Journal of Symplectic Geometry

Volume 18 (2020)

Number 6

Homological Berglund-Hübsch mirror symmetry for curve singularities

Pages: 1515 – 1574



Matthew Habermann (Department of Mathematics, University College London, United Kingdom)

Jack Smith (St John’s College, Cambridge University, Cambridge, United Kingdom)


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–Hübsch 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.

Received 10 June 2019

Accepted 10 March 2020

Published 2 February 2021