More on Chiral Higher Spin Gravity and Convex Geometry

Recently, a unique class of local Higher Spin Gravities with propagating massless fields in $4d$ - Chiral Higher Spin Gravity - was given a covariant formulation both in flat and $(A)dS_4$ spacetimes at the level of equations of motion. We unfold the corresponding homological perturbation theory as to explicitly obtain all interaction vertices. The vertices reveal a remarkable simplicity after an appropriate change of variables. Similarly to formality theorems the $A_\infty/L_\infty$ multi-linear products can be represented as integrals over a configuration space, which in our case is the space of convex polygons. The $A_\infty$-algebra underlying Chiral Theory is of pre-Calabi-Yau type. As a consequence, the equations of motion have the Poisson sigma-model form.