Genus-One String Amplitudes from Conformal Field Theory

We explore and exploit the relation between non-planar correlators in ${\cal N}=4$ super-Yang-Mills, and higher-genus closed string amplitudes in type IIB string theory. By conformal field theory techniques we construct the genus-one, four-point string amplitude in AdS${}_5\times S^5$ in the low-energy expansion, dual to an $\mathcal{N}=4$ super-Yang-Mills correlator in the 't Hooft limit at order $1\mathrm{/}{c}^2$ in a strong coupling expansion. In the flat space limit, this maps onto the genus-one, four-point scattering amplitude for type II closed strings in ten dimensions. Using this approach we reproduce several results obtained via string perturbation theory. We also demonstrate a novel mechanism to fix subleading terms in the flat space limit of AdS amplitudes by using string/M-theory.