Deformations of bordered Riemann surfaces and associahedral polytopes

TL;DR

This paper studies the moduli space of bordered Riemann surfaces with boundary and marked points, introducing a combinatorial framework to classify when these spaces form convex polytopes, and relating new polytopes to associahedra.

Contribution

It classifies all moduli spaces of bordered Riemann surfaces that can be realized as convex polytopes and introduces a new polytope based on cube truncations, connecting it to associahedral structures.

Findings

Classified all such moduli spaces as convex polytopes.

Introduced a new polytope from cube truncations.

Connected new polytope to associahedra and multiplihedra.

Abstract

We consider the moduli space of bordered Riemann surfaces with boundary and marked points. Such spaces appear in open-closed string theory, particularly with respect to holomorphic curves with Lagrangian submanifolds. We consider a combinatorial framework to view the compactification of this space based on the pair-of-pants decomposition of the surface, relating it to the well-known phenomenon of bubbling. Our main result classifies all such spaces that can be realized as convex polytopes. A new polytope is introduced based on truncations of cubes, and its combinatorial and algebraic structures are related to generalizations of associahedra and multiplihedra.