This paper is a representation-theoretic extension of Part I. It has been
inspired by three recent developments: surface cluster algebras studied by
Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials
initiated by Derksen-Weyman-Zelevinsky, and string modules associated to arcs
on unpunctured surfaces by Assem-Brustle-Charbonneau-Plamondon.