$$ \frac{\mathrm{SL}\left(2,\mathrm{\mathbb{R}}\right)\times \mathrm{U}(1)}{\mathrm{U}(1)} $$ CFT, NS5+F1 system and single trace $$ T\overline{T} $$

In this paper we prove the equivalence among (i) the weakly coupled worldsheet string theory described by the coset sigma model $$ \frac{\mathrm{SL}{\left(2,\mathrm{\mathbb{R}}\right)}_k\times \mathrm{U}(1)}{\mathrm{U}(1)} $$ SL 2 ℝ k × U 1 U 1 × S3 × T4 with SL(2, ℝ) WZW level k ≥ 2, (ii) the full near horizon theory of the NS5 branes with k NS5 branes wrapping T4 × S1, p » 1 F1 strings wrapping S1 and n units of momentum along the S1 and (iii) the single trace $$ T\overline{T} $$ T T ¯ deformation of string theory in AdS3 × S3 × T4. As a check we compute the spectrum (continuous) of the spacetime theory by performing BRST quantization of the coset description of the worldsheet theory and show that it matches exactly with the one derived in the case of single trace $$ T\overline{T} $$ T T ¯ deformed string theory in AdS3. Secondly, we compute the two-point correlation function of local operators of the spacetime theory using the worldsheet coset approach and reproduce the same two-point function from the supergravity approach.