Instabilities of the small black hole:: a view from N=4 SYM

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.