Warped $AdS_6\times S^2$ in Type IIB supergravity I: Local solutions

TL;DR

This work constructs the general local 16-supercharge Type IIB supergravity solutions with $SO(2,5) imes SO(3)$ symmetry, realized as $AdS_6 imes S^2$ warped over a Riemann surface $Σ$. The local solutions are encoded entirely in two locally holomorphic functions ${ m extcal A}_ plus$ and ${ m extcal A}_ minus$ on $Σ$, with explicit expressions for the metric factors, axion-dilaton, and fluxes derived from these data under positivity/regularity constraints. A sequence of variable changes decouples the nonlinear BPS system, yielding closed-form local solutions and clarifying the role of $SU(1,1)$ duality in organizing the holomorphic data. The results provide a robust framework to seek globally regular solutions, with connections to $(p,q)$ 5-brane/7-brane webs and the holographic duals of five-dimensional SCFTs, and include a concrete check by recovering the T-dual of the D4/D8 system as a local, singular example. Future work will address global regularity, monodromies, and complete holographic constructions of brane-web duals in this class.

Abstract

We investigate the existence of solutions with 16 residual supersymmetries to Type IIB supergravity on a space-time of the form $AdS_6 \times S^2$ warped over a two-dimensional Riemann surface $Σ$. The $SO(2,5) \times SO(3)$ isometry extends to invariance under the exceptional Lie superalgebra $F(4)$. In the present paper, we construct the general Ansatz compatible with these symmetries, derive the corresponding reduced BPS equations, and obtain their complete local solution in terms of two locally holomorphic functions ${\cal A}_\pm$ on $Σ$, subject to certain positivity and regularity conditions. Globally, $({\cal A}_+, {\cal A}_-)$ are allowed to be multiple-valued on $Σ$ and be holomorphic sections of a holomorphic bundle over $Σ$ with structure group contained in $SU(1,1) \times { C}$. Globally regular solutions are expected to provide the near-horizon geometry of $(p,q)$ 5-brane and 7-brane webs which are holographic duals to five-dimensional conformal field theories. A preliminary analysis of the positivity and regularity conditions will be presented here, leaving the construction of globally regular solutions to a subsequent paper.