Complex structures and zero-curvature equations for σ -models

Abstract

We construct zero-curvature representations for the equations of motion of a class of σ-models with complex homogeneous target spaces, not necessarily symmetric. We show that in the symmetric case the proposed flat connection is gauge-equivalent to the conventional one. * Emails: dmitri.bykov@aei.mpg.de, dbykov@mi.ras.ru 1 Generalizations to non-simple groups G are possible, but will be considered elsewhere.

Complex structure-induced deformations of σ-models

“…In this case To finalize this discussion, let us point out that, although the models of [1,2] are obtained from (1.3) as particular limits, some generalizations thereof considered in [3] cannot be obtained in this way. In [3] it was pointed out that an action of the type (2.16) defines a model with a Lax connection for any complex homogeneous target space, whose Killing metric is compatible with the complex structure. As an example, one can consider…”

“…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 R-matrix of the latter models as a complex structure on the target space of the σ-model.…”

“…Let us explain this. First of all, the models of [3] do not have any free parameters -in this JHEP03(2017)130 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).…”

“…• 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).…”

“…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 R-matrix is a complex structure on this group -then the η-deformed model interpolates between the model of [3] and the principal chiral model. In this case the deformation amounts to adding a B-field proportional to the (non-closed) Kähler form on the target space.…”