  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. It can also be considered as an analogue of a
$G_2$-instanton. In this paper, we see that some important observations that
appear in other geometric problems are also found in the dDT case as follows.
  (1) A dDT connection exists if a 7-manifold has full holonomy $G_2$ and the
$G_2$-structure is ``sufficiently large". (2) The dDT equation is described as
the zero of a certain multi-moment map. (3) The gradient flow equation of a
Chern-Simons type functional of Karigiannis and Leung, whose critical points
are dDT connections, agrees with the ${\rm Spin}(7)$ version of the dDT
equation on a cylinder with respect to a certain metric on a certain space.
This can be considered as an analogue of the observation in instanton Floer
homology for 3-manifolds.
