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

TL;DR

The authors construct globally regular Type IIB supergravity solutions with $AdS_6\times S^2$ warped over a Riemann surface $\Sigma$, encoding the local data in two holomorphic functions $\mathcal{A}_\pm$ and enforcing positivity/regularity to obtain physical global geometries. By exploiting an electrostatics-like method, they build meromorphic data on $\Sigma$ with poles on the boundary, corresponding to external $(p,q)$ 5-branes, and demonstrate that global solutions exist when $\partial\Sigma\neq\emptyset$, with detailed analysis for the upper half-plane and annulus topologies. Explicit three- and four-pole disk solutions are presented, illustrating their relation to fully localized 5-brane webs and providing a moduli-count that matches brane-web data in the conformal limit. The work also outlines the general framework for higher-genus surfaces and discusses the challenges and prospects for including monodromies (e.g., from seven-branes) and computing holographic observables. Overall, the paper advances a concrete holographic program for five-dimensional SCFTs via globally regular warped $AdS_6$ solutions and brane-web realizations, while highlighting open topological questions and potential generalizations.

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 $Σ$. In earlier work, the general local solutions were expressed in terms of two locally holomorphic functions $\mathcal A_\pm$ on $Σ$ and global solutions were sketched when $Σ$ 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 $Σ$. Global solutions exist only when $Σ$ has a non-empty boundary $\partial Σ$. The differentials $\partial \mathcal A_\pm$ are allowed to have poles only on $\partial Σ$ 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.

Figures (9)

• Figure 1: The upper-half plane $\Sigma= {\mathbb H}$; its boundary $\partial \Sigma = {\mathbb R}$; an array of zeros of the function $\lambda$ at the points $s_n$ and poles at $\bar{s}_n$, with $n=1,\cdots, N$ and ${\rm Im \,} (s_n) >0$; and an array of poles of $\partial_w {\cal A}_\pm$ at the points $p_\ell \in {\mathbb R}$ with $\ell =1, \cdots, L$.
• Figure 2: Integration contour (a) when $w \in [p_{\ell'}, p_\ell]$ and (b) when $w \in [-\infty , p_{\ell'}]$.
• Figure 3: The three-pole solution: the metric factors $f_2^2,f_6^2$ and $\rho^2$, the real and imaginary parts of the two-form potential ${\cal C}$ and axion and dilaton corresponding to the parameters given in (\ref{['3j2']}), as functions of $w=x+iy$.
• Figure 4: The four-pole solution: the metric factors $f_2^2,f_6^2$, and $\rho^2$, the real and imaginary parts of the two-form potential ${\cal C}$ and of the axion and dilaton corresponding to the parameters given in (\ref{['3h6']}), with $N=2$, $M=3$. All supergravity functions are regular throughout $\Sigma$ except for the poles, where their behavior is precisely as discussed in sec. \ref{['sec:asympt']}.
• Figure 5: The $N$-junction, a $(p,q)$ 5-brane intersection with three external 5-brane stacks, obtained by taking $N$ copies of the basic junction of a D5 and an NS5 brane.