Chiral Higher Spin Gravity From Strong Homotopy Algebra
Richard van Dongen
TL;DR
This thesis develops a Covariant, unfolding-based formulation of Chiral Higher Spin Gravity (HiSGRA) by encoding its dynamics in an underlying $L_ infty$-algebra, built from an $A_ infty$-algebra of pre-Calabi–Yau type. The self-dual subtheories SDYM and SDGR are analyzed as minimal models, and their FDA/jet-space data are extended to higher spins using Homological Perturbation Theory to generate all higher-order vertices. A Stokes-theorem framework is established to prove the $A_ infty$-relations and connect the construction to deformation quantization and Kontsevich formality, with integration spaces linked to convex polygons and swallowtails (positive Grassmannians). The results show that a covariant formulation of chiral HiSGRA is compatible with AdS/CFT expectations, implying a closed chiral subsector of $O(N)$ vector models, and suggest a generalized formality theorem beyond Kontsevich in a non-commutative setting. Altogether, the work provides a complete algebraic and geometric program for all-order interactions in Chiral HiSGRA and lays groundwork for covariant analysis of higher-spin holography and non-commutative deformation quantization.
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.