A note on defect Mellin amplitudes

TL;DR

This work extends the Mellin representation to generic flat defect CFTs of arbitrary co-dimension, analyzing the analytic structure of defect Mellin amplitudes and their OPE-implied poles. It defines a multi-variable Mellin transform for mixed bulk/defect correlators, derives bulk and defect pole structures, and decomposes two-point bulk correlators into bulk and defect conformal blocks, including a co-dimension one simplification with cross-ratios $\xi$ and $\cos\phi$. The paper also constructs Witten diagrams in the defect setting, providing explicit results for contact diagrams $W_{n,m}$ and detailed expressions for $W_{1,0}$, $W_{1,1}$, and $W_{2,0}$, along with bulk and defect exchange diagrams via spectral representations. These results pave the way for defect Mellin-space bootstrap, holographic calculations, and a deeper understanding of defect-related observables in AdS/CFT, with several avenues for generalization to higher-point functions and derivative interactions.

Abstract

We generalize the Mellin representation for a generic co-dimension flat defect CFT. We study the analytic structure of the Mellin amplitudes. We also compute Witten diagrams for a generic co-dimension flat defect CFT.