Comments on the symmetry of AdS$_6$ solutions in String/M-theory and Killing spinor equations

TL;DR

The paper reveals a hidden $SL(3,\mathbb{R})/SO(2,1)$ symmetry in AdS$_6$ solutions of type IIB supergravity by reducing to a bosonic $D=4$ action with five scalars organized as a nonlinear sigma model. It shows the Killing spinor equations can be written covariantly, and that the scalar potential is governed by a single diagonal coset component, with integrability connecting to the four-dimensional equations of motion. The authors generalize the construction to arbitrary $SL(n,\mathbb{R})/SO(p,q)$ cosets, obtaining a broad class of 4D theories with a diagonal-component-controlled potential and a residual symmetry $sl(n-1,\mathbb{R})\ltimes \mathbb{R}^{n-1}$, plus a simple $n=2$ example. They illustrate the framework with explicit IIB AdS$_6$ backgrounds (including abelian and non-abelian T-duals) and discuss implications for solution generation, as well as parallel AdS$_6$ structures in M-theory, highlighting a unified coset-based perspective across dimensions.

Abstract

It was recently pointed out in \cite{Kim:2015hya} that AdS$_6$ solutions in IIB theory enjoy an extended symmetry structure and the consistent truncation to $D=4$ internal space leads to a nonlinear sigma model with target $SL(3,\mathbb{R})/SO(2,1)$. We continue to study the purely bosonic $D=4$ effective action, and elucidate how the addition of scalar potential term still allows Killing spinor equations in the absence of gauge fields. In particular, the potential turns out to be a single diagonal component of the coset representative. Furthermore, we perform a general analysis of the integrability conditions of Killing spinor equations and establish that the effective action can be in fact generalized to arbitrary sizes and signatures, e.g. with target $SL(n,\mathbb{R})/SO(p,n-p)$ and the scalar potential expressible by a single diagonal component of the coset representative. We also comment on a similar construction and its generalizations of effective $D=5$ purely bosonic non-linear sigma model action related to AdS$_6$ in M-theory.