Ungraded matrix factorizations as mirrors of non-orientable Lagrangians

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.