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We compute a one-loop effective action for the constant modes of the scalarsand 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 potentialdevelops new unstable directions accompanied by new saddle points which onlypreserve an SO(5) subgroup of the SO(6) global R-symmetry. We identify thisphenomenon as the weak coupling version of the well known Gregory-Laflammelocalization instability in the gravity dual of the strongly coupled fieldtheory: 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 tolocalization on S^5. Our effective potential, in a specific Lorentziancontinuation, can provide a qualitative holographic description of the decay ofthe "topological black hole'' into the AdS bubble of nothing.Comment: 39 pages, 6 figures, uses JHEP3.cls, references adde

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