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

TL;DR

This work advances the protected sector of 6d ${\cal N}=(2,0)$ SCFTs by identifying a chiral ${\cal W}_{\mathfrak g}$-algebra structure for theories of type ${\mathfrak g} \in \{A_{N-1},D_N\}$ via a holomorphic twist of mixed four-point functions of 1/2-BPS operators. By reducing the 6d superconformal Ward identities to a meromorphic 2d CFT problem and employing both the VOA bootstrap and Headrick’s holomorphic bootstrap, the authors obtain closed-form expressions for the twisted four-point functions $\mathcal{F}_{k_1k_2k_3k_4}(\chi)$ up to $\{k_i\} = 4$ and extract explicit 6d OPE coefficients. They map 6d data $\lambda_{k_1k_2\mathcal X}\lambda_{k_3k_4\mathcal X}$ to specific ${\cal W}_{\mathfrak g}$ structure constants, resolve ambiguities with crossing data, and analyze Regge trajectories in the protected sector. The results are shown to be consistent with perturbative M-theory on $AdS_7\times S^4/\mathbb{Z}_{\mathfrak{o}}$, including $R$, $R^4$, and $D^6R^4$ terms, and offer a concrete platform for future numerical bootstrap studies of higher-point correlators and defects in 6d (2,0) theories.

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 $AdS_7\times S^4/\mathbb{Z}_\mathfrak{o}$.