Modular transformation and boundary states in logarithmic conformal field theory

TL;DR

This paper investigates the $c=-2$ logarithmic conformal field theory with boundaries, constructing boundary states using symplectic fermions and exploring their modular properties despite complex module structures.

Contribution

It demonstrates the existence of consistent boundary states in the $c=-2$ logarithmic CFT, addressing challenges posed by indecomposable modules and modular invariance.

Findings

Boundary states with consistent modular properties are constructed.

The vacuum representation is a sub-representation of a larger indecomposable module.

The Verlinde formula fails in this model, indicating unusual modular behavior.

Abstract

We study the $c=-2$ model of logarithmic conformal field theory in the presence of a boundary using symplectic fermions. We find boundary states with consistent modular properties. A peculiar feature of this model is that the vacuum representation corresponding to the identity operator is a sub-representation of a ``reducible but indecomposable'' larger representation. This leads to unusual properties, such as the failure of the Verlinde formula. Despite such complexities in the structure of modules, our results suggest that logarithmic conformal field theories admit bona fide boundary states.