  We construct a central extension of the smooth Deligne cohomology group of a
compact oriented odd dimensional smooth manifold, generalizing that of the loop
group of the circle. While the central extension turns out to be trivial for a
manifold of dimension 3, 7, 11,..., it is non-trivial for 1, 5, 9,.... In the
case where the central extension is non-trivial, we show an analogue of the
Segal-Witten reciprocity law.
