Eisenstein Series in String Theory

Abstract

We discuss the relevance of Eisenstein series for representing certain G(Z)-invariant string theory amplitudes which receive corrections from BPS states only. The Eisenstein series are constructed using G(Z)-invariant mass formulae and are manifestly invariant modular functions on the symmetric space K\G(R) of non-compact type, with K the maximal compact subgroup of G(R). In particular, we show how Eisenstein series of the T-duality group SO(d,d,Z) can be used to represent one- and g-loop amplitudes in compactified string theory. We also obtain their non-perturbative extensions in terms of the Eisenstein series of the U-duality group E_{d+1(d+1)}(Z).

11 pages, Latex, submitted to Proceedings of Strings '99, published version

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• M-theory and E10: Billiards, Branes, and Imaginary Roots
• The ${\hat{G}}^4{\lambda }^{16}$ Term in IIB Supergravity
• Duality completion of higher derivative corrections
• Two-Loop Superstrings VII, Cohomology of Chiral Amplitudes
• S-matrix elements from T-duality
• The D^6 R^4 term in type IIB string theory on T^2 and U-duality
• Supersymmetry constraints on the R^4 multiplet in type IIB on T^2
• Ramond-Ramond corrections to type II supergravity at order ${\alpha }^{\prime 3}$
• On S-duality of D${}_3$-brane S-matrix
• Sphere-level Ramond-Ramond couplings in Ramond-Neveu-Schwarz formalism
• S-duality of tree-level S-matrix elements in D${}_3$-brane effective action
• U-duality from Matrix Membrane Partition Function
• Gauge Threshold Corrections for N = 2 Heterotic Local Models with Flux, and Mock Modular Forms
• Tree-level S-matrix elements from S-duality
• S-duality invariant dilaton couplings at order ${\alpha }^{\prime 3}$
• S-duality of color-ordered amplitudes
• S-duality constraint on higher-derivative couplings
• Higher derivative type II string effective actions, automorphic forms and E11
• Parameters, limits and higher derivative type II string corrections
• Poisson equations, higher derivative automorphic forms and string parameter limits
• Symmetries of non-maximal supergravities with higher-derivative corrections