Ungraded matrix factorizations as mirrors of non-orientable Lagrangians

TL;DR

This paper introduces ungraded matrix factorizations as mirrors of non-orientable Lagrangian submanifolds, establishing a new correspondence in homological mirror symmetry, especially for non-orientable cases like $ ext{RP}^2$ in $ ext{CP}^2$.

Contribution

It defines ungraded matrix factorizations over characteristic 2 fields and demonstrates their role as mirrors to non-orientable Lagrangians, extending homological mirror symmetry.

Findings

Constructed mirror ungraded matrix factorization for $ ext{RP}^2$ in $ ext{CP}^2$

Proved a version of Homological Mirror Symmetry in this non-orientable setting

Illustrated the correspondence with explicit examples

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.