Formally integrable complex structures on higher dimensional knot spaces

Let $S$ be a compact oriented finite dimensional manifold and $M$ a finite dimensional Riemannian manifold, let $\operatorname{Imm}_f (S,M)$ the space of all free immersions $\varphi : S \to M$ and let $B^{+}_{i,f} (S,M)$ the quotient space $\operatorname{Imm}_f (S,M) / \operatorname{Diff}^{+} (S)$, where $\operatorname{Diff}^{+} (S)$ 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 $\chi \in \Omega^r (M,T M)$ and $\operatorname{dim}S=r -1$ then $B^{+}_{i,f} (S,M)$ is a formally Kähler 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.

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The research of H.V.L. was supported by GAČR-project 18-00496S and RVO:67985840.

Received 18 December 2019

Accepted 14 October 2020

Published 21 July 2021