I'll describe a perturbative classical field theory defined on 11-manifolds with a rank 6 transversely holomorphic foliation and a transverse Calabiâ€“Yau structure. The theory has an infinite-dimensional algebra of gauge symmetries preserving the trivial background, which is $L_\infty$ equivalent to a Lie 2-extension of the infinite-dimensional exceptional simple super Lie algebra $E(5|10)$. Conjecturally, this theory describes the minimal twist of eleven-dimensional supergravity. After describing this conjecture, and evidence for it, I'll describe twisted avatars of the $AdS_7\times S^4$ and $AdS_4\times S^7$ backgrounds, and how two other infinite-dimensional exceptional simple super Lie algebras $E(3|6)$ and $E(1|6)$ appear as asymptotic symmetries. Enumerating gravitons on such backgrounds naturally leads to refinements of generating functions of representation-theoretic significance, such as the MacMahon function. Time permitting, I'll explain how our results combined with holographic techniques can be used to produce enhancements of familiar vertex algebras such as the Heisenberg and Virasoro algebras, to holomorphic factorization algebras in three complex dimensions, and furnish geometric constructions of representations thereof. This talk is based on joint work with Ingmar Saberi and Brian Williams.