A formally Kahler structure on a knot space of a G2-manifold

TL;DR

This paper proves that the space of knots in a holonomy G2 manifold possesses a formally Kahler structure, extending previous results from Riemannian threefolds to G2-manifolds.

Contribution

It establishes a formal Kahler structure on the knot space of a G2-manifold, generalizing Brylinski's work from threefolds to G2 geometry.

Findings

Knot space in a G2-manifold is formally Kahler.

Extension of Brylinski's result to G2-manifolds.

Provides a new geometric structure on knot spaces in special holonomy manifolds.

Abstract

A knot space in a manifold M is a space of oriented immersions from a circle S^1 to M up to Diff(S^1). Brylinski has shown that a knot space of a Riemannian threefold is formally Kahler. We prove that a space of knots in a holonomy G2 manifold is formally Kahler.