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We study modular properties of the AdS3 WZNW model. Although the Euclideanpartition function is modular invariant, the characters on the Euclidean torusare ill-defined and their modular transformations are unknown. We reconsiderthe characters defined on the Lorentzian torus, focusing on their structure asdistributions. We find a generalized S matrix, depending on the sign of thereal modular parameters, which has two diagonal blocks and one off-diagonalblock, mixing discrete and continuous representations, that we fully determine.We then explore the relations among the modular transformations, the fusionalgebra and the boundary states. We explicitly construct Ishibashi states forthe maximally symmetric D-branes and show that the generalized S matrix definesthe one-point functions associated to point-like and H2 branes as well as thefusion rules of the degenerate representations of SL(2,R) appearing in the openstring spectrum of the point-like D-branes, through a generalized Verlindetheorem.

On modular properties of the AdS3 CFT