On instantons on nearly Kähler 6-manifolds
type: Item
instance of: scholarly article
title: On instantons on nearly Kähler 6-manifolds
zbMATH Open document ID: 1209.53042
author: Feng Xu
published in: The Asian Journal of Mathematics
publication date: 2010-08-13T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ajm/1275671456
review text: 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 struc…
Mathematics Subject Classification ID: 53C38
Mathematics Subject Classification ID: 53C07
zbMATH DE Number: 5770016
zbMATH Keywords: nearly Kähler
zbMATH Keywords: Yang-Mills connections
zbMATH Keywords: instantons
reviewed by: Simon Chiossi
MaRDI profile type: Publication
OpenAlex ID: W2073053224
DOI: 10.4310/AJM.2009.V13.N4.A5
description: scientific article; zbMATH DE number 5770016