Level Repulsion in $\mathcal{N}=4$ super-Yang-Mills via Integrability, Holography, and the Bootstrap

We combine supersymmetric localization with the numerical conformal bootstrap to bound the scaling dimension and OPE coefficient of the lowest-dimension operator in $\mathcal{N}=4$ $\text{SU}(N)$ super-Yang-Mills theory for a wide range of $N$ and Yang-Mills couplings $g_{\text{YM}}$. We find that our bounds are approximately saturated by weak coupling results at small $g_{\text{YM}}$. Furthermore, at large $N$ our bounds interpolate between integrability results for the Konishi operator at small $g_{\text{YM}}$ and strong-coupling results, including the first few stringy corrections, for the lowest-dimension double-trace operator at large $g_{\text{YM}}$. In particular, our scaling dimension bounds describe the level splitting between the single- and double-trace operators at intermediate coupling.