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arXiv · 2205.01046

Ungraded matrix factorizations as mirrors of non-orientable Lagrangians

Lino Amorim·Cheol-Hyun Cho
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Abstract

We introduce the notion of ungraded matrix factorization as a mirror of non-orientable Lagrangian submanifolds. An ungraded matrix factorization of a polynomial $W$, with coefficients in a field of characteristic 2, is a square matrix $Q$ of polynomial entries satisfying $Q^2 = W \cdot \mathrm{Id}$. We then show that non-orientable Lagrangians correspond to ungraded matrix factorizations under the localized mirror functor and illustrate this construction in a few instances. Our main example is the Lagrangian submanifold $\mathbb{R}P^2 \subset \mathbb{C}P^2$ and its mirror ungraded matrix factorization, which we construct and study. In particular, we prove a version of Homological Mirror Symmetry in this setting.

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BibTeXRIS

Lino Amorim, Cheol-Hyun Cho. 2022-05-02. Ungraded matrix factorizations as mirrors of non-orientable Lagrangians. https://arxiv.org/abs/2205.01046

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