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 ${\mathrm{AdS}}_3\times {\mathrm{S}}^3\times {\mathbb{T}}^4$ 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 ${\mathrm{AdS}}_3\times {\mathrm{S}}^3$ 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 $\mathcal{N}=2$ generators of BVW (whose cohomology defines the physical states) does not give rise to an $\mathcal{N}=2$ 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 $Z$.

arXiv comment28 pages

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Cited by in corpus (7)

• BPS Correlators for ${\text{AdS}}_3/{\text{CFT}}_2$
• Perturbing the symmetric orbifold from the worldsheet
• Symmetries and Covering Maps for the Minimal Tension String on $\mathbf{Ad}{\mathbf{S}}_3\times {\mathbf{S}}^3\times {\mathbf{T}}^4$
• DDF operators in the Hybrid Formalism
• Free field realisation of boundary vertex algebras for Abelian gauge theories in three dimensions
• String theory in twistor space and minimal tension holography
• Gromov-Witten/Hilbert versus AdS3/CFT2 Correspondence