Deformation theory of deformed Donaldson-Thomas connections for $\mathrm{Spin}(7)$-manifolds

Abstract

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 ${\mathcal{M}}_{\mathrm{Spin}(7)}^{\prime }\subset {\mathcal{M}}_{\mathrm{Spin}(7)}$. (1) Deformations of elements of ${\mathcal{M}}_{\mathrm{Spin}(7)}^{\prime }$ are controlled by a subcomplex of the canonical complex introduced 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 ${\mathcal{M}}_{\mathrm{Spin}(7)}^{\prime }$ 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, ${\mathcal{M}}_{\mathrm{Spin}(7)}^{\prime }$ is smooth if we perturb the initial $\mathrm{Spin}(7)$-structure generically. (4) The space ${\mathcal{M}}_{\mathrm{Spin}(7)}^{\prime }$ admits a canonical orientation if all deformations are unobstructed.

51 pages, v2: minor corrections, final version. arXiv admin note: text overlap with arXiv:2004.00532