Bulk from boundary in finite CFT by means of pivotal module categories

TL;DR

This paper develops a mathematical framework using modular tensor categories to reconstruct the bulk content of finite conformal field theories from boundary data, applicable to logarithmic CFTs.

Contribution

It introduces a boundary-to-bulk reconstruction method in finite CFTs using pivotal module categories and internal natural transformations, extending to non-semisimple cases.

Findings

Explicit structures for bulk reconstruction from boundary fields.

Validation of consistency conditions including genus-zero constraints.

Framework applicable to logarithmic conformal field theories.

Abstract

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.