Coset Graphs in Bulk and Boundary Logarithmic Minimal Models

The logarithmic minimal models are not rational but, in the W-extended picture, they resemble rational conformal field theories. We argue that the W-projective representations are fundamental building blocks in both the boundary and bulk description of these theories. In the boundary theory, each W-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 A^{(2)}_{p} and their coset graphs A^{(2)}_{p,p'}=A^{(2)}_{p} x A^{(2)}_{p'}/Z_2. This provides compact formulas for the conformal partition functions with W-projective boundary conditions. On the torus, we propose modular invariant partition functions as sesquilinear forms in W-projective and rational minimal characters and observe that they are encoded by the same coset fusion graphs.