Eisenstein series and scattering amplitudes

Scattering amplitudes of superstring theory are strongly constrained by the requirement that they be invariant under dualities generated by discrete subgroups, $En(Z)$, of simply-laced Lie groups in the $En$ series ($n\le 8$). In  articular, expanding the four-supergraviton amplitude at low energy gives a series of higher derivative corrections to Einstein’s theory,  with coefficients that are automorphic functions. Boundary conditions supplied by string and supergravity perturbation theory, together with a chain of relations between successive groups in the $En$ series, constrain the constant terms of these coefficients in three distinct parabolic subgroups. Using this information we are able to determine the expressions for the first two higher derivative interactions in terms of specific Eisenstein series. The coefficient of the third term in this expansion  is an automorphic function that satisfies an inhomogeneous Laplace equation and has constant terms in certain parabolic subgroups that contain information about all the preceding terms.

In this talk we will present the  construction of these  automorphic forms. By analyzing the constant term expansion we will exhibit nested relations between the series for various groups leading to peculiar analytic properties of combinations of Eisenstein series. We will as well discuss  the relation to integrals of theta function over the moduli space of Riemann surfaces of genus g arising from string perturbation. This talks is based on the papers [arXiv:1001.2535 [4]]  and [arXiv:1004.0163 [5]].