$Ad{S}_2\times S^6$ versus $Ad{S}_6\times S^2$ in Type IIB supergravity

Abstract

We obtain the complete local solutions with 16 supersymmetries to Type IIB supergravity on a space-time of the form $Ad{S}_2\times S^6$ warped over a Riemann surface $\Sigma$ in terms of two locally holmorphic functions on $\Sigma$. We construct the general Ansatz for the bosonic supergravity fields and supersymmetry generators compatible with the $SO(2,1)\oplus SO(7)$ isometry algebra of space-time, which extends to the corresponding real form of the exceptional Lie superalgebra $F(4)$. We reduce the BPS equations to this Ansatz, obtain their general local solutions, and show that these local solutions solve the full Type IIB supergravity field equations and Bianchi identities. We contrast the $Ad{S}_2\times S^6$ solution with the closely related $Ad{S}_6\times S^2$ case and present our results for both in parallel. Finally, we present a preliminary analysis of positivity and regularity conditions for $Ad{S}_2\times S^6$, but postpone the construction of globally regular solutions to a subsequent paper.

arXiv comment50 pages

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• Type IIB 7-branes in warped $Ad{S}_6$: partition functions, brane webs and probe limit
• New AdS${}_2$ supergravity duals of 4d SCFTs with defects
• AdS${}_2$ geometries and non-Abelian T-duality in non-compact spaces
• Warped $Ad{S}_2$ and $SU(1,1|4)$ symmetry in Type IIB
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• Global half-BPS $Ad{S}_2\times S^6$ solutions in Type IIB
• Non-integrability in AdS${}_3$ vacua
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• Mink${}_4\times S^2$ Solutions of 10 and 11 Dimensional Supergravity
• Non-supersymmetric AdS${}_6$ and the swampland
• Exact solutions to complex Type IIB supergravity for complex superalgebra $F(4)$ and its real forms
• New AdS${}_3$/CFT${}_2$ pairs in massive IIA with $(0,4)$ and $(4,4)$ supersymmetries