Gauge and integrable theories in loop spaces Original Research Article

We propose an integral formulation of the equations of motion of a large class of field theories which leads in a quite natural and direct way to the construction of conservation laws. The approach is based on generalized non-abelian Stokes theorems for p-form connections, and its appropriate mathematical language is that of loop spaces. The equations of motion are written as the equality of a hyper-volume ordered integral to a hyper-surface ordered integral on the border of that hyper-volume. The approach applies to integrable field theories in image dimensions, Chern–Simons theories in image dimensions, and non-abelian gauge theories in image and image dimensions. The results presented in this paper are relevant for the understanding of global properties of those theories. As a special byproduct we solve a long standing problem in image-dimensional Yang–Mills theory, namely the construction of conserved charges, valid for any solution, which are invariant under arbitrary gauge transformations.