Braid group actions on branched coverings and full exceptional sequences

TL;DR

The paper establishes canonical links between three perspectives on full exceptional sequences: exceptional dissections of marked surfaces, simple branched coverings with matching paths, and Hurwitz systems. By applying Birman–Hilden theory and Hurwitz theory, it analyzes the natural braid-group action on full exceptional sequences, concluding that transitivity holds in several classical settings but fails in positive genus, providing explicit counterexamples to the Bondal–Polishchuk conjecture. The authors construct equivalences that translate braid mutations into arc mutations and Hurwitz moves, enabling a geometric understanding of mutations in Fukaya categories of graded surfaces and their derived gentle-algebra models. The results illuminate the limitations of transitivity for braid actions and motivate further study of autoequivalences and semi-orthogonal decompositions in broader triangulated-category contexts.

Abstract

We relate full exceptional sequences in Fukaya categories of surfaces or equivalently in derived categories of graded gentle algebras to branched coverings over the disk, building on a previous classification result of the first and third author. This allows us to apply tools from the theory of branched coverings such as Birman--Hilden theory and Hurwitz systems to study the natural braid group action on exceptional sequences. As an application, counterexamples are given to a conjecture of Bondal--Polishchuk on the transitivity of the braid group action on full exceptional sequences in a triangulated category.

Figures (8)

• Figure 1: A collection of matching paths $b_1,\ldots,b_n$ from $-1$ to the branch points $p_1,\ldots,p_n$.
• Figure 2: Copy of the disk $D$ cut along paths $c_1,c_2,\ldots,c_n$ avoiding the matching paths (dashed).
• Figure 3: Possible intersections of $a_{1}$ and $a_{2}$ in an ordered exceptional pair $(a_1,a_2)$, where case II is depicted in the universal covering. This figure also illustrates the mutations of the ordered exceptional pair $(a_1,a_2)$ (in blue).
• Figure 4: The action of the generator $\sigma_i$ in the braid group $\mathfrak{B}_n$ on matching paths.
• Figure 5: Example of a full formal arc system with four arcs on a genus one curve with two boundary components. The boundary paths are $\alpha_i,\beta_i,\gamma_i$, $i\in\{1,2\}$, and their concatenations.

Theorems & Definitions (39)

• Conjecture 1.1: Bondal--Polishchuk
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• proof