Homological mirror symmetry of Fermat polynomials

We discuss homological mirror symmetry of Fermat polynomials in terms of derived Morita equivalence between derived categories of coherent sheaves and Fukaya-Seidel categories (a.k.a. perfect derived categories of directed Fukaya categories), and some related aspects such as stability conditions, (kinds of) modular forms, and Hochschild homologies.

💡 Research Summary

The paper investigates homological mirror symmetry (HMS) for the family of Fermat polynomials (W_n = x_1^n + \dots + x_n^n) regarded as Landau‑Ginzburg (LG) potentials. The authors establish a derived Morita equivalence between the B‑model side, described by the derived category of coherent sheaves on the projective hypersurface defined by (W_n) (or equivalently by the category of matrix factorizations (\operatorname{MF}(W_n))), and the A‑model side, given by the Fukaya‑Seidel category (\mathcal{F}!S(W_n)) associated to the same potential.

The construction proceeds as follows. On the B‑side, the authors recall that (\operatorname{MF}(W_n)) is equivalent to the derived category of modules over the finite‑dimensional algebra (A_n = \mathbb{C}