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In this paper we prove the equivalence among (i) the weakly coupledworldsheet string theory described by the coset sigma model$\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}\times S^3 \times T^4$ with$SL(2,\mathbb{R})$ WZW level $k\geq 2$, (ii) the full near horizon theory ofthe NS5 branes with $k$ NS5 branes wrapping $T^4\times S^1$, $p\gg1$ F1 stringswrapping $S^1$ and $n$ units of momentum along the $S^1$ and (iii) the singletrace $T\bar{T}$ deformation of string theory in $AdS_3\times S^3\times T^4$.As a check we compute the spectrum of the spacetime theory by performing BRSTquantization of the coset description of the worldsheet theory and show that itmatches exactly with the one derived in the case of single trace $T\bar{T}$deformed string theory in $AdS_3$. Secondly, we compute the two-pointcorrelation function of local operators of the spacetime theory using theworldsheet coset approach and reproduce the same two-point function from thesupergravity approach.

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