Atiyah-Hitchin metrics, none of the locally realized extended world-sheet supersymmetries can be preserved under non-Abelian duality

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We present a Lagrangian description of the SU(2)/U(1) coset model perturbed by its first thermal operator. This is the simplest perturbation that changes sign under Krammers-Wannier duality. The resulting theory, which is a 2-component generalization of the sine-Gordon model, is then taken in Minkowski space. For negative values of the coupling constant g, it is classically equivalent to the O(4) non-linear σ-model reduced in a certain frame. For g > 0, it describes the relativistic motion of vortices in a constant external field. Viewing the classical equations of motion as a zero curvature condition, we obtain recursive relations for the infinitely many conservation laws by the abelianization method of gauge connections. The higher spin currents are constructed entirely using an off-critical generalization of the W ∞ generators. We give a geometric interpretation to the corresponding charges in terms of embeddings. Applications to the chirally invariant U(2) Gross-Neveu model are also discussed.