Correlators of long strings on AdS$_3\times$S$^3\times$T$^4$

In this work, we calculate correlators of long strings on AdS$_3\times$S$^3\times$T$^4$ with pure NS-NS flux. We first construct physical vertex operators that correspond to long strings. Due to the GSO projection, they depend on the parity of the spectral flow parameter $w$. For a given $w$, we construct the physical operators that have the lowest space-time weights in both the NS and R sector. Then, we calculate three point correlators for each possible type of parities of spectral flows. We find that the recursion relations of correlators in the bosonic SL$(2,\mathbb{R})$ WZW model can be understood from the equivalence of these superstring correlators with different picture choices. Furthermore, after carefully mapping the vertex operators to appropriate operators in the dual CFT, we find that once the fermionic contributions together with the picture changing effects are correctly taken into account, some mathematical identities of covering maps lead to the matching of the correlators of the two sides. We check this explicitly at the leading order in the conformal perturbation computation and conjecture that this remains correct to all orders.