  The duality symmetries of the STU-model of Sen and Vafa are very restrictive.
This is utilized to determine the holomorphic function that encodes its
two-derivative Wilsonian effective action and its couplings to the square of
the Weyl tensor to fifth order in perturbation theory. At fifth order some
ambiguities remain which are expected to resolve themselves when proceeding to
the next order. Subsequently, a corresponding topological string partition
function is studied in an expansion in terms of independent invariants of $S$,
$T$ and $U$, with coefficient functions that depend on an effective duality
invariant coupling constant $u$, which is defined on a Riemann surface
$\mathbb{C}$. The coefficient function of the invariant that is independent of
$S$, $T$ and $U$ is determined to all orders by resummation. The other
functions can be solved as well, either algebraically or by solving
differential equations whose solutions have ambiguities associated with
integration constants. This determination of the topological string partition
function, while interesting in its own right, reveals new qualitative features
in the result for the Wilsonian action, which would be difficult to appreciate
otherwise. It is demonstrated how eventually the various ambiguities are
eliminated by comparing the results for the effective action and the
topological string. While we only demonstrate this for the leading terms, we
conjecture that this will hold in general for this model.
