Communications in Number Theory and Physics

Volume 4 (2010)

Number 3

Eisenstein series for higher-rank groups and string theory amplitudes

Pages: 551 – 596



Michael B. Green (Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom)

Stephen D. Miller (Department of Mathematics, Rutgers University, New Jersey, U.S.A.)

Jorge G. Russo (Institució Catalana de Recerca i Estudis Avançats (ICREA), University de Barcelona, Spain)

Pierre Vanhove (Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France)


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.

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