标题： 无等级矩阵分解作为非可定向拉格朗日子的镜像 显示英文标题

标题： Ungraded matrix factorizations as mirrors of non-orientable Lagrangians

摘要： 我们引入了无等级矩阵分解的概念，作为非可定向拉格朗日子流形的镜像。 一个多项式$W$的无等级矩阵分解，在特征为 2 的域上，是一个满足$Q^2 = W \cdot \mathrm{Id}$的多项式项的平方矩阵$Q$。 然后我们证明在局部化镜像函子下，非可定向拉格朗日子流形对应于无等级矩阵分解，并在几个实例中说明这一构造。 我们的主要例子是拉格朗日子流形$\mathbb{R}P^2 \subset \mathbb{C}P^2$及其镜像无等级矩阵分解，我们构造并研究了它。 特别是，我们在这种情况下证明了同调镜像对称的一个版本。 显示英文摘要

摘要： 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.