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Description
We apply the conformal bootstrap to holographic four-dimensional N=2 superconformal field theories in the planar limit. We employ a dispersion relation that reconstructs the correlator from its double discontinuity, which naturally isolates the single-trace spectrum. Supersymmetric localization provides an additional exact constraint that makes the bootstrap sensitive to the gauge coupling. Combining these two ingredients, we obtain rigorous bounds on the scaling dimension of single-trace operators at any value of the coupling. As examples we consider the stress tensor correlator in N=4 SYM dual to closed string scattering, and the flavour multiplet correlator in a certain N=2 theory dual to open string scattering. Our bounds are saturated at both weak and strong coupling and smoothly interpolate between the two.