Homological mirror symmetry for the symmetric squares of punctured spheres

TL;DR

This paper proves a homological mirror symmetry equivalence between the wrapped Fukaya category of symmetric squares of punctured spheres and derived categories of coherent sheaves on certain singular surfaces, using categorical resolutions and localizations.

Contribution

It establishes the HMS for symmetric squares of punctured spheres with explicit constructions of the involved categories and conjectures a general framework for higher symmetric powers.

Findings

Wrapped Fukaya category is quasi-equivalent to derived category of coherent sheaves.

Constructs explicit toric Landau-Ginzburg models related to punctured surfaces.

Proposes a general homological mirror symmetry conjecture for higher symmetric powers.

Abstract

For an appropriate choice of a $\mathbb{Z}$-grading structure, we prove that the wrapped Fukaya category of the symmetric square of a $(k+3)$-punctured sphere, i.e. the Weinstein manifold given as the complement of $(k+3)$ generic lines in $\mathbb{C}P^2$ is quasi-equivalent to the derived category of coherent sheaves on a singular surface $\mathcal{Z}_{2,k}$ constructed as the boundary of a toric Landau-Ginzburg model $(\mathcal{X}_{2,k}, \mathbf{w}_{2,k})$. We do this by first constructing a quasi-equivalence between certain categorical resolutions of both sides and then localising. We also provide a general homological mirror symmetry conjecture concerning all the higher symmetric powers of punctured spheres. The corresponding toric LG-models $(\mathcal{X}_{n,k},\mathbf{w}_{n,k})$ are constructed from the combinatorics of curves on the punctured surface and are related to small toric resolutions of the singularity $x_1\ldots x_{n+1}= v_1\ldots v_k$.