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Parallel spinors and connections with skew-symmetric torsion in string theory

Parallel spinors and connections with skew-symmetric torsion in string theory,Thomas Friedrich,Stefan Ivanov

Parallel spinors and connections with skew-symmetric torsion in string theory   (Citations: 129)
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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 solutions of the type II string in dimension $n=5,6$ and 7.
Published in 2001.
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    • ...By a routine calculcation, onw shows that d(e2 ∧ e3) = d(e4 ∧ e5) = 0, hence dF ± = 0 and the general expression for the torsion of an almost contact metric structure [15, Theorem 8.2] is reduced to T c = η ∧ dη, and hence yields the formula stated in (1)...
    • ...However, one checks that the Nijenhuis tensor of ϕ+ is not a 3-form, hence, by [15, Theorem 8.2], it does not admit an invariant metric connection with skew-symmetric torsion...

    Ilka Agricolaet al. On the topology and the geometry of SO(3)-manifolds

    • ...[14]), while in the quasi-Kähler (or (1,2)-symplectic) case it coincides with the so called second canonical Hermitian connection (see [15])...

    Antonio J. Di Scalaet al. Chern-flat and Ricci-flat invariant almost Hermitian structures

    • ...Moreover, its torsion form is equal to Id!. Proof. See [3], [9]...
    • ...An orthogonal connection is uniquely determined by its torsion (see for example [9])...

    Ruxandra Moraruet al. Stable bundles on hypercomplex surfaces

    • ...Proof. The first Bianchi identity [Agr06, Thm 2.6], [FrI02] states for flat ∇...
    • ...(1) dT(X,Y,Z,V ) − �T(X,Y,Z,V ) + (∇V T)(X,Y,Z) = 0. By the general formula [FrI02, Cor...
    • ...The expression for the curvature follows from this and the general identity [Agr06, Thm A.1], [FrI02]...
    • ...Recall (see [FrI02, Thm 4.8]) that a 7-dimensional Riemannian manifold (M 7,g) with a fixed G2 structure ! ∈ �3(M 7) admits a ‘characteristic’ connection ∇c (i.e., a metric G2 connection with antisymmetric torsion) if and only if it is of Fernandez-Gray type X1 ⊕ X3 ⊕ X4 (see [FG82] and [Agr06, p. 53] for this notation)...
    • ...A more elaborate argument shows that they cannot even be cocalibrated (type X1 ⊕ X3): by [FrI02, Thm 5.4], a cocalibrated G2 structure on a 7-dimensional manifold is Ric∇-flat if and only if its torsion T is harmonic...

    Ilka Agricolaet al. A note on flat metric connections with antisymmetric torsion

    • ...There is manifest interest on co-calibrated structures, defined by the weaker condition �� = 0, as we see from recent work of Th. Friedrich and S. Ivanov in [15]...

    Rui Albuquerque. On the G 2 bundle of a Riemannian 4-manifold

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