The free field realisation of the BVW string

TL;DR

This paper explores the free field realization of the BVW string formalism in AdS3/CFT2 correspondence, addressing issues with the N=2 algebra and proposing solutions via ghost fields.

Contribution

It demonstrates that the free field realization at level k=1 is contaminated by a Z-field and introduces ghost fields to properly implement the quotienting process.

Findings

The free field realization does not naturally produce an N=2 algebra.

Additional ghost fields can correct the realization to match the expected algebra.

The approach clarifies the structure of the worldsheet theory in the minimal flux case.

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 $Z$.