  Chiral higher spin gravity is defined in terms of a strong homotopy algebra
of pre-Calabi-Yau type (noncommutative Poisson structure). All structure maps
are given by the integrals over the configuration space of concave polygons and
the first two maps are related to the (Shoikhet-Tsygan-)Kontsevich Formality.
As with the known formality theorems, we prove the $A_\infty$-relations via
Stokes' theorem by constructing a closed form and a configuration space whose
boundary components lead to the $A_\infty$-relations. This gives a new way to
formulate higher spin gravities and hints at a construct encompassing the known
formality theorems.
