Sutured Manifolds and Polynomial Invariants from Higher Rank Bundles

TL;DR

This paper confirms predictions about generalized Donaldson invariants for certain surfaces and explores their application in defining Floer homology for sutured 3-manifolds, advancing the understanding of higher rank bundle invariants.

Contribution

It rigorously verifies Mari o and Moore's predictions for specific surfaces and develops a Floer homology theory for sutured 3-manifolds based on higher rank bundle invariants.

Findings

Confirmation of predictions for N=3 on specific surfaces

Development of sutured 3-manifold Floer homology

Application of invariants to 3-manifold topology

Abstract

For each integer $N\geq 2$, Mari\~no and Moore defined generalized Donaldson invariants by the methods of quantum field theory, and made predictions about the values of these invariants. Subsequently, Kronheimer gave a rigorous definition of generalized Donaldson invariants using the moduli spaces of anti-self-dual connections on hermitian vector bundles of rank $N$. In this paper, Mari\~no and Moore's predictions are confirmed for simply connected elliptic surfaces without multiple fibers and certain surfaces of general type in the case that $N=3$. The primary motivation is to study 3-manifold instanton Floer homologies which are defined by higher rank bundles. In particular, the computation of the generalized Donaldson invariants are exploited to define a Floer homology theory for sutured 3-manifolds.