Homological mirror symmetry for nodal stacky curves

TL;DR

This paper proves homological mirror symmetry for certain nodal stacky curves by relating the A-model to quotients of Milnor fibres and generalising previous approaches to include non-trivial stabilisers.

Contribution

It establishes homological mirror symmetry for nodal stacky curves with non-trivial stabilisers, confirming a conjecture and extending existing methods to more general cases.

Findings

Homological mirror symmetry proven for specific nodal stacky curves.

A-model is shown to be symplectomorphic to a quotient of the Milnor fibre.

Generalisation of previous techniques to include non-trivial stabilisers.

Abstract

In this paper, we establish homological mirror symmetry where the A-model is a finite quotient of the Milnor fibre of an invertible curve singularity, proving a conjecture of Lekili and Ueda from arXiv:1806.04345 in this dimension. Our strategy is to view the B--model as a cycle of stacky projective lines and generalise the approach of Lekili and Polishchuk in arXiv:1705.06023 to allow the irreducible components of the curve to have non-trivial generic stabiliser, a result which might also be of independent interest. We then prove that the A--model which results from this strategy is graded symplectomorphic to the corresponding quotient of the Milnor fibre.