From boundary to bulk in logarithmic CFT

The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy `identity brane'). We apply the general method to the c_1,p triplet models and reproduce the previously known bulk theory for p=2 at c=-2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-product we obtain a logarithmic version of the Verlinde formula for the c_1,p triplet models.Comment: 35 pages, 2 figures; v2: minor corrections, version to appear in J.Phys.

Integrable Boundary Conditions and W-Extended Fusion in the Logarithmic Minimal Models LM(1,p)

We consider the logarithmic minimal models LM(1,p) as `rational' logarithmic conformal field theories with extended W symmetry. To make contact with the extended picture starting from the lattice, we identify 4p-2 boundary conditions as specific limits of integrable boundary conditions of the underlying Yang-Baxter integrable lattice models. Specifically, we identify 2p integrable boundary conditions to match the 2p known irreducible W-representations. These 2p extended representations naturally decompose into infinite sums of the irreducible Virasoro representations (r,s). A further 2p-2 reducible yet indecomposable W-representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank-2 Virasoro representations. The fusion rules in the extended picture are deduced from the known fusion rules for the Virasoro representations of LM(1,p) and are found to be in agreement with previous works. The closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach.Comment: 15 page