Flat surfaces and stability structures

Abstract

We identify spaces of half-translation surfaces, equivalently complex curves with quadratic differential, with spaces of stability structures on Fukaya-type categories of punctured surfaces. This is achieved by new methods involving the complete classification of objects in these categories, which are defined in an elementary way. We also introduce a number of tools to deal with surfaces of infinite area, where structures similar to those in cluster algebra appear.

67 pages, 7 figures, v2: expanded from previous version following feedback, new title

References in corpus (3)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Stability conditions and the $A_2$ quiver
• Topological Fukaya category and mirror symmetry for punctured surfaces

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