$AdS_2 \times S^6$ versus $AdS_6 \times S^2$ in Type IIB supergravity

TL;DR

This paper classifies all local supersymmetric solutions of Type IIB supergravity with an $AdS_2 imes S^6$ geometry warped over a Riemann surface, extending known solutions and contrasting with the $AdS_6 imes S^2$ case.

Contribution

It provides the complete local solutions for $AdS_2 imes S^6$ in Type IIB supergravity using holomorphic functions, extending the understanding of such geometries and their supersymmetry.

Findings

Derived the general local solutions in terms of holomorphic functions.

Showed solutions satisfy full supergravity equations and Bianchi identities.

Compared $AdS_2 imes S^6$ with $AdS_6 imes S^2$ solutions.

Abstract

We obtain the complete local solutions with 16 supersymmetries to Type IIB supergravity on a space-time of the form $AdS_{2}\times S^{6}$ 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 $AdS_{2}\times S^{6}$ solution with the closely related $AdS_6\times S^2$ case and present our results for both in parallel. Finally, we present a preliminary analysis of positivity and regularity conditions for $AdS_{2}\times S^{6}$, but postpone the construction of globally regular solutions to a subsequent paper.