Automorphic properties of low energy string amplitudes in various dimensions

Abstract

This paper explores the moduli-dependent coefficients of higher derivative interactions that appear in the low-energy expansion of the four-graviton amplitude of maximally supersymmetric string theory compactified on a d-torus. These automorphic functions are determined for terms up to order D^6R^4 and various values of d by imposing a variety of consistency conditions. They satisfy Laplace eigenvalue equations with or without source terms, whose solutions are given in terms of Eisenstein series, or more general automorphic functions, for certain parabolic subgroups of the relevant U-duality groups. The ultraviolet divergences of the corresponding supergravity field theory limits are encoded in various logarithms, although the string theory expressions are finite. This analysis includes intriguing representations of SL(d) and SO(d,d) Eisenstein series in terms of toroidally compactified one and two-loop string and supergravity amplitudes.

80 pages. 1 figure. v2:Typos corrected, footnotes amended and small clarifications. v3: minor corrections. Version to appear in Phys Rev D

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• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
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• E7(7) symmetry in perturbatively quantised N=8 supergravity
• $R^4$ couplings and automorphic unipotent representations
• Automorphic Forms and Fermion Masses
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• Tropical Amplitudes
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• Loops in exceptional field theory
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• D-instantons, String Field Theory and Two Dimensional String Theory
• Normalization of Type IIB D-instanton Amplitudes
• Shedding black hole light on the emergent string conjecture
• Minimal unitary representations from supersymmetry
• A master exceptional field theory
• D-instantons in Type IIA string theory on Calabi-Yau threefolds
• Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality
• Cosmological consequences of a principle of finite amplitudes
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• The two D6R4 type invariants and their higher order generalisation
• Higher Derivative Supergravities in Diverse Dimensions
• SUSY Ward identities, Superamplitudes, and Counterterms
• Poisson equation for the three loop ladder diagram in string theory at genus one
• Species scale, worldsheet CFTs and emergent geometry
• The D^{2k} R^4 Invariants of N=8 Supergravity
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• The D^6 R^4 term from three loop maximal supergravity
• Infrared divergences and harmonic anomalies in the two-loop superstring effective action
• Supersymmetry constraints on the R^4 multiplet in type IIB on T^2
• Supervertices and Non-renormalization Conditions in Maximal Supergravity Theories
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• On higher spin realizations of K(E10)
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• Arithmetic and Hyperbolic Structures in String Theory
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• No-boundary solutions are robust to quantum gravity corrections
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• Perturbative type II amplitudes for BPS interactions
• Higher derivative type II string effective actions, automorphic forms and E11
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• Lectures in Quantum Gravity
• Perturbative higher-derivative terms in N=6 asymmetric orbifolds
• Non--analytic terms from nested divergences in maximal supergravity
• Uniformization, Unipotent Flows and the Riemann Hypothesis
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• Constraining non--BPS interactions from counterterms in three loop maximal supergravity
• Inflationary constraints on the moduli-dependent species scale in modular invariant theories
• Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors
• Kac--Moody groups and automorphic forms in low dimensional supergravity theories
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• Mysterious Triality and the Exceptional Symmetry of Loop Spaces
• Symmetries of non-maximal supergravities with higher-derivative corrections
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