Th. Friedrich, S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian Journ. Math. 6 (2002), 303-336.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....4.4) The article ends with a thourough discussion of an example, namely, the naturally reductive metrics on the 5 dimensional Stiefel manifold. Although we rarely refer to it, this paper is in spirit very close (and in some sense complementary) to a recent article by Friedrich and Ivanov ([FI01]) There, the authors study metric connections with totally skew symmetric torsion preserving a given geometry. Thanks. I am grateful to Thomas Friedrich (Humboldt Universit at zu Berlin) for many valuable discussions on the topic of this paper. My thanks are also due to the Erwin Schr odinger ....

....For the rst claim, one deduces from equation (2) that X r X T = 0. Then it follows for the orthonormal basis Z i ; Z n of m that Z i r Z i T = 0 : In particular, the divergence of T with respect to r coincides with its Riemannian divergence (a more general fact, see [FI01]) 1=2 = 0. Hence we shall drop the superscript, as we did in the statement of the lemma. The second claim follows from Lemma 2.4 by a simple algebraic computation. Remark 2.2. We nish this section with a remark about the connection between the torsion and the Lie algebra ....

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Th. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Tech. report, SFB 288 preprint Nr. 492, 2001, math.DG/0102142.

.... of interest in supergravity and string theories (e.g. FO99a] FO99b] BFOHP01] Whereas in the pure Riemannian case the situation is quite well understood ( Wan89] Wan93] MS00] and the study is extended to other connections than the Levi Civita connection ( Mor96] Buc00b] Buc00a] [FI01]) the question of the existence of parallel spinor elds even for the Levi Civita connection in the Lorentzian and the general pseudo Riemannian case is widely open and the occuring geometries are known only for irreducible manifolds ( BK99] The diculty of the problem in the inde nite case is ....

T. Friedrich and S. Ivanov. Parallel spinors and connections with skewsymmetric torsion in string theory. math.DG/0102142, 2001.

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Th. Friedrich, S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian Journ. Math. 6 (2002), 303-336.

No context found.

Th. Friedrich, S. Ivanov, Parallel spinors and connections with skew symmetric torsion in string theory, math.DG/0102142.

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Th. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, math.dg/0102142, to appear in Asian Journ. Math.

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Th. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, preprint 492, SFB 288, 2001.

....equipped with a G 2 structure. Since the group G 2 is the isotropy group of a 3 form of general type, a G 2 structure is a triple (M ; g; consisting of a 7 dimensional Riemannian manifold and a 3 form of general type at any point. We decompose the G 2 representation (see [11]) 27 and, consequently, there are four basic types of non integrable G 2 structure. In this way we obtain the Fernandez Gray classi cation of G 2 structures (see [7] The di erent types of G 2 structures can be characterized by di erential equations. For example, a G 2 structure ....

....27 (cocalibrated structures) are characterized by the condition that the 3 form is coclosed, 0. In general, the di erential equations for any type of G 2 structure involving the 3 form were derived in [7] In the spirit of the approach of this paper one can nd the computations in [11]. 2.4. Spin(7) structures in dimension 8. Let us consider Spin(7) structures on 8 dimensional Riemannian manifolds. The subgroup Spin(7) SO(8) is the real Spin(7) representation 7 = R . The complement m = R is the standard 7 dimensional representation and the Spin(7) structures on an ....

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Th. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, math.dg/0102142, to appear in Asian Journ. Math.

.... any hermitian manifold with skew symmetric Nijenhuis tensor in even dimensions, any cocalibrated G 2 manifold in dimension seven and any Spin(7) manifold in dimension eight admit a unique metric connection with skew symmetric torsion and preserving the additional geometric structure (see [10] and [9] The torsion forms of these connections are models for the B eld in the string equations and their parallel spinor elds are the supersymmetries of the models. From the mathematical point of view, the basic role of these connections is closely related to the fact that many of the ....

....operator contains all r parallel spinor. Proof. By Theorem 2.1, one of the integrability conditions for a r parallel spinor eld is 3 dT 2 T 2 (T) Scal = 0 : If the torsion form T is r parallel, the formulas for the Casimir operator simplify. Indeed, in this case we have (see [10]) dT = 2 T ; T) 0 : Moreover, the Ricci tensor Ric is symmetric and we have Scal = Scal 2 Using the formula of Proposition 2.1 as well as the formulas of Theorem 2.1 and Theorem 2.2, we obtain a simpler expressions for the Casimir operator. Proposition 3.2. The Casimir operator of a ....

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Th. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian Journ. Math. 6 (2002), 303-336.

....is a 6 tuple (M ; g; r; T ; consisting of a Riemannian metric g, a metric connection r with totally skew symmetric torsion form T , a dilation function and a spinor eld . If the dilation function is constant, the string equations can be written in the following form (see [Stro] and [IP, FI1]) Ric = 0; T ) 0; r = 0; T = 0 : Therefore, an interesting problem is the investigation of metric connections with totally skewsymmetric torsion. In [FI1] we proved that several non integrable geometric structures (almost contact metric structures, almost complex structures, G 2 ....

....spinor eld . If the dilation function is constant, the string equations can be written in the following form (see [Stro] and [IP, FI1] Ric = 0; T ) 0; r = 0; T = 0 : Therefore, an interesting problem is the investigation of metric connections with totally skewsymmetric torsion. In [FI1] we proved that several non integrable geometric structures (almost contact metric structures, almost complex structures, G 2 structures) admit a unique connection r preserving it with totally skew symmetric torsion. Moreover, we computed the corresponding torsion form T and we studied the ....

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Th. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, math.dg/0102142.

....too. The canonical connection of a naturally reductive Riemannian space is a rst example (see [1] Moreover, we know many non integrable geometric structures on Riemannian manifolds admitting a unique metric connection preserving the structure and with non vanishing skew symmetric torsion (see [15], 14] Following Cartan as well as the idea that torsion forms are candidates for the so called B eld in string theory, the geometry of these connections deserves systematic investigation. Basically, there are no general results concerning the holonomy group of connections with torsion. The ....

....We remark that there exist indeed metric connections with skew symmetric torsion and parallel 2 forms. Indeed, consider an almost hermitian manifold with totally skewsymmetric Nijenhuis tensor. Then there is a unique connection r preserving the hermitian structure with skew symmetric torsion (see [15]) The fundamental form of the hermitian structure is r parallel. A second example are Sasakian manifolds. For these, the di erential of the contact form is parallel with respect to the unique connection preserving the Sasakian structure. 6. Schr odinger Lichnerowicz type formulas for Dirac ....

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Th. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian Journ. Math. 6 (2002), 303-336.

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T. Friedrich, S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 No 2 (2002), 303-336.

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T. Friedrich and S. Ivanov. Parallel spinors and connections with skewsymmetric torsion in string theory. math.DG/0102142, 2001.

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