Moduli spaces of quadratic differentials: Abel-Jacobi map and deformation

Technical Summary: Moduli Spaces of Quadratic Differentials: Abel-Jacobi Map and Deformation

Problem Statement
The paper investigates the topology of moduli spaces of framed quadratic differentials with prescribed singularities on Riemann surfaces. Specifically, it addresses the calculation of the (orbifold) fundamental groups of these spaces, a problem that has remained challenging with few known cases beyond genus 2 and 3 or specific quotient groups. The study focuses on the case where finite singularities are simple zeros, while infinite singularities include double poles and higher-order poles. These singularities are encoded via a "decorated marked surface with punctures" (DMSp), denoted $S^\mathbb{A}_\Delta$, where decorations correspond to simple zeros, punctures to double poles, and boundary components to higher-order poles.

The central problem is to characterize the fundamental group of the moduli space $\text{FQuad}(S^\mathbb{A})$ and its partial compactifications. The author seeks to relate this topological invariant to the kernel of a topological Abel-Jacobi (AJ) map and to the space of stability conditions on associated 3-Calabi-Yau (3-CY) categories. Furthermore, the paper explores how allowing the collision of simple zeros and double poles (partial compactification) and introducing orbifold structures (via signs of residues) deforms these fundamental groups, potentially recovering non-exceptional spherical and Euclidean Artin braid groups.

Methodology
The methodology combines techniques from low-dimensional topology, the theory of quadratic differentials, and the representation theory of algebras (specifically categorification via stability conditions).

1. Topological Framework (Mixed Twist Groups):
   The author introduces mixed twist groups $MT(S^\mathbb{A}_\Delta, K)$, subgroups of the surface braid group $SBr(S^\mathbb{A}_\Delta)$. These groups are generated by braid twists (associated with simple closed arcs) and $L$-twists (point-pushing diffeomorphisms associated with arcs enclosing specific subsets of punctures $K$). The paper establishes that these mixed twist groups are precisely the kernels of relative topological Abel-Jacobi maps $AJ_K: SBr(S^\mathbb{A}_\Delta) \to H_1(S_L)$, where $L$ is the complement of $K$.

2. Flip Graphs and Groupoids:
   The study analyzes the fundamental groups of oriented flip graphs of decorated triangulations. The author defines flip groupoids $\mathcal{FG}(S^\mathbb{A})$ and identifies their fundamental groups as flip twist groups $FT(T)$. A crucial step involves analyzing the relations in these graphs, which include:

   • Squares and pentagons (lifted from unoriented exchange graphs).
   • Short and long dumbbell relations.
   • Symmetric hexagon relations.
     The paper demonstrates that the higher braid relation ($Br_4$) decomposes into a combination of two long dumbbell hexagons and two symmetric hexagons, a decomposition distinct from non-symmetric hexagon relations found in cluster theory.

3. Categorification via Stability Conditions:
   The moduli space is related to the space of stability conditions $\text{Stab}^\circ(\mathcal{D}_3(S^\mathbb{A}_\Delta))$ on a 3-Calabi-Yau category constructed from a quiver with potential associated with the surface. The author proves that the space of stability conditions is simply connected and serves as the universal cover of the moduli space of framed quadratic differentials. This establishes a correspondence between the fundamental group of the moduli space and the group of autoequivalences (specifically spherical/braid twists) of the category. The author notes that this result supports the belief that the $K(\pi, 1)$ conjecture holds for these strata, as the contractibility of the stability space is closely related to the conjecture.

4. Partial Compactification and Orbifolding:
   The paper studies the deformation of these structures when punctures in a subset $L$ are allowed to collide with simple zeros. This leads to a partial compactification $\text{FQuad}(S_{K,L})$. By further introducing signs of residues at double poles (vortices), the author constructs an orbifold $\text{Quad}(S_{K,Y})$. The fundamental groups of these spaces are calculated by quotienting the original fundamental group by loops corresponding to $L$-flips (for compactification) and their squares (for orbifolding).

Key Contributions and Results

• Identification of the Fundamental Group: The paper proves that the fundamental group of the moduli space of framed quadratic differentials $\text{FQuad}(S^\mathbb{A})$ is isomorphic to the kernel of the topological Abel-Jacobi map from the surface braid group to the first homology of the underlying surface:
  $$\pi_1 \text{FQuad}(S^\mathbb{A}) \cong \ker(AJ: SBr(S^\mathbb{A}_\Delta) \to H_1(S))$$
  This kernel is identified as the mixed twist group $MT(S^\mathbb{A}_\Delta)$.

• Universal Cover via Stability Conditions: It is shown that the space of stability conditions $\text{Stab}^\circ(\mathcal{D}_3(S^\mathbb{A}_\Delta))$ is simply connected and constitutes the universal cover of $\text{FQuad}(S^\mathbb{A})$. This result provides strong evidence for the $K(\pi, 1)$ conjecture for these specific strata of moduli spaces (where all finite singularities are simple zeros).

• Deformation and Compactification:

  • The fundamental group of the partially compactified space (allowing collision of simple zeros and double poles in $L$) is the quotient of the original fundamental group by the subgroup generated by $L$-flips (loops around the collision locus).
  • The fundamental group of the orbifolded space (incorporating signs of residues) is the quotient by the squares of these loops.
  • This deformation process yields an isomorphism between the orbifold moduli space and the quotient of the stability space of a deformed 3-CY category.

• Realization of Artin Braid Groups: The construction provides a topological realization of any non-exceptional spherical and Euclidean Artin braid groups. Specifically, for Euclidean types, the fundamental group of the orbifolded moduli space (with appropriate choices of $K$ and $L$) recovers groups such as $\widetilde{A}_n$, $\widetilde{B}_n$, and $\widetilde{D}_n$.

• New Relations in Flip Graphs: The paper introduces and analyzes new relations in the flip groupoids of punctured surfaces, specifically the long dumbbell and symmetric hexagon relations, which are necessary to describe the fundamental group in the presence of punctures and $L$-twists.

Significance
The paper claims significance in several areas:

1. Topological Classification: It provides a calculation of the fundamental groups for a class of moduli spaces of quadratic differentials with simple zeros, double poles, and higher poles, extending previous results limited to specific genera or singularity types.
2. Bridge between Fields: It solidifies the connection between the topology of moduli spaces of differentials, the combinatorics of flip graphs, and the representation theory of 3-Calabi-Yau categories. It demonstrates that the "categorification" of the moduli space via stability conditions is not just a structural analogy but a tool for computing topological invariants.
3. Deformation Theory: The work offers a precise topological interpretation of the deformation of 3-CY categories (via quivers with potential) corresponding to the partial compactification of moduli spaces. It shows how "colliding" singularities in the geometric setting corresponds to "deforming" the potential in the categorical setting.
4. Braid Group Realizations: By producing non-exceptional Artin braid groups as fundamental groups of these moduli spaces, the paper offers a new geometric and topological perspective on these groups, linking them to the collision of singularities in quadratic differentials and the orbifolding of stability spaces.

The author notes that while the injectivity of the map from the moduli space fundamental group to the braid group kernel is the most challenging part of the proof, the established isomorphism relies on the simple connectivity of the stability space and the specific decomposition of higher braid relations in the flip graphs. The results are presented as a generalization of previous work on braid twist groups and cluster algebras, specifically addressing the complexities introduced by punctures and double poles.