6d holographic anomaly match as a continuum limit

TL;DR

The paper tests a holographic duality between a family of AdS$_7\times S^3$ solutions in massive IIA (the crescent-roll geometry from D8-branes) and 6d linear quiver CFTs built from NS5-D6-D8 brane systems. The authors compute the Weyl anomaly $a$ on the field-theory side, finding a leading term proportional to $\sum_{i,j} C^{-1}_{ij} r_i r_j$, and independently compute $a$ from the gravity side via the warped AdS$_7$ solution, showing the two agree in the holographic limit $N\to\infty$. They make this precise by identifying the discrete quiver data with a continuum function $r(x)$, so the inverse Cartan matrix $C^{-1}$ plays the role of a discrete Green's function for a second derivative, and the gravity integral reduces to the same functional of $r(x)$. This establishes a clean continuum/discretization correspondence: the field-theory partitions encode the geometry and the gravity data discretizes to the quiver partitions, providing strong evidence for the gaiotto-t-6d duality in this setup.

Abstract

An infinite class of analytic AdS_7 x S^3 solutions has recently been found. The S^3 is distorted into a "crescent roll" shape by the presence of D8-branes. These solutions are conjectured to be dual to a class of "linear quivers", with a large number of gauge groups coupled to (bi-)fundamental matter and tensor fields. In this paper we perform a precise quantitative check of this correspondence, showing that the a Weyl anomalies computed in field theory and gravity agree. In the holographic limit, where the number of gauge groups is large, the field theory result is a quadratic form in the gauge group ranks involving the inverse of the A_N Cartan matrix C. The agreement can be understood as a continuum limit, using the fact that C is a lattice analogue of a second derivative. The discrete data of the field theory, summarized by two partitions, become in this limit the continuous functions in the geometry. Conversely, the geometry of the internal space gets discretized at the quantum level to the discrete data of the two partitions.

Figures (7)

• Figure 1: The general structure of a linear quiver.
• Figure 2: In \ref{['fig:quiver']}, an example of linear quiver. As described in the text, round nodes represent gauge groups, square nodes flavor symmetries. Links represent hypermultiplets; horizontal links also have tensor multiplets associated to them. In \ref{['fig:r-plot']} we plot the numbers of colors $r_i$, as a function of the position $i$ in the quiver. We added a linear interpolation to guide the eye. The bigger dots indicate points where the slope changes; these are the positions where flavors are present, and the change in slope equals the number of flavors. In \ref{['fig:s-plot']} we plot the $s_i=r_i-r_{i-1}$; this can be thought of as the derivative of the linear interpolation in \ref{['fig:r-plot']}. We have filled in the plot with boxes, that define two Young diagrams $\rho_{\rm L}$, $\rho_{\rm R}$.
• Figure 3: In \ref{['fig:D8-in']} and \ref{['fig:D8-out']} we see two versions of the brane system that engineers the particular quiver in figure \ref{['fig:quiver']}, related by Hanany--Witten moves. In both cases, round dots represent NS5-branes; horizontal lines represent D6's; vertical lines represent D8's. In \ref{['fig:D8-in']} we see the system in a configuration where the quiver can be read off easily: the segment between the $i$-th and $(i+1)$-th NS5-branes contains $r_i$ D6-branes, and $f_i$ D8-branes intersecting them. The two Young diagrams can be read off intuitively from both pictures \ref{['fig:D8-in']} and \ref{['fig:D8-out']}. Focusing for example on $\rho_{\rm L}$, in \ref{['fig:D8-in']} we see that there are 1 D8 in the first segment, 2 in the second, 1 in the fifth: these represent the drops $s_i-s_{i+1}=f_i$ in the Young diagram. In \ref{['fig:D8-out']} we see even more directly that there are 1 D8 with $\mu=1$ D6-branes ending on it, 2 D8's with $\mu=2$ D6's ending on them, 0 D8's with $\mu=3$, 0 D8's with $\mu=4$, 1 D8's with $\mu=5$; these are the $f_i= s_i-s_{i+1}$ associated to $\rho_{\rm L}$. Finally in \ref{['fig:M3-D8']} we see an artist's impression of the shape of the internal $M_3$ in the AdS$_7$ solution. The D8's are represented by the black lines. There are as many D8-brane stacks as in the brane pictures (the two D8's with $\mu=2$ are on top of each other). These D8 stacks are in correspondence with the flavors in figure \ref{['fig:quiver']}.
• Figure 4: The effect of the map (\ref{['eq:hor-stretch']}) on the plot in figure \ref{['fig:s-plot']}, for $n=2$.
• Figure 5: A theory that is dual to the massless solution in the holographic limit. From the top left, anticlockwise, we show: the Young diagrams, the quiver, a sketch of the internal space $M_3$, and the brane configuration; cf. the general case in figures \ref{['fig:s-plot']}, \ref{['fig:quiver']}, \ref{['fig:M3-D8']}, \ref{['fig:D8-out']}. The brane picture is shown in the version that follows from the general correspondence reviewed in section \ref{['sub:ads']}, as well as in an alternative version, using the equivalence of a D8-brane with one D6 attached and a semi-infinite D6 gaiotto-witten-1. Taking the general correspondence literally, one would see in the gravity solution two D8 stacks with D6-charges $\pm1$, but in the holographic limit these become so small as to be indistinguishable from a D6 and an anti-D6 stack.