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

TL;DR

This paper constructs local solutions in Type IIB supergravity with $AdS_6 imes S^2$ warped over a Riemann surface, revealing potential holographic duals to five-dimensional conformal field theories.

Contribution

It derives the general local form of solutions with 16 supersymmetries, expressed through holomorphic functions on a Riemann surface, extending the understanding of $AdS_6$ backgrounds.

Findings

Solutions characterized by two holomorphic functions on $\\Sigma$

Invariance under $F(4)$ superalgebra

Framework for globally regular solutions

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 $\Sigma$. 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 $\Sigma$, subject to certain positivity and regularity conditions. Globally, $({\cal A}_+, {\cal A}_-)$ are allowed to be multiple-valued on $\Sigma$ and be holomorphic sections of a holomorphic bundle over $\Sigma$ 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.