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Homological Berglund–Hübsch–Henningson mirror symmetry for curve singularities

Part of: Representation theory of rings and algebras Symplectic geometry, contact geometry

Published online by Cambridge University Press:  28 April 2025

MATTHEW HABERMANN
MATTHEW HABERMANN*
Affiliation:
Department of Mathematics and Statistics, University of Lancaster, Bailrigg LA1 4YW, UK e-mails: [email protected], [email protected]
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Abstract

In this paper, 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 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 Fan–Jarvis–Ruan–Witten (FJRW) A–model, a result which might be of independent interest for its implications in the Landau–Ginzburg analogue of Dubrovin’s conjecture.

MSC classification

Primary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category
Secondary: 16G50: Cohen-Macaulay modules
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 58
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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