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

TL;DR

This work constructs and analyzes the general local half-BPS solutions of Type IIB supergravity with space-time geometry $AdS_{2}\times S^{6}$ warped over a Riemann surface $\boldsymbol{\u03a3}$, governed by the $SO(2,1)\times SO(7)$ isometry and related to a real form of $F(4)$. The authors derive a complete reduction of the BPS equations and solve them in terms of two locally holomorphic functions $\mathcal{A}_{\pm}$ on $\boldsymbol{\u03a3}$, providing explicit expressions for the metric factors, axion-dilaton, and flux potentials, and showing these locally solve the full Type IIB field equations. They compare the results to the closely related $AdS_{6}\times S^{2}$ case, highlighting subtle sign differences captured by a unified framework using a parameter $K$, and examine reality, positivity, and regularity constraints, demonstrating obstacles to global regular solutions on compact $\boldsymbol{\u03a3}$ or with boundary in the present setup. A discussion of double analytic continuation reveals that globally regular supersymmetric continuations from the $AdS_{6}\times S^{2}$ solutions do not yield regular $AdS_{2}\times S^{6}$ solutions, motivating independent construction of globally regular $AdS_{2}\times S^{6}$ backgrounds and suggesting potential brane-web interpretations in the dual 2D SCFT context.

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 $Σ$ in terms of two locally holmorphic functions on $Σ$. 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.