Central extensions of gauge transformation groups of higher abelian gerbes

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,…3,7,11,…, it is non-trivial for 1,5,9,…1,5,9,…. In the case where the central extension is non-trivial, we show an analogue of the Segal–Witten reciprocity law.