A formula for the block expansion in free CFTs and applications to ${\cal N}=4$ SYM at strong coupling

TL;DR

This paper introduces an explicit formula for conformal block decomposition in free CFTs with various supersymmetries and applies it to analyze four-point correlators in ${ m f N}=4$ SYM at strong coupling, supporting AdS/CFT.

Contribution

The authors derive a new analytic formula for conformal blocks in free CFTs and demonstrate its application to strong coupling correlators in ${ m f N}=4$ SYM, providing novel checks of AdS/CFT.

Findings

Explicit conformal block formula for free scalar and BPS diagrams.

Verification of the tree-level Witten diagram representation in ${ m f N}=4$ SYM.

Method applicable to a broader class of multi-particle correlators.

Abstract

An explicit analytic formula is presented that computes the conformal (super-)block decomposition of any free scalar or half-BPS diagram in 1d, 2d or 4d CFTs, with various supersymmetries, including none. We prove our formula by exploiting a connection between conformal blocks and symmetric polynomials. Then we give a direct application of our result to the study of four-point correlators in ${\cal N}=4$ SYM at strong coupling. In particular, we give a CFT proof of the tree-level Witten diagram representation of $\langle {\cal O}_2^2{\cal O}_2^2 {\cal O}_q{\cal O}_q\rangle$ on AdS$_5\times$S$^5$, providing new and highly non-trivial checks of the AdS/CFT correspondence. Our method works for a more general class of multi-particle correlators and can be used to bootstrap new results at strong coupling.