From boundary to bulk in logarithmic CFT

TL;DR

This paper constructs the bulk spectrum of rational logarithmic conformal field theories from boundary conditions, verifies modular invariance, and develops a logarithmic Verlinde formula, advancing understanding of these complex models.

Contribution

It introduces a method to reconstruct the bulk spectrum from boundary conditions in logarithmic CFTs, including explicit models and a new Verlinde formula.

Findings

Reproduced known bulk theory for p=2 at c=-2

Verified modular invariance for general p

Constructed complete set of boundary states

Abstract

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.