Logarithmic conformal field theory, log-modular tensor categories and modular forms

TL;DR

This paper explores the extension of modular tensor category concepts and Verlinde's formula to logarithmic conformal field theories, providing a categorical and modular framework supported by explicit examples like triplet algebras and symplectic fermions.

Contribution

It introduces a logarithmic variant of the Verlinde formula and connects modular data with categorical structures in logarithmic CFTs, expanding the theoretical foundation.

Findings

Logarithmic conformal field theories have modular structures involving trace functions with intertwiners.

Categorical structures are finite ribbon tensor categories with specific double isomorphisms.

Explicit relations between logarithmic Hopf links and modular transformations are established.

Abstract

The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters on the one hand and its interpretation of modules as objects in a modular tensor category on the other one. Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and $C_2$-cofinite vertex operator algebras. We suggest that their modular pillar are trace functions with insertions corresponding to intertwiners of the projective cover of the vacuum, and that the categorical pillar are finite tensor categories $\mathcal C$ which are ribbon and whose double is isomorphic to the Deligne product $\mathcal C\otimes \mathcal C^{opp}$. Overarching these pillars is then a logarithmic variant of Verlinde's formula. Numerical data realizing this are the modular $S$-matrix and modified traces of open Hopf links. The representation categories of $C_2$-cofinite and logarithmic conformal field theories that are fairly well understood are those of the $\mathcal W_p$-triplet algebras and the symplectic fermions. We illustrate the ideas in these examples and especially make the relation between logarithmic Hopf links and modular transformations explicit.