Warped $AdS_6\times S^2$ in Type IIB supergravity III: Global solutions with seven-branes

TL;DR

This work extends the class of globally regular $AdS_6\times S^2$ solutions in Type IIB supergravity by incorporating isolated punctures with parabolic $SL(2,\mathbb{R})$ monodromy, interpreted as $[p,q]$ 7-branes. Using disc topology for $\Sigma$ and a refined holomorphic data framework, the authors construct explicit solutions, derive regularity conditions, and identify punctures with 7-branes while preserving the $F(4)$ symmetry. They provide concrete 3-pole examples with D7/D5-branes and their S-duals, demonstrating that all supergravity fields have the expected near-branch-point behavior and that the global geometry remains regular away from punctures. The results yield candidate holographic duals to UV fixed points of five-dimensional gauge theories realized on 5-brane intersections with additional 7-branes, offering a platform for quantitative holography of 5d SCFTs and insights into brane web dynamics in the presence of 7-branes.

Abstract

We extend our previous construction of global solutions to Type IIB supergravity that are invariant under the superalgebra $F(4)$ and are realized on a spacetime of the form $AdS_6 \times S^2$ warped over a Riemann surface $Σ$ by allowing the supergravity fields to have non-trivial $SL(2,{\mathbb R})$ monodromy at isolated punctures on $Σ$. We obtain explicit solutions for the case where $Σ$ is a disc, and the monodromy generators are parabolic elements of $SL(2,{\mathbb R})$ physically corresponding to the monodromy allowed in Type IIB string theory. On the boundary of $Σ$ the solutions exhibit singularities at isolated points which correspond to semi-infinite five-branes, as is familiar from the global solutions without monodromy. In the interior of $Σ$, the solutions are everywhere regular, except at the punctures where $SL(2,{\mathbb R})$ monodromy resides and which physically correspond to the locations of $[p,q]$ seven-branes. The solutions have a compelling physical interpretation corresponding to fully localized five-brane intersections with additional seven-branes, and provide candidate holographic duals to the five-dimensional superconformal field theories realized on such intersections.

Figures (6)

• Figure 1: Branch cuts for ${\cal F}$ are drawn as black dashed lines and do not intersect each other or poles on the real line. The cuts shown correspond to $\gamma_i=-1$, $\gamma_j=1$ and $\gamma_k=e^{i\pi/3}$. An integration contour for ${\cal I}$, which does not intersect any of the branch cuts, is shown in red.
• Figure 2: Integration contour $C=C_1\cup C_2$, where $C_1$ denotes the left half of the contour shown in red and $C_2$ the right half.
• Figure 3: On the left hand side the allowed locations for the branch point $w_1$ in the upper half plane, for the solution (\ref{['eq:3pole-ex']}) with a single puncture corresponding to a D7-brane and $n_1=1$. On the right hand side the imaginary part of the charges along the curve shown on the left. At $\arg(w_1)=\pi$ the curves are, from top to bottom, ${\rm Im \,}({\cal Y}_+^1)$, ${\rm Im \,}({\cal Y}_+^2)$ and ${\rm Im \,}({\cal Y}_+^3)$.
• Figure 4: 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 for the 3-pole solution with $[1,0]$ branch point.
• Figure 5: 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 for the 3-pole solution with $[0,1]$ monodromy.