Geometry of the ends of the moduli space of anti-self-dual connections

TL;DR

This paper proves that under certain conditions, the moduli space of anti-self-dual connections on a four-manifold has finite volume and diameter, using bubble-tree compactification techniques inspired by harmonic maps and Yang-Mills theory.

Contribution

It establishes the finiteness of volume and diameter of the moduli space for specific gauge groups and topological conditions, confirming conjectures by Groisser, Parker, and Donaldson.

Findings

Finite volume and diameter of the moduli space under specified conditions

Development of bubble-tree compactification for anti-self-dual connections

Application of ideas from harmonic maps and Yang-Mills sequences

Abstract

Let $X$ be a closed, four-dimensional, oriented, smooth manifold with a Riemannian metric, $g$, let $G$ be a compact Lie group, and $P$ be a principal $G$ bundle over $X$. D. Groisser and T. Parker (1987, 1989) and S. K. Donaldson (1990) conjectured that the moduli space of $g$-anti-self-dual connections on $P$, endowed with the $L^2$ metric, has finite volume and diameter. The purpose of this article is to prove this conjecture under the following additional hypotheses. Suppose that $g$ is generic and $X$ is simply-connected. If (i) $G=SU(2)$ or $SO(3)$ and $b^+(X)=0$ or (ii) $G=SO(3)$ and $w_2(P)\neq 0$, where $w_2(P)$ is the second Stiefel-Whitney class of $P$, then we prove that the moduli space of $g$-anti-self-dual connections on $P$ has finite volume and diameter with respect to the $L^2$ metric. Our development of the bubble-tree compactification of the moduli space of $g$-anti-self-dual connections --- based on ideas of Sacks and Uhlenbeck for sequences of harmonic maps from the two-sphere (1981), Taubes (1988) for sequences of Yang-Mills connections, and Parker and Wolfson (1993, 1996) for sequences of pseudo-holomorphic maps --- provides one of the key technical tools used in the proof.