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

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.