On instantons on nearly Kähler 6-manifolds
An \(\omega\)-anti-selfdual instanton on an almost Hermitian \(6\)-manifold \((M, \omega, g)\) is a connection on a principal \(U(3)\)-bundle whose curvature two-form \(F\) is primitive and of type \((1,1)\), so that \(\omega\wedge F=-*F\), \(*\) being the Hodge operator. For overviews and a general discussion see [\textit{S. K. Donaldson} and \textit{R. P. Thomas}, in: The geometric universe: science, geometry, and the work of Roger Penrose. Proceedings of the symposium on geometric issues in the foundations of science, Oxford, UK, June 1996 in honour of Roger Penrose in his 65th year. Oxford: Oxford University Press. 31--47 (1998; Zbl 0926.58003)] and [\textit{G. Tian}, Ann. Math. (2) 151, No.~1, 193--268 (2000; Zbl 0957.58013)]. The crucial point is to find interesting examples. \textit{R. L. Bryant} showed [Asian J. Math. 10, No.~3, 561--605 (2006; Zbl 1114.53026)] that so-called quasi-integrable \(U(3)\)-structures on \(M^6\) give non-trivial instantons, essentially because the governing differential system is involutive. This class is also known as being of `type 1+3+4', in the argot of intrinsic torsion. This paper considers \(\omega\)-anti-selfdual instantons \(A\) on nearly Kähler \(SU(3)\)-manifolds, and shows that the local differential system for an \(\omega\)-anti-selfdual instanton is involutive. The main tool is Cartan-Kähler theory. As \(\omega\)-anti-selfdual instantons are critical for the Chern-Simons functional \(\int \text{tr}(F^2)\wedge \omega\), variational methods may be employed towards their study. \textit{M. Verbitsky}'s article [Pac. J. Math. 235, No.~2, 323--344 (2008; Zbl 1156.53045)] is another valuable reference for a similar point of view. Non-homogeneous examples with additional \(SO(4)\)-symmetry are discussed, together with deformations of \(\omega\)-anti-selfdual instantons on the six-sphere. For the latter see also the treatise of [\textit{A. Moroianu, P.-A. Nagy} and \textit{U. Semmelmann}, ``Deformations of nearly Kähler structures'', Pac. J. Math. 235, No.~1, 57--72 (2008; Zbl 1152.58015)] and references. A parallel discussion is carried out for \(\Omega\)-anti-selfdual instantons on \(G_2\)-manifolds \(X^7\) (with fundamental form \(\Omega\)), precisely \(G_2\)-cones over nearly Kähler manifolds \(M\), and the mutual relationship is established.
scientific article; zbMATH DE number 5770016