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Decorated Marked Surfaces with vortices: Cluster braid group vs. braid twist group

Yu Qiu, Yu Zhou

TL;DR

The paper introduces decorated marked surfaces with vortices, enforcing a Z2-symmetric collision-path relation $B_{s}^{4}=1$ to define a vortex-enhanced surface $ extbf{S}^{igtriangleup}_{ig/Y}$. It proves that King–Qiu’s cluster braid group, the braid twist group of the decorated vortex surface, and the fundamental group of Bridgeland–Smith’s moduli space of $ extbf{S}$-framed GMN differentials are all isomorphic, and provides finite presentations for these groups. By developing a detailed surface-based cluster- and exchange-graph framework, including signed and tagged triangulations, dual triangulations, and Galois coverings, the work unifies cluster-theoretic and geometric approaches to braid groups in the presence of vortices. The results are then applied to the topology of moduli spaces of GMN differentials, establishing a clear link between the combinatorics of decorated surfaces and the geometry of quadratic differential moduli, with implications for K(π,1) type questions and cluster-algebraic symmetries. Overall, the paper extends cluster braid theory to vortex decorations, yielding new algebraic presentations and a deeper topological understanding of moduli spaces of differentials.

Abstract

Let $\mathbf{S}$ be a marked surface with vortices (=punctures with extra $\mathbb{Z}_2$ symmetry). We study the decorated version $\mathbf{S}_\bigtriangleup$, where the $\mathbb{Z}_2$ symmetry lifts to the relation that the fourth power of the braid twist of any collision path (connecting a decoration in $\bigtriangleup$ and a vortex) is identity. We prove the following three groups are isomorphic: King-Qiu's cluster braid group associated to $\mathbf{S}$, the braid twist group of $\mathbf{S}_\bigtriangleup$ and the fundamental group of Bridgeland-Smith's moduli space of $\mathbf{S}$-framed GMN differentials. Moreover, we give finite presentations of such groups.

Decorated Marked Surfaces with vortices: Cluster braid group vs. braid twist group

TL;DR

The paper introduces decorated marked surfaces with vortices, enforcing a Z2-symmetric collision-path relation to define a vortex-enhanced surface . It proves that King–Qiu’s cluster braid group, the braid twist group of the decorated vortex surface, and the fundamental group of Bridgeland–Smith’s moduli space of -framed GMN differentials are all isomorphic, and provides finite presentations for these groups. By developing a detailed surface-based cluster- and exchange-graph framework, including signed and tagged triangulations, dual triangulations, and Galois coverings, the work unifies cluster-theoretic and geometric approaches to braid groups in the presence of vortices. The results are then applied to the topology of moduli spaces of GMN differentials, establishing a clear link between the combinatorics of decorated surfaces and the geometry of quadratic differential moduli, with implications for K(π,1) type questions and cluster-algebraic symmetries. Overall, the paper extends cluster braid theory to vortex decorations, yielding new algebraic presentations and a deeper topological understanding of moduli spaces of differentials.

Abstract

Let be a marked surface with vortices (=punctures with extra symmetry). We study the decorated version , where the symmetry lifts to the relation that the fourth power of the braid twist of any collision path (connecting a decoration in and a vortex) is identity. We prove the following three groups are isomorphic: King-Qiu's cluster braid group associated to , the braid twist group of and the fundamental group of Bridgeland-Smith's moduli space of -framed GMN differentials. Moreover, we give finite presentations of such groups.

Paper Structure

This paper contains 53 sections, 38 theorems, 95 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Fundamental groups and $K(\pi,1)$-conjecture
  4. Cluster braid groups
  5. Topological realizations of Z2-symmetry
  6. New $\mathbb{Z}_{2}$-symmetry realization
  7. Context and results
  8. Comparison with other works
  9. Braid groups associated to quivers with potential
  10. DMS with mixed punctures and vortices
  11. Cluster Weyl groups
  12. Preliminaries
  13. Decorated surfaces with puncture
  14. Braid twists and L-twists
  15. The surface braid group
  16. ...and 38 more sections

Key Result

Lemma 2.3

Let $s\in\mathrm{DV}(\operatorname{S}^{}_\bigtriangleup)$ and $\delta=s^\circlearrowleft$. Then $\operatorname{B}_{s}^2=\mathrm{L}_{\delta}$ in $\operatorname{MCG}(\operatorname{S}^{}_\bigtriangleup)$.

Figures (26)

  • Figure 1.1: Applying $\operatorname{B}_{s}^2$: from left to right
  • Figure 2.1: The braid twist and L-twist
  • Figure 2.2: The completion of a collision path
  • Figure 2.3: Generators for $\operatorname{SBr}(\operatorname{S}^{}_\bigtriangleup)$
  • Figure 2.4: Alternative generators for $\operatorname{SBr}(\operatorname{S}^{}_\bigtriangleup)$, where $\xi_{-j}=\varepsilon_{2j-1}$, $\,\xi_j=\varepsilon_{2j}$, $\zeta_k=\varepsilon_{2g+k}$ for $1\leq j\leq g$ and $1\leq k\leq b-1$
  • ...and 21 more figures

Theorems & Definitions (96)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4: cf. BG
  • Definition 2.5
  • Theorem 2.6
  • Lemma 2.7: Q24
  • Proposition 2.8: QZ3
  • Definition 3.1
  • ...and 86 more