Gauge-theoretic methods in the geometry of G2 manifolds.

• Applicant: Professor Dr. Andriy Haydys
• Subject Area: Mathematics
• Term: from 2020 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 441893240

G2 manifolds constitute an important class of Ricci-flat seven-manifolds and are by now known to exist in abundance. This project is concerned with the construction of invariants of compact G2 manifolds by gauge-theoretic means. More precisely, the intended invariant is based on the count of the so called G2 instantons. However, unlike in lower dimensions, the number of G2 instantons is not expected to be invariant with respect to deformations of the background parameters, in this case the G2 metric. Conjecturally, an invariant, which remains invariant along isotopies of G2 metrics, can be obtained by counting G2 instantons together with certain Seiberg-Witten monopoles on distinguished 3-submanifolds of the ambient G2 manifold.The focus of this project is on the interplay between G2 instantons and the Seiberg-Witten monopoles. A good understanding of the compactifications of the moduli spaces of G2 instantons and the Seiberg-Witten monopoles is the key to establishing such a relationship.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity