Correspondences between WZNW models and CFTs with W-algebra symmetry

Thomas Creutzig,Yasuaki Hikida,Peter B Rønne

doi:10.1007/jhep02(2016)048

Abstract

We study theories with W-algebra symmetries and their relation to WZNW models on (super-)groups. Correlation functions of the WZNW models are expressed in terms of correlators of CFTs with W-algebra symmetry. The symmetries of the theories involved in these correspondences are related by the Drinfeld-Sokolov reduction of Lie algebras to W-algebras. The W-algebras considered in this paper are the Bershadsky-Polyakov algebra for sl(3) and the quasi-superconformal algebra for generic sl(N|M). The quantum W-algebras obtained from affine sl(N) are constructed using embeddings of sl(2) into sl(N), and these can in turn be characterized by partitions of N. The above cases correspond to \underline{N+2} = \underline{2} + N \underline{1} and its supergroup extension. Finally, sl(2N) and the correspondence corresponding to \underline{2N} = N \underline{2} is also analyzed.

Highlights

Spin gravity theory on AdS3 can be identified with W -symmetry
Correlation functions of the WZNW models are expressed in terms of correlators of CFTs with W -algebra symmetry
We relate correlation functions of Wess-Zumino-NovikovWitten (WZNW) models1 on the groups SL(3) and SL(2N ) to correlators of theories with W -symmetry, and we consider the correspondence between WZNW models on the supergroups SL(N |M ) and theories with super W -algebra symmetry

Summary

A relation to Bershadsky-Polyakov algebra

We would like to derive the relation between correlation functions, so we begin by considering a correlator in the SL(3) WZNW model. We remove the function B3 from the action by the following transformation: γ′a (uB3). This will change the background charge of φ, and the remaining β1′ , γ′1, β2′ , γ′2 ghost systems are dimension (1/2, 1/2) systems. This gives extra terms of the form δS. The extra terms can put in the form of vertex operators if we bosonize the β, γ systems as follows β1′ = ∂Y1e−X1+Y1 , γ′1 = eX1−Y1 ,. Note that the central charge for one pair (pa, θa) is −2

Correspondence to a quasi-superconformal theory

The reduced action

Conclusion and discussions

A Conventions

B Higher genus extension for quasi-superconformal algebra

Nb φIN