Recursion Relations for Anomalous Dimensions in the 6d $(2,0)$ Theory

TL;DR

This paper develops a spin-truncated conformal bootstrap approach to compute anomalous dimensions of double-trace operators in the 6d $(2,0)$ theory, by expanding crossing symmetry in the inverse central charge and projecting onto a hypergeometric basis to obtain recursion relations. A bosonic 6d toy model is analyzed first, establishing that spin-truncated recursions depend on $(L+2)(L+4)/8$ free parameters, interpreted as coefficients of bulk higher-derivative interactions in AdS$_7\times S^4$. The method is then extended to the supersymmetric $(2,0)$ theory, yielding explicit recursion relations and solutions for spin-0 and spin-2 truncations, with their large-twist behavior encoding the derivative order of the dual bulk vertices (e.g., $\mathcal{R}^4$, $D^6\mathcal{R}^4$). The results reproduce known Heslop et al. findings and provide a practical, computer-friendly framework (including a Mathematica notebook) for obtaining OPE data without requiring explicit four-point functions, thereby illuminating higher-derivative corrections in M-theory on AdS$_7\times S^4$.

Abstract

We derive recursion relations for the anomalous dimensions of double-trace operators occurring in the conformal block expansion of four-point stress tensor correlators in the 6d $(2,0)$ theory, which encode higher-derivative corrections to supergravity in $AdS_7 \times S^4$ arising from M-theory. As a warm-up, we derive analogous recursion relations for four-point functions of scalar operators in a toy non-supersymmetric 6d conformal field theory.