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Generalised \(G_2\)-manifolds. (English) Zbl 1154.53014

The author introduces new Riemannian structures on \(7\)-manifolds by a differential form of mixed degree which is the critical point of a (possibly constrained) variational problem over a fixed cohomology class. The unconstrained critical points generalise the notion of a manifold of holonomy \(G _2\), while the constrained ones give rise to a new geometry without a classical counterpart. The new geometry may be defined by an even or odd form that is considered as a spinor for the orthogonal bundle \(T\oplus T^*\) (where \(T\) and \(T^*\) denote the tangent and cotangent bundle, respectively) with its natural inner product of split signature. The author considers the case of a stable spinor whose stabiliser is conjugate to \(G_2\times G_2\), and a generalised \(G_2\)-structure is a (topological) reduction from \(\mathbb R^*\times \text{Spin}(7,7)\) to \(G_2 \times G_2\).
For a generalised \(G_2\)-structure, he introduces the notion of strong integrability, interpreting the condition in terms of two linear metric connections with skew-symmetric, closed torsion. He shows that the integrability conditions are equivalent to the supersymmetry equations on spinor in type II supergravity theory with bosonic background fields.
Moreover, he constructs explicit examples of generalised \(G_2\)-structures by introducing the device of \(T\)-duality. Furthermore, the notion of generalised \(\text{Spin}(7)\)-structure is briefly considered (more details can be found in his Ph.D. thesis, see “Special metric structures and closed forms”, preprint,
url{arxiv:math.DG/0502443}).

MSC:

53C10 \(G\)-structures
53C29 Issues of holonomy in differential geometry
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References:

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