AdS 2 D - Branes in Lorentzian AdS 3

The boundary states for AdS 2 D-branes in Lorentzian AdS 3 space-time are presented. AdS 2 D-branes are algebraically defined by twisted Dirichlet boundary conditions and are located on twisted conjugacy classes of SL(2,R). Using free field representation of symmetry currents in the SL(2,R) WZNW model, the twisted Dirichlet gluing conditions among currents are translated to matching conditions among free fields and then to boundary conditions among the modes of free fields. The Ishibashi states are written as coherent states on AdS 3 in the free field formalism and it is shown that twisted Dirichlet boundary conditions are satisfied on them. The tree-level amplitude of propagation of closed strings between two AdS 2 D-branes is evaluated and by comparing which to the characters of sl(2,R) Kac-Moody algebra it is shown that only states in the principal continuous series representation of sl(2,R) Kac-Moody algebra contributes to the amplitude and thus they are the only ones that couple to AdS 2 D-branes. The form of the character of sl(2,R) principal continuous series and the boundary condition among the zero modes are used to determine the physical boundary states for AdS 2 D-branes.