Chiral Higher Spin Gravity From Strong Homotopy Algebra

Abstract: In this thesis, we derive the equations of motion of Chiral Higher Spin Gravity (HiSGRA) in terms of its underlying $L_\infty$-algebra. Chiral HiSGRA contains self-dual Yang-Mills and self-dual gravity as closed subsectors, which themselves form closed subsectors of Yang-Mills and general relativity. We begin by constructing a covariant formulation for self-dual Yang-Mills and self-dual gravity, and subsequently extend this construction to the full Chiral Higher Spin Gravity. Remarkably, the $L_\infty$-algebra is constructed from an $A_\infty$-algebra of pre-Calabi-Yau type, suggesting a deep connection to non-commutative deformation quantization. The structure maps of the resulting $L_\infty$-algebra are expressed as integrals of a simple exponential over convex polygons in $\mathbb{R}^2$. The existence of this covariant and coordinate independent formulation of chiral HiSGRA demonstrates, via the AdS/CFT correspondence, that $O(N)$ vector models possess a closed chiral subsector. Finally, we prove that the $A_\infty$-algebra follows from Stokes' theorem -- a crucial feature of the known formality theorems. To this end, we construct integration spaces that generalize convex polygons to $\mathbb{R}^3$, and are intimately connected to positive Grassmanians. This Stokes-based derivation points towards a novel generalization of Kontsevich' formality theorem to the non-commutative setting.