A construction of the generalized higher cluster category arising from an $(m+2)$-angulation of a marked surface

TL;DR

This paper constructs a generalized higher cluster category from (m+2)-angulations of marked surfaces, linking geometric surface configurations with algebraic structures like graded quivers and superpotentials.

Contribution

It introduces a new framework connecting surface angulations with higher cluster categories, including counting arcs and associating graded quivers with superpotentials.

Findings

Counted the number of arcs in (m+2)-angulated surfaces.

Established compatibility between surface flips and algebraic mutations.

Linked geometric configurations with algebraic structures like graded quivers.

Abstract

In this article, we study the $(m+2)$-angulations on a Riemann surface, characterized with its boundary components, punctures, and gender. We count the number of arcs in such a surface, and associate a graded quiver with superpotential associated with an $(m2)$-angulation. We show the compatibility between the flip of an $(m+2)$-angulation and the flip in the unpunctured case.