= Breitenlohner-Freedman bound
{c}
The <Breitenlohner-Freedman bound> permits a <scalar field> in <Anti-de Sitter spacetime> to have a moderately negative squared <mass> without the instability that the same parameter would imply in flat <spacetime>. In $\mathrm{AdS}_{d+1}$ of <curvature> radius $L$, the bound is $m^2L^2\geq-d^2/4$ for the standard scalar <dynamical stability> problem with appropriate <boundary conditions>.
The near-boundary exponents satisfy $\Delta(\Delta-d)=m^2L^2$, hence $\Delta_\pm=d/2\pm\sqrt{d^2/4+m^2L^2}$. The bound keeps these exponents real. The <spacetime> geometry and <boundary conditions> are part of the <dynamical stability> statement; a negative squared <mass> cannot be classified without them.<Stability in gauged extended supergravity by Peter Breitenlohner and Daniel Z. Freedman>[Original scalar-field stability result; explicit AdS dimension and curvature-radius convention]{ref}
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