Sectorial Decompositions of Symmetric Products of Surfaces

TL;DR

This paper investigates the symplectic topology of symmetric products of Riemann surfaces using Liouville sectorial techniques, providing new decompositions and a geometric proof of Homological Mirror Symmetry for the pair of pants.

Contribution

It introduces a method to induce sectorial decompositions of symmetric products from those of Riemann surfaces, with applications to mirror symmetry.

Findings

Sectorial decompositions of symmetric products are constructed from surface decompositions.

A new geometric proof of Homological Mirror Symmetry for the pair of pants is provided.

Examples demonstrate the applicability of the techniques in symplectic geometry.

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.