4d N=2 superconformal linear quivers with type IIA duals

TL;DR

The paper develops a concrete holographic framework for 4d ${ m N}=2$ SCFTs realized by D4–NS5–D6 brane systems, identifying type IIA supergravity backgrounds that are weakly coupled and curved in controlled regions away from branes. Central to the construction is the reduction of the 11d ${ m SU}(2,2|2)$-symmetric Toda system to a 3d axially symmetric Laplace problem, whose boundary data (a line-charge density $ ilde ho( u)$ and, for multiple stacks, a set of cuts) encode the quiver data, ranks, and exactly marginal couplings of the SCFTs. The authors provide explicit solutions for one-stack NS5 configurations and general (multi-stack) extensions, analyze asymptotics near brane sources, and derive the corresponding dual gauge theories, including conditions under which the Type IIA description is valid. They show how flux quantization and linking numbers map to gauge-theory data, and discuss the limitations and regimes of validity, including the role of large $N$ and large quiver length in achieving a reliable weakly curved dual. Overall, the work furnishes a practical, boundary-value approach to constructing and interpreting strong-coupling holographic duals of a broad class of 4d ${ m N}=2$ SCFTs and outlines avenues to compute strong-coupling observables via explicit Laplace solutions.

Abstract

We discuss the gravity duals of 4d N=2 superconformal field theories (SCFTs) arising from the low-energy limit of brane configurations of D4-branes stretched between and intersecting NS5-branes and D6-branes. This gives rise to a product of SU(N_i) groups, with bi-fundamental matter between adjacent groups, and extra fundamental hypermultiplets. The most general configuration in 11d (or type IIA) supergravity that is dual to a 4d N=2 SCFT (when the dual of this SCFT is a weakly curved background) was written down by Gaiotto and Maldacena, but finding it explicitly involves solving a complicated Toda equation. This equation simplifies only when the solution can be reduced to type IIA supergravity, so we ask for which SCFTs of this type is there a type IIA dual that is weakly coupled and weakly curved (away from NS5-branes and D6-branes). We find that such solutions (a special case of which was analyzed by Reid-Edwards and Stefanski) exist when there is a large number of gauge groups, with large ranks, and with large 't Hooft couplings for all but a finite number of groups. The general solution of this type is given by solving an axially symmetric Laplace equation in three dimensions, with specific boundary conditions. We match the parameters of the 4d SCFTs, including the exactly marginal coupling constants, with the boundary conditions for the Laplace equation.

Figures (7)

• Figure 1: a) Canonical description of a general brane configuration. Horizontal lines denote D4-branes, vertical lines denote NS5-branes and diagonal broken lines denote D6-branes. b) Linear quiver describing the gauge theory corresponding to the brane configuration in a). Circular nodes denote special unitary gauge groups $SU(k_n)$. Horizontal lines between consecutive circular nodes denote bifundamental hypermultiplets. Square nodes denote a unitary global symmetry group $U(d_n)$. A line connecting a square and a circular node denotes $d_n$ hypermultiplets in the fundamental representation of $SU(k_n)$.
• Figure 2: a) The quiver diagram for our first example, with varying ranks. b) The quiver diagram for our second example, with equal ranks.
• Figure 3: Typical consistent profile for the line charge density $\lambda(\eta)$.
• Figure 4: The Riemann surface $\Sigma'$. We depict the first choice for the surface $\Sigma_1$ on which ${\cal F}_1$ is well-defined, and for the surface $\Sigma_2$ on which ${\cal F}_2$ is well-defined, separated by the curve $\gamma$.
• Figure 5: a) Simple example of a line charge density $\lambda(\eta)$ describing one stack of $2N$ D6-branes. b) Simple example of a line charge density $\lambda(\eta)$ describing two stacks of $N$ D6-branes.