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Berglund–Hübsch–Kontsevich homological mirror symmetry conjecture for invertible polynomials

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Let WW be an invertible polynomial in nn variables, let R=C[x1,…,xn]/(W)R=\mathbb{C}[x_1,\dots,x_n]/(W), and let W∗W^\ast be its transpose polynomial. Write Dsingb(R)D^b_{\mathrm{sing}}(R) for the bounded stable derived category of RR, and let \FukW∗\Fuk W^\ast be the directed A∞A_\infty-category of vanishing cycles of an exact Lefschetz fibration associated with W∗W^\ast. Berglund–Hübsch–Kontsevich homological mirror symmetry conjecture. There is an equivalence of triangulated categories

Dsingb(R)≅Db\FukW∗.D^b_{\mathrm{sing}}(R)\cong D^b\Fuk W^\ast.

This conjecture combines transposition mirror symmetry with Kontsevich's homological mirror symmetry, relating categories of matrix factorizations to Fukaya categories. The source presents it as a conjecture and gives no resolution status.

References

Primary source

Masahiro Futaki and Kazushi Ueda, “A note on homological mirror symmetry for singularities of type D”, arXiv:1004.0078 (2010).

derived-categorieshomological-mirror-symmetrysingularities
Posted

Progress summary

Refreshed
Claimed progress

Important special cases are proved, but the conjecture remains open for general invertible polynomials in arbitrary dimensions.

The conjecture predicts an equivalence between a matrix-factorization or singularity category and the directed Fukaya category of the transpose polynomial. It is discussed in the framework of Takahashi, Lekili, and Ueda, but no general resolution is reported.

Known results

  • Futaki and Ueda proved the equivalence for Fermat polynomials in arbitrary dimension.
  • The conjecture is known for Brieskorn–Pham polynomials in arbitrary dimension.
  • Equivalences are proved for transpose polynomials that are disconnected sums of type AA or DD polynomials.
  • Habermann and Smith proved all two-variable invertible cases, in an equivariant or maximally graded formulation.

June 2023 two-variable proof

Habermann and Smith’s revised paper gives a quasi-equivalence for every two-variable invertible polynomial with maximal symmetry group and states that the two-variable conjecture was already established. This does not settle arbitrary dimension or the ungrouped category in the stated problem.

Current status (as of August 2026): Special families, including Fermat and type AA/DD cases, and all two-variable invertible polynomials are settled, while the full conjecture for arbitrary nn remains open.

Sources

Solutions 0

No solutions have been posted yet.