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On the holonomy of connections with skew-symmetric torsion. (English) Zbl 1055.53031

The authors study linear metric connections with skew-symmetric torsion. Such connections recently appeared in string theory and on Riemannian manifolds they have the same geodesics as the Levi-Civita connection. The basic case of the Eulidean space with a constant skew-symmetric torsion form is investigated in detail. Therewith to every exterior form \(T\) corresponds a holonomy algebra which is the Lie algebra of the infinitesimal holonomy group of the spinorial covariant derivative \(\nabla^T\). This algebra is a Lie subalgebra of the Clifford algebra endowed with a commutator operation. For \(T\) is a three-form (this case corresponds to a metric connection with the skew-symmetric torsion \(T\)) it is proved that the holonomy algebra is semisimple and perfect, i.e. coincides with its derived algebra. Moreover it is proved that the adjoint representation of any compact semisimple Lie algebra is realized as the holonomy algebra of certain \(3\)-form.
If \(T \neq 0\) then there are no nontrivial \(\nabla^T\)-parallel spinors on the Euclidean space. This result is generalized for general Riemannian manifolds as follows. Assuming that a Riemannian manifold \(M\) is compact, spin and has nonpositive sectional curvature and that \(dT\) acts on spinors as a nonpositive endomorphism, the arguments based on the Schrödinger-Lichnerowicz formula allow to prove that the existence of a nontrivial parallel spinor implies that the torsion and the scalar curvature vanish and the spinor is parallel with respect to the Levi-Civita connection. It is also established that for a linear \(s\)-pencil of connections with the torsion \(sT\) in a generic case such a connection has a nontrivial parallel spinor if and only if \(s\) meets some polynomial equation.
The authors also construct and analyze in detail a two-parameter family of metrics on the Aloff-Wallach spaces which admit parallel spinors and has many other interesting geometric properties. For a general \(3\)-Sasakian seven-dimensional manifold the authors prove that it admits a \({\mathbb P}^2\)-parameter family of metric connections with skew-symmetric torsion and parallel spinors and the holonomy groups lying in \(G_2\).

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
53C29 Issues of holonomy in differential geometry
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