Quilted strips, graph associahedra, and $A_\infty$ $n$–modules

Abstract

We consider moduli spaces of quilted strips with markings. By identifying each compactified moduli space with the nonnegative real part of a projective toric variety, we conclude that it is homeomorphic under the moment map to the moment polytope. The moment polytopes in these cases belong to a certain class of graph associahedra, which include the associahedra and permutahedra as special cases. In fact, these graph associahedra are precisely the polytopes whose facet combinatorics encode the $A_{\infty }$ equations of $A_{\infty }$ $n$–modules.