Symmetric Boundary Conditions in Boundary Critical Phenomena

It has been an extremely fruitful idea to study a conformal field theory by putting it on various surfaces, with or without boundaries. Apart from the sphere, that has been considered first, prime examples of non–trivial geometries include the torus [1] and the cylinder [2,3]. They serve to probe different facets of a given conformal theory. However the data specific of these surfaces are inextricably related to each other, and this fact provides very stringent constraints on the theory itself, allowing for example to determine its field content. For minimal conformal theories, the problem on the torus for single–valued fields has been resolved in [4]: consistent models have a periodic partition function that can be associated in a unique way with a pair (A,G) of simple Lie algebras of ADE type. The solution of the analogous problem for the cylinder is much more recent, even if early calculations in either specific models or with specific boundary conditions have been carried out in [2,3,5]. The recent discovery in [6] of a new conformally invariant boundary condition in the 3–state Potts model triggered a renewal of interest in the problem. For minimal models, its solution was given in [7,8], and shown to be encoded in the same Dynkin graphs that specify the torus partition function. When a model has a symmetry, necessarily discrete in this context, fields can be multiple–valued on the torus, so that non–periodic sectors exist. Furthermore, the fields transform under the symmetry group, and, upon diagonalization, can be assigned charges. All this information is encoded in frustrated partition