Some observations on deformed Donaldson–Thomas connections

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 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.