Normal form of equivariant maps in infinite dimensions

TL;DR

This paper develops new local normal form theorems for smooth equivariant maps in infinite-dimensional manifolds, extending classical results and applying them to important moduli spaces in geometry and gauge theory.

Contribution

It introduces novel normal form results for equivariant maps in infinite dimensions, utilizing a Slice Theorem for Fréchet manifolds, and applies these to key moduli spaces.

Findings

Normal form theorems are valid even in finite dimensions.

Moduli spaces can be locally modeled as Kuranishi spaces.

Applications include anti-self-dual instantons, Seiberg-Witten, and pseudoholomorphic curves.

Abstract

Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov-Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a Slice Theorem for Fr\'echet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg-Witten moduli space and the moduli space of pseudoholomorphic curves.