Bubble tree compactification of instanton moduli spaces

Abstract

We study the compactification of moduli spaces defined by the anti-self-dual (ASD) Yang-Mills equations on SU(2) or SO(3) bundles over closed oriented Riemannian 4-manifolds and 4-orbifolds. In general the ASD moduli space is not compact since the curvature of a sequence of connections may blow up at some points in the manifold. There is a widely-used version of compactification called Uhlenbeck compactification. The idea is to add “ideal connections” to the ASD moduli space as limit points of sequences of ASD connections with curvature blow up. One downside of Uhlenbeck compactification is that the space near those added “ideal connections” (which are called lower strata) are very singular. But under certain circumstances, e.g., the KotschickMorgan conjecture, we want the compact moduli space to be smooth (smooth manifold or orbifold) so lower strata contributions can be extracted. In the first part of this thesis, we review the background and set-up of the ASD moduli spaces and Uhlenbeck compactification. In the second part of this thesis, we introduce the main technical machinery: bubble tree compactification, which gives us a smooth orbifold structure on ASD moduli spaces around lower strata. In the third part of this thesis, we apply the technique to a 4-orbifold with finite singularities and cyclic-group-actions around singularities and get a bubble tree compactification of the ASD moduli space on SU(2) or SO(3) bundles over this particular kind of 4-orbifold. Then we calculate an example on the bubble tree compactification of the instanton moduli space on the complex projective space CP2 and the weighted complex projective space CP2 [a0,a1,a2] . In the fourth part, we give the definition of the Donaldson invariant through bubble tree compactification, extend it to the orbifold case and get an “equivariant localised Donaldson invariant”. In the fifth part, as an application of bubble tree compactification, we prove the Kotschick-Morgan conjecture. Finally, we discuss the future directions that the bubble-tree-technique may apply to, including an orbifold-version of the Kotschick-Morgan conjecture, K-theoretic Donaldson invariant, bubble tree compactification for non-abelian monopole moduli spaces and potential application to the Witten conjecture relating Donaldson invariants and Seiberg-Witten invariants. Show more