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

Bo Le, Bin Zhu

TL;DR

The paper builds a geometric and algebraic bridge between $d$-angulations of unpunctured marked surfaces and quivers with superpotentials, assigning to each $D$ a quiver-with-potential $(Q_D,W_D)$ whose mutations mirror flips of $D$ via Oppermann-type mutations of the Ginzburg dg algebra. Under the generalized higher cluster framework, it proves that the geometric flip operation is compatible with dg-algebra mutations, yielding a geometric model for the generalized $(d-2)$-cluster categories. Consequently, certain almost complete $(d-2)$-cluster tilting objects in these higher cluster categories have exactly $d-1$ complements, with detailed treatment of non-self-folded vs self-folded cases. The results extend Labardini’s mutation-link to higher Calabi–Yau dimensions and provide a concrete geometric realization of mutation dynamics in generalized cluster settings, contributing to the understanding of exchange graphs and tilting theory in higher dimensions.

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.

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

TL;DR

The paper builds a geometric and algebraic bridge between -angulations of unpunctured marked surfaces and quivers with superpotentials, assigning to each a quiver-with-potential whose mutations mirror flips of via Oppermann-type mutations of the Ginzburg dg algebra. Under the generalized higher cluster framework, it proves that the geometric flip operation is compatible with dg-algebra mutations, yielding a geometric model for the generalized -cluster categories. Consequently, certain almost complete -cluster tilting objects in these higher cluster categories have exactly complements, with detailed treatment of non-self-folded vs self-folded cases. The results extend Labardini’s mutation-link to higher Calabi–Yau dimensions and provide a concrete geometric realization of mutation dynamics in generalized cluster settings, contributing to the understanding of exchange graphs and tilting theory in higher dimensions.

Abstract

In this paper, we associate a quiver with superpotential to each -angulation of a (unpunctured) marked surface. We show that, under quasi-isomorphisms, the flip of a -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 -cluster categories associated to this quiver with superpotential, we prove that some certain almost complete -cluster tilting objects in the higher cluster category have exactly complements.
Paper Structure (9 sections, 12 theorems, 31 equations, 9 figures)

This paper contains 9 sections, 12 theorems, 31 equations, 9 figures.

Table of Contents

  1. Preliminaries
  2. Generalized higher cluster categories
  3. Quivers with superpotentials
  4. $d$-angulation of a marked surface
  5. $\Gamma_{1}$ is quasi-isomorphic to $\Gamma_ {2}$
  6. Proofs of Theorem \ref{['thm2.16']} and Theorem \ref{['mainthm']}
  7. Proof of Theorem \ref{['thm2.16']}
  8. Proof of Theorem \ref{['mainthm']}
  9. Non-self-folded case

Key Result

Lemma 1.1

Let $A$ be a homologically smooth dg algebra and be an object in $\mathscr{D}(A^{e})$. Then for any $L$ in $\mathscr{D}(A)$ and any $M$ in $\mathscr{D}_{fd}(A)$, there is a canonical isomorphism

Figures (9)

  • Figure 1: Example of 3-angulation and 4-angulation on the same marked surface $(S,M)$
  • Figure 2: Flip of an arc $i$
  • Figure 3: Flip of a self-folded arc $i$
  • Figure 5:
  • Figure 7: The $d$-gon contain $\alpha_t,\alpha_{t+1}$
  • ...and 4 more figures

Theorems & Definitions (42)

  • Lemma 1.1: guo1keller08
  • Definition 1.2: amiotguo1
  • Theorem 1.3: amiotguo1
  • Definition 1.4
  • Definition 1.5: oppermann
  • Definition 1.6: oppermann
  • Proposition 1.7
  • Theorem 1.8: AI
  • Theorem 1.9: oppermann
  • Definition 2.1
  • ...and 32 more