Gauge-invariant coordinates on gauge-theory orbit space

A gauge-invariant field is found which describes physical configurations, i.e., gauge orbits, of non-Abelian gauge theories. This is accomplished with non-Abelian generalizations of the Poincaré-Hodge decomposition formula for one-forms. In a particular sense, the new field is dual to the gauge field. Using this field as a coordinate, the metric and intrinsic curvature are discussed for Yang-Mills orbit space for the (2+1)- and (3+1)-dimensional cases. The sectional, Ricci, and scalar curvatures are all formally non-negative. An expression for the new field in terms of the Yang-Mills connection is found in 2+1 dimensions. The measure on Schrödinger wave functionals is found in both 2+1 and 3+1 dimensions; in the former case, it resembles the Karabali, Kim, and Nair measure. We briefly discuss the form of the Hamiltonian in terms of the dual field and comment on how this is relevant to the mass gap for both the (2+1)- and (3+1)-dimensional cases.