8 - Cohomology and Wess-Zumino terms: an introduction

Summary

The relation between mechanics and cohomology already sketched in chapter 3 is re-analysed here from the Lagrangian point of view, itself initially introduced in the 1-jet bundle framework. It is seen how quasi-invariance for Lagrangians is related to group (or Lie algebra) central extensions. This point of view is extended by considering J(J¹(E))-valued cocycles in such a way that the problem of quasi-invariant Lagrangians is described in terms of one-cocycles as well as two-cocycles, and they are seen to be related by the cohomological descent procedure. This general scheme will appear again in chapter 10 in the context of non-abelian, consistent, chiral gauge anomalies.

The complementary aspect of obtaining physical actions (or terms in them) from non-trivial cohomology is also studied. This leads to the concept of Wess-Zumino term on a group manifold. The cohomological descent is then applied to this picture, which exhibits the different role of the left and right versions of the symmetry Lie algebra involved.

Examples of these two aspects are given by using the Galilei group (also studied in chapter 3) and the supersymmetric extended objects (which may be omitted by readers not interested in supersymmetry). Finally, a Lagrangian action for the monopole (chapter 2) is studied as a different kind of Wess-Zumino term associated with quasi-invariance under gauge (rather than rigid) transformations.

A short review of the variational principle and of the Noether theorem in Newtonian mechanics

In the variational formulation of dynamical systems the starting point is the definition of the Lagrangian.