AdS2D-branes in LorentzianAdS3

Abstract

The boundary states for two dimensional anti--de Sitter $({\mathrm{AdS}}_{2})$ Dirichlet-branes (D-branes) in Lorentzian ${\mathrm{AdS}}_{3}$ space-time are presented. ${\mathrm{AdS}}_{2}$ D-branes are algebraically defined by twisted Dirichlet boundary conditions and are located on twisted conjugacy classes of $\mathrm{SL}(2,R).$ Using the free-field representation of symmetry currents in the $\mathrm{SL}(2,R)$ Wess-Zumino-Novikov-Witten 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 ${\mathrm{AdS}}_{3}$ 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 ${\mathrm{AdS}}_{2}$ D-branes is evaluated and by comparing it with the characters of $\mathrm{sl}^(2,R)$ Kac-Moody algebra it is shown that only states in the principal continuous series representation of $\mathrm{sl}^(2,R)$ Kac-Moody algebra contribute to the amplitude and thus they are the only ones that couple to ${\mathrm{AdS}}_{2}$ D-branes. The form of the character of $\mathrm{sl}^(2,R)$ principal continuous series and the boundary condition among the zero modes are used to determine the physical boundary states for ${\mathrm{AdS}}_{2}$ D-branes.