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.

MSC:

 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.)
Full Text: