Perturbation Theory From Automorphic Forms

TL;DR

This paper develops a group-theoretical framework to derive automorphic forms relevant to string theory, providing explicit formulas for perturbative and non-perturbative contributions across different groups and dimensions.

Contribution

It introduces a method to compute automorphic forms for various groups, including special cases like E_{n+1} and SO(5,5), with explicit string theory applications and new predictions.

Findings

Explicit formulas for automorphic forms for E_{n+1} groups.

Construction of SO(5,5) automorphic form using vector representation.

Identification of a constrained automorphic form compatible with string theory.

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.