The real Fourier-Mukai transform of Cayley cycles (Q2071464)

This paper concerns Spin(7)- and \(G_{2}\)-structures. As is well known, a manifold supporting a Spin(7)- (resp. \(G_{2}\)-)structure has dimension 8 (resp. 7). These structures are closely related to each other and appear in Berger's list of possible holonomy groups of Riemannian manifolds. In the paper under review the authors give an alternative definition of the notion of a deformed Donaldson-Thomas connection for a manifold with a Spin(7)-structure, checking that it is compatible with a deformed Donaldson-Thomas connection for a manifold with a \(G_{2}\)-structure. It is also compatible with a deformed Hermitian Yang-Mills connection for a Calabi-Yau 4-manifold. As essential tool of the work, the authors use the real Fourier-Mukai transform, which has been previously used by \textit{N. C. Leung} et al. [Adv. Theor. Math. Phys. 4, No. 6, 1319--1341 (2000; Zbl 1033.53044)] to introduce deformed Hermitian Yang-Mills connections for Kähler manifolds and by \textit{J.-H. Lee} and \textit{N. C. Leung} [Adv. Theor. Math. Phys. 13, No. 1, 1--31 (2009; Zbl 1171.81417)] to introduce deformed Donaldson-Thomas connections for Spin(7)- and \(G_{2}\)-manifolds. Besides, the theories of deformations of deformed Donaldson-Thomas connections for a manifold with a Spin(7)-structure are developed in the present paper.