Sectorial Decompositions of Symmetric Products of Surfaces
Xinle Dai
TL;DR
The work shows that sectorial decompositions of a surface induce corresponding decompositions of its second symmetric product, enabling a local-to-global approach to the wrapped Fukaya category via Liouville sectors. By constructing quadratic Stein structures and explicit local models, the authors obtain a global sectorial decomposition of $Sym^2(\Sigma)$ and analyze the fibers and completions of the sectorial pieces. This framework yields a geometric proof of Homological Mirror Symmetry for the 2D pair of pants $Sym^2(\mathbb{P}^1-4\text{ pts})$, matching its wrapped Fukaya category with the derived category of coherent sheaves on the complex variety $\{xyz=0\}\subset\mathbb{C}^3$. The results provide a robust method to glue simple Liouville sectors to recover the symplectic and categorical structure of symmetric products, with potential extensions to higher symmetric powers and broader mirror-symmetry contexts.
Abstract
Symmetric products of Riemann surfaces play a crucial role in symplectic geometry and low-dimensional topology. Symmetric products of punctured surfaces are Liouville manifolds of interest e.g. for Heegaard Floer theory. We study the symplectic topology of these spaces using Liouville sectorial techniques, along with examples and applications of these decompositions in the context of homological mirror symmetry. More precisely, we show that a sectorial decomposition of a Riemann surface along a union of arcs induces a sectorial decomposition of its second symmetric product and as an application, we give a new geometric proof of Homological Mirror Symmetry for the complex two dimensional pair of pants.