Holographic correlators in AdS$_3$ without Witten diagrams

TL;DR

This work derives holographic four-point functions of chiral primary operators in the AdS$_3$/CFT$_2$ D1-D5 setting by leveraging HHLL gravity results and reframing the problem in Mellin space to capture single-trace exchanges. The authors obtain explicit expressions for the $s$-channel contributions $\widetilde{\mathcal{G}}^{(s)}_{k,l}$ and translate them into compact Mellin amplitudes $\widetilde{\mathcal{M}}^{(s)}_{k,l}$, then propose a complete, symmetry-consistent formula $\widetilde{\mathcal{M}}_{k,l}$ including $t$- and $u$-channel exchanges when the flavours coincide. A key outcome is a set of closed forms, e.g. for $k=l$ and for $k$ or $l=1$, including the simple rational parts and the structure encoded in $\hat{D}$-functions and polynomials $Y(n;\sigma,\tau)$, with poles reflecting single-trace exchanges. The results align with a hidden-symmetry approach and provide a Witten-diagram-free route to AdS$_3\times S^3$ correlators, offering a robust framework for AdS$_3$/CFT$_2$ holography and future generalizations.

Abstract

We present a formula for the holographic 4-point correlators in AdS$_3 \times S^3$ involving four single-trace operators of dimension $k, k, l, l$. As an input we use the supergravity results for the Heavy-Heavy-Light-Light correlators that can be derived by studying the linear fluctuations around known asymptotically AdS$_3 \times S^3$ 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.