AdS5 compactifications with punctures in massive IIA supergravity

TL;DR

<p>The paper constructs AdS$_5$ solutions in massive IIA that are holographically dual to compactifications of six-dimensional ${\cal N}=(1,0)$ theories on punctured Riemann surfaces. It simplifies a prior PDE system using a separable Ansatz, yielding three classes of solutions with O8–D8, D6, and D4 branes, where D4s are smeared to represent punctures. The central charge decomposes into a 6d-origin term $a\sim (g-1)N^3M^2$ from compactifying a massive E-string theory and puncture contributions $\sim nN^2M$ (or $\sim n^{5/2}$ in a D6-free class), consistent with the brane content; in a certain limit a 5d origin is suggested by $a\sim n^{5/2}$. The authors also connect these AdS$_5$ vacua to AdS$_7$ gravity duals of the underlying 6d theories, clarifying the field-theory interpretation in terms of massive E-strings and quiver tails. The work advances the holographic understanding of ${\cal N}=(1,0)$ SCFTs in four dimensions and their higher-dimensional origins.

Abstract

We find AdS5 solutions holographically dual to compactifications of six-dimensional N=(1,0) supersymmetric field theories on Riemann surfaces with punctures. We simplify a previous analysis of supersymmetric AdS5 IIA solutions, and with a suitable Ansatz we find explicit solutions organized in three classes, where an O8--D8 stack, D6- and D4-branes are simultaneously present, localized and partially localized. The D4-branes are smeared over the Riemann surface and this is interpreted as the presence of a uniform distribution of punctures. For the first class we identify the corresponding six-dimensional theory as an E-string theory coupled to a quiver gauge theory. The second class of solutions lacks D6-branes and its central charge scales as $n^{5/2}$, suggesting a five-dimensional origin for the dual field theory. The last class has elements of the previous two.

Figures (6)

• Figure 1: A cartoon of the internal space $M_3$ corresponding to the range $z\in [0,z_0]$. Roughly, $z$ runs vertically and $k$ horizontally. At $\{z=z_0,\, k=1\}$ the solution has a D6-brane stack. There is an O8-plane (with $n_8$ D8-brane pairs on top) at $k=0$ and $z=0$. In fact, the locus $z=0$ is revealed to be a "blowup" of a D4-brane stack inside an O8-plane, in the sense that near that locus, $k$ becomes one of the angular coordinates of the sphere transverse to the D4-branes. Finally, at $z=z_0$ the metric is regular.
• Figure 2: A cartoon of $M_3$ for the range $z\in [0,z_1]$. Compared to the solution in figure \ref{['fig:0z0']}, the solution has no D6-branes; the upper range for $z$ is at $z=z_1$, where a distribution of D4-branes appears.
• Figure 3: A cartoon of $M_3$ for the range $z\in [z_1,z_0]$. Compared to the solution in figure \ref{['fig:0z0']}, the lower range for $z$ is at $z=z_1$, where a distribution of D4-branes appears; in other words, the D4-branes in figure \ref{['fig:0z0']} have spread out in the $k, \psi$ directions as well.
• Figure 4: A summary of the various solutions we found for $\kappa=0,-1,1$. The cartoons refer to figures \ref{['fig:0z0']}, \ref{['fig:0z1']} and \ref{['fig:z1z0']}. The grey dot for $\kappa=1$ is the solution discussed in appendix \ref{['app:z1=z0']}. The white dots for $\kappa=-1$ both represent the punctureless solution of afpt, whose six-dimensional origin we discuss in section \ref{['sec:6d']}.
• Figure 5: In figure \ref{['fig:O8-NS5-D6']}, the brane diagram whose near-horizon limit produces the solution (\ref{['eq:ads7']}) for $n_0=2$ and $N=3$ is depicted. The vertical line represents an O8-plane with $n_8=8-n_0$ ($=6$, in this case) D8-brane pairs. The nodes denote NS5-branes, and the horizontal lines D6-branes. In figure \ref{['fig:E7-6']}, the quiver diagram of the corresponding field theory for this particular case is depicted. The empty node represents the E-string theory, as explained in the main text.