The free field realisation of the BVW string

Abstract

Abstract The symmetric orbifold of 𝕋4 was recently shown to be exactly dual to string theory on AdS3 × S3 × 𝕋4 with minimal (k = 1) NS-NS flux. The worldsheet theory is best formulated in terms of the hybrid formalism of Berkovits, Vafa & Witten (BVW), in terms of which the AdS3× S3 factor is described by a $$ \mathfrak{psu} $$ psu (1, 1|2)k WZW model. At level k = 1, $$ \mathfrak{psu} $$ psu (1, 1 2)1 has a free field realisation that is obtained from that of $$ \mathfrak{u} $$ u (1, 1 2)1 upon setting a $$ \mathfrak{u} $$ u (1) field, often called Z, to zero. We show that the free field version of the $$ \mathcal{N} $$ N = 2 generators of BVW (whose cohomology defines the physical states) does not give rise to an $$ \mathcal{N} $$ N = 2 algebra, but is rather contaminated by terms proportional to the Z-field. We also show how to overcome this problem by introducing additional ghost fields that implement the quotienting by Z.