Genus-One String Amplitudes from Conformal Field Theory
Luis F. Alday, Agnese Bissi, Eric Perlmutter
TL;DR
This work develops a conformal bootstrap framework to connect non-planar ${\cal N}=4$ SYM four-point functions to genus-one Type IIB string amplitudes, constructing the genus-one AdS$_5\times S^5$ four-point amplitude in the low-energy expansion and relating it to the flat-space genus-one amplitude ${\mathcal A}^{(g=1)}$ in ten dimensions. By analyzing the double-discontinuity via Lorentzian inversion, the authors extract one-loop anomalous dimensions and organize stringy corrections with a basis of special functions $S_L^{(q)}$, revealing the analytic structure governed by UV divergences and higher-derivative vertices like $\mathcal{R}^4$ and $\partial^4\mathcal{R}^4$. They show that subleading flat-space terms can be constrained and in some cases fixed by matching to string/M-theory amplitudes, establishing a concrete link between AdS curvature corrections and higher-dimensional loop amplitudes. The results yield all-orders predictions for certain discontinuities and demonstrate a coherent map between AdS loop data and the perturbative string theory, offering a path toward systematic higher-genus extensions. The approach provides insights into how AdS/CFT encodes UV completions and how mixing among double-trace operators shapes the one-loop stringy corrections.
Abstract
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 ${\cal N}=4$ super-Yang-Mills correlator in the 't Hooft limit at order $1/c^2$ 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.