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

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