  We study five-dimensional minimally supersymmetric gauge theory compactified
on a torus down to three dimensions, and its embedding into string/M-theory
using geometric engineering. The moduli space on the Coulomb branch is
hyperkaehler equipped with a metric with modular transformation properties. We
determine the one-loop corrections to the metric and show that they can be
interpreted as worldsheet and D1-brane instantons in type IIB string theory.
Furthermore, we analyze instanton corrections coming from the solitonic BPS
magnetic string wrapped over the torus. In particular, we show how to compute
the path-integral for the zero-modes from the partition function of the M5
brane, or, using a 2d/4d correspondence, from the partition function of N=4 SYM
theory on a Hirzebruch surface.
