White Noise Analysis on manifolds and irreducibility of the energy representation of a gauge group

The energy representation of a gauge group on a Riemannian manifold has been discussed by several authors, for instance, in Refs. 1, 2, 4, 5, 7 and 13. Their methods are essentially to reduce the problem to the estimate of the support of a Gaussian measure in an infinite dimensional space. The best result in this direction seems to be the one in Ref. 13. After all these contributions, the irreducibility in two dimensions has remained unsettled yet. One of the difficulties is the conformal invariance of the energy representation, i.e., when we transform the Riemannian metric g(x) into eg(x), the energy representation remains unchanged. For a historical survey of this line of research, we refer the reader to Ref. 1. In contrast to the above, Y. Shimada has recently shown the irreducibility of a gauge group on a compact Riemannian manifold in Ref. 12, applying White Noise Analysis. For a compact, oneor two-dimensional Riemannian manifold, a new result has been obtained there. We extend this result to include some class of noncompact manifolds in the presence of a weight function. Moreover, this approach is expected to be of importance to random fields on manifolds.