We give a physical derivation of generalized Kähler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri [10] regarding the equivalence between generalized Kähler geometry and the bi-hermitean geometry of Gates-Hull-Roček [9]. When cast in the language of supersymmetric sigma models, this relation maps precisely to that between the Lagrangian and the Hamiltonian formalisms. We also discuss topological twist in this context.

Starting from first principles, a constructive method is presented to obtain boundary states in conformal field theory. It is demonstrated that this method is well suited to compute the boundary states of logarithmic conformal field theories. By studying the logarithmic conformal field theory with central charge c = −2 in detail, we show that our method leads to consistent results. In particular, it allows to define boundary states corresponding to both, indecomposable representations as well as their irreducible subrepresentations.

We propose the definition of (twisted) generalized hyperkähler geometry and its relation to supersymmetric non-linear sigma models. We also construct the corresponding twistor space. * Andreas.Bredthauer@teorfys.uu.se

We investigate the set of boundary states in the symplectic fermion description of the logarithmic conformal field theory with central charge c = −2. We show that the thus constructed states correspond exactly to those derived under the restrictions of the maximal chiral symmetry algebra for this model , W(2, 3, 3, 3). This connects our previous work to the coherent state approach of Kawai and Wheater. * Andreas.Bredthauer@itp.uni-hannover.de

We study the conditions under which N = (1, 1) generalized sigma models support an extension to N = (2, 2). The enhanced supersymmetry is related to the target space complex geometry. Concentrating on a simple situation, related to Poisson sigma models we develop a language that may help us analyze more complicated models in the future. In particular, we uncover a geometrical framework which contains generalized complex geometry as a special case.