论文标题
变形的Hermitian Yang-Mills连接和变形的Donaldson-Thomas Connections的变形理论
Deformation theory of deformed Hermitian Yang-Mills connections and deformed Donaldson-Thomas connections
论文作者
论文摘要
变形的Donaldson-Thomas（DDT）连接是在满足某个非线性PDE的$ G_2 $ -Manifold $ x $上的Hermitian系列束的Hermitian连接。在镜像对称性的背景下，这被认为是（CO）关联循环的镜像。 DDT连接是最近对最近进行了广泛研究的变形的Hermitian Yang-Mills（Dhym）连接的类似物。 在本文中，我们研究了DDT和DHYM连接的模量空间。在前半部分中，我们证明了DDT连接的变形由Canonical Complex的一个子复合物（由ReyesCarrión定义的椭圆形复合体）的子复合，通过引入新的Coclosed $ G_2 $结构来控制。如果变形毫无结构，我们还表明连接的组件是$ b^{1} $ - 尺寸圆环，其中$ b^{1} $是$ x $的第一个betti号。还给出了模量空间上的规范取向。我们还证明，如果我们在某些温和的假设下，通常将变形的障碍物消失。 在后半部分，我们证明了Dhym连接的模量空间，如果是非空的，则是$ b^{1} $ - 尺寸的圆环，尤其是它是连接和定向的。如果满足某些必要的条件，我们还沿着基础结构的变形证明了模量空间家族的存在。
A deformed Donaldson-Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a $G_2$-manifold $X$ satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. The dDT connection is an analogue of a deformed Hermitian Yang-Mills (dHYM) connection which is extensively studied recently. In this paper, we study the moduli spaces of dDT and dHYM connections. In the former half, we prove that the deformation of dDT connections is controlled by a subcomplex of the canonical complex, an elliptic complex defined by Reyes Carrión, by introducing a new coclosed $G_2$-structure. If the deformation is unobstructed, we also show that the connected component is a $b^{1}$-dimensional torus, where $b^{1}$ is the first Betti number of $X$. A canonical orientation on the moduli space is also given. We also prove that the obstruction of the deformation vanishes if we perturb the $G_2$-structure generically under some mild assumptions. In the latter half, we prove that the moduli space of dHYM connections, if it is nonempty, is a $b^{1}$-dimensional torus, especially, it is connected and orientable. We also prove the existence of a family of moduli spaces along a deformation of underlying structures if some necessary conditions are satisfied.
