The W-algebra bootstrap of 6d N=(2,0) theories

6d (2,0) SCFTs of type g 𝔤 have protected subsectors that were conjectured in [C. Beem et al., J. High Energy Phys. 05, 017 (2015)] to be captured by W_g 𝒲𝔤 algebras. We write down the crossing equations for mixed four-point functions Sk_1 Sk_2Sk_3Sk_4 ⟨Sk1Sk2Sk3Sk4⟩ of 1/2-BPS operators Sk_i Ski 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) W 𝒲 -algebras of type g = \{AN-1,D_N\} 𝔤={AN−1,DN} as truncations of W1+∞ 𝒲1+∞ and solve OPE associativity conditions for their structure constants, both using OPEdefs and the holomorphic bootstrap of [M. Headrick et al., J. High Energy Phys. 07, 059 (2015)]. With this, we solve the multi-correlator bootstrap for twisted 6d four-point correlators Fk_1k_2k_3k_4 ℱk1k2k3k4 involving all Sk_i Ski up to \{k_i\}=4 {ki}=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× S^4/Z_o AdS7×S4/ℤ𝔬 .