Parallel spinors and connections with skew-symmetric torsion in string theory. (English) Zbl 1127.53304

We describe all almost contact metric, almost Hermitian and \(G_2\)-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the \(\nabla\)-parallel spinors. In particular, we obtain partial solutions of the type II string equations in dimensions \(n = 5\), 6 and 7.


53C27 Spin and Spin\({}^c\) geometry
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C05 Connections (general theory)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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