There is a natural correspondence between holomorphic
bundles over complex manifolds and flat bundles over affine
manifolds. More specifically, an elliptic K3 surface can be viewed as
a torus fibration over P^1, and away from the singular fibers a torus
invariant holomorphic bundle reduces to a flat bundle over punctured
P^1. In this talk I will describe and solve the reduction of the
Hermitian-Yang-Mills equations to a flat bundle on this Riemann
surface, and discuss its relation to twisted harmonic metrics and
mirror symmetry. This is joint work with T.C. Collins and S.-T. Yau.