论文标题
未分级的矩阵因素化作为不可取向拉格朗日的镜子
Ungraded matrix factorizations as mirrors of non-orientable Lagrangians
论文作者
论文摘要
我们介绍了未分级矩阵分解的概念，作为不可取向拉格朗日submanifolds的镜像。多项式$ w $的未分级矩阵分解，具有特征2领域的系数，是满足$ q^2 = w \ cdot \ cdot \ mathrm {id} $的方形矩阵$ q $。然后，我们表明，不可方向的拉格朗日人对应于局部镜像函子下未分级的矩阵因子化，并在少数情况下说明了这种结构。我们的主要示例是Lagrangian submanifold $ \ Mathbb {r} p^2 \ subset \ mathbb {c} p^2 $及其镜像未分级的矩阵分解，我们构建和研究。特别是，在这种情况下，我们证明了同源镜像的一种版本。
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.
