$R^4$ couplings and automorphic unipotent representations

TL;DR

This paper proposes that the automorphic form governing certain string theory couplings in various dimensions is the minimal theta series of the U-duality group, supported by multiple lines of evidence including BPS charge support and decompactification limits.

Contribution

It introduces the minimal theta series as the automorphic form for four-graviton couplings, providing evidence from representation theory and non-perturbative effects, and conjectures similar forms for other couplings.

Findings

Matching of infinitesimal characters supports the proposal.

Fourier coefficients support 1/2-BPS charge structure.

Decompactification limits align with the minimal theta series.

Abstract

Four-graviton, eight-derivative couplings in the low energy effective action of toroidal type II string compactifications are tightly constrained by U-duality invariance and by supersymmetry. In this note, we revisit earlier proposals for the automorphic form governing these couplings in dimension D=3,4,5,6, and propose that the correct automorphic form is the minimal theta series for the corresponding U-duality group. Evidence for this proposal comes from i) the matching of infinitesimal characters, ii) the fact that the Fourier coefficients have support on 1/2-BPS charges and iii) decompactification limits. In particular, we show that non-perturbative effects can be interpreted as 1/2-BPS instantons, or 1/2-BPS particles in one dimension higher (together with Taub-NUT instantons in the D=3 case). Based on similar considerations, we also conjecture the form of 1/4-BPS saturated couplings such as $\nabla^4 R^4$ couplings in the same dimensions.