Higher order integrability in generalized holonomy

Batrachenko, A.; Liu, James T.; Varela, Oscar; Wen, Weiling

doi:10.1016/j.nuclphysb.2006.10.017

Cited by 4 publications

(11 citation statements)

References 22 publications

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“…This computation can be simplified in various ways. First, it is not necessary to compute both D α σ I and D ᾱσ I because since the spinors ǫ are real the equations derived from D α ǫ are complex conjugate to those of D ᾱǫ and so are not independent 12 . In addition, since D 0 ǫ is real only half of the relations are independent.…”

Section: Killing Spinor Equations In Canonical Basismentioning

confidence: 99%

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Systematics of M-theory spinorial geometry

Gran

Papadopoulos

Roest

2005

Class. Quantum Grav.

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We reduce the classification of all supersymmetric backgrounds in eleven dimensions to the evaluation of the supercovariant derivative and of an integrability condition, which contains the field equations, on six types of spinors. We determine the expression of the supercovariant derivative on all six types of spinors and give in each case the field equations that do not arise as the integrability conditions of Killing spinor equations. The Killing spinor equations of a background become a linear system for the fluxes, geometry and spacetime derivatives of the functions that determine the spinors. The solution of the linear system expresses the fluxes in terms of the geometry and specifies the restrictions on the geometry of spacetime for all supersymmetric backgrounds. We also show that the minimum number of field equations that is needed for a supersymmetric configuration to be a solution of eleven-dimensional supergravity can be found by solving a linear system. The linear systems of the Killing spinor equations and their integrability conditions are given in both a timelike and a null spinor basis. We illustrate the construction with examples.

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Section: Killing Spinor Equations In Canonical Basismentioning

confidence: 99%

“…One may have to consider higher order integrability conditions[12] 4. We denote the tenth direction with ♮.…”

mentioning

confidence: 99%

Systematics of M-theory spinorial geometry

Gran

Papadopoulos

Roest

2005

Class. Quantum Grav.

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“…Because of this, we use only the Spin(9, 1) gauge group to bring the Killing spinor into a canonical form. We find that there are three cases to be considered which are distinguished by the stability subgroup of the Killing spinors in Spin (9,1). These are Spin(7) ⋉ R 8 , SU(4) ⋉ R 8 and G 2 .…”

Section: Introductionmentioning

confidence: 99%

The spinorial geometry of supersymmetric IIB backgrounds

Gran

Gutowski

Papadopoulos

2005

Class. Quantum Grav.

201

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We investigate the Killing spinor equations of IIB supergravity for one Killing spinor. We show that there are three types of orbits of Spin(9, 1) in the space of Weyl spinors which give rise to Killing spinors with stability subgroups Spin(7) ⋉ R 8 , SU (4) ⋉ R 8 and G 2 . We solve the Killing spinor equations for the Spin(7) ⋉ R 8 and SU (4) ⋉ R 8 invariant spinors, give the fluxes in terms of the geometry and determine the conditions on the spacetime geometry imposed by supersymmetry. In both cases, the spacetime admits a null, self-parallel, Killing vector field. We also apply our formalism to examine a class of SU (4) ⋉ R 8 backgrounds which admit one and two pure spinors as Killing spinors and investigate the geometry of the spacetimes.

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“…One difference with respect to the uncorrected spaces, however, is that although a decomposition of the supersymmetry parameter representation that includes a singlet is a necessary condition for supersymmetry preservation, it is not sufficient. There can be yet higher order integrability conditions for the Killing spinor equation that need to be checked as well [24].…”

Section: Generalized Structure Groups and Generalized Holonomymentioning

confidence: 99%

Generalized holonomy in String-corrected Spacetimes

Stelle¹

2006

J. Phys.: Conf. Ser.

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Abstract. The quartic-curvature corrections derived from string theory have a specific impact on the geometry of target-space manifolds of special holonomy. In the cases of Calabi-Yau manifolds, D = 7 manifolds of G2 holonomy and D = 8 manifolds of Spin 7 holonomy, string theory α corrections conspire to preserve the unbroken supersymmetry of these backgrounds despite the fact that the α corrections cause the Riemannian holonomy to lose its special character. We show how this supersymmetry preservation is expressed in the language of generalized holonomy for the Killing spinor operator. IntroductionThe effective action for the massless modes of a given string theory consists at the lowest order of the action for the corresponding supergravity theory, which is then modified by an infinite sequence of higher derivative corrections. These correction terms occur individually with finite coefficients but are of similar structure to those that occur with divergent coefficients in the corresponding quantized supergravity [1,2,3]. In this sense, the string theory may be viewed as a regularization of the corresponding supergravity. The string tension α plays the rôle of the dimensionful cutoff parameter and gives the scale at which the microscopic physics of string theory begins to take over from the effective field theory of the massless modes. At the (α ) 3 level in type II theories, one encounters corrections quartic in Riemann curvatures; these are the first such corrections whose variations do not vanish subject to the leading order effective supergravity field equations, so they play a particularly important rôle in the onset of string-theory effects. This subject has had a revival of interest as appreciation has grown of the importance of noncompact Calabi-Yau spaces or G 2 manifolds as the underlying Ricci-flat geometries of brane spacetimes. In the compact cases, the relation between the quartic curvature corrections and Calabi-Yau compactifications has been studied in Ref. [4], while the non-compact case has been studied in [5].This review, based on Refs [6,7,8,9], summarizes the geometrical impact that the quartic corrections have on Calabi-Yau, G 2 and Spin 7 holonomy manifolds and on the preservation of supersymmetry in the face of the α corrections, extending the earlier analysis given in Refs [10,11]. It then proceeds to show how this supersymmetry preservation is expressed in the language of generalized structure groups and generalized holonomy for the α -corrected Killing spinor operator.

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Higher order integrability in generalized holonomy

Cited by 4 publications

References 22 publications

Systematics of M-theory spinorial geometry

Systematics of M-theory spinorial geometry

The spinorial geometry of supersymmetric IIB backgrounds

Generalized holonomy in String-corrected Spacetimes

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