Perturbation Theory From Automorphic Forms

Abstract

Using our previous construction of Eisenstein-like automorphic forms we derive formulae for the perturbative and non-perturbative parts for any group and representation. The result is written in terms of the weights of the representation and the derivation is largely group theoretical. Specialising to the E_{n+1} groups relevant to type II string theory and the representation associated with node n+1 of the E_{n+1} Dynkin diagram we explicitly find the perturbative part in terms of String Theory variables, such as the string coupling g_d and volume V_n. For dimensions seven and higher we find that the perturbation theory involves only two terms. In six dimensions we construct the SO(5,5) automorphic form using the vector representation. Although these automorphic forms are generally compatible with String Theory, the one relevant to R^4 involves terms with g_d^{-6} and so is problematic. We then study a constrained SO(5,5) automorphic form, obtained by summing over null vectors, and compute its perturbative part. We find that it is consistent with String Theory and makes precise predictions for the perturbative results. We also study the unconstrained automorphic forms for E_6 in the 27 representation and E_7 in the 133 representation, giving their perturbative part and commenting on their role in String Theory.

Typos fixed and other minor corrections. A 'note added' included

References in corpus (10)

• Automorphic properties of low energy string amplitudes in various dimensions
• New Higher-Derivative $R^4$ Theorems
• $R^4$ couplings and automorphic unipotent representations
• The D^4 R^4 term in type IIB string theory on T^2 and U-duality
• Enhanced Coset Symmetries and Higher Derivative Corrections
• Duality Groups, Automorphic Forms and Higher Derivative Corrections
• Aspects of higher curvature terms and U-duality
• The D^{10} R^4 term in type IIB string theory
• Constraints on Automorphic Forms of Higher Derivative Terms from Compactification
• U-Duality and the Compactified Gauss-Bonnet Term

Cited by in corpus (12)

• $R^4$ couplings and automorphic unipotent representations
• Eisenstein series and automorphic representations
• Eisenstein series for infinite-dimensional U-duality groups
• Constraints on Automorphic Forms of Higher Derivative Terms from Compactification
• Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories
• Higher derivative type II string effective actions, automorphic forms and E11
• Uniformization, Unipotent Flows and the Riemann Hypothesis
• Parameters, limits and higher derivative type II string corrections
• Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors
• Poisson equations, higher derivative automorphic forms and string parameter limits
• Kac--Moody groups and automorphic forms in low dimensional supergravity theories
• Modular realizations of hyperbolic Weyl groups