Instanton Floer homology and Milnor Fibers

TL;DR

This paper explores the relationship between Floer homology and Milnor fibers, providing combinatorial formulas for singularities and linking them to invariants in gauge theories and knot theory.

Contribution

It offers a Milnor fiber description of Floer homology for certain singularities and connects these to Casson invariants and Heegaard Floer invariants.

Findings

Derived combinatorial formulas for Floer homology of singularities

Proved equality of Casson invariants in different gauge theories

Expressed Heegaard Floer d-invariant in terms of Milnor fibers

Abstract

In the early days of the Floer theory, Atiyah asked if there is a Milnor fiber description of the Floer homology of the links of singularities. We answer this question for the Brieskorn-Hamm complete intersection singularities. The resulting combinatorial formulas lead to an independent proof of the equality of the Casson invariants in the Donaldson and Seiberg-Witten theories. We use similar techniques to express the Heegaard Floer d-invariant of torus knots in terms of their Milnor fibers.