A formally Kähler structure on a knot space of a \(G_2\)-manifold
Similar to \(C^{\infty}(M)\) one can define a Fréchet structure on the space of smooth sections of a vector bundle over a finite-dimensional smooth manifold \(M\). The space \(\text{Imm}(S^1,M)\) of smooth immersions of the circle \(S^1\) in \(M\) turns out to be a Fréchet manifold. Put \(\text{Knot}(S^1,M)=\text{Imm}(S^1,M)/\text{Diff}(S^1)\), then it becomes a Fréchet orbifold. Put \(X=\text{Imm}(S^1,M)\). The tangent space \(TX\) is the sheaf of continuous derivations which supports a Lie algebra structure with a bracket of derivations. As usual, an almost complex structure is an endomorphism \(J:TX\rightarrow TX\) with \(J^2=-\text{id}\). When \(TX\otimes{ \mathbb C}=T^{1,0}X\oplus T^{0,1}X\) is the eigenvalue decomposition, if \([T^{1,0}X,T^{1,0}X]\subset T^{1,0}X\), then \(J\) is integrable, or a complex structure on \(X\). \((X,J)\) is then called a complex Fréchet manifold. Given a Hermitian structure \(g\) on \((X,J)\), \((X,g,J)\) is said to be a formally Kähler (Fréchet) manifold if the fundamental \(2\)-form is closed. In this paper, the author constructs a \(CR\)-twistor space from a holonomy \(G_2\)-manifold \(M\). Using this \(CR\)-structure bundle, \(\text{Knot}(S^1,M)\) gets a complex structure \(J\) for which it becomes a complex Fréchet manifold. Moreover, the author observes that Movshev's \(2\)-form \(\omega\) gives a Hermitian form on \((\text{Knot}(S^1,M),J)\). One of the main theorems of this paper is the following one. Theorem. Let \(M\) be a holonomy \(G_2\)-manifold. Then \((\text{Knot}(S^1,M), \omega,J)\) is formally Kähler.
scientific article; zbMATH DE number 6085753
scientific article