  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.

High Energy Physics - Theory