Lagrangians, SO(3)-instantons and mixed equation

TL;DR

This paper investigates the mixed equation combining gauge theory and symplectic geometry, establishing regularity, Fredholm properties, and compactness of solution moduli spaces, with applications to the Atiyah-Floer conjecture.

Contribution

It introduces and analyzes the mixed equation, proving key analytical properties and compactness results crucial for future studies related to the Atiyah-Floer conjecture.

Findings

Established regularity and Fredholm properties for solutions.

Proved compactness of moduli spaces combining Uhlenbeck and Gromov compactness.

Provided foundational results for future work on the Atiyah-Floer conjecture.

Abstract

The mixed equation, defined as a combination of the anti-self-duality equation in gauge theory and Cauchy-Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah-Floer conjecture.