Abstract

We investigate all N = 2 supersymmetric IIB supergravity backgrounds with non-vanishing five-form flux. The Killing spinors have stability subgroups Spin ( 7 ) ⋉ R 8 , SU ( 4 ) ⋉ R 8 and G 2 . In the SU ( 4 ) ⋉ R 8 case, two different types of geometry arise depending on whether the Killing spinors are generic or pure. In both cases, the backgrounds admit a null Killing vector field which leaves invariant the SU ( 4 ) ⋉ R 8 structure, and an almost complex structure in the directions transverse to the lightcone. In the generic case, the twist of the vector field is trivial but the almost complex structure is non-integrable, while in the pure case the twist is non-trivial but the almost complex structure is integrable and associated with a relatively balanced Hermitian structure. The G 2 backgrounds admit a time-like Killing vector field and two spacelike closed one-forms, and the seven directions transverse to these admit a co-symplectic G 2 structure. The Spin ( 7 ) ⋉ R 8 backgrounds are pp-waves propagating in an eight-dimensional manifold with holonomy Spin ( 7 ) . In addition we show that all the supersymmetric solutions of simple five-dimensional supergravity with a time-like Killing vector field, which include the AdS 5 black holes, lift to SU ( 4 ) ⋉ R 8 pure Killing spinor IIB backgrounds. We also show that the LLM solution is associated with a co-symplectic co-homogeneity one G 2 manifold which has principal orbit S 3 × S 3 .

Highlights

  • Supersymmetric IIB backgrounds with “active” five-form fluxes and with vanishing oneand three-form field strengths have been extensively investigated in the context of string theory, branes and black holes

  • Motivated by the above widespread applications, we present a systematic investigation of all supersymmetric IIB supergravity backgrounds with active five-form flux F

  • This is based on our solution [9, 10] of the Killing spinor equations of IIB supergravity [11, 1, 12] for one Killing spinor, using the spinorial geometry technique of [13]

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Summary

Introduction

Supersymmetric IIB backgrounds with “active” five-form fluxes and with vanishing oneand three-form field strengths have been extensively investigated in the context of string theory, branes and black holes. The geometry rather depends on the embedding of the Killing spinor bundle into the spinor bundle of IIB supergravity up to a Spin(9, 1) rotation as has been explained in [16] In both cases, the spacetime admits a null Killing vector field X with non-vanishing twist or rotation, and an almost complex structure with compatible (4,0)-form leading to an SU(4) structure in the directions transverse to the lightcone. We solve the Killing spinor equations of backgrounds that admit a pure Killing spinor, and show that D3-branes and their intersections as well as the uplifts of supersymmetric solutions of minimal fivedimensional gauged supergravity admitting a timelike Killing vector field are examples. In appendix B and C, we summarize some results on null and G2 structures in ten-dimensions

Geometry and supersymmetry
Conditions on the geometry
Fluxes
Special cases
Geometry and fluxes
D3-branes and intersecting branes
Uplifted Five-Dimensional Solutions
Geometry
Co-calibrated G2 manifolds
D3 brane
Bubbling AdS Solutions
Concluding remarks
Solution of the linear system
Spin connection
Solution of the linear system of G2 backgrounds
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