Coset graphs in bulk and boundary logarithmic minimal models

The logarithmic minimal models are not rational but, in the WW-extended picture, they resemble rational conformal field theories. We argue that the WW-projective representations are fundamental building blocks in both the boundary and bulk description of these theories. In the boundary theory, each WW-projective representation arising from fundamental fusion is associated with a boundary condition. Multiplication in the associated Grothendieck ring leads to a Verlinde-like formula involving A  -type twisted affine graphs Ap(2) and their coset graphs Ap,p′(2)=Ap(2)⊗Ap′(2)/Z2. This provides compact formulas for the conformal partition functions with WW-projective boundary conditions. On the torus, we propose modular invariant partition functions as sesquilinear forms in WW-projective and rational minimal characters and observe that they are encoded by the same coset fusion graphs.