Quadratic differentials as stability conditions: collapsing subsurfaces

Abstract

We introduce a new class of triangulated categories, which are Verdier quotients of three-Calabi-Yau categories from (decorated) marked surfaces, and show that its spaces of stability conditions can be identified with moduli spaces of framed quadratic differentials on Riemann surfaces with arbitrary order zeros and arbitrary higher order poles. A main tool in our proof is a comparison of two exchange graphs, obtained by tilting hearts in the quotient categories and by flipping mixed angulations associated with the quadratic differentials.

Figures (8)

• Figure 1: The figure shows several local horizontal strip decompositions on $\mathbf{S}_\mathbf{w}$ with fixed weighted decorations, depending on a quadratic differential $q$. Here the black vertices are marked points on $\partial\mathbf{S}_\mathbf{w}$, the red vertices are weighted zeros of $q$, the green arcs are geodesics, the black arcs are separating trajectories. The blue lines define $\mathbf{w}$-mixed-angulations of $\mathbf{S}_\mathbf{w}$. The red solid arcs are simple saddle connections, (except for the thick one in the top small octagon, which is a saddle trajectory) and represent the duel graphs of the $\mathbf{w}$-mixed-angulations. The picture in the middle represents crossing a wall of second kind, resulting in a forward flip.
• Figure 3: Horizontal foliation before and after rotating
• Figure 4: A collapse with $\kappa_{11}=5,\kappa_{12}=4$.
• Figure 5: The braid twist $\operatorname{B}_{\eta}$
• Figure 6: Refinement of a flip $\mathbb{A}\xrightarrow{\gamma}\mathbb{A}^\sharp_\gamma$ (collision case)

Theorems & Definitions (80)

• Theorem 1
• Corollary 1
• Lemma 1
• Theorem 2
• Definition 1
• Definition 2
• Remark
• Lemma 2
• Lemma 1
• Proposition 1