Local Cohomology in Field Theory

This is an introductory survey of the theory of p-form conservation laws in field theory. It is based upon a series of lectures given at the Second Mexican School on Gravitation and In these lectures I will provide a survey of some aspects of the theory of local coho-mology in field theory and provide some illustrative applications, primarily taken from the Einstein equations of the general theory of relativity. There are many facets to this branch of mathematical physics, and I will only concentrate on a relatively small portion of the subject. A sample of some of the recent literature on the subject can be found in [1]. For the purposes of these lectures I will restrict the development of local cohomology to the theory of p-form conservation laws for field equations. This theory can be considered to be a generalization of Noether's theory of conserved currents to differential forms of any degree. Rather than presenting the theory in its full generality, I will illustrate each point via examples taken from relativistic field theory, with the principal example being the vacuum Einstein equations. In so doing I will summarize the results of a classification of all p-form conservation laws that can be locally constructed from an Einstein metric and its derivatives to any finite order [2]. Finally, I show how the theory and techniques used to analyze p-form conservation laws can be used to give a useful derivation of asymptotic conservation laws (e.g., ADM energy), when they exist, for any field theory. Many of the basic results on p-form conservation laws can be obtained from a variety of points of view [1]. Virtually all of the results presented here were obtained in collaboration with Ian Anderson. We are currently preparing an in-depth exposition of a theory of p-form conservation laws [2], relative to which these lectures can be considered an introductory survey. An outline of the topics to be covered is as follows: • p-form conservation laws in field theory: introduction and examples. • Jet bundle of metrics and the Euler-Lagrange complex. • Jet bundle of Einstein metrics. • p-form conservation laws and local cohomology. • Identically closed p-forms: cohomology of the free Euler-Lagrange complex. • Identically closed p-forms locally constructed from a metric: gravitational kink number. • (n − 1)-form conservation laws and infinitesimal symmetries of the field equations. 1 • Lower-degree conservation laws and infinitesimal gauge symmetries of …