Holographic duals of 6d RG flows

TL;DR

This paper investigates the relationship between nilpotent elements and RG flows in 6d SCFTs using holographic duals, confirming the conjecture that the partial order of nilpotent elements matches RG flow hierarchy.

Contribution

It provides a holographic analysis of 6d SCFTs with ADE groups, demonstrating the correspondence between nilpotent element ordering and RG flows via AdS$_7$ solutions.

Findings

Supersymmetric AdS$_7$ vacua are classified by nilpotent elements.

Domain wall solutions connect vacua if and only if the nilpotent elements are ordered.

The partial ordering of nilpotent elements matches the RG flow hierarchy.

Abstract

A notable class of superconformal theories (SCFTs) in six dimensions is parameterized by an integer $N$, an ADE group $G$, and two nilpotent elements $\mu_\mathrm{L,R}$ in $G$. Nilpotent elements have a natural partial ordering, which has been conjectured to coincide with the hierarchy of renormalization-group flows among the SCFTs. In this paper we test this conjecture for $G=\mathrm{SU}(k)$, where AdS$_7$ duals exist in IIA. We work with a seven-dimensional gauged supergravity, consisting of the gravity multiplet and two $\mathrm{SU}(k)$ non-Abelian vector multiplets. We show that this theory has many supersymmetric AdS$_7$ vacua, determined by two nilpotent elements, which are naturally interpreted as IIA AdS$_7$ solutions. The BPS equations for domain walls connecting two such vacua can be solved analytically, up to a Nahm equation with certain boundary conditions. The latter admit a solution connecting two vacua if and only if the corresponding nilpotent elements are related by the natural partial ordering, in agreement with the field theory conjecture.