Landau-Ginzburg Description of Boundary Critical Phenomena in Two Dimensions

The Virasoro minimal models with boundary are described in the Landau-Ginzburg theory by introducing a boundary potential, function of the boundary field value. The ground state field configurations become non-trivial and are found to obey the soliton equations. The conformal invariant boundary conditions are characterized by the reparametrization-invariant data of the boundary potential, that are the number and degeneracies of the stationary points. The boundary renormalization group flows are obtained by varying the boundary potential while keeping the bulk critical: they satisfy new selection rules and correspond to real deformations of the Arnold simple singularities of A_k type. The description of conformal boundary conditions in terms of boundary potential and associated ground state solitons is extended to the N=2 supersymmetric case, finding agreement with the analysis of A-type boundaries by Hori, Iqbal and Vafa.