Generalized Kähler geometry and manifest N = (2,2)supersymmetric nonlinear sigma-models

Generalized complex geometry is a new mathematical framework that is useful for describing the target space of N ≤ (2,2) nonlinear sigma-models. The most direct relation is obtained at the N ≤ (1,1) level when the sigma model is formulated with an additional auxiliary spinorial field. We revive a formulation in terms of N ≤ (2,2) semi-(anti)chiral multiplets where such auxiliary fields are naturally present. The underlying generalized complex structures are shown to commute (unlike the corresponding ordinary complex structures) and describe a Generalized Kähler geometry. The metric, B-field and generalized complex structures are all determined in terms of a potential K.