Integrability and Holographic Aspects of Six-Dimensional ${\cal N}=(1,0)$ Superconformal Field Theories

TL;DR

The paper addresses the dynamics of six-dimensional $ \mathcal{N}=(1,0)$ SCFTs, which lack a Lagrangian description, by developing a holographic dictionary linking Massive IIA backgrounds, Hanany–Witten quivers, and CFT observables. The authors construct general holographic expressions for brane charges, linking numbers, and entanglement entropy in terms of a single governing function $\alpha(z)$ and explore a special background with constant AdS warp factor, showing classical integrability via a Lax pair and a connection to the $\lambda$-deformed WZW model. They demonstrate that, for the integrable background, the string sigma-model splits into an integrable AdS$_7$ sector and an integrable $\mathcal M^3$ sector, providing a concrete example of integrability in a non-Lagrangian 6d CFT. A field-theoretic interpretation in the smeared D8-brane limit yields central charge and entanglement entropy scalings consistent with a holographic CFT, offering a practical program to extract dynamical information for these theories. Overall, the work delivers a detailed holographic-CFT dictionary, identifies an integrable sector, and lays groundwork for exploring RG flows and lower-dimensional reductions within six-dimensional $\mathcal N=(1,0)$ holography.

Abstract

In the framework of six-dimensional conformal field theories with ${\cal N}=(1,0)$ supersymmetry we develop the map between the holographic description, the field theoretical description and the associated Hanany-Witten set-ups. General expressions that calculate various observables are presented. The study of string solitons singles out a special background of Massive IIA on which we show (by explicitly finding a Lax pair) that the Neveu-Schwarz part of the string sigma model is classically integrable. We study the particular dual conformal field theory and compute some of its observables.

Figures (8)

• Figure 1: The Hanany-Witten set-up for the field theory. The vertical lines denote individual Neveu-Schwarz branes extended on the $(x_4,x_5)$ space. The horizontal ones D6-branes, that extend on $x_6$, in between fivebranes. The crossed-circles represent D8-branes, that extend on the $(x_7,x_8,x_9)$ directions. All the branes share the Minkowski directions. This realises the isometries $SO(1,5) \times SO(3)$.
• Figure 2: The quiver corresponding to the Hanany-Witten set-up above.
• Figure 3: The rank function $R(z)$ corresponding to the field theory.
• Figure 4: The Hanany-Witten set-up corresponding to the generic field theory studied here.
• Figure 5: The quiver corresponding to the generic field theory studied here.