Holographic correlators in AdS₃ without Witten diagrams

We present a formula for the holographic 4-point correlators in AdS₃ × S³ involving four single-trace operators of dimension k, k, l, l. As an input we use the super-gravity results for the Heavy-Heavy-Light-Light correlators that can be derived by studying the linear fluctuations around known asymptotically AdS₃ × S³ geometries. When the operators of dimension k and l are in the same multiplet there are contributions due to the exchange of single-trace operators in the t and u-channels, which are not captured by the approach mentioned above. However by rewriting the s-channel results in Mellin space we obtain a compact expression for the s-channel contribution that makes it possible to conjecture a formula for the complete result. We discuss some consistency checks that our proposal meets.