The free field realisation of the BVW string
Matthias R. Gaberdiel, Kiarash Naderi, Vit Sriprachyakul
Abstract
A bstract The symmetric orbifold of π 4 was recently shown to be exactly dual to string theory on AdS 3 Γ S 3 Γ π 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 Γ S 3 factor is described by a $$ \mathfrak{psu} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>psu</mml:mi> </mml:math> (1 , 1 | 2) k WZW model. At level k = 1, $$ \mathfrak{psu} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>psu</mml:mi> </mml:math> (1 , 1 2) 1 has a free field realisation that is obtained from that of $$ \mathfrak{u} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> (1 , 1 2) 1 upon setting a $$ \mathfrak{u} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> (1) field, often called Z , to zero. We show that the free field version of the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 generators of BVW (whose cohomology defines the physical states) does not give rise to an $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 .