$A_\infty$-structures associated with pairs of 1-spherical objects and noncommutative orders over curves

Abstract: We show that pairs $(X,Y)$ of 1-spherical objects in $A_\infty$-categories, such that the morphism space ${\rm Hom}(X,Y)$ is concentrated in degree 0, can be described by certain noncommutative orders over (possibly stacky) curves. In fact, we establish a more precise correspondence at the level of isomorphism of moduli spaces which we show to be affine schemes of finite type over ${\Bbb Z}$.