# Warped $AdS_6\times S^2$ in Type IIB supergravity II: Global solutions and five-brane webs

## TL;DR

This paper constructs and analyzes global solutions in Type IIB supergravity with $AdS_6 imes S^2$ geometry warped over Riemann surfaces, providing insights into holographic duals of five-dimensional superconformal theories.

## Contribution

It extends previous local solutions to global solutions on arbitrary Riemann surfaces with boundaries, linking poles of holomorphic functions to five-brane configurations.

## Key findings

- Global solutions exist only when the Riemann surface has a boundary.
- Poles of the differentials correspond to semi-infinite $(p,q)$ five-branes.
- Conditions for solutions are detailed for surfaces with multiple boundaries and higher genus.

## Abstract

Motivated by the construction of holographic duals to five-dimensional superconformal quantum field theories, we obtain global solutions to Type IIB supergravity invariant under the superalgebra $F(4)$ on a space-time of the form $AdS_6 \times S^2$ warped over a two-dimensional Riemann surface $\Sigma$. In earlier work, the general local solutions were expressed in terms of two locally holomorphic functions $\mathcal A_\pm$ on $\Sigma$ and global solutions were sketched when $\Sigma$ is a disk. In the present paper, the physical regularity conditions on the supergravity fields required for global solutions are implemented on $\mathcal A_\pm$ for arbitrary $\Sigma$. Global solutions exist only when $\Sigma$ has a non-empty boundary $\partial \Sigma$. The differentials $\partial \mathcal A_\pm$ are allowed to have poles only on $\partial \Sigma$ and each pole corresponds to a semi-infinite $(p,q)$ five-brane. The construction for the disk is carried out in detail and the conditions for the existence of global solutions are articulated for surfaces with more than one boundary and higher genus.