Type IIB 7-branes in warped $AdS_6$: partition functions, brane webs and probe limit

TL;DR

This work extends the Type IIB $AdS_6$ holographic program to include punctures with $SL(2,\mathbb{R})$ monodromy, interpreted as 7-branes in 5-brane webs. It establishes a precise link between punctures and 7-branes by matching infinitesimal monodromy to probe D7-brane physics via $\\kappa$-symmetry, and then constructs fully backreacted 7-brane solutions with fixed 5-brane charges. The authors compute sphere partition functions (via entanglement entropy) for these backreacted backgrounds, study their branch-cut orientation dependence, and explore explicit 3- and 4-pole solutions, revealing nontrivial puncture-position dependence that discriminates between distinct brane-web realizations. The results thus provide a consistent brane-web interpretation of punctured $AdS_6$ solutions and demonstrate how partition functions encode the detailed web structure, including Hanany-Witten effects and the distribution of 7-branes within faces of the web.

Abstract

We study Type IIB supergravity solutions with spacetime of the form $AdS_6\times S^2$ warped over a Riemann surface $Σ$, where $Σ$ includes punctures around which the supergravity fields have non-trivial $SL(2,R)$ monodromy. Solutions without monodromy have a compelling interpretation as near-horizon limits of $(p,q)$ 5-brane webs, and the punctures have been interpreted as additional 7-branes in the web. In this work we provide further support for this interpretation and clarify several aspects of the identification of the supergravity solutions with brane webs. To further support the identification of the punctures with 7-branes, we show that punctures with infinitesimal monodromy match a probe 7-brane analysis using $κ$-symmetry. We then construct families of solutions with fixed 5-brane charges and punctures with finite monodromy, corresponding to fully backreacted 7-branes. We compute the sphere partition functions of the dual 5d SCFTs and use the results to discuss concrete brane web interpretations of the supergravity solutions.

Figures (7)

• Figure 1: On the left hand side a disc representation of the 3-pole solutions discussed in sec. \ref{['sec:D5-NS5x2-D7']}. The D7 brane can be placed on the horizontal diameter of the disc. On the right hand side a disc representation of the 4-pole solutions discussed in sec. \ref{['sec:4-pole']}.
• Figure 2: Integration contours for the constraint in (\ref{['eq:D5-NS5-D7-constr-2']}), depending on whether or not the branch cut intersects the boundary in the interval $(p_3,p_1)$.
• Figure 3: On the left hand side a plot of ${\cal J}_0$, which yields the partition function for the 3-pole solutions via (\ref{['eq:cJ0']}). On the right hand side similar plots for ${\cal J}_1$ and ${\cal J}_2$, which yield the partition functions of the 4-pole solutions via (\ref{['eq:cJ12']}).
• Figure 4: Fig. \ref{['fig:3-pole-web-1a']} shows a possible 5-brane web corresponding to the class of supergravity solutions illustrated in fig. \ref{['fig:3-pole-disc']}, with a puncture corresponding to two D7 branes. The brane web shows a general deformation of the SCFT, not the fixed point. Fig. \ref{['fig:3-pole-web-1b']} and \ref{['fig:3-pole-web-1c']} show two options for 5-brane webs with the same external 5-brane charges but 7-branes in a different face of the web. The web in fig. \ref{['fig:3-pole-web-1b']} is related to the web in fig. \ref{['fig:3-pole-web-1a']} by 7-brane moves, the web in fig. \ref{['fig:3-pole-web-1c']} is not.
• Figure 5: Starting from the web shown in fig. \ref{['fig:3-pole-web-1a']} and moving the 7-branes out of the web along their branch cuts produces 5-brane prongs stretching between the 7-branes and the 5-branes of the web, with avoided intersections due to the $s$-rule shown as broken lines.