Duality Groups, Automorphic Forms and Higher Derivative Corrections

TL;DR

This paper explores how higher derivative corrections in superstring theories are linked to automorphic forms invariant under duality groups, providing a systematic construction method and evidence for U-duality symmetry.

Contribution

It introduces a systematic method to construct automorphic forms from duality groups and subgroups, connecting these forms to scalar field functions in superstring theories.

Findings

Automorphic forms contain weights of G, including E_8.

Dimensional reduction reveals weights of E_8, supporting U-duality.

Constructed automorphic forms from G(Z) and H, based on non-linear realizations.

Abstract

We study the higher derivative corrections that occur in type II superstring theories in ten dimensions or less. Assuming invariance under a discrete duality group G(Z) we show that the generic functions of the scalar fields that occur can be identified with automorphic forms. We then give a systematic method to construct automorphic forms from a given group G(Z) together with a chosen subgroup H and a linear representation of G(Z). This construction is based on the theory of non-linear realizations and we find that the automorphic forms contain the weights of G. We also carry out the dimensional reduction of the generic higher derivative corrections of the IIB theory to three dimensions and find that the weights of E_8 occur generalizing previous results of the authors on M-theory. Since the automorphic forms of this theory contain the weights of E_8 we can interpret the occurrence of weights in the dimensional reduction as evidence for an underlying U-duality symmetry.