2022/03/25 by Arnav Tripathy, Tripathy, Arnav, Max Zimet +1
Mathematics · #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2203.13730
openalex publication_date 2022/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose an infinite-dimensional generalization of Kronheimer's construction of families of hyperkahler manifolds resolving flat orbifold quotients of ℝ4. As in [Kro89], these manifolds are constructed as hyperkahler quotients of affine spaces. This leads to a study of singular equivariant instantons in various dimensions. In this paper, we study singular equivariant Nahm data to produce the family of D2 asymptotically locally flat (ALF) manifolds as a deformation of the flat orbifold (ℝ3 × S1)/Z2. We furthermore introduce a notion of stability for Nahm data and prove a Donaldson-Uhlenbeck-Yau type theorem to relate real and complex formulations. We use these results to construct a canonical Ehresmann connection on the family of non-singular D2 ALF manifolds. In the complex formulation, we exhibit explicit relationships between these D2 ALF manifolds and corresponding A1 ALE manifolds. We conjecture analogous constructions and results for general orbifold quotients of ℝ4-r × Tr with 2 ≤ r ≤ 4. The case r = 4 produces K3 manifolds as hyperkahler quotients.