A deformed Donaldson–Thomas connection for a manifold with a \(\mathrm{Spin}(7)\)-structure, which we call a \(\mathrm{Spin}(7)\)-dDT connection, is a Hermitian connection on a Hermitian line bundle L over a manifold with a \(\mathrm{Spin}(7)\)-structure defined by fully nonlinear PDEs. It was first introduced by Lee and Leung as a mirror object of a Cayley cycle obtained by the real Fourier–Mukai transform and its alternative definition was suggested in our other paper. As the name indicates, a \(\mathrm{Spin}(7)\)-dDT connection can also be considered as an analogue of a Donaldson–Thomas connection (\(\mathrm{Spin}(7)\)-instanton). In this paper, using our definition, we show that the moduli space \(\mathcal {M}_{\mathrm{Spin}(7)}\) of \(\mathrm{Spin}(7)\)-dDT connections has similar properties to these objects. That is, we show the following for an open subset \(\mathcal {M}'_{\mathrm{Spin}(7)} \subset \mathcal {M}_{\mathrm{Spin}(7)}\). (1) Deformations of elements of \(\mathcal {M}'_{\mathrm{Spin}(7)}\) are controlled by a subcomplex of the canonical complex defined by Reyes Carrión by introducing a new \(\mathrm{Spin}(7)\)-structure from the initial \(\mathrm{Spin}(7)\)-structure and a \(\mathrm{Spin}(7)\)-dDT connection. (2) The expected dimension of \(\mathcal {M}'_{\mathrm{Spin}(7)}\) is finite. It is \(b^1\), the first Betti number of the base manifold, if the initial \(\mathrm{Spin}(7)\)-structure is torsion-free. (3) Under some mild assumptions, \(\mathcal {M}'_{\mathrm{Spin}(7)}\) is smooth if we perturb the initial \(\mathrm{Spin}(7)\)-structure generically. (4) The space \(\mathcal {M}'_{\mathrm{Spin}(7)}\) admits a canonical orientation if all deformations are unobstructed.

具有\(\mathrm{Spin}(7)\) -结构的流形的变形 Donaldson-Thomas 连接，我们称之为\(\mathrm{Spin}(7)\) -dDT 连接，是 Hermitian 连接在具有由完全非线性偏微分方程定义的\(\mathrm{Spin}(7)\)结构的流形上的厄米线丛L上。它首先由 Lee 和 Leung 引入，作为通过实傅立叶-穆凯变换获得的凯莱循环的镜像对象，并且在我们的另一篇论文中提出了它的替代定义。顾名思义，\(\mathrm{Spin}(7)\) -dDT 连接也可以被认为是 Donaldson-Thomas 连接的类似物 ( \(\mathrm{Spin}(7)\)-瞬间）。在本文中，使用我们的定义，我们证明了\(\mathrm{Spin}(7)\) -dDT 连接的模空间\(\mathcal {M}_{\mathrm{Spin}(7)}\)具有与这些对象相似的属性。也就是说，我们对一个开放子集显示以下内容\(\mathcal {M}'_{\mathrm{Spin}(7)} \subset \mathcal {M}_{\mathrm{Spin}(7)}\) . (1) \(\mathcal {M}'_{\mathrm{Spin}(7)}\)元素的变形由 Reyes Carrión 通过引入新的\(\mathrm{ Spin}(7)\) -结构来自初始\(\mathrm{Spin}(7)\) -结构和\(\mathrm{Spin}(7)\) -dDT 连接。(2) 期望维度\(\mathcal {M}'_{\mathrm{Spin}(7)}\)是有限的。如果初始\(\mathrm{Spin}(7)\)结构是无扭转的，则它是\(b^1\)，即基础流形的第一个 Betti 数。(3) 在一些温和的假设下，\(\mathcal {M}'_{\mathrm{Spin}(7)}\)是平滑的，如果我们扰动初始\(\mathrm{Spin}(7)\) -结构一般。(4) 空间\(\mathcal {M}'_{\mathrm{Spin}(7)}\)如果所有变形都畅通无阻，则承认一个规范取向。