  Given an N=2 supersymmetric field theory in four dimensions, its dimensional
reduction on S^1 is a sigma model with hyperkahler target space M. We describe
a canonical line bundle V on M, equipped with a hyperholomorphic connection.
The construction of this connection is similar to the known construction of the
metric on M itself: one begins with a simple "semiflat" connection and then
improves it by including contributions weighed by the degeneracies of BPS
particles going around S^1. We conjecture that V describes the physics of the
theory dimensionally reduced on NUT space.
