Elliptic operators in the functional quantisation for gauge field theories

Given a gauge theory with gauge group G acting on a path space X, G and X being both infinite dimensional manifolds modelled on spaces of sections of vector bundles on a compact riemannian manifold without boundary, it is shown that when the action of G on X is smooth, free and proper, the same ellipticity condition on an operator naturally given by the geometry of the problem yields both the existence of a principal fibre bundle structure induced by the canonical projection π:X → X/G and the existence of the Faddeev-Popov determinant arising in the functional quantisation of the gauge theory. This holds for certain gauge theories with anomalies like bosonic closed string theory in non-critical dimension and also holds for a class of gauge theories which includes Yang-Mills theory. (orig.)