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Modular transformation and boundary states in logarithmic conformal field theory

Kawai, S, JOHN F. WHEATER

Published Mar 22, 2001• 36 citations
OpenAlex

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.

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We study the c=−2c=-2c=−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.
This lecture note covers topics on boundary conformal field theory, modular transformations and the Verlinde formula, and boundary logarithmic CFT. An introductory review on CFT with boundary and a discussion of its applications to logarithmic cases are given. LCFT at c=−2c=-2c=−2 is mainly discussed.
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.
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 c1,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 c1,p triplet models.
We present explicit mathematical structures that allow for the reconstruction of the field content of a full local conformal field theory from its boundary fields. Our framework is the one of modular tensor categories, without requiring semisimplicity, and thus covers in particular finite rigid logarithmic conformal field theories. We assume that the boundary data are described by a pivotal module category over the modular tensor category, which ensures that the algebras of boundary fields are Frobenius algebras. Bulk fields and, more generally, defect fields inserted on defect lines, are given by internal natural transformations between the functors that label the types of defect lines. We use the theory of internal natural transformations to identify candidates for operator products of defect fields (of which there are two types, either along a single defect line, or accompanied by the fusion of two defect lines), and for bulk-boundary OPEs. We show that the so obtained OPEs pass various consistency conditions, including in particular all genus-zero constraints in Lewellen's list.
We introduce ( p − 1) pseudocharacters in the space of (1, p ) model vacuum torus amplitudes to complete the distinguished basis in the 2 p -dimensional fusion algebra to a basis in the whole (3 p − 1)-dimensional space of torus amplitudes, and the structure constants in this basis are (not necessarily non-negative) integer numbers. We obtain a generalized Verlinde formula that gives these structure constants. In the context of theories with boundaries, we identify the space of vacuum torus amplitudes with the space of Ishibashi states. Then, we propose (3 p − 1) boundary states satisfying the Cardy condition.