Aspects of Spinorial Geometry

Gran, Ulf; Gutowski, J.; Papadopoulos, G.; Roest, Diederik

doi:10.1142/s0217732307022517

Cited by 7 publications

(3 citation statements)

References 30 publications

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“…This is because the gauge group of the Killing spinor equations in type II supergravities is a proper subgroup of the holonomy group of the supercovariant connection. This, and its consequences, have been explained in detail in the conclusions of [44] and we shall not repeat the analysis here. Nevertheless the results of this paper can be adapted to solve the algebraic Killing spinor equations of type II supergravities provided that a solution of the gravitino Killing spinor equation is known.…”

Section: Discussionmentioning

confidence: 87%

Geometry of all supersymmetric type I backgrounds

Gran

Papadopoulos

Sloane

et al. 2007

J. High Energy Phys.

195

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We find the geometry of all supersymmetric type I backgrounds by solving the gravitino and dilatino Killing spinor equations, using the spinorial geometry technique, in all cases. The solutions of the gravitino Killing spinor equation are characterized by their isotropy group in Spin(9, 1), while the solutions of the dilatino Killing spinor equation are characterized by their isotropy group in the subgroup Σ(P) of Spin(9, 1) which preserves the space of parallel spinors P. Given a solution of the gravitino Killing spinor equation with L parallel spinors, L = 1, 2, 3, 4, 5, 6, 8, the dilatino Killing spinor equation allows for solutions with N supersymmetries for any 0 < N ≤ L. Moreover for L = 16, we confirm that N = 8, 10, 12, 14, 16. We find that in most cases the Bianchi identities and the field equations of type I backgrounds imply a further reduction of the holonomy of the supercovariant connection. In addition, we show that in some cases if the holonomy group of the supercovariant connection is precisely the isotropy group of the parallel spinors, then all parallel spinors are Killing and so there are no backgrounds with N < L supersymmetries.

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Section: Discussionmentioning

confidence: 87%

Geometry of all supersymmetric type I backgrounds

Gran

Papadopoulos

Sloane

et al. 2007

J. High Energy Phys.

195

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“…In §5, we introduce a fermion representation of SO( 7) spinors and simplify general spinors by choosing a special local Lorentz frame. 29), 30) The bilinear forms of the Killing spinors are introduced in §3. These forms are invariant under some group G (⊂ SO (7)) and give G-invariant forms defining the G-structure.…”

Section: §1 Introductionmentioning

confidence: 99%

Supersymmetric AdS3 Solutions in Heterotic Supergravity

Kunitomo¹,

Ohta²

2009

Progress of Theoretical Physics

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We analyze the AdS 3 ×M 7 -type supersymmetric solutions, including nontrivial fluxes, of Killing spinor equations in heterotic supergravity. We classify these solutions using their Gstructures and intrinsic torsions, for the cases that the number of seven-dimensional Killing spinors N are equal to 1, 2, 3 and 4. We find that the solutions cannot have a nontrivial warp factor and the seven-dimensional manifold M 7 is characterized by G 2 (SU(3))-structures for the N = 1 (2) case and an SU(2)-structure for the N = 3 and 4 cases. They are further classified using their nontrivial intrinsic torsions. It is shown, including the leading-order α -corrections, that if we impose the Bianchi identities, the integrability conditions of the Killing spinor equations imply that all the field equations are satisfied.Subject Index: 111, 121, 122 §1. IntroductionIt is important to investigate the classical solutions of supergravity that preserve some supersymmetries. In particular, solutions including AdS-space are interesting from the viewpoint of the AdS/CFT correspondence. 1) They provide us with a deeper understanding of the dynamics of both gauge theory and gravity.In particular, the AdS 3 /CFT 2 correspondence in heterotic string theory has recently been studied, 2)-5) in which it was conjectured to exist as a CFT dual of a geometry describing a fundamental heterotic string. A mini-black-string solution 6), 7) was considerd in the five-dimensional heterotic supergravity, obtained by T 5 -compactification, with R 2 correction terms. 8) Without R 2 corrections, the solution has a zero horizon area and thus there is no room for the AdS-space to appear. However, once the corrections are included, the horizon is stretched 9) and its nearhorizon geometry becomes the AdS 3 -space. Unfortunately, however, such a miniblack-string solution is yet unknown in the ten-dimensional heterotic supergravity, which is required for more general compactifications. It is interesting, therefore, to study supersymmetric classical solutions in the form of AdS 3 × M 7 , admitting a warp factor in general, taking into account the R 2 corrections.In general, manifolds described by supersymmetric classical solutions with nontrivial fluxes are characterized by G-structure, which are generalizations of those with special holonomy. 10) As is well known, if we set all the fluxes to zero, supersymmetric solutions yield some special holonomy manifolds. However, if the fluxes * )

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“…Let the manifold under consideration be spin, and take a connection on the spinor bundle S on M . This connection and its curvature are locally described by elements in the exterior algebra of M , the so called k-form potentials and fluxes; see for example [14,15,16]. The duality relation presented here may be a candidate to generalize the duality for metric connections on the base manifold M .…”

Section: Discussionmentioning

confidence: 99%

Generalized duality for k-forms

Klinker

2011

Journal of Geometry and Physics

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Abstract. We give the definition of a duality that is applicable to arbitrary k-forms. The operator that defines the duality depends on a fixed form Ω. Our definition extends in a very natural way the Hodge duality of n-forms in 2n dimensional spaces and the generalized duality of two-forms. We discuss the properties of the duality in the case where Ω is invariant with respect to a subalgebra of so(V ). Furthermore, we give examples for the invariant case as well as for the case of discrete symmetry.

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Aspects of Spinorial Geometry

Cited by 7 publications

References 30 publications

Geometry of all supersymmetric type I backgrounds

Geometry of all supersymmetric type I backgrounds

Supersymmetric AdS3 Solutions in Heterotic Supergravity

Generalized duality for k-forms

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