A Lie algebraic pattern behind logarithmic CFTs

TL;DR

This paper presents a Lie algebraic framework for logarithmic conformal field theories, enabling uniform construction of principal W-algebras and establishing new character formulas and simplicity results.

Contribution

It introduces a Lie algebraic formalization of the Feigin--Tipunin construction, extending previous results to a broader class of Lie algebras and superalgebras.

Findings

Constructed principal W-algebras for any simple Lie algebra and Lie superalgebra.

Established Weyl-type character formulas for these W-algebras.

Proved simplicity theorems extending prior work.

Abstract

We introduce a purely Lie algebraic formalization of the Feigin--Tipunin's geometric construction of logarithmic CFTs/VOAs. After reformulating the geometric representation theory of FT construction under this new setting, within this framework, we uniformly construct the (multiplet) principal W-algebras at positive integer level associated with any simple Lie algebra $\mathfrak{g}$ and Lie superalgebra $\mathfrak{osp}(1|2r)$, thereby establishing Weyl-type character formulas and simplicity theorems that extend the second author's previous results.