  Motivated by precision counting of BPS black holes, we analyze six-derivative
couplings in the low energy effective action of three-dimensional string vacua
with 16 supercharges. Based on perturbative computations up to two-loop,
supersymmetry and duality arguments, we conjecture that the exact coefficient
of the $\nabla^2(\nabla\phi)^4$ effective interaction is given by a genus-two
modular integral of a Siegel theta series for the non-perturbative Narain
lattice times a specific meromorphic Siegel modular form. The latter is
familiar from the Dijkgraaf-Verlinde-Verlinde (DVV) conjecture on exact
degeneracies of 1/4-BPS dyons. We show that this Ansatz reproduces the known
perturbative corrections at weak heterotic coupling, including tree-level, one-
and two-loop corrections, plus non-perturbative effects of order
$e^{-1/g_3^2}$. We also examine the weak coupling expansions in type I and type
II string duals and find agreement with known perturbative results, as well as
new predictions for higher genus perturbative contributions. In the limit where
a circle in the internal torus decompactifies, our Ansatz predicts the exact
$\nabla^2 F^4$ effective interaction in four-dimensional CHL string vacua,
along with infinite series of exponentially suppressed corrections of order
$e^{-R}$ from Euclideanized BPS black holes winding around the circle, and
further suppressed corrections of order $e^{-R^2}$ from Taub-NUT instantons. We
show that instanton corrections from 1/4-BPS black holes are precisely weighted
by the BPS index predicted from the DVV formula, including the detailed moduli
dependence. We also extract two-instanton corrections from pairs of 1/2-BPS
black holes, demonstrating consistency with supersymmetry and wall-crossing,
and estimate the size of instanton-anti-instanton contributions.

High Energy Physics - Theory