Floer Field Philosophy

TL;DR

This survey introduces Floer field theory, a framework for constructing 3-manifold invariants via categorical methods, and discusses its potential 4-dimensional extensions and connections to gauge theories.

Contribution

It provides an accessible introduction to the categorical language of Floer field theory and explores its applications to gauge theoretic invariants and conjectures.

Findings

Categorical framework for Floer field theory explained

Connections between Floer homology and gauge theories discussed

Foundations for potential 4D extensions outlined

Abstract

Floer field theory is a construction principle for e.g. 3-manifold invariants via decomposition in a bordism category and a functor to the symplectic category, and is conjectured to have natural 4-dimensional extensions. This survey provides an introduction to the categorical language for the construction and extension principles and provides the basic intuition for two gauge theoretic examples which conceptually frame Atiyah-Floer type conjectures in Donaldson theory as well as the relations of Heegaard Floer homology to Seiberg-Witten theory.