On Bosonic and Supersymmetric Current Algebras for Non-Semi-Simple Groups

TL;DR

This paper develops a systematic method for constructing current algebras based on non-semi-simple groups, highlighting the non-integer Virasoro central charges and the importance of different bilinear forms in the algebra.

Contribution

It introduces a novel approach to current algebra construction for non-semi-simple groups, including the associated action and the role of bilinear forms.

Findings

Virasoro central charges can be non-integer.

Different bilinear forms are used in the algebra and index raising/lowering.

A specific action realizing these current algebras is provided.

Abstract

We present a systematic approach to constructing current algebras based on non-semi-simple groups. The Virasoro central charges corresponding to these current algebras are not, in general, given by integer numbers. The key point in this construction is that the bilinear form appearing in the current algebra can be different from the bilinear form used to raise and lower group indices. The action which realises this current algebra as its symmetry is also found.