Mellin amplitudes for $AdS_3 \times S^3$

TL;DR

This paper develops a new method to compute four-point functions in two-dimensional holographic conformal field theories using Mellin amplitudes, overcoming previous technical challenges and applying it to the D1-D5 CFT.

Contribution

It introduces a simple fix linking parity-odd and parity-even conformal blocks in two dimensions, enabling the use of Mellin space techniques for 2D CFTs, and applies this to derive new correlator formulas in the D1-D5 theory.

Findings

Reproduces known results for Kaluza-Klein modes of the tensor multiplet.

Provides new compact formulas for correlators involving graviton multiplet modes.

Establishes a practical method for Mellin amplitude calculations in 2D CFTs.

Abstract

There are holographic superconformal theories in all dimensions between two and six which allow arbitrary tree-level four-point functions to be fixed by basic consistency conditions. Although Mellin space is usually the most efficient setting for imposing these contraints, four-point functions in two dimensions have thus far been an exception due to their more intricate dependence on the conformal cross-ratios. In this paper, we introduce a simple fix which exploits the relation between a parity-odd conformal block in two dimensions and a parity-even conformal block in four dimensions. We then apply the resulting toolkit to a study of the paradigmatic holographic theory in two dimensions which is the D1-D5 CFT. For correlators involving Kaluza-Klein modes of the tensor multiplet, this analysis reproduces results which were previously obtained using hidden conformal symmetry. With four Kaluza-Klein modes of the graviton multiplet, it yields new results including a compact formula for the correlators of all pairwise identical operators,