From non-semisimple Hopf algebras to correlation functions for logarithmic CFT

TL;DR

This paper explores the structure of bulk correlation functions in logarithmic conformal field theories using factorizable ribbon Hopf algebras, providing new algebraic constructions and invariance properties.

Contribution

It introduces a novel algebraic framework for bulk fields in logarithmic CFTs using factorizable ribbon Hopf algebras and derives explicit formulas for partition functions.

Findings

Constructed a candidate space of bulk fields as a commutative symmetric Frobenius algebra.

Derived expressions for bulk partition functions involving the Cartan matrix.

Established invariance of correlation functions under mapping class group actions.

Abstract

We use factorizable finite tensor categories, and specifically the representation categories of factorizable ribbon Hopf algebras H, as a laboratory for exploring bulk correlation functions in local logarithmic conformal field theories. For any ribbon Hopf algebra automorphism omega of H we present a candidate for the space of bulk fields and endow it with a natural structure of a commutative symmetric Frobenius algebra. We derive an expression for the corresponding bulk partition functions as bilinear combinations of irreducible characters; as a crucial ingredient this involves the Cartan matrix of the category. We also show how for any candidate bulk state space of the type we consider, correlation functions of bulk fields for closed oriented world sheets of any genus can be constructed that are invariant under the natural action of the relevant mapping class group.