Nonsemisimple Sugawara Constructions

TL;DR

This paper unifies and extends the understanding of nonsemisimple Sugawara constructions, showing they factorize into standard and nonsemisimple parts, with implications for conformal field theory and Lie algebra representations.

Contribution

It provides a unifying formalism for all known nonsemisimple Sugawara constructions and proves their factorization into standard and generalized nonsemisimple components.

Findings

All nonsemisimple Sugawara constructions factorize into standard and nonsemisimple parts.

The central charge in these constructions is always an integer equal to the Lie algebra's dimension.

The formalism applies to Lie algebras with invariant metrics, unifying previous examples.

Abstract

By a Sugawara construction we mean a generalized Virasoro construction in which the currents are primary fields of conformal weight one. For simple Lie algebras, this singles out the standard Sugawara construction out of all the solutions to the Virasoro master equation. Examples of nonsemisimple Sugawara constructions have appeared recently. They share the properties that the Virasoro central charge is an integer equal to the dimension of the Lie algebra and that they can be obtained by high-level contraction of reductive Sugawara constructions: they thus correspond to free bosons. Exploiting a recent structure theorem for Lie algebras with an invariant metric, we are able to unify all the known constructions under the same formalism and, at the same time, to prove several results about the Sugawara constructions. In particular, we prove that all such constructions factorize into a standard (semisimple) Sugawara construction and a nonsemisimple one (with integral central charge) of a form which generalizes the nonsemisimple examples known so far.