Homological Berglund-Hübsch-Henningson mirror symmetry for curve singularities

Abstract

In this article, we establish homological Berglund--Hübsch mirror symmetry for curve singularities where the A--model incorporates equivariance, otherwise known as homological Berglund--Hübsch--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.

48 pages, 12 figures. Comments welcome. V2 minor updates. V3 name change, expanded and clarified arguments and added section on deformations. To appear in Mathematical Proceedings of the Cambridge Philosophical Society