Orbifold resolution via hyperkahler quotients: the $D_2$ ALF manifold

TL;DR

This paper generalizes Kronheimer's hyperkahler resolution construction to infinite dimensions, producing a family of $D_2$ ALF manifolds via hyperkahler quotients and studying their complex and real structures.

Contribution

It introduces a new infinite-dimensional framework for hyperkahler resolutions, constructs $D_2$ ALF manifolds from singular equivariant Nahm data, and establishes stability and connection results.

Findings

Constructed $D_2$ ALF manifolds as hyperkahler quotients.

Proved a Donaldson-Uhlenbeck-Yau type theorem for Nahm data.

Established relationships between $D_2$ ALF and $A_1$ ALE manifolds.

Abstract

We propose an infinite-dimensional generalization of Kronheimer's construction of families of hyperkahler manifolds resolving flat orbifold quotients of $\mathbb{R}^4$. As in [Kro89], these manifolds are constructed as hyperkahler quotients of affine spaces. This leads to a study of \emph{singular equivariant instantons} in various dimensions. In this paper, we study singular equivariant Nahm data to produce the family of $D_2$ asymptotically locally flat (ALF) manifolds as a deformation of the flat orbifold $(\mathbb{R}^3 \times S^1)/Z_2$. 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 $D_2$ ALF manifolds. In the complex formulation, we exhibit explicit relationships between these $D_2$ ALF manifolds and corresponding $A_1$ ALE manifolds. We conjecture analogous constructions and results for general orbifold quotients of $\mathbb{R}^{4-r} \times T^r$ with $2 \le r \le 4$. The case $r = 4$ produces K3 manifolds as hyperkahler quotients.