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Parallel spinors and connections with skewsymmetric torsion in string theory
Parallel spinors and connections with skewsymmetric torsion in string theory,Thomas Friedrich,Stefan Ivanov
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Parallel spinors and connections with skewsymmetric torsion in string theory
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Citations: 129
)
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Thomas Friedrich
,
Stefan Ivanov
We describe all almost contact metric, almost hermitian and $G_2$structures admitting a connection with totally skewsymmetric 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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Citation Context
(78)
...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 3form, hence, by [
15, Theorem 8.2
], it does not admit an invariant metric connection with skewsymmetric torsion...
Ilka Agricola
,
et al.
On the topology and the geometry of SO(3)manifolds
...[
14
]), while in the quasiKähler (or (1,2)symplectic) case it coincides with the so called second canonical Hermitian connection (see [15])...
Antonio J. Di Scala
,
et al.
Chernflat and Ricciflat 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 Moraru
,
et 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 7dimensional 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 FernandezGray 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 7dimensional manifold is Ric∇flat if and only if its torsion T is harmonic...
Ilka Agricola
,
et al.
A note on flat metric connections with antisymmetric torsion
...There is manifest interest on cocalibrated 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 4manifold
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Thomas Friedrich
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Citations
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Riemannian manifolds with structure group PSU(3)
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(
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Lars Schäfer
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On the topology and the geometry of SO(3)manifolds
Ilka Agricola
,
Julia BeckerBender
,
Thomas Friedrich
Journal:
Annals of Global Analysis and Geometry  ANN GLOB ANAL GEOM
, vol. 40, no. 1, pp. 6784, 2011
IIB black hole horizons with fiveform flux and KT geometry
U. Gran
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J. Gutowski
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Chernflat and Ricciflat invariant almost Hermitian structures
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, vol. 40, no. 1, pp. 2145, 2011