We compute a one-loop effective action for the constant modes of the scalars
and the Polyakov loop matrix of N=4 SYM on S^3 at finite temperature and weak
't Hooft coupling. Above a critical temperature, the effective potential
develops new unstable directions accompanied by new saddle points which only
preserve an SO(5) subgroup of the SO(6) global R-symmetry. We identify this
phenomenon as the weak coupling version of the well known Gregory-Laflamme
localization instability in the gravity dual of the strongly coupled field
theory: The small AdS_5 black hole when viewed as a ten dimensional,
asymptotically AdS_5 X S^5 solution smeared on the S^5 is unstable to
localization on S^5. Our effective potential, in a specific Lorentzian
continuation, can provide a qualitative holographic description of the decay of
the "topological black hole'' into the AdS bubble of nothing.Comment: 39 pages, 6 figures, uses JHEP3.cls, references adde
