Ungraded matrix factorizations as mirrors of non-orientable Lagrangians

Abstract

We introduce the notion of ungraded matrix factorization as a mirror of nonorientable 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 • 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 RP 2 ⊂ CP 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.