The free field realisation of the BVW string

Abstract: 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 $Z$.