Formally integrable complex structures on higher dimensional knot spaces

Abstract

Let $S$ be a compact oriented finite dimensional manifold and $M$ a finite dimensional Riemannian manifold, let ${\rm Imm}_f(S,M)$ the space of all free immersions $\varphi:S \to M$ and let $B^+_{i,f}(S,M)$ the quotient space ${\rm Imm}_f(S,M)/{\rm Diff}^+(S)$, where ${\rm Diff}^+(S)$ denotes the group of orientation preserving diffeomorphisms of $S$. In this paper we prove that if $M$ admits a parallel $r$-fold vector cross product $\varphi \in \Omega ^r(M, TM)$ and $\dim S = r-1$ then $B^+_{i,f}(S,M)$ is a formally K\"ahler manifold. This generalizes Brylinski's, LeBrun's and Verbitsky's results for the case that $S$ is a codimension 2 submanifold in $M$, and $S = S^1$ or $M$ is a torsion-free $G_2$-manifold respectively.