Ungraded Matrix Factorizations as Mirrors of Non-orientable Lagrangians

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 Q² = W · 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 ℝP² ⊂ ℂP² and its mirror ungraded matrix factorization, which we construct and study. In particular, we prove a version of Homological Mirror Symmetry in this setting.