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

Understand

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 $\mathcal C\otimes \mathcal C^{opp}$.