Complex structure-induced deformations of σ-models

We describe a deformation of the principal chiral model (with an evendimensional target space G) by a B-field proportional to the Kähler form on the target space. The equations of motion of the deformed model admit a zero-curvature representation. As a simplest example, we consider the case of G = S × S. We also apply a variant of the construction to a deformation of the AdS3 × S × S (super-)σ-model. The main goal of this paper is to relate the σ-models introduced by the author [1], [2], [3] to the so-called η-deformed models [4], which have recently attracted considerable interest [5], [6]. The relation is based on the interpretation of the Rmatrix of the latter models as a complex structure on the target space of the σ-model. In this case the R-matrix has eigenvalues ±i and is therefore non-degenerate. This is in contrast to the R-matrix utilized in [4], [5], where it has a nontrivial null space. The relation that we find between the two classes of models is not one-to-one. Let us explain this. First of all, the models of [3] do not have any free parameters – in this sense they are not deformations of any simpler models. However, in certain cases they may be obtained as limits of the η-deformed models for special values of η (In standard normalization, this is η = ±i). For instance, the η-deformed model with target space SU(N) degenerates in this limit to a σ-model with target space SU(N)/S(U(1)) – the complete flag manifold. Two remarks are in order: • The target spaces of σ-models obtained in such limit are always of the type G/H with abelian H . On the other hand, the models of [3] are defined for arbitrary complex homogeneous spaces, irrespective of whether H is abelian. They are only well-defined, however, for Euclidean worldsheets (From the point of view of the limit, the reason is that η needs to be taken complex). • The inverse procedure does not exist in general, i.e. there is in general no ηdeformation of the flag manifold σ-model. The reason is that the limit η → ±i in general irreversibly modifies the target space of the model. When it does not, the R-matrix deformation provides generalizations of the models of [3]. This is so, for instance, when the target space is a group manifold, and the ∗Emails: [email protected], [email protected]