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Abstract: G2-instantons on 7-dimensional manifolds generalize both flat connections in dimension 3, and anti self-dual connections in dimension 4. Donaldson-Segal program expects a certain count of G2-instantons and other objects could yield a topological invariant for 7--manifolds called the prospective G2-Casson invariant.

Related to the compactification/boundary of the moduli space, and based on the gluing construction by Sá Earp-Walpuski on twisted connected sums, Walpuski proposed to construct singular G2-instantons via gluing. This requires a package of analysis adapted to the underlying geometry, which is essentially related to the spectrum of the cross-sectional operator on the standard 5-dimensional unit sphere.

In this talk, we report some work on the Bochner formula for the cross-sectional operator, spectral theory, and a partial analytic obstruction. It is the cohomology of a particular endomorphism bundle on $CP^{2}$, and is related to a version of the Atiyah classes that govern the movement of singularities in this case. It vanishes if and only if the tangent connection is a particular one on the tangent bundle of $CP^{2}$, which is the only case we wish to carry out the analysis. Under model data, we identify a “leading term” in the solution of the linearized equation with the vector fields that “move” the singular locus. This is a joint project with Henrique Sá Earp and Grégoire Menet.