# The free field realisation of the BVW string

### Matthias R. Gaberdiel, Kiarash Naderi, Vit Sriprachyakul

**23/2/22**Published in : arXiv:2202.11392

The symmetric orbifold of \mathbb{T}^4 was recently shown to be exactly dual to string theory on {\rm AdS}_3\times {\rm S}^3 \times \mathbb{T}^4 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 {\rm AdS}_3\times {\rm S}^3 factor is described by a \mathfrak{psu}(1,1|2)_k WZW model. At level k=1, \mathfrak{psu}(1,1|2)_1 has a free field realisation that is obtained from that of \mathfrak{u}(1,1|2)_1 upon setting a \mathfrak{u}(1) field, often called Z, to zero. We show that the free field version of the {\cal N}=2 generators of BVW (whose cohomology defines the physical states) does not give rise to an {\cal 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.