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Large $N$ factorization ensures that, for low-dimension gauge-invariantoperators in the half-BPS sector of ${\cal N}=4$ SYM, products of holomorphictraces have vanishing correlators with single anti-holomorphic traces. Thisvanishing is necessary to consistently map trace operators in the CFT$_4$ to aFock space of graviton oscillations in the dual AdS$_5$. We investigate theregimes at which the CFT correlators do not vanish but become of order one inthe large $N$ limit, which we call a factorization threshold. Quite generally,we find the threshold to be when the product of the two holomorphic operatordimensions is of order $N\log N$. Our analysis considers extremal andnon-extremal correlators and correlators in states dual to LLM backgrounds, andwe observe intriguing similarities between the the energy-dependent runningcoupling of non-abelian gauge theories and our threshold equations. Finally, wediscuss some interpretations of the threshold within the bulk AdS spacetime.

Thresholds of Large N Factorization in CFT4 : Exploring bulk spacetime in AdS5