Communications in Number Theory and Physics
Volume 4 (2010)
Eisenstein series for higher-rank groups and string theory amplitudes
Pages: 551 – 596
Scattering amplitudes of superstring theory arestrongly\break constrained by the requirement that they beinvariant under dualities generated by discrete subgroups,$E_n(\IZ)$, of simply laced Lie groups in the $E_n$ series($n\le 8$). In particular, expanding the four-supergravitonamplitude at low energy gives a series of higher derivativecorrections to Einstein’s theory, with coefficients thatare automorphic functions with a rich dependence on themoduli. Boundary conditions supplied by string andsupergravity perturbation theory, together with a chain ofrelations between successive groups in the $E_n$ series,constrain the constant terms~of these coefficients in threedistinct parabolic subgroups. Using this information we areable to determine the expressions for the first two higherderivative interactions (which are BPS-protected) in termsof specific Eisenstein series. Further, we determine keyfeatures~of the coefficient of the third term in thelow-energy expansion of~the four-supergraviton amplitude(which is also BPS-protected) in the $E_8$ case. This is anautomorphic function that satisfies an inhomogeneousLaplace equation and has constant terms in certainparabolic subgroups that contain information about all thepreceding terms.