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\(G_2\)-manifolds with parallel characteristic torsion. (English) Zbl 1141.53019

The author studies seven-dimensional Riemannian manifolds \((M,g)\) admitting a \(G_2\) structure \(\varphi\), which is cocalibrated, i.e., \(d*\varphi=0\). Then there exists a characteristic connection \(\nabla^c\) with totally skew symmetric torsion \(T^c=\frac16(d\varphi,*\varphi)-*d\varphi\) which preserves the \(G_2\) structure. The author investigates those characteristic connections for which the torsion is \(\nabla^c\)-parallel. In this case the manifold \(M\) admits a \(\nabla^c\)-parallel spinor field \(\Psi\) such that \(T^c\cdot\Psi=a\cdot \Psi\), where the factor \(a\) depends on an algebraic type of the torsion form.
The author gives a classification of cocalibrated \(G_2\)-manifolds with parallel characteristic torsion and non-abelian holonomy algebra. There are eight non-abelian Lie subalgebras of the Lie algebra \({\mathfrak g}_2\). The author describes for each of these sub-algebras the set of admissible torsion forms \(T^c\) and the geometry of the corresponding \(G_2\)-manifolds. For example, a \(G_2\)-manifold with parallel torsion and \({\mathfrak{hol}}(\nabla^c)= {\mathfrak{su}}(3)\) turns out to be homothetic to an \(\eta\)-Einstein Sasakian manifold or a product of a nearly Kähler 6-manifold by \(\mathbb{R}\). A regular \(G_2\)-manifold with parallel torsion and \({\mathfrak{hol}}(\nabla^c)= {\mathfrak u}(2)\) is proved to be a principal \(S^1\)-bundle over the projective space \(\mathbb{CP}^3\) or a flag manifold \(\mathbb{F}(1,2)\) equipped with their standard nearly Kähler structures coming from the twistor construction or a principal \(S^1\)-bundle over the Kähler manifold satisfying some further conditions. In the case \({\mathfrak{hol}}(\nabla^c)= {\mathfrak{so}}_{ir}(3)\) the author gives the complete classification result: A complete, simply-connected and cocalibrated \(G_2\)-manifold with parallel characteristic torsion and \({\mathfrak{hol}}(\nabla^c)= {\mathfrak{so}}_{ir}(3)\) is isometric to \(\text{SO}(5)/\text{SO}_{ir}(3)\). Also in the case \({\mathfrak{hol}}(\nabla^c)= {\mathfrak{su}}_c(2)\) the author gives a complete classification.
It should be noted that \(G_2\)-manifolds with a parallel characteristic torsion are solutions of the common sector of type II superstring theory.

MSC:

53C10 \(G\)-structures
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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