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We investigate N=(2,2) supersymmetric nonlinear sigma-models in the presenceof a boundary. We restrict our attention to the case where the bulk geometry isdescribed by chiral and twisted chiral superfields corresponding to abihermitian bulk geometry with two commuting complex structures. The D-braneconfigurations preserving an N=2 worldsheet supersymmetry are identified.Duality transformations interchanging chiral for twisted chiral fields and viceversa while preserving all supersymmetries are explicitly constructed. Weillustrate our results with various explicit examples such as the WZW-model onthe Hopf surface S3xS1. The duality transformations provide e.g new examples ofcoisotropic A-branes on Kahler manifolds (which are not necessarilyhyper-Kahler). Finally, by dualizing a chiral and a twisted chiral field to asemi-chiral multiplet, we initiate the study of D-branes in bihermitiangeometries where the cokernel of the commutator of the complex structures isnon-empty.

An N=2 worldsheet approach to D-branes in bihermitian geometries: I. Chiral and twisted chiral fields