arXiv subjects

Matthew Habermann

Publications and source records attributed to Matthew Habermann.

4 recordsLinked to original sources

arXiv2022

Homological Berglund-H\"ubsch-Henningson mirror symmetry for curve singularities

In this article, we establish homological Berglund--H\"ubsch mirror symmetry for curve singularities where the A--model incorporates equivariance, otherwise known as homological Berglund--H\"ubsch--Henningson mirror symmetry, including for certain deformations of categories. More precisely, we prove a conjecture of Futaki and Ueda in arXiv:1004.0078 which posits that the equivariance in the A-model can be incorporated by pulling back the superpotential to the total space of the corresponding crepant resolution. Along the way, we show that the B--model category of matrix factorisations has a tilting object whose length is the dimension of the state space of the FJRW A--model, a result which might be of independent interest for its implications in the Landau--Ginzburg analogue of Dubrovin's conjecture.

math.SG↗

arXiv2021

Homological mirror symmetry for nodal stacky curves

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.

math.AG↗

arXiv2020

Homological mirror symmetry for invertible polynomials in two variables

In this paper, we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the $B$-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and as an application, we prove derived equivalences between certain nodal stacky curves, some of whose irreducible components have non-trivial generic stabiliser.

math.SG↗

arXiv2019

Homological Berglund-H\"ubsch mirror symmetry for curve singularities

Given a two-variable invertible polynomial, we show that its category of maximally-graded matrix factorisations is quasi-equivalent to the Fukaya-Seidel category of its Berglund-H\"ubsch transpose. This was previously shown for Brieskorn-Pham and $D$-type singularities by Futaki-Ueda. The proof involves explicit construction of a tilting object on the B-side, and comparison with a specific basis of Lefschetz thimbles on the A-side.

math.SG↗