Deformation Theory of Deformed Donaldson–Thomas Connections for $${\text {Spin}(7)}$$-manifolds

Abstract

A deformed Donaldson-Thomas connection for a manifold with a ${\rm Spin}(7)$-structure, which we call a ${\rm Spin}(7)$-dDT connection, is a Hermitian connection on a Hermitian line bundle $L$ over a manifold with a ${\rm Spin}(7)$-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 ${\rm Spin}(7)$-dDT connection can also be considered as an analogue of a Donaldson-Thomas connection (${\rm Spin}(7)$-instanton). In this paper, using our definition, we show that the moduli space $\mathcal{M}_{{\rm Spin}(7)}$ of ${\rm Spin}(7)$-dDT connections has similar properties to these objects. That is, we show the following for an open subset $\mathcal{M}'_{{\rm Spin}(7)} \subset \mathcal{M}_{{\rm Spin}(7)}$. (1) Deformations of elements of $\mathcal{M}'_{{\rm Spin}(7)}$ are controlled by a subcomplex of the canonical complex introduced by Reyes Carri\'on by introducing a new ${\rm Spin}(7)$-structure from the initial ${\rm Spin}(7)$-structure and a ${\rm Spin}(7)$-dDT connection. (2) The expected dimension of $\mathcal{M}'_{{\rm Spin}(7)}$ is finite. It is $b^1$, the first Betti number of the base manifold, if the initial ${\rm Spin}(7)$-structure is torsion-free. (3) Under some mild assumptions, $\mathcal{M}'_{{\rm Spin}(7)}$ is smooth if we perturb the initial ${\rm Spin}(7)$-structure generically. (4) The space $\mathcal{M}'_{{\rm Spin}(7)}$ admits a canonical orientation if all deformations are unobstructed.