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 pro spective G2-Casson invariant.
Related to the compactification/boundar y of the moduli space, and based on the gluing construction by Sá Earp-Walp uski on twisted connected sums, Walpuski proposed to construct singular G2- instantons via gluing. This requires a package of analysis adapted to the u nderlying geometry, which is essentially related to the spectrum of the cro ss-sectional operator on the standard 5-dimensional unit sphere.
In t his talk, we report some work on the Bochner formula for the cross-sectiona l 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 singularitie s in this case. It vanishes if and only if the tangent connection is a part icular one on the tangent bundle of $CP^{2}$, which is the only case we wis h 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 “mo ve” the singular locus. This is a joint project with Henrique Sá Earp and G régoire Menet.
CONTACT:Yuanqi Wang (University of Kansas) DTSTAMP:20260828T185000 DTSTART;TZID=America/New_York:20251111T110000 DTEND;TZID=America/New_York:20251111T120000 SEQUENCE:0 TRANSP:OPAQUE END:VEVENT END:VCALENDAR