Invariant tubular neighborhoods in infinite-dimensional Riemannian geometry, with applications to Yang-Mills theory

TL;DR

This paper develops a new method for constructing invariant tubular neighborhoods in infinite-dimensional Riemannian manifolds, facilitating advanced geometric and topological analysis in gauge theory and moduli spaces.

Contribution

It introduces a novel construction of invariant tubular neighborhoods in infinite-dimensional Riemannian manifolds, applicable to gauge theory and moduli space analysis.

Findings

Constructed G-invariant tubular neighborhoods for submanifolds with trivial normal bundle.

Applied the construction to Morse strata in Yang-Mills theory.

Enabled calculations of gauge-equivariant cohomology for moduli spaces.

Abstract

We present a new construction of tubular neighborhoods in (possibly infinite dimensional) Riemannian manifolds M, which allows us to show that if G is an arbitrary group acting isometrically on M, then every G-invariant submanifold with locally trivial normal bundle has a G-invariant total tubular neighborhood. We apply this result to the Morse strata of the Yang-Mills functional over a closed surface. The resulting neighborhoods play an important role in calculations of gauge-equivariant cohomology for moduli spaces of flat connections over non-orientable surfaces.