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A bstract The symmetric orbifold of 𝕋 4 was recently shown to be exactly dual to string theory on AdS 3 × S 3 × 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 AdS 3 × S 3 factor is described by a psu psu (1, 1 | 2) k WZW model. At level k = 1, psu psu (1, 1 2) 1 has a free field realisation that is obtained from that of u u (1, 1 2) 1 upon setting a u u (1) field, often called Z, to zero. We show that the free field version of the N N = 2 generators of BVW (whose cohomology defines the physical states) does not give rise to an 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.
Gaberdiel et al. (Fri,) studied this question.