Boundary States, Extended Symmetry Algebra and Module Structure for   certain Rational Torus Models
  The massless bosonic field compactified on the circle of rational $R^2$ is reexamined in the presense of boundaries. A particular class of models corresponding to $R^2=\frac{1}{2k}$ is distinguished by demanding the existence of a consistent set of Newmann boundary states. The boundary states are constructed explicitly for these models and the fusion rules are derived from them. These are the ones prescribed by the Verlinde formula from the S-matrix of the theory. In addition, the extended symmetry algebra of these theories is constructed which is responsible for the rationality of these theories. Finally, the chiral space of these models is shown to split into a direct sum of irreducible modules of the extended symmetry algebra. 
Ioannis Smyrnakis