Holography for (1,0) theories in six dimensions

TL;DR

<3-5 sentence high-level summary>This work constructs and tests a holographic correspondence between six-dimensional ${\cal N}=(1,0)$ SCFTs, engineered from D8--D6--NS5 brane systems, and AdS$_7$ vacua in massive IIA supergravity. The authors refine the AdS$_7$ classification to match brane configurations and Higgs-branch RG flows, showing a one-to-one map between field-theoretic CFT data and gravity data, including a precise bound on the NS-NS flux $N$ and an ordering constraint for D8 stacks. They compute the degrees of freedom of these theories via holographic free energy in multiple regimes, obtaining the characteristic $F_0\sim N^3 k^2$ scaling for massless configurations and controlled corrections in massive cases. The results illuminate how Higgs-branch deformations and Nahm-pole boundary conditions in the brane picture translate into near-horizon AdS$_7$ geometries with $M_3\cong S^3$, offering quantitative probes of 6d holography and a bridge between higher-dimensional SCFTs and their gravity duals.

Abstract

M-theory and string theory predict the existence of many six-dimensional SCFTs. In particular, type IIA brane constructions involving NS5-, D6- and D8-branes conjecturally give rise to a very large class of N=(1,0) CFTs in six dimensions. We point out that these theories sit at the end of RG flows which start from six-dimensional theories which admit an M-theory construction as a M5 stack transverse to $R^4/Z_k \times R$. The flows are triggered by Higgs branch expectation values and correspond to D6's opening up into transverse D8-branes via a Nahm pole. We find a precise correspondence between these CFT's and the AdS$_7$ vacua found in a recent classification in type II theories. Such vacua involve massive IIA regions, and the internal manifold is topologically $S^3$. They are characterized by fluxes for the NS three-form and RR two-form, which can be thought of as the near-horizon version of the NS5's and D6's in the brane picture; the D8's, on the other hand, are still present in the AdS$_7$ solution, in the form of an arbitrary number of concentric shells wrapping round $S^2$'s.

Figures (11)

• Figure 1: Brane configurations realizing some of the theories we consider in this paper. The top left configuration represents the theory $T^{A_N}_{A_k}$, describing the interaction of $N$ NS5-branes with $k$ D6's; the D8-branes on the two sides enforce Dirichlet boundary conditions for the fields on the D6's, thus decoupling the seven-dimensional degrees of freedom from the six-dimensional ones. The other brane configurations are induced by Higgs branch deformations (vertical arrows) or tensor branch deformations (horizontal arrows).
• Figure 2: Diagrams describing the field theories corresponding to the brane configurations in figure \ref{['fig:branes']}. (In that figure there are $k=3$ D6's. The lower-right quiver is appropriate for $k=3$ and the particular $\rho$, $\rho'$ corresponding to the lower-right configuration in figure \ref{['fig:branes']}.) In each quiver, a small node corresponds to a vector multiplet; a link between small nodes to a bi-fundamental hypermultiplet; a bigger node between two small nodes describes a non-abelian tensor theory $T$ whose flavor symmetries are gauged at the small nodes. The gauge coupling at each small node is promoted to a tensor multiplet scalar. Again the vertical arrows represent RG flows induced by Higgs branch deformations, and the horizontal ones by tensor branch deformations.
• Figure 3: Solution with four D8-brane stacks, placed asymmetrically around the equator. In \ref{['fig:6d8s-a']}, we plot the radius of the transverse $S^2$ (orange), the warping factor $e^{2A}$ (black; multiplied by $1/40$, also in all subsequent figures), and the string coupling $e^\phi$ (green). The slopes are given, from left to right, by $\mu_i= \frac{n_{{\rm D6},i}}{n_{{\rm D8},i}}=\{1,4;-3,-2\}$. The central area around the equator has $F_0=0$, as described in the text. In \ref{['fig:6d8s-b']}, we see the corresponding brane configuration; the central circle represents a stack of 18 NS5-branes. (Here and in all the figures that follows, the number of D8 branes in the gravity solution is actually 5 times the number of D8's in the brane pictures. We do this so that the dilaton is small; increasing $n_{\rm D8}$ would make it even smaller. See discussion around (\ref{['eq:nF0']}).)
• Figure 4: Solution with six D8-brane stacks placed symmetrically around the equator. The slopes are $\mu_i= \frac{n_{{\rm D6},i}}{n_{{\rm D8},i}}=\{2,3,5;-5,-3,-2\}$. The central area around the equator again has $F_0=0$. In \ref{['fig:6d8s-b']}, we see the corresponding brane configuration. The central circle represents a stack of 20 NS5-branes.
• Figure 5: An artist's impression of a system with a few D8--D6 "spikes"; the central sphere represents the NS5 stack. As one takes a near-horizon limit near the $N$ NS5's, they dissolve as usual leaving behind $N$ quanta of $H$ flux; the D8--D6 spikes should then become the D8--D6 bound states in the gravity solutions depicted in figures \ref{['fig:4d8as-a']}, \ref{['fig:6d8s-a']}. Intuitively, one can think of the intersections between the spikes and the central sphere as the location of the D8--D6 sources in the internal $S^3$ of the gravity solutions AdS$_7 \times S^3$.