Integrable and Conformal Boundary Conditions for s`(2) A-D-E Lattice Models and Unitary Minimal Conformal Field Theories

Integrable boundary conditions are studied for critical A{D{E and general graph-based lattice models of statistical mechanics. In particular, using tech-niques associated with the Temperley-Lieb algebra and fusion, a set of bound-ary Boltzmann weights which satisfles the boundary Yang-Baxter equation is obtained for each boundary condition. When appropriately specialised, these boundary weights, each of which depends on three spins, decompose into more natural two-spin edge weights. The specialised boundary conditions for the A{D{E cases are naturally in one-to-one correspondence with the confor-mal boundary conditions of ŝ‘(2) unitary minimal conformal fleld theories. Supported by this and further evidence, we conclude that, in the continuum scaling limit, the integrable boundary conditions provide realisations of the complete set of conformal boundary conditions in the corresponding fleld the-ories.