Deformation theory of nearly G₂-structures and nearly G₂ instantons

We study two different deformation theory problems on manifolds with a nearly G₂-structure. The first involves studying the deformation theory of nearly G₂ manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly G₂-structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly G₂-structure on the Aloff–Wallach space are all obstructed to second order. We also completely describe the de Rham cohomology of nearly G₂ manifolds. In the second problem we study the deformation theory of G₂ instantons on nearly G₂ manifolds. We make use of the one-to-one correspondence between nearly parallel G₂-structures and real Killing spinors to formulate the deformation theory in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to explicitly describe the deformation space of the canonical connection on the four normal homogeneous nearly G₂ manifolds. We also describe the infinitesimal deformation space of the SU(3) instantons on Sasaki–Einstein 7-folds which are nearly G₂ manifolds with two Killing spinors. A Sasaki–Einstein structure on a 7-dimensional manifold is equivalent to a 1-parameter family of nearly G₂-structures. We show that the deformation space can be described as an eigenspace of a twisted Dirac operator