The $\mathcal{W}$-algebra bootstrap of 6d $\mathcal{N}=(2,0)$ theories

TL;DR

This paper develops a bootstrap approach for 6d (2,0) theories using $ ext{W}$-algebras, deriving crossing equations, solving for structure constants, and connecting results to M-theory corrections.

Contribution

It formulates crossing equations for 6d (2,0) theories, identifies relevant $ ext{W}$-algebras, and solves the multi-correlator bootstrap to obtain explicit OPE data.

Findings

Derived crossing equations for 6d (2,0) four-point functions.

Solved for structure constants of $ ext{W}$-algebra truncations.

Connected CFT data to M-theory graviton scattering corrections.

Abstract

6d (2,0) SCFTs of type $\mathfrak{g}$ have protected subsectors that were conjectured in arxiv:1404.1079 to be captured by $\mathcal{W}_\mathfrak{g}$ algebras. We write down the crossing equations for mixed four-point functions $\langle S_{k_1} S_{k_2}S_{k_3}S_{k_4}\rangle$ of 1/2-BPS operators $S_{k_i}$ in 6d (2,0) theories and detail how a certain twist reduces this system to a 2d meromorphic CFT multi-correlator bootstrap problem. We identify the relevant 6d (2,0) $\mathcal{W}$-algebras of type $\mathfrak{g} = \{A_{N-1},D_N\}$ as truncations of $\mathcal{W}_{1+\infty}$ and solve OPE associativity conditions for their structure constants, both using $\texttt{OPEdefs}$ and the holomorphic bootstrap of arxiv:1503.07111. With this, we solve the multi-correlator bootstrap for twisted 6d four-point correlators $\mathcal{F}_{k_1k_2k_3k_4}$ involving all $S_{k_i}$ up to $\{k_i\}=4$ and extract closed-form expressions for 6d OPE coefficients. We describe the implications of our CFT data on conformal Regge trajectories of the (2,0) theories and finally, demonstrate the consistency of our results with protected higher-derivative corrections to graviton scattering in M-theory on $AdS_7\times S^4/\mathbb{Z}_\mathfrak{o}$.