Non-semisimple CFT/TFT correspondence I: General setup

TL;DR

This paper develops a topological field theory framework to construct and analyze non-semisimple logarithmic conformal field theories, extending previous semisimple models and exploring non-invertible symmetries.

Contribution

It generalizes the TFT construction of CFT correlators to finite logarithmic CFTs with non-semisimple algebraic data, providing explicit examples and analyzing topological defects.

Findings

Constructed full CFTs from non-semisimple modular tensor categories.

Explicit example of the Cardy case with transparent surface defect.

Identified non-invertible and non-semisimple topological symmetries.

Abstract

We extend the TFT construction of CFT correlators of [arXiv:hep-th/0204148] to so-called finite logarithmic CFTs for which the algebraic input data is no longer semisimple but still finite. More specifically, starting from the data of a chiral CFT given in the form of a not necessarily semisimple modular tensor category C we use a three dimensional topological field theory with surface defects based on the surgery TFT of [arXiv:1912.02063] to construct a full CFT as a braided monoidal oplax natural transformation. We make our construction explicit in the example of the transparent surface defect, resulting in the so-called Cardy case. In particular, we consider topological line defects and their action on bulk fields in these logarithmic CFTs, providing a source of examples for non-invertible and non-semisimple topological symmetries.