The quiver with superpotentials of a $d$-angulation of a marked surface

TL;DR

This paper constructs quivers with superpotentials from $d$-angulations of marked surfaces and demonstrates their mutation compatibility, advancing the understanding of higher cluster categories and their tilting objects.

Contribution

It introduces a new association between $d$-angulations and quivers with superpotentials, extending mutation compatibility results to higher cluster categories.

Findings

Quivers with superpotentials are associated to $d$-angulations.

Flip operations correspond to mutations of Ginzburg algebras.

Certain almost complete $(d-2)$-cluster tilting objects have exactly $d-1$ complements.

Abstract

In this paper, we associate a quiver with superpotential to each $d$-angulation of a (unpunctured) marked surface. We show that, under quasi-isomorphisms, the flip of a $d$-angulation is compatible with Oppermann's mutation of (the Ginzburg algebra of) the corresponding quiver with superpotential, thereby partially generalizing the result in [LF09]. Applying to the generalized $(d-2)$-cluster categories associated to this quiver with superpotential, we prove that some certain almost complete $(d-2)$-cluster tilting objects in the higher cluster category have exactly $d-1$ complements.