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A bstract 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 ×  S  3  ×  T  4  with SL(2 ,  ℝ) WZW level  k  ≥ 2, (ii) the full near horizon theory of the NS5 branes with  k  NS5 branes wrapping  T  4  ×  S  1 ,  p  » 1 F1 strings wrapping  S  1  and  n  units of momentum along the  S  1  and (iii) the single trace    $$ T\overline{T} $$  T T ¯ deformation of string theory in  AdS  3  ×  S  3  ×  T  4 . 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  AdS  3 . 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.

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